The Local Times for Additive Brownian Sheets and the Intersection Local Times for Independent Brownian Sheets

Abstract

A new class of Gaussian random fields is introduced in this article, described as additive Brownian sheets (ABSs), which can be regarded as a type of generalized Brownian sheet encompassing Brownian motions, Brownian sheets, and additive Brownian motions. The existence, joint continuity and the Hölder law of the local times for ABSs are derived under certain conditions, and some results of the intersection local times for two independent Brownian sheets are also given as special cases. Furthermore, the intersection local times for two independent Brownian sheets in a Hida distribution is proved through white noise analysis, and the Wiener chaos expansion of the intersection local times is expressed in terms of S-transform. Additionally, the large deviations for the intersection local time of two independent Brownian sheets are established. The multi-parameter Gaussian random fields have become a core tool for complex system analysis due to their flexible multidimensional modeling capabilities. With the improvement of computational efficiency and interdisciplinary integration, the ABS constructed in this article will unleash greater potential in fields such as metaverse simulation, financial mathematics, climate science, precision medicine, quantum physics, and string theory.

1. Introduction

The stable process has not only been attractive in its own right but also a source of insight in various branches of mathematical analysis and physics. The most important and representative stable process is Brownian sheet. The Brownian sheet $W=\left\{W\left(t\right)\in {\mathbb{R}}^d;t\in {\mathbb{R}}_{+}^N\right\}$ is defined as the ${\mathbb{R}}^d$-valued N-parameter Gaussian random field. To be more precise, W is a centered Gaussian process, whose $d⩾1$ components $W^1,\dots ,W^d$ are independent Gaussian random fields with zero mean and common covariance function such that

$$\begin{array}{c}\hfill \mathrm{Cov}\left(W^i\left(s\right),W^j\left(t\right)\right)=\left\{\begin{array}{cc}{\displaystyle \prod _{k=1}^N}\left(s_k\wedge t_k\right),\hfill & \mathrm{if}1⩽i=j⩽d,\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

where $s=\left(s_1,\dots ,s_N\right),t=\left(t_1,\dots ,t_N\right)$.

A Brownian sheet is a natural multi-parameter generalization of Brownian motion, which has been considered by many authors as one of the fundamental Gaussian random fields (cf. [1,2,3,4]). As one of the natural extensions of Brownian motion to higher-dimensional time, some properties of Brownian sheets are straightforward analogs of properties of Brownian motion. More specifically, a Brownian sheet is a temporally inhomogeneous Markov process, and has almost surely continuous sample functions and independent and stationary increments. Besides its own specific properties, the Brownian sheet has been studied by many authors as well; see, e.g., the works done by Pyke [5], Adler [6], Dalang and Walsh [7], Florit and Nualart [8], Khoshnevisan [9], Xiao et al. [10], Lin and Wang [11], among many others. Special to note is that the Brownian sheet with an N-dimensional “time” parameter is also known as the N-parameter Wiener process, which first appeared in the existing literature as a term expressed by Kitagawa [12]. He introduced a function space, which is the collection of the two variables’ continuous functions on the unit square, and investigated the integration on this space. Then, Yeh developed the measure of this space and made a logical foundation on this space in [13]. Meanwhile, Wiener random fields with several parameters have been studied by Chentsov [14]. In recent decades, more and more scholars in probability theory and related fields have joined the research efforts to make the theory and application of the Brownian sheet richer. There is abundant literature on the research on Brownian sheets. We mention some important and interesting results below. Firstly, Berman [15] extended a classical theorem on the increments of the Brownian motion by Lévy [16] to Brownian sheets. Additionally, the result in [15] also represented a generalization of Baxter’s theorem [17]. The sample functions of the Brownian sheet (e.g., continuity and recurrence properties) were investigated by Orey and Pruitt [2], and subsequently in [1] by Pyke. The Brownian sheet with two parameters is nowhere differentiable in any direction on the open unit square, which was proved by Csörgö and Révész [18], who also presented a “modulus of nondifferentiability”. Walsh [19] showed certain singularities of the Brownian sheet, which have the property that they propagated parallel to the coordinate axis, which is used to give an intuitive explanation of the fact that the Brownian sheet does not satisfy Lévy’s sharp Markov property. Furthermore, Epstein [20] studied some properties of functionals for the Brownian sheet. In particular, he constructed square-integrable functionals of ${\mathbb{R}}^d$-valued Brownian sheets. In [21], a small ball estimate for the Brownian sheet when the ball was given by a certain Hölder norm has been obtained by Kuelbs and Li. Tallagrand [22] also discussed the small ball problem of the Brownian sheet and derived the logarithm of the probability for the Brownian sheet. Khoshnevisan [23] showed that the image of a two-dimensional set under a d-dimensional, two-parameter Brownian sheet can have a positive Lebesgue measure if and only if the set in question has positive $(d/2)$-dimensional Bessel–Riesz capacity. And then he analyzed the potential theory of the Brownian sheet together with Shi [24]. Dalang and Mountford [25] presented several results concerning level sets, bubbles and excursions of the Brownian sheet. Xiao [26] applied the properties of strong local nondeterminism—an important technical tool that extended the concept of local nondeterminism first introduced by Berman for Gaussian processes [27]—to describing the sample path properties of several classes of anisotropic Gaussian random fields including fractional Brownian sheets. In addition, the most important results on sample path properties a Brownian sheet were then obtained by Ehm via the local times of a Brownian sheet. More exactly, Ehm established the almost sure behavior of the sample functions of more generalized random fields of a Brownian sheet, which is analogous to the well-known stable processes in [28].

Early interest in Brownian sheets came from limit theorems in multivariate statistics by Pyke [29]. From a practical point of view in statistics, the Brownian sheet has been more realistic and interesting in studying statistics. In fact, the research of the Brownian sheet began with work of Kitagawa [12] in connection with applications to statistical problems. The initial motivation in statistics on the study of the Brownian sheet can be found in Adler [30] and the references therein. Indeed, the statistical motivation for this came from the well-known relationship in limit theorems between the empirical processes and the Brownian sheets. There exists a great deal of literature on this subject; see, e.g., [31,32]. We need to mention here that Li and Pritchard [33] derived a functional central limit theorem, with a limiting Gaussian process closely related to the Brownian sheet using tools from the theory of empirical processes. Furthermore, motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, Deheuvels, Peccati, and Yor [34] obtained the law of quadratic functionals of the (weighted) Brownian sheet. The Brownian sheet, as the most important Gaussian random field, has been extensively studied in probability theory and widely used in many fields of natural science, social science and engineering technology, including statistics, physics, hydrology, biology, economics, finance, etc. For more comprehensive information about Brownian sheets, we mention Wolfgang and Jörg [35], Khoshnevisan [36], Kuznetsov et al. [37], Mansuy and Yor [38], and refer the reader to their detailed investigations for further references. Their conclusions show that a useful Brownian sheet can profit from the progress of the related theories of the Brownian sheet, while the research of the Brownian sheet can also be enriched by its application. With this literature in mind, we catch sight of an interesting process when the behavior of the intersection of two independent Brownian sheets $W_1$ and $W_2$ has been investigated. More specifically, that is formally structured as

$$\begin{array}{c}\hfill \mathbb{Y}\left(t_1,t_2\right)=W_1\left(t_1\right)-W_2\left(t_2\right),\end{array}$$

where $W_1$ and $W_2$ are independent Brownian sheets in ${\mathbb{R}}^d$ with parameters $N_1,N_2$, respectively.

Quite obviously, the process $\mathbb{Y}\left(t_1,t_2\right)$ is neither a Brownian sheet, a classical generalized Brownian sheet, additive Brownian motion, or Brownian motion. More to the point, $\mathbb{Y}\left(t_1,t_2\right)$ is defined as the ${\mathbb{R}}^d$-valued $\left(N_1+N_2\right)$-parameter random field with zero mean and special covariance function

$$\begin{array}{c}\hfill \mathrm{Cov}\left({\mathbb{Y}}^i\left(s_1,s_2\right),{\mathbb{Y}}^j\left(t_1,t_2\right)\right)=\left\{\begin{array}{cc}{\displaystyle \sum _{n=1}^2}{\displaystyle \prod _{k=1}^{N_n}}\left(s_{nk}\wedge t_{nk}\right),\hfill & \mathrm{if}1⩽i=j⩽d,\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

where $s_n=\left(s_{n1},\dots ,s_{nN}\right),t_n=\left(t_{n1},\dots ,t_{nN}\right),n=1,2$.

More generally, for any $k\in \mathbb{N}$, let $W_1,\dots ,W_k$ be independent standard Brownian sheets in ${\mathbb{R}}^d$ with parameters $N_1,\dots ,N_k$, respectively. Then, we can define a $\left(N_1,\dots ,N_k;d\right)$ Gaussian random field

$$\begin{array}{c}\hfill \mathbb{X}\left(t_1,\dots ,t_k\right)=W_1\left(t_1\right)+\cdots +W_k\left(t_k\right)\in {\mathbb{R}}^d.\end{array}$$

The $\left(N_1,\dots ,N_k;d\right)$ Gaussian random field $\mathbb{X}\left(t_1,\dots ,t_k\right)$ is called $\left(N_1,\dots ,N_k;d\right)-$$additive$ $Brownian$ $sheets$ (ABS in abbreviation) in view of the additive composition of $\mathbb{X}$ in this paper. Using tensor notation, we will often write $\mathbb{X}={⨁}_{n=1}^k{W}_n$ for brevity. Note that if $N_1=\cdots =N_k=1$, then $\mathbb{X}={⨁}_{n=1}^k{W}_n$ is the $(K,d)$-additive Brownian motions; if $k=1$, then $\mathbb{X}={⨁}_{n=1}^k{W}_n$ is $(N_1,d)$-Brownian sheet. Therefore, ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ could be regarded as a special generalization of Brownian motion to additive Brownian motions, as well as a generalization of the Brownian sheet.

The concept of an additive process originates from the study of an additive functional for Markov processes, which can be traced back to the research of Markov processes by Dynkin [39], Blumenthal and Getoor [40], and further development in [41,42,43,44]. Specifically, Dynkin [45] put forward the necessity and importance of additive Brownian motions in the research of Brownian sheets, and then he studied additive functionals of several independent time-reversible Markov processes in [46]. It is well-known that additive one-parameter processes locally resemble multi-parameter processes (e.g., additive Brownian motions locally resemble a Brownian sheet, additive Lévy processes locally resemble a Lévy sheet, and additive stable processes locally resemble a multi-parameter stable process), and there is a lot of research done on the corresponding research on the additive one-parameter processes. Additionally, multi-parameter potential theory provides another probabilistic motivation for the study of the additive processes. In fact, additive one-parameter processes are connected with a natural class of energy forms and their corresponding capacities; see [23,24,47] for a detailed discussion. At this point, we have to mention that Hirsch and Song [48] introduced a new Skorohod topology for functions of several variables. They defined the notion of complete N-parameter symmetric Markov processes and proved a maximal inequality, which implies the continuity of additive functionals associated with finite energy measures for these processes and their Bochner subordinates. It is our firm conviction that there are external motivations for the study of additive multi-parameter processes, and it has been actively investigated from different points of view; see Khoshnevisan, Xiao, and Zhong [49,50] for detailed discussion and the bibliography for further works in this area. First of all, additive processes play an important role in the study of other more interesting multi-parameter processes since they locally resemble multi-parameter processes, such as Brownian sheet, fractional Brownian sheet and stable sheet, and also because they are more amenable to analysis. For example, locally and with time suitable rescaling, the Brownian sheet closely resembles additive Brownian motions (see e.g., [7,51]). They also arise in the theory of intersection and self-intersection local times of Brownian processes (see, e.g., [47,52]). In view of the available literature and current research, only additive one-parameter processes have been defined and studied, such as additive Brownian motion [51,53], additive Lévy processes [47,49], and additive stable processes [54]. Another caveat worth mentioning is that “additive Brownian sheets” are introduced for the first time in the existing literature. Thus, we expect that something may still be gained from new derivations of existing results and would prefer some of which are improved here.

On the basis of the above definition of $\mathrm{ABS}$ $\mathbb{X}={⨁}_{n=1}^k{W}_n$, the global and the local sample function properties of $\mathbb{X}={⨁}_{n=1}^k{W}_n$ could be investigated in the primary aim. Some of our work is motivated by Symanzik [55], Westwater [56] and Wolpert [57], as they focused on studying the quantum field. In particular, the local times would seem to be ideally suited to the study of the nondifferentiable functions and, consequently, provides a useful tool in the analysis of the sample paths of additive Brownian sheets. In this, and a companion paper, we focus on the local times for additive Brownian sheets. The main objective of our work is twofold. We study the existence and joint continuity of the local times for additive Brownian sheets $\mathbb{X}={⨁}_{n=1}^k{W}_n$ in the first place. Secondly, we establish the Hölder law for the increment of the local times. Furthermore, the main theorems of our paper are conducive to solving and establishing the random geometrical surfaces and fractal characteristics generated by Brownian motions and Brownian sheets. In tackling local times for ABSs, we take advantage of the powerful Fourier analytic methods improved by Berman (cf. [27,58]). Our standard reference is the monograph [28]. Due to the complex dimensions of the parameter, there are certain challenges and difficulties in exploring the additive Brownian sheets $\mathbb{X}={⨁}_{n=1}^k{W}_n$. To obtain more details on those, see Section 3 below. Our argument is based on the additive structure of additive Brownian sheets and the independent increment property of Brownian sheets. The new results on additive Brownian sheets derived in this paper include the main results on Brownian motions and Brownian sheets in some previous papers.

Within the analysis of our paper, an additional purpose of this work is an extension of the results presented in [52,59] to K independent Brownian sheets in ${\mathbb{R}}^d$ with different parameters. In a sense, this approach should technically have the advantage that the underlying probability space does not depend on any index of parameters in regular space ${\mathbb{R}}^d$ ($d⩽3$). As a consequence, one may study the intersection local time of any two independent Brownian sheets in ${\mathbb{R}}^d$ ($d⩽3$), without any restriction on the corresponding parameters. For more details on intersection local time, see the last part of Section 2. Specifically, the problem of intersection (confluences) is closely related to that of multiple points for a single process. Then, following the result of ([60] Theorem 1.1), we show a new simple indirect approach to prove the existence of the double points of a Brownian sheet. We hope this work supports and helps researchers in constructing reasonable models in polymer physics and quantum field theory.

The rest of the paper is arranged as follows. We present some sample function properties of additive Brownian sheets in Section 2, including the local times and its Hölder law. Some applications are also indicated for intersection behaviors of two independent Brownian sheets. These conclusions give precise information about the intersection local times for independent Brownian sheets. The basic lemmas and the proof of the main results are given in Section 3.

2. Preliminaries and Main Results

Let us begin with some standard notations in this section. Throughout this paper, we shall study with the additive Brownian sheets $\mathbb{X}\left(t_1,\dots ,t_k\right)$, which are constructed in Section 1 on the complete probability space $\left(\Omega ,\mathcal{F},\mathbb{P}\right)$. For convenience, sometimes we also use $\mathbb{X}\left(\mathrm{t}\right)$ to denote the abbreviated version of $\mathbb{X}\left(t_1,\dots ,t_k\right)$, where $\mathrm{t}=\left(t_1,\dots ,t_k\right)$. For any $k\in \mathbb{N}$, there is underlying parameter space, which is ${\mathbb{R}}_{+}^{{⨁}_{n=1}^k{N}_n}={\displaystyle \prod _{n=1}^k}{[0,\infty )}^{N_n},N_n\in \mathbb{N},n=1,\dots ,k$, throughout. A typical parameter, $\mathrm{t}\in {\mathbb{R}}^{{⨁}_{n=1}^k{N}_n}$ is written as $\mathrm{t}=\left(t_1,\dots ,t_k\right)$, where $t_n=\left(t_{n,1},\dots ,t_{n,N_n}\right),n=1,\dots ,k$, coordinate-wise. Sometimes we also write $\mathrm{t}$ as $\langle c\rangle$, if $t_{n,1}=\cdots =t_{n,N_n}=c\in \mathbb{R},n=1,\dots ,k$. Particularly, there is another underlying parameter space ${\mathbb{R}}_{>}^{{⨁}_{n=1}^k{N}_n}=\left\{\mathrm{t}:\mathrm{t}=\left(t_1,\dots ,t_k\right)\in {\mathbb{R}}_{+}^{{⨁}_{n=1}^k{N}_n},t_{n,i}>0,i=1,\dots ,N_n,n=1,\dots ,k\right\}$. There is a natural partial ordering, “≼”, on ${\mathbb{R}}^{{⨁}_{n=1}^k{N}_n}$. Namely, $s\preccurlyeq t$ if and only if $s_{n,i}⩽t_{n,i}$ for all $n=1,\dots ,k$ and $i=1,\dots ,N_n$. When $s\preccurlyeq t$, we define the closed interval $[\mathrm{s},\mathrm{t}]={\displaystyle \prod _{n=1}^k}[s_n,t_n]$, where $[s_n,t_n]={\displaystyle \prod _{i=1}^{N_n}}[s_{n,i},t_{n,i}],n=1,\dots ,k$. Throughout, we let $A_k$ denote the class of all ${⨁}_{n=1}^k{N}_n$-dimensional intervals $Q\subset {\mathbb{R}}_{+}^{{⨁}_{n=1}^k{N}_n}$ of the type $Q=(\mathrm{s},\mathrm{t}]={\displaystyle \prod _{n=1}^k}(s_n,t_n]$, where $(s_n,t_n]={\displaystyle \prod _{i=1}^{N_n}}(s_{n,i},t_{n,i}]$ with $0⩽s_{n,i}<t_{n,i},n=1,\dots ,k,i=1,\dots ,N_n$. In some parts, ${}_Q\mathrm{t}$ (resp. $\mathrm{t}^Q$) is used to denote the lower (resp. upper) vertex of an interval $Q\in A_k$ with respect to “≼”, and so that $Q\in A_k$ can be rewritten as $Q=({}_Q\mathrm{t},\mathrm{t}^Q]$. The m-dimensional Lebesgue measure is denoted by $\lambda$, no matter the value of the integer m. Following Orey and Pruitt [2], we use $\delta (\mathrm{s},\mathrm{t})$ to denote the Lebesgue measure $\lambda \left(\right(0,\mathrm{s}]\Delta (0,\mathrm{t}\left]\right)$ of the symmetric difference of two intervals $(0,\mathrm{s}],(0,\mathrm{t}]\in A_k$, and simply by $\delta \left(\mathrm{t}\right)$ in case $\mathrm{s}=0$ (i.e., $\delta \left(\mathrm{t}\right)=\lambda \left(\right(0,\mathrm{t}\left]\right)$). The state space ${\mathbb{R}}^d$ is endowed with the $L^2$ Euclidean norm $\parallel ·\parallel$ and the corresponding dot product $x·y={\displaystyle \sum _{i=1}^d}{x}_i{y}_i(x,y\in {\mathbb{R}}^d)$.

Before we present our main results, we sketch the general definition of the local times for $\mathbb{X}={⨁}_{n=1}^k{W}_n$. Because $\mathbb{X}={⨁}_{n=1}^k{W}_n:{\mathbb{R}}_{+}^{{⨁}_{n=1}^k{N}_n}\to {\mathbb{R}}^d$ is a Borel vector field, then for Borel set $Q\subset {\mathbb{R}}_{+}^{{⨁}_{n=1}^k{N}_n}$, we can define the occupation measure ${\mu }_Q$ on ${\mathbb{R}}^d$ by

$$\begin{array}{c}\hfill {\mu }_Q\left(B\right)=\lambda \left({\mathbb{X}}^{-1}\left(B\right)\cap Q\right),B\in B\left({\mathbb{R}}^d\right).\end{array}$$

If ${\mu }_Q$ is absolutely continuous with respect to $\lambda$, we say that $\mathbb{X}$ has local time on Q and define the local time, $L(Q,x)$, as the Radon-Nikodým derivative of ${\mu }_Q$ with respect to $\lambda$, i.e.,

$$\begin{array}{c}\hfill L(Q,x)={\displaystyle \frac{\mathrm{d}{\mu }_Q}{\mathrm{d}\lambda }}\left(x\right),x\in {\mathbb{R}}^d.\end{array}$$

Remark 1. 

Formally, the local times for $\mathbb{X}={⨁}_{n=1}^k{W}_n$ can be defined by

$$\begin{array}{c}\hfill L(Q,x)={\int }_Q{\delta }_x\left(\mathbb{X}\left(\mathrm{t}\right)\right)\mathrm{d}\mathrm{t},\end{array}$$

where ${\delta }_x(·)$ is the Dirac’s delta function at x. The fact that $\mathbb{X}$ is a random field, i.e., $\mathbb{X}=\mathbb{X}(\mathrm{t},⊋)$, where ω is a point in the probability space $\left(\Omega ,\mathcal{F},\mathbb{P}\right)$, we say that $\mathbb{X}$ has a local time on Q, if $\mathbb{X}(·,\omega )$ has a local time on Q for almost all ω. We suppress ω in our formulas. Our standard reference about local times is the monograph [61], to which the reader is referred for details on local times.

Let us recall the original idea, which is the initial motivation for structuring the additive Brownian sheets $\mathbb{X}={⨁}_{n=1}^k{W}_n$ in this paper. That is, the confluence behavior between two independent Brownian sheets $W_1,W_2$. Naturally, the problem is studying the set of “confluences” $x\in {\mathbb{R}}^d$ where two strands intersect, and also the set of two-tuples of times $\mathrm{t}=\left(t_1,t_2\right)\in {\mathbb{R}}_{+}^{N_1+N_2}$ at which confluences occur, that is $x=W_1\left(t_1\right)=W_2\left(t_2\right)$. There is a natural measure tool, which is called the intersection local time, that provides an appropriate approach to study the above problems. In fact, the intersection local times of $W_1$ and $W_2$ are formally defined as

$$\begin{array}{c}\hfill I\left(W_1,W_2,Q_1\times Q_2\right)=\underset{Q_1\times Q_2}{\int \int }{\delta }_0\left(W_1\left(t_1\right)-W_2\left(t_2\right)\right)\mathrm{d}{t}_1\mathrm{d}{t}_2.\end{array}$$

According to Lévy’s point of view, the intersection local times $I\left(W_1,W_2,Q_1\times Q_2\right)$ can be regarded as the “amount of time spent by $W_1-W_2$ at 0 during $Q_1\times Q_2$”. Specifically, $I\left(W_1,W_2,Q_1\times Q_2\right)$ can be formally rewritten as the following quantity in quantum field theory (cf. [52]),

$$\begin{array}{c}\hfill I\left(W_1,W_2,Q_1\times Q_2\right)={\int }_{{\mathbb{R}}^d}\left({\int }_{Q_1}{\delta }_x\left(W_1\left(t_1\right)\right)\mathrm{d}{t}_1{\int }_{Q_2}{\delta }_x\left(W_2\left(t_2\right)\right)\mathrm{d}{t}_2\right)\mathrm{d}x.\end{array}$$

(1)

In view of the above representation, $I\left(W_1,W_2,Q_1\times Q_2\right)$ could be described as a measure of the amount of time of the mutual intersection of the independent paths for $W_1$ and $W_2$ in the time interval $Q_1\times Q_2$. Notice that the intersection local times of $W_1$ and $W_2$ are exactly the local time of confluent Brownian sheets $W_1-W_2$ at zero. Furthermore, it is easy to see that $W_1-W_2$ and $W_1+W_2$ have the same finite dimensional distributions. As pointed out above, the local time of additive Brownian sheets $\mathbb{X}={⨁}_{n=1}^2{W}_n$ at zero would be equivalent to the intersection local time of $W_1$ and $W_2$. If properly defined, there is $I\left(W_1,W_2,Q_1\times Q_2\right)=L\left(Q_1\times Q_2,0\right)$ under the condition $d<2\left(N_1+N_2\right)$ for the reasons above. Consequently, some results for intersection local time of two independent Brownian sheets can be obtained as specific examples. We shall come back to these with detailed descriptions in the last part of this section. Additionally, we establish the large deviations for the intersection local time of two independent Brownian sheets in Proposition 4. It is worthwhile pointing out that the large deviations of the intersection local times for independent multi-parameter processes remain essentially open, and to the best of the author’s knowledge, this is the first result of Brownian sheets.

Remark 2. 

Notice that there is an analogy between the collision local time and intersection local time for two independent Brownian sheets. Specifically, the collision local time of two independent Brownian sheets $W_1=\left\{W_1\left(t\right)\in {\mathbb{R}}^d;t\in {\mathbb{R}}_{+}^N\right\}$ and $W_2=\left\{W_2\left(t\right)\in {\mathbb{R}}^d;t\in {\mathbb{R}}_{+}^N\right\}$ is formally defined by

$$\tilde{L}(W_1,W_2,Q)={\int }_Q{\delta }_0\left(W_1\left(t\right)-W_2\left(t\right)\right)\mathrm{d}t.$$

It is a measure of the amount of time for which the trajectories of $W_1$ and $W_2$ collide on the time interval Q. There is scarce research on the collision local time of two independent multi-parameter processes other than two independent one-parameter processes [62,63,64,65]. By chance, we can use similar methods to study the collision local time of two independent Brownian sheets $W_1$ and $W_2$, and some related results are shown in elsewhere to shorten the length of this paper.

Starting from now, we briefly give the main results of local times for additive Brownian sheets. The purposes of Theorems 1 and 2 are to present the basic properties about the existence and joint continuity of the local times for additive Brownian sheets.

Theorem 1. 

Let $\mathbb{X}={⨁}_{n=1}^k{W}_n$ be $\left(N_1,\dots ,N_k;d\right)-\mathrm{ABS}$, if $d<2{\displaystyle \sum _{n=1}^k}{N}_n$, then for any $Q\in A_k$, $\mathbb{X}={⨁}_{n=1}^k{W}_n$ have local times $\left\{L(Q,x),x\in {\mathbb{R}}^d\right\}$ on Q, $L(Q,x)\in {\mathcal{L}}^2(\mathrm{d}\lambda ,{\mathbb{R}}^d)a.s.$ and admits the following representation:

$$\begin{array}{c}\hfill L(Q,x)={\left(2\pi \right)}^{-d}{\int }_{{\mathbb{R}}^d}exp\left\{-\mathbf{i}y·x\right\}\mathrm{d}y{\int }_{Q}exp\left\{\mathbf{i}y·\mathbb{X}\left(\mathrm{t}\right)\right\}\mathrm{d}\mathrm{t}.\end{array}$$

(2)

Remark 3. 

The result of Theorem 1 is sharp. When $N_1=\cdots =N_k=1$ and $k=1$, they contain the corresponding results for the additive Brownian motions and Brownian sheet, respectively. Specifically, the sufficient condition $d<8$ for the existence of intersection local time for two independent ${\mathbb{R}}^d$-valued two-parameter Brownian sheets in ([20] Corollary 3.2) is the special case in Theorem 1 (i.e., $k=2,N_1=N_2=2$).

With the technical assistance of the Fourier analysis, it is not difficult to obtain the sufficient condition of the existence of the local times for $\mathrm{ABS}$. However, it is difficult to find a necessary and sufficient condition for the existence of the local times for $\mathrm{ABS}$ in the general case. Although it is only a sufficient condition in the general case, it can be strengthened as a sufficient and necessary condition in a particular case. Indeed, when $N_1=\cdots =N_k=1$ for any $k\in \mathbb{N}$, then for any $Q\in A_k$, $\mathbb{X}={⨁}_{n=1}^k{W}_n$ have local times $L(Q,x)\in {\mathcal{L}}^2(\mathrm{d}\lambda ,{\mathbb{R}}^d)a.s.$ if and only if $d<2k$ (see Khoshnevisan, Xiao and Zhong [49]). In a more special case, if $\left(N_1,\dots ,N_k;d\right)-\mathrm{ABS}$ $\mathbb{X}={⨁}_{n=1}^k{W}_n$ run with nonlinear clocks [66], that is $W_n\left(t_n\right)=W_n\left(F_{n,1}\left(t_{n,1}\right),\dots ,F_{n,N_n}\left(t_{n,N_n}\right)\right)$ for $n=1,\dots ,k$, where $F_{n,l}:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}(n=1,\dots ,k,l=1,\dots ,N_n)$ is non-negative nondecreasing functions that satisfy the following bi-Lipschitz conditions: there exist constants $K_{n,l}>0$ such that

$${\displaystyle \frac{1}{K_{n,l}}}|t_{n,l}-s_{n,l}| ⩽ |F_{n,l}\left(t_{n,l}\right)-F_{n,l}\left(s_{n,l}\right)| ⩽K_{n,l}|t_{n,l}-s_{n,l}|,\forall t_{n,l},s_{n,l}\in {\mathbb{R}}_{+},$$

then for any $Q\in A_k$, it has local times $L(Q,x)\in L^2(\mathrm{d}\lambda ,{\mathbb{R}}^d)a.s.$ if and only if $d<2{\displaystyle \sum _{n=1}^k}{N}_n$, which is followed with the same arguments of Wu [66] and we omit the details here.

For $Q\in A_k$ and $d=1$, there is an interesting formula for (2), which is similar to the Newton–Leibniz formula in calculus. In fact, using the similar approach of Berman [58], we have

$$\underset{R\to \infty }{lim}\mathbb{E}{\left|L_R(Q,x)-L(Q,x)\right|}^2=0,x\in \mathbb{R},$$

where $L_R(Q,x)={\displaystyle \frac{1}{2\pi }}{\int }_{-R}^{R}exp\left\{-\mathbf{i}y·x\right\}\mathrm{d}y{\int }_{Q}exp\left\{\mathbf{i}y·\mathbb{X}\left(\mathrm{t}\right)\right\}\mathrm{d}\mathrm{t}.$

Then, for any $a,b\in \mathbb{R}$, it holds

$$\underset{R\to \infty }{lim}{\int }_a^b{L}_R(Q,x)\mathrm{d}x={\mu }_Q\left(b\right)-{\mu }_Q\left(a\right),$$

where ${\mu }_Q\left(x\right)={\int }_Q{\mathbf{1}}_x\left(\mathbb{X}\left(\mathrm{t}\right)\right)\mathrm{d}\mathrm{t},x\in \mathbb{R}$.

More generally, let us fix an interval $Q\in A_k,B\in B\left({\mathbb{R}}^d\right)$ and consider the occupation measure ${\mu }_Q\left(B\right)$ of ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$, i.e., ${\mu }_Q\left(B\right)={\int }_Q{\mathbf{1}}_B\left(\mathbb{X}\left(\mathrm{t}\right)\right)\mathrm{d}\mathrm{t}$. Applying Parseval’s identity, Theorem 1 follows that ${\mu }_Q\left(B\right)={\int }_{B}L(Q,x)\mathrm{d}x$ under the condition $d<2{\displaystyle \sum _{n=1}^k}{N}_n$. By means of simple calculus, we arrive at

$$\mathbb{E}\left(L(Q,0)\right)={\int }_Q{\left(2\pi {\displaystyle \sum _{n=1}^k}{\displaystyle \prod _{i=1}^{N_n}}{t}_{ni}\right)}^{-d/2}\mathrm{d}\mathrm{t}>0.$$

Therefore, $L(Q,0)>0$ with positive probability. Since the mapping: $Q\mapsto L(Q,0)$ is supported in $\mathbb{X}\left(0\right)$, with positive probability, $Q\cap \mathbb{X}\left(0\right)\ne ⌀$. An application of Kolmogorov’s $0-1$ law shows that $\mathbb{X}\left(0\right)\ne ⌀a.s.$ Then, that result can be extended to the following Proposition, which is motivated by the work of Khoshnevisan [9].

Proposition 1. 

Let $\mathbb{X}={⨁}_{n=1}^k{W}_n$ be $\left(N_1,\dots ,N_k;d\right)-\mathrm{ABS}$, then for any $a\in {\mathbb{R}}^d$,

$$\begin{array}{c}\hfill \mathbb{P}\left(\exists \mathrm{t}\in {\mathbb{R}}_{+}^{{⨁}_{n=1}^k{N}_n}:\mathbb{X}\left(\mathrm{t}\right)=a\right)=\left\{\begin{array}{cc}\hfill 1,& \mathrm{if}d<2{\displaystyle \sum _{n=1}^k}{N}_n,\hfill \\ \hfill 0,& \mathrm{if}d⩾2{\displaystyle \sum _{n=1}^k}{N}_n.\hfill \end{array}\right.\end{array}$$

(3)

Remark 4. 

The proof of Proposition 1 follows from the similar approach given by Khoshnevisan [9] and Kaufman [67], and so we omit the details here. The singletons $\left\{a\right\}$ in Proposition 1 are said to be polar in the language of Markov processes. Proposition 1 improves the characterizations of polar sets for Brownian sheets obtained by Khoshnevisan ([9] Theorem 1). Additionally, there is an interesting result in the special situation of Proposition 1. In the case $k=2, a=0$, we observe that

$$\begin{array}{c}\hfill \mathbb{P}\left(\exists (t_1,t_2)\in {\mathbb{R}}_{+}^{N_1+N_2}:W_1\left(t_1\right)=W_2\left(t_2\right)\right)=\left\{\begin{array}{cc}\hfill 1,& \mathrm{if}d<2(N_1+N_2),\hfill \\ \hfill 0,& \mathrm{if}d⩾2(N_1+N_2),\hfill \end{array}\right.\end{array}$$

(4)

by the fact that $W_1-W_2$ and $W_1+W_2$ have the same finite dimensional distributions. This means that two independent Brownian sheets $W_1$ and $W_2$ intersect with probability one.

In a rather special situation with $N_1=N_2=N$, it is clear from the proof process of Theorem 1 that $\mathbb{E}{\int }_{{\mathbb{R}}^d}{\int }_Q{\int }_{Q}exp\left\{\mathbf{i}\gamma ·(\mathbb{X}\left({\mathrm{t}}^1\right)-\mathbb{X}\left({\mathrm{t}}^2\right))\right\}\mathrm{d}{\mathrm{t}}^1\mathrm{d}{\mathrm{t}}^2\mathrm{d}\gamma <\infty$ remains holding if we put $Q\in A_2^{*}$ in it, where $A_2^{*}$ denote the class of intervals in $A_2$, which do not touch the boundary and has the disjoint subinterval (e.g., $A_2^{*}=\left\{Q=(\mathrm{s},\mathrm{t}]\in A_2:0<s_{1,i}<t_{1,i}<s_{2,i}<t_{2,i},i=1,\dots ,N\right\}$). Following with the same argument as that of Khoshnevisan [9], Kaufman [67] and Kahane [68], we have a proposition as below.

Proposition 2. 

Let $W_1$ and $W_2$ be independent N-parameter Brownian sheets in ${\mathbb{R}}^d$, then

$$\begin{array}{c}\hfill \mathbb{P}\left(\exists (t_1,t_2)\in 𝒯:W_1\left(t_1\right)=W_2\left(t_2\right)\right)=\left\{\begin{array}{cc}\hfill 1,& \mathrm{if}d<4N,\hfill \\ \hfill 0,& \mathrm{if}d⩾4N,\hfill \end{array}\right.\end{array}$$

(5)

where $𝒯=\left\{(s,t)\in {\mathbb{R}}_{>}^{2N}:s_i\ne t_i,i=1,\dots ,N\right\}$.

Remark 5. 

There is a similar proposition concerning the existence of intersections for two independent Brownian sheets given by Chen [69]. That is, if $d⩾4N$, then

$$\begin{array}{c}\hfill \mathbb{P}\left(W_1\left({\mathbb{R}}_{+}^N\right)\cap W_2\left({\mathbb{R}}_{>}^N\right)=Ø\right)=1.\end{array}$$

It is worth reviewing an interesting result, which is closely related to (5) in Dalang et al. ([60] Theorem 1.1). That is, choose and fix a Borel set $A\subseteq {\mathbb{R}}^d$, then

$$\begin{array}{c}\hfill \mathbb{P}\left(\exists (u_1,u_2)\in 𝒯:W\left(u_1\right)=W\left(u_2\right)\in A\right)>0\end{array}$$

(6)

if and only if

$$\begin{array}{c}\hfill \mathbb{P}\left(\exists (u_1,u_2)\in 𝒯:W_1\left(u_1\right)=W_2\left(u_2\right)\in A\right)>0,\end{array}$$

(7)

where $W,W_1,W_2$ are ${\mathbb{R}}^d$-valued Brownian sheets with N-parameter, $W_1$ and $W_2$ are independent unrelated to W. Let $A={\mathbb{R}}^d$ in (6) and (7), we conclude the following

$$\begin{array}{c}\hfill \mathbb{P}\left(\exists (u_1,u_2)\in 𝒯:W\left(u_1\right)=W\left(u_2\right)\right)>0\end{array}$$

(8)

if and only if

$$\begin{array}{c}\hfill \mathbb{P}\left(\exists (u_1,u_2)\in 𝒯:W_1\left(u_1\right)=W_2\left(u_2\right)\right)>0.\end{array}$$

(9)

This implies that two independent Brownian sheets $W_1$ and $W_2$ intersecting with positive probability are equivalent to one Brownian sheet W self-intersecting with positive probability. As a consequence, we obtain a new approach to prove the existence of the double points of the Brownian sheet. Actually, combining (5), (8) and (9) to deduce the following by dint of ([60] Theorem 1.1) and Taylor’s theorem ([36] pp. 523–525).

Corollary 1. 

A ${\mathbb{R}}^d$-valued N-parameter Brownian sheet W has double points if and only if $d<4N$.

Remark 6. 

(1) 

In fact, there are interesting results in the special case of $d<4N$. If $d>2N$, we have

$$\mathbb{P}(M_2\cap A\ne Ø)>0\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\mathrm{Cap}}_{2(d-2N)}\left(A\right)>0,$$

where $A\in B\left({\mathbb{R}}^d\right)$, $M_2=\left\{x\in {\mathbb{R}}^d:W\left(u_1\right)=W\left(u_2\right),u_1,u_2\in {\mathbb{R}}_{>}^N\right\}$ and ${\mathrm{Cap}}_{2(d-2N)}(·)$ denotes the Bessel–Riesz capacity in dimension $2(d-2N)\in \mathbb{R}$. Moreover, if $d=2N$, it holds $\mathbb{P}(M_2\cap A\ne Ø)>0$ if and only if there is a probability measure μ, which is compactly supported in A and satisfies that

$${\int }_{{\mathbb{R}}^{2d}}{\left|{log}_{+}{\left(\right|x-y|}^{-1})\right|}^2\mu \left(\mathrm{d}x\right)\mu \left(\mathrm{d}y\right),$$

where ${log}_{+}\left(z\right)=1\vee log\left(z\right)$ for $z⩾0$. For more details, we refer the reader to Dalang, Khoshnevisan, Nualart, Wu and Xiao [60], Dalang and Mueller [70].

(2) 

In the case of $N=1$, Corollary 1 indicates that Brownian motions have double points if and only if $d⩽3$. In fact, there is abundant literature on double points for Brownian motions, and the reader could be cross-referenced to the monographs [16,71,72,73,74,75].

(3) 

For any $N>1$, a ${\mathbb{R}}^d$-valued N-parameter Brownian sheet W has double points when $d<4N$, and has no double points when $d>4N$. This is known to all and much simpler to derive; however, there does not seem to be an elementary way to prove that a Brownian sheet does not have double points in the critical case $d=4N$. Corollary 1 asserts that the Brownian sheet has no double points in the critical case that $d=4N$, which provides an indirect and alternative approach to solving this old and difficult problem.

Subsequently, there is a natural question that is whether there exists a version of the local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ which is jointly continuous in both state space and time parameter variables. The answer to this question is in the affirmative, but it is significantly difficult to solve due to the fact that ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ does not have the property of local nondeterminism (LND), which is an important part of a sufficient condition for the existence of a jointly continuous version of the local times for some general Gaussian fields (cf. Berman [27], Pitt [76] and Cuzick [77]). Fortunately, partially inspired by Ehm [28], we establish a sufficient condition for the existence of jointly continuous local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$. In part of the proving, we also refer to Rosen [78,79], and Geman, Horowitz and Rosen [59], in which the local time of other stochastic processes are discussed by some methods without applying LND.

Theorem 2. 

Let $\mathbb{X}={⨁}_{n=1}^k{W}_n$ be $\left(N_1,\dots ,N_k;d\right)-\mathrm{ABS}$, if $d<2{\displaystyle \sum _{n=1}^k}{N}_n$, then for any $Q\in A_k$, $\mathbb{X}={⨁}_{n=1}^k{W}_n$ have jointly continuous local times $\left\{L(Q,x),x\in {\mathbb{R}}^d\right\}$ on $Qa.s.$

Remark 7. 

The result of Theorem 2 recovers the corresponding results for additive Brownian motions and Brownian sheets in the condition of $N_1=\cdots =N_k=1$ and $k=1$, respectively. Although a necessary and sufficient condition for joint continuity is gained by Barlow [80] in the case of $N=1,k=1$, the method of it is impossible to apply in the ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$. It seems more difficult to find a necessary and sufficient condition for the joint continuity of the local times for additive Brownian sheets.

In particular, setting $N_1=\cdots =N_k=N$ for any $k\in \mathbb{N}$ and $Q={\displaystyle \prod _{n=1}^k\prod _{i=1}^N}(s_{n,i},t_{n,i}]\in A_k$. Then, by Theorem 2 and the scaling property of the ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$, for any $d<2kN$ and any integrable function $f:{\mathbb{R}}^d\to \mathbb{R}$, we have the following convergence in the space $C\left({\mathbb{R}}_{+}^{kN}\right)$,

$$\begin{array}{ccc}\hfill m^{dN/2-kN}{\int }_{mQ}f\left({\mathbb{X}}_{\mathrm{t}}\right)\mathrm{d}\mathrm{t}& =& m^{dN/2}{\int }_{Q}f\left({\mathbb{X}}_{m\mathrm{t}}\right)\mathrm{d}\mathrm{t}\hfill \\ & \stackrel{\mathrm{law}}{=}& m^{dN/2}{\int }_{Q}f\left(m^{N/2}{\mathbb{X}}_{\mathrm{t}}\right)\mathrm{d}\mathrm{t}\hfill \\ & =& m^{dN/2}{\int }_{{\mathbb{R}}^d}f\left(m^{N/2}x\right)L(Q,x)\mathrm{d}x\hfill \\ & =& {\int }_{{\mathbb{R}}^d}f\left(x\right)L(Q,{\displaystyle \frac{x}{m^{N/2}}})\mathrm{d}x\hfill \\ & \stackrel{a.s.}{⟶}& L(Q,0){\int }_{{\mathbb{R}}^d}f\left(x\right)\mathrm{d}x,\hfill \end{array}$$

(10)

where $mQ={\displaystyle \prod _{n=1}^k\prod _{i=1}^N}(m{s}_{n,i},m{t}_{n,i}]$ and the last line we applied the continuity of $L(Q,x)$.

Obviously, for any $d<2kN$, (10) indicates the following asymptotic result on an additive functional of ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ driven by f

$$\begin{array}{c}\hfill m^{dN/2-kN}{\int }_{mQ}f\left({\mathbb{X}}_{\mathrm{t}}\right)\mathrm{d}\mathrm{t}\stackrel{\mathrm{law}}{⟶}L(Q,0){\int }_{{\mathbb{R}}^d}f\left(x\right)\mathrm{d}x,\forall Q\in A_k,\end{array}$$

where $\stackrel{\mathrm{law}}{⟶}$ denotes the convergence law as $m\to \infty$.

Additionally, there are some interesting asymptotic behaviors for special forms of ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$. For instance, applying the inverse Itô formula and the similar arguments of Peszat and Russo [81], Peszat and Talarczyk [82], if ${\int }_{\mathbb{R}}{\left|x\right|\parallel f\parallel }_{\mathrm{Var}}\mathrm{d}x<\infty$ and as $m\to \infty$, then the following holds

$$\begin{array}{c}\hfill m^{1/4}\left(m^{-1/2}{\int }_0^{ms}f\left({\tilde{\mathbb{X}}}_t\right)\mathrm{d}t-\tilde{L}((0,s),0){\int }_{\mathbb{R}}f\left(x\right)\mathrm{d}x\right)\stackrel{\mathrm{law}}{⟶}2\sqrt{c_f}{B}_{\tilde{L}((0,s),0)},\end{array}$$

where the real-valued processes ${\tilde{\mathbb{X}}}_t$ can be the following two forms

$${\displaystyle \sum _{n=1}^k}{W}_n\left({\left({\displaystyle \frac{t}{k}}\right)}^{1/N_n},\cdots ,{\left({\displaystyle \frac{t}{k}}\right)}^{1/N_n}\right)$$

or

$${\displaystyle \sum _{n=1}^k}{W}_n(\underset{N_n^l}{\underbrace{1,\cdots ,1}},{\displaystyle \frac{t}{k}},\underset{N_n^r}{\underbrace{1,\cdots ,1}})$$

here $0⩽N_n^l,N_n^r⩽N_n-1,N_n^l+N_n^r=N_n-1,$ $\tilde{L}$ is the local time of ${\tilde{\mathbb{X}}}_t$, $c_f={\int }_{\mathbb{R}}{\left({\int }_{-\infty }^{x}f\left(y\right)\mathrm{d}y-{\mathbf{1}}_{(0,+\infty )}{\int }_{\mathbb{R}}f\left(y\right)\mathrm{d}y\right)}^2\mathrm{d}x$ and B is a real-valued standard Brownian motion, which is independent of $\tilde{\mathbb{X}}$.

Moreover, let μ be a complex-valued measure with ${\parallel \mu \parallel }_{\mathrm{Var}}<\infty$, $\overline{\mu }(-B)=\mu \left(B\right),B\in B\left(\mathbb{R}\right)$, $\mu \left(\right\{0\left\}\right)=0$, and for any $n\in \mathbb{N}$ satisfies

$${\int }_{U_n}{\displaystyle \prod _{l=1}^n}{\left|{\displaystyle \sum _{i=1}^l}{x}_i\right|}^{-2}{\parallel \mu \parallel }_{\mathrm{Var}}^{\otimes n}\mathrm{d}x<\infty ,$$

where $U_n=\left\{x\in {\mathbb{R}}^n:{\displaystyle \sum _{i=1}^l}{x}_i\ne 0,l=1,\dots ,n\right\}$.

Then for any $f\left(x\right)={\int }_{\mathbb{R}}exp\left\{\mathbf{i}xy\right\}\mu \left(\mathrm{d}y\right)$, and as $m\to \infty$,

$$m^{-1/2}{\int }_0^{ms}f\left({\tilde{\mathbb{X}}}_t\right)\mathrm{d}t\stackrel{\mathrm{law}}{⟶}\sqrt{C_f}{B}_t,$$

where $C_f={\int }_{\{x+y=0\}}{\displaystyle \frac{2}{{\left|x\right|}^2}}\mu \left(\mathrm{d}x\right)\mu \left(\mathrm{d}y\right)={\int }_{\{x=y\}}{\displaystyle \frac{2}{{\left|x\right|}^2}}\mu \left(\mathrm{d}x\right)\overline{\mu \left(\mathrm{d}y\right)}$.

The case ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ has jointly continuous local times $\left\{L(Q,x),x\in {\mathbb{R}}^d\right\}$ on $Q\in A_{k}a.s.$, so it follows from Adler [83] that $L(·,x)$ could be extended to be a finite Borel measure supported on the level set

$${\mathbb{X}}_Q^{-1}\left(x\right)=\left\{\mathrm{t}\in Q:\mathbb{X}\left(\mathrm{t}\right)=x\right\}.$$

This enabled the local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ not only to be of interest but also to be a useful tool in investigating fractal properties of the sample paths for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$. As a special case of the ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$, the exact Hausdorff measure function for level sets of ${\mathbb{R}}^d$-valued N-parameters, the Brownian sheet is given by Lin [84]. From this, along with the similar argument of [84], we can obtain the exact Hausdorff measure function for level sets of ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ when $d<2{\displaystyle \sum _{n=1}^k}{N}_n$. That is $\varphi \left(r\right)=r^{{\displaystyle \sum _{n=1}^k}{N}_n-d/2}{\left(loglog(1/r)\right)}^{d/2}$, and we omit the detail to save pages.

Furthermore, it is natural to ask whether there are smoothness properties of local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$. The answer is in the affirmative. Indeed, by a result of Berman [58] and entirely analogous to the proof of Theorem 2, we can discover that the local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ become smooth in higher dimensional time parameters. In detail, if $d+2v<2{\displaystyle \sum _{n=1}^k}{N}_n$ for some $v\in \mathbb{N}$, then for any $Q\in A_k$, the local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ have all partial derivatives $\left\{{\displaystyle \frac{{\partial }^{P_1}\dots {\partial }^{P_d}}{{\partial }^{P_1}{x}_1\dots {\partial }^{P_d}{x}_d}}L(Q,x),x\in {\mathbb{R}}^d\right\}$ with order ${\displaystyle \sum _{l=1}^d}{P}_l⩽v$ and they are jointly continuous in $(Q,x)\in A_k\times {\mathbb{R}}^{d}a.s.$

Applying the similar estimation methods of Ehm [28] and Lévy’s dyadic expansion [85], we obtain the following result, which is concerned with the Hölder laws of local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$.

Theorem 3. 

Let $\mathbb{X}={⨁}_{n=1}^k{W}_n$ be $\left(N_1,\dots ,N_k;d\right)-\mathrm{ABS}$, and let $L(Q,x)$ be its continuous local times. If $d<2{\displaystyle \sum _{n=1}^k}{N}_n$, then there is a finite constant $c_1>0$ such that, for every $\mathrm{s}\in {\mathbb{R}}_{>}^{{⨁}_{n=1}^k{N}_n}$,

$$\begin{array}{c}\hfill \underset{r\to 0}{lim\; sup}\underset{x\in {\mathbb{R}}^d}{sup}{\displaystyle \frac{L\left(\left(\mathrm{s}-\langle r\rangle ,\mathrm{s}+\langle r\rangle \right],x\right)}{r^{{\displaystyle \sum _{n=1}^k}{N}_n-d/2}{\left(loglog{r}^{-1}\right)}^{d/2}}}⩽c_1\delta {\left(\mathrm{s}\right)}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}a.s.\end{array}$$

(11)

Remark 8. 

(1) 

Theorem 3 characterizes the irregular and intricate property of the ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ sample paths. Additionally, we can obtain another interesting law of the iterated logarithm of the local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ that follows along similar lines of (11). More precisely, if $d<2{\displaystyle \sum _{n=1}^k}{N}_n$, then

$$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim}\underset{\begin{array}{c}Q\in A_k,\\ Q\subset T,\lambda \left(Q\right)<\epsilon \end{array}}{sup}\underset{x\in {\mathbb{R}}^d}{sup}{\displaystyle \frac{L(Q,x)}{\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}{(log\lambda {\left(Q\right)}^{-1})}^{d/2}}}⩽c_2\delta {\left({}_T\mathrm{t}\right)}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}a.s.\end{array}$$

(12)

for every $T\in A_k$, here $c_2>0$. Observe that (11) and (12) are seen as the moduli of continuity of the local time increments for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$, which have a close relationship with the behaviors of the sample functions for ABS. Indeed, by (11) and (12), we see that

$$\begin{array}{c}\hfill \underset{r\to 0}{lim\; inf}\underset{\mathrm{s}\preccurlyeq \mathrm{t}\preccurlyeq \mathrm{s}+\langle r\rangle }{sup}{\displaystyle \frac{\parallel \mathbb{X}\left(\mathrm{t}\right)-\mathbb{X}\left(\mathrm{s}\right)\parallel }{{\left(r/loglog{r}^{-1}\right)}^{1/2}}}⩾c_1^{-1/d}\delta {\left(\mathrm{s}\right)}^{(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)/2}>0a.s.\end{array}$$

(13)

and

$$\begin{array}{c}\hfill \underset{r\to 0}{lim\; inf}\underset{\mathrm{s}\in T}{inf}\underset{\mathrm{s}\preccurlyeq \mathrm{t}\preccurlyeq \mathrm{s}+\langle r\rangle }{sup}{\displaystyle \frac{\parallel \mathbb{X}\left(\mathrm{t}\right)-\mathbb{X}\left(\mathrm{s}\right)\parallel }{{\left(r/log{r}^{-1}\right)}^{1/2}}}⩾c_2^{-1/d}{\left({\displaystyle \sum _{n=1}^k}{N}_n\right)}^{-1/2}\delta {\left(\mathrm{s}\right)}^{(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)/2}>0a.s.\end{array}$$

(14)

where we have used the fact that

$$\lambda \left(Q\right)={\int }_{{\mathbb{R}}^d}L(Q,x)\mathrm{d}x⩽\underset{x\in {\mathbb{R}}^d}{sup}L(Q,x){\left(\underset{\mathrm{s},\mathrm{t}\in Q}{sup}\parallel \mathbb{X}\left(\mathrm{s}\right)-\mathbb{X}\left(\mathrm{t}\right)\parallel \right)}^d.$$

Note that (14), for instance, reveals that the ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ are almost surely nowhere differentiable in ${\mathbb{R}}_{>}^{{⨁}_{n=1}^k{N}_n}$.

(2) 

Unfortunately, we only achieved the upper bounds for the moduli of continuity of local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$. The lower bounds about them are used to establish which will have implications on the fractal properties of sample paths for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$, and we will be looking at those very issues and items over the next several articles. More specifically, when $N_1=\cdots =N_k=1$ for any $k\in \mathbb{N}$, we state the following coarse Hölder estimates on the smoothness of the local times, which are inspired by the argument of Khoshnevisan, Xiao and Zhong [49]. Let $\mathrm{s}\in {\mathbb{R}}_{>}^k$ and $T\in A_k$ be fixed. If $d<2k$, then there exist positive constants $c_3$ and $c_4$ such that

$$\begin{array}{c}\hfill c_3^{-1}⩽\underset{r\to 0}{lim\; sup}\underset{x\in {\mathbb{R}}^d}{sup}{\displaystyle \frac{L\left(\left(\mathrm{s}-\langle r\rangle ,\mathrm{s}+\langle r\rangle \right],x\right)}{r^{k-d/2}{\left(loglog{r}^{-1}\right)}^{d/2}}}⩽c_{3}a.s.\end{array}$$

and

$$\begin{array}{c}\hfill c_4^{-1}⩽\underset{\epsilon \to 0}{lim}\underset{\begin{array}{c}Q\in A_k,\\ Q\subset T,\lambda \left(Q\right)<\epsilon \end{array}}{sup}\underset{x\in {\mathbb{R}}^d}{sup}{\displaystyle \frac{L(Q,x)}{\lambda {\left(Q\right)}^{1-d/\left(2k\right)}{(log\lambda {\left(Q\right)}^{-1})}^{d/2}}}⩽c_{4}a.s.\end{array}$$

Furthermore, following Chen [54], we also can obtain the law of the iterated logarithm in the above special case

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}{\displaystyle \frac{L\left({[0,t]}^k,x\right)}{t^{k-d/2}{\left(loglogt\right)}^{d/2}}}={\rho }_k{\left(2\pi \right)}^{-d}{\left({\displaystyle \frac{2}{d}}\right)}^{d/2}{\left(1-{\displaystyle \frac{d}{2k}}\right)}^{-(k-d/2)}a.s.\end{array}$$

and

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}\underset{x\in {\mathbb{R}}^d}{sup}{\displaystyle \frac{L\left({[0,t]}^k,x\right)}{t^{k-d/2}{\left(loglogt\right)}^{d/2}}}={\rho }_k{\left(2\pi \right)}^{-d}{\left({\displaystyle \frac{2}{d}}\right)}^{d/2}{\left(1-{\displaystyle \frac{d}{2k}}\right)}^{-(k-d/2)}a.s.,\end{array}$$

where

$$\begin{array}{c}\hfill {\rho }_k=\underset{{\parallel f\parallel }_2=1}{sup}{\int }_{{\mathbb{R}}^d}{\left[{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{f(x+y)f\left(y\right)}{\sqrt{1+{(x+y)}^2/2}\sqrt{1+y^2/2}}}\mathrm{d}y\right]}^k\mathrm{d}x,\end{array}$$

(15)

${\parallel f\parallel }_2={\left({\int }_{{\mathbb{R}}^d}{f}^2\left(x\right)\mathrm{d}x\right)}^{1/2}$ and $0<{\rho }_k⩽{\int }_{{\mathbb{R}}^d}{\displaystyle \frac{1}{{(1+x^2/2)}^k}}\mathrm{d}x<\infty$ as $d<2k$.

In the following, we begin to pay close attention to the intersection local times for two independent Brownian sheets. In fact, the mathematical research of various intersection local times were motivated by the models of statistical mechanics (see, e.g., [86,87]). Intersection local times for one-parameter processes (e.g., Brownian motions and random walks) have been studied for a long time in probability theory (see, e.g., [52,88,89]). However, for multi-parameter processes, and even for additive Brownian motions, results on intersection local times are much less complete. With the aid of the above methods in analyzing local times for $\mathrm{ABS}$, the relationship in the ‘space-time’ dimension of the intersection for two independent Brownian sheets can be characterized as reasonable. Actually, given any pair of parameters $N_1,N_2$, such that $d<2(N_1+N_2)$, according to the fact that $W_1-W_2$ and $W_1+W_2$ have the same finite dimensional distributions, the results in the case $k=2$ stated in the above theorems can naturally turn into the conclusions of the intersection local times for two independent Brownian sheets with the parameters $N_1,N_2$. It is well known that intersection local times can be used as a powerful tool in studying the intersection behaviors for some independent stochastic processes, yet the intersection behaviors have strong dimension dependence. In view of the following Corollary 2, we have come to the conclusion that for $d⩽3$, all intersection local times for two independent Brownian sheets are well-defined for all possible parameters $N_1$ and $N_2$.

Corollary 2. 

Let $W_1=\left\{W_1\left(t_1\right)\in {\mathbb{R}}^d;t_1\in {\mathbb{R}}_{+}^{N_1}\right\}$ and $W_2=\left\{W_2\left(t_2\right)\in {\mathbb{R}}^d;t_2\in {\mathbb{R}}_{+}^{N_2}\right\}$ be two independent Brownian sheets, if $d<2(N_1+N_2)$, then for any $Q_1\times Q_2\in A_2$, $W_1$ and $W_2$ have the intersection local times $I\left(W_1,W_2,Q_1\times Q_2\right)$ and admits the following similar ${\mathcal{L}}^2$ representation:

$$\begin{array}{c}\hfill I\left(W_1,W_2,Q_1\times Q_2\right)={\left(2\pi \right)}^{-d}{\int }_{{\mathbb{R}}^d}\mathrm{d}y{\int }_{Q_1\times Q_2}exp\left\{\mathbf{i}y·(W_1\left(t_1\right)-W_2\left(t_2\right))\right\}\mathrm{d}{t}_1\mathrm{d}{t}_2.\end{array}$$

(16)

Furthermore, under the same condition, $W_1$ and $W_2$ have almost surely continuous intersection local times on $Q_1\times Q_2\in A_2$.

Remark 9. 

In particular, the condition in the above corollary can be strengthened as a sufficient and necessary condition in the case of two independent Brownian motions (for more details, refer to Wu and Xiao [90]).

As an expression of the intersection local times for independent Brownian sheets $W_1$ and $W_2$, (16) provides a good description of the numerical characteristics of the intersection local times $I\left(W_1,W_2,Q_1\times Q_2\right)$. With more details, we give the following statement.

Proposition 3. 

Let $W_1=\left\{W_1\left(t_1\right)\in {\mathbb{R}}^d;t_1\in {\mathbb{R}}_{+}^{N_1}\right\}$ and $W_2=\left\{W_2\left(t_2\right)\in {\mathbb{R}}^d;t_2\in {\mathbb{R}}_{+}^{N_2}\right\}$ be two independent Brownian sheets, and let $I\left(W_1,W_2,Q_1\times Q_2\right)$ be their continuous intersection local times for any $Q_1\times Q_2\in A_2$ with $\delta {(}_{Q_1\times Q_2}\mathrm{t})\ne 0$. If $d<2(N_1+N_2)$, then it holds that

$$\begin{array}{c}\hfill \mathbb{E}I\left(W_1,W_2,Q_1\times Q_2\right)={\left(2\pi \right)}^{-d/2}{\int }_{Q_1\times Q_2}{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{-d/2}\mathrm{d}{t}_1\mathrm{d}{t}_2\end{array}$$

(17)

and

$$\begin{array}{cc}\hfill \mathrm{Var}I\left(W_1,W_2,Q_1\times Q_2\right)⩽& C\delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{(1-1/(N_1+N_2))d}\lambda {(Q_1\times Q_2)}^{2-d/(N_1+N_2)}\hfill \\ \hfill & -{\left(2\pi \right)}^{-d}{\left({\int }_{Q_1\times Q_2}{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{-d/2}\mathrm{d}{t}_1\mathrm{d}{t}_2\right)}^2,\hfill \end{array}$$

(18)

where $C=3^{2(N_1+N_2)}{\left(2\pi \right)}^{-2d}{\left({\displaystyle \frac{2}{\Gamma (1+2(1-d/\left(2(N_1+N_2)\right)))}}\right)}^{(N_1+N_2)}{\left({\int }_{{\mathbb{R}}^d}exp\{-{\displaystyle \frac{N_1+N_2}{2}}{\parallel \gamma \parallel }^2\}\mathrm{d}\gamma \right)}^2.$

It is obvious that (17) presents the explicit expression of the first moment for the intersection local times $I\left(W_1,W_2,Q_1\times Q_2\right)$ under certain conditions, and (18) implies that

$$\begin{array}{cc}\hfill \underset{a\in \mathbb{R}}{inf}\mathbb{E}{\left(I\left(W_1,W_2,Q_1\times Q_2\right)-a\right)}^2⩽& C\delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{(1-1/(N_1+N_2))d}\lambda {(Q_1\times Q_2)}^{2-d/(N_1+N_2)}\hfill \\ \hfill & -{\left(2\pi \right)}^{-d}{\left({\int }_{Q_1\times Q_2}{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{-d/2}\mathrm{d}{t}_1\mathrm{d}{t}_2\right)}^2,\hfill \end{array}$$

where $Q_1\times Q_2\in A_2$, $\delta {(}_{Q_1\times Q_2}\mathrm{t})\ne 0$ and $d<2(N_1+N_2)$.

However, (18) only gives a sharp upper bound estimate of variance for the intersection local times $I\left(W_1,W_2,Q_1\times Q_2\right)$. In order to investigate a rigorous meaning of the intersection local times for independent Brownian sheets $W_1$ and $W_2$, we consider a canonical approximate the Dirac’s delta function by the heat kernel

$$p_{ϵ,d}(x,y)={\displaystyle \frac{1}{{\left(2\pi ϵ\right)}^{d/2}}}exp\left\{-{\displaystyle \frac{{|y-x|}^2}{2ϵ}}\right\},x,y\in {\mathbb{R}}^d,ϵ>0.$$

Let $d<2(N_1+N_2)$, using the similar argument of Imkeller and Weisz [91], then for any $Q_1\times Q_2\in A_2$, there exists

$$\begin{array}{c}\hfill I\left(W_1,W_2,Q_1\times Q_2\right)=\underset{ϵ\to 0}{lim}{I}_{ϵ}(W_1,W_2,Q_1\times Q_2),\end{array}$$

(19)

where $I_{ϵ}(W_1,W_2,Q_1\times Q_2)={\int }_{Q_1\times Q_2}{p}_{ϵ,d}\left(W_1\left(t_1\right),W_2\left(t_2\right)\right)\mathrm{d}{t}_1\mathrm{d}{t}_2$.

In fact, similar to the method of Nualart and Ortiz-Latorre [92], we can show that for $d⩾4$ with all possible parameters $N_1$ and $N_2$,

$$\underset{ϵ\to 0}{lim}\mathbb{E}{I}_{ϵ}(W_1,W_2,Q_1\times Q_2)=+\infty$$

and

$$\underset{ϵ\to 0}{lim}\mathrm{Var}{I}_{ϵ}(W_1,W_2,Q_1\times Q_2)=+\infty ,$$

where $Q_1\times Q_2\in A_2$ just satisfy the condition $\delta {(}_{Q_1\times Q_2}\mathrm{t})=0$.

Additionally, using the semigroup property of the heat kernel $p_{ϵ,d}(x,y)$, we have

$$p_{ϵ,d}\left(W_1\left(t_1\right),W_2\left(t_2\right)\right)={\int }_{{\mathbb{R}}^d}{p}_{ϵ/2,d}\left(W_1\left(t_1\right),x\right)p_{ϵ/2,d}\left(W_2\left(t_2\right),x\right)\mathrm{d}x$$

and so

$$\begin{array}{ccc}& & I\left(W_1,W_2,Q_1\times Q_2\right)\hfill \\ & =& \underset{ϵ\to 0}{lim}{\int }_{Q_1\times Q_2}\left({\int }_{{\mathbb{R}}^d}{p}_{ϵ/2,d}\left(W_1\left(t_1\right),x\right)p_{ϵ/2,d}\left(W_2\left(t_2\right),x\right)\mathrm{d}x\right)\mathrm{d}{t}_1\mathrm{d}{t}_2\hfill \\ & =& \underset{ϵ\to 0}{lim}{\int }_{{\mathbb{R}}^d}\left({\int }_{Q_1}{p}_{ϵ/2,d}\left(W_1\left(t_1\right),x\right)\mathrm{d}{t}_1\right)\left({\int }_{Q_2}{p}_{ϵ/2,d}\left(W_2\left(t_2\right),x\right)\mathrm{d}{t}_2\right)\mathrm{d}x\hfill \\ & =& {\int }_{{\mathbb{R}}^d}{L}_1(Q_1,x)L_2(Q_2,x)\mathrm{d}x,\hfill \end{array}$$

(20)

where $L_1(Q_1,x)$ and $L_2(Q_2,x)$ denote the local time of independent Brownian sheets $W_1$ and $W_2$, respectively. Obviously, (20) induce (1) exactly.

In the following, let us provide another perspective to scan a part of expression (16) for the intersection local times $I\left(W_1,W_2,Q_1\times Q_2\right)$. In fact, it is natural to regard the Brownian sheet as functionals of the standard white noises by the white noise analysis. Namely, we can define a ${\mathbb{R}}^d$-valued N-parameter Brownian sheet W as

$$W\left(t\right):=\left(w_1\left({\otimes }_{i=1}^N[0,t_i]\right),\dots ,w_d\left({\otimes }_{i=1}^N[0,t_i]\right)\right),t_i⩾0,i=1,\dots ,N,$$

where $t=(t_1,\dots ,t_N)$ and $w_j,j=1,\dots ,d$ are independent one-dimensional white noises spread over ${\mathbb{R}}^N$. For $j=1,\dots ,d$, the process $w_j$ is a well defined vector-valued random measure with values in $L^N\left(\mathbb{P}\right)$ (cf. [36] for more details). In this way, applying white noise analysis and the similar approach of Oliveira et al. [93], we can conclude that the following Bochner integral, which is in the part of (16)

$${\mathrm{I}}_{\mathrm{Bochner}}\left(W_1\left(t_1\right)-W_2\left(t_2\right)\right):={\left(2\pi \right)}^{-d}{\int }_{{\mathbb{R}}^d}exp\left\{\mathbf{i}y·(W_1\left(t_1\right)-W_2\left(t_2\right))\right\}\mathrm{d}y$$

is a Hida distribution when $d<2(N_1+N_2)$.

Remark 10. 

Using the similar methods of [93], we also can show $I_{ϵ}(W_1,W_2,Q_1\times Q_2)$ is a Hida distribution under the same condition as above, that is $d<2(N_1+N_2)$.

The classical approach to proving the statement of the above result was initiated by Hida [94]. More specifically, the above Bochner integral contains white noise calculus, which gives rise to a powerful approach of asymptotic and renormalization properties of intersection local times (see, e.g., [95]). In particular, give a two-tuple of one-dimensional white noises $(w_{1,1},\dots ,w_{1,d})$ and $(w_{2,1},\dots ,w_{2,d})$, which spread over ${\mathbb{R}}^{N_1}$ and ${\mathbb{R}}^{N_2}$, respectively. For $n=1,2$, we define

$$\widetilde{W_n}\left(t_n\right):=\left(w_{n,1}\left({[0,t_n^{1/N_n}]}^{\otimes N_n}\right),\dots ,w_{n,d}\left({[0,t_n^{1/N_n}]}^{\otimes N_n}\right)\right)$$

or

$$\widetilde{W_n}\left(t_n\right):=\left({\tilde{w}}_{n,1}\left(t_n\right),\dots ,{\tilde{w}}_{n,d}\left(t_n\right)\right),$$

where ${\tilde{w}}_{n,j}\left(t_n\right)=w_{n,j}(\underset{N_{n,j}^l}{\overset{[0,1]\times \cdots \times [0,1]}{︸}}\times [0,t_n]\times \underset{N_{n,j}^r}{\overset{[0,1]\times \cdots \times [0,1]}{︸}}),0⩽N_{n,j}^l,N_{n,j}^r⩽N_n-1,N_{n,j}^l+N_{n,j}^r=N_n-1,j=1,\dots ,d.$

Then, it follows from the similar arguments of [93], if $d⩽3$ and for any $N_1,N_2\in \mathbb{N}$, we derive that

$$\begin{array}{c}\hfill {\mathrm{I}}_{\mathrm{Bochner}}\left({\tilde{W}}_1\left(t_1\right)-{\tilde{W}}_2\left(t_2\right)\right)\end{array}$$

(21)

is a Hida distribution.

Remark 11. 

Moreover, we can obtain the following S-transform formula of (21)

$$\begin{array}{ccc}& & S{\mathrm{I}}_{\mathrm{Bochner}}\left({\tilde{W}}_1\left(t_1\right)-{\tilde{W}}_2\left(t_2\right)\right)\left(\mathrm{f}\right)\hfill \\ & =& {\left(2\pi (t_1+t_2)\right)}^{-d/2}exp\left\{-{\displaystyle \frac{1}{2(t_1+t_2)}}{\displaystyle \sum _{j=1}^d}{\left[{\int }_{\mathbb{R}}\left(f_{1,j}\left(x\right){\mathbf{1}}_{[0,t_1]}-f_{2,j}\left(x\right){\mathbf{1}}_{[0,t_2]}\right)\mathrm{d}x\right]}^2\right\},\hfill \end{array}$$

where $\mathrm{f}=(f_{1,1},\dots ,f_{1,d},f_{2,1},\dots ,f_{2,d})\in S_{2d}\left(\mathbb{R}\right)$, $S_{2d}\left(\mathbb{R}\right)$ denote the Schwartz space of all the vector-valued test functions. The S-transform can characterize the space of Hida distributions perfectly, and the readers can refer to [93,96,97] for more details.

It is well known that the renormalized intersection local times of Brownian motion is a Hida distribution, which is shown by Watanabe [98,99]. It is natural to ask if there is a similar result of intersection local time for independent Brownian sheets. The answer is in the affirmative and also includes the subtracted counterparts of intersection local time for independent Brownian sheets. More specifically, let ${exp}_{\left(M\right)}\left(x\right)$ be the truncated exponential series, namely, ${exp}_{\left(M\right)}\left(x\right)={\displaystyle \sum _{m=M}^{\infty }}{\displaystyle \frac{x^m}{m!}}$. Then, one can follow the arguments of ([93] Theorem 7) to conclude that the Bochner integral

$$\begin{array}{cc}& {\mathrm{I}}_{\mathrm{Bochner}}^{\left(M\right)}\left({\tilde{W}}_1\left(t_1\right)-{\tilde{W}}_2\left(t_2\right)\right)\hfill \\ \hfill =& {\left(2\pi \right)}^{-d}{\int }_0^T\mathrm{d}{t}_1{\int }_0^T\mathrm{d}{t}_2{\int }_{{\mathbb{R}}^d}\mathrm{d}y{exp}_{\left(M\right)}\left\{\mathbf{i}y·({\tilde{W}}_1\left(t_1\right)-{\tilde{W}}_2\left(t_2\right))\right\},T>0\hfill \end{array}$$

(22)

is also a Hida distribution with additional condition $d<2(M+2)$. Notice that, (22) is the Taylor expansion of the intersection local time $I\left({\tilde{W}}_1,{\tilde{W}}_2,[0,T]\times [0,T]\right)$ if we take $M=0$. In this sense, we can regard each intersection local time as Hida distributions under condition $d⩽3$. However, what about in the situation $d⩾4$? In fact, in virtue of the theoretical study of regularity for renormalized self-intersection local times by Hu and Nualart [100,101], we can show that

$$I_{ϵ}\left({\tilde{W}}_1,{\tilde{W}}_2,[0,T]\times [0,T]\right)$$

is a Hida distribution for all dimensions $d⩾1$, where $I_{ϵ}$ is the same definition as the previous one in (19).

Furthermore, from similar arguments as in the proof of ([93] Proposition 8), we can derive the Wiener chaos expansion of ${\mathrm{I}}_{\mathrm{Bochner}}^{\left(M\right)}\left({\tilde{W}}_1\left(t_1\right)-{\tilde{W}}_2\left(t_2\right)\right)$ under the same condition $d<2(M+2)$ and omit the proof,

$$\begin{array}{c}\hfill {\mathrm{I}}_{\mathrm{Bochner}}^{\left(M\right)}\left({\tilde{W}}_1\left(t_1\right)-{\tilde{W}}_2\left(t_2\right)\right)(\overrightarrow{w_1},\overrightarrow{w_2})=\sum _{\mathrm{m}}\sum _{\mathrm{k}}〈:{\overrightarrow{w_1}}^{\otimes \mathrm{m}}:\otimes :{\overrightarrow{w_2}}^{\otimes \mathrm{k}}:,F_{\mathrm{m},\mathrm{k}}〉,\end{array}$$

(23)

where $\mathrm{m}=(m_1,\dots ,m_d)$, $\mathrm{k}=(k_1,\dots ,k_d)$ and the kernel functions

$$\begin{array}{c}\hfill F_{\mathrm{m},\mathrm{k}}=\left\{\begin{array}{cc}& {\pi }^{-d/2}{2}^{-(m+k+d)/2}{\displaystyle \frac{{(-1)}^{(m+3k)/2}}{\left(\frac{\mathrm{m}+\mathrm{k}}{2}\right)!}}\left(\begin{array}{c}\mathrm{m}+\mathrm{k}\hfill \\ \mathrm{m}\hfill \end{array}\right){\int }_0^T{\int }_0^T({\displaystyle \underset{j=1}{\overset{d}{⨂}}}\left({\mathbf{1}}_{[0,t_1]}^{\otimes m_j}\otimes {\mathbf{1}}_{[0,t_2]}^{\otimes k_j}\right)\hfill \\ & \times {(t_1+t_2)}^{-{\displaystyle \sum _{j=1}^d}(m_j+k_j+1)/2})\mathrm{d}{t}_1\mathrm{d}{t}_2,\mathrm{if}m+k⩾2M,\mathrm{and}\hfill \\ & \mathrm{mod}(m_j+k_j,2)=0,\hfill \\ & j=1,\dots ,d,\hfill \\ & 0,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

(24)

Combining (17), (22) and (24), for $d⩽3$, we arrive at the following result, which provides more quantitative information on the moment of the Wiener chaos expansion of ${\mathrm{I}}_{\mathrm{Bochner}}^{\left(M\right)}\left({\tilde{W}}_1\left(t_1\right)-{\tilde{W}}_2\left(t_2\right)\right)$ with $M=0$.

$$\begin{array}{cc}\hfill \mathbb{E}I\left({\tilde{W}}_1,{\tilde{W}}_2,[0,T]\times [0,T]\right)& =\mathbb{E}{\mathrm{I}}_{\mathrm{Bochner}}^{\left(0\right)}\left({\tilde{W}}_1\left(t_1\right)-{\tilde{W}}_2\left(t_2\right)\right)\hfill \\ \hfill & =F_{0,0}\hfill \\ \hfill & ={\int }_0^T{\int }_0^T{(t_1+t_2)}^{-d/2}\mathrm{d}{t}_1\mathrm{d}{t}_2<\infty .\hfill \end{array}$$

Remark 12. 

The above conclusions bring us significant insight into the Wiener chaos expansion for intersection local times. It is worthwhile pointing out that the chaos expansions are extremely helpful to the study of the complex polymer model; for instance, Hou, Luo, Rozovskii, et al. [102] applied the Wiener chaos expansion to solve the stochastic Burgers and Navier–Stokes equations driven by Brownian motion; Crestaux, Olivier and Martinez [103] make the computation of Sobol’s sensitivity indices by using the polynomial chaos expansions to approximate the model output. For more research and applications, see [104,105,106] and the references therein. Moreover, one can prove that the intersection local times $I\left(W_1,W_2,Q_1\times Q_2\right)$ for independent Brownian sheets $W_1$ and $W_2$ belong to the Sobolev space ${\mathbb{D}}^{N_1+N_2-1/2-ϵ,2}$ with the benefit of the similar arguments of Nualart and Vives [107], where $Q_1\times Q_2=(0,{\mathrm{t}}^{Q_1\times Q_2}]$ with $\delta \left({\mathrm{t}}^{Q_1\times Q_2}\right)⩽1$.

Then, using the similar argument of (10), we have the below statement.

Corollary 3. 

Let $W_1=\left\{W_1\left(t_1\right)\in {\mathbb{R}}^d;t_1\in {\mathbb{R}}_{+}^N\right\}$ and $W_2=\left\{W_2\left(t_2\right)\in {\mathbb{R}}^d;t_2\in {\mathbb{R}}_{+}^N\right\}$ be two independent Brownian sheets, and let $I\left(W_1,W_2,Q_1\times Q_2\right)$ be their continuous intersection local time for any $Q_1\times Q_2\in A_2$. If $d<4N$, then for any integrable function $f:{\mathbb{R}}^d\to \mathbb{R}$ and as $m\to \infty$, we have the following convergence in law in the space $C\left({\mathbb{R}}_{+}^{2N}\right)$,

$$\begin{array}{c}\hfill m^{dN/2-2N}{\int }_{m(Q_1\times Q_2)}f(W_1\left(t_1\right)-W_2\left(t_2\right))\mathrm{d}{t}_1\mathrm{d}{t}_2\stackrel{\mathrm{law}}{⟶}I\left(W_1,W_2,Q_1\times Q_2\right){\int }_{{\mathbb{R}}^d}f\left(x\right)\mathrm{d}x,\end{array}$$

where $m(Q_1\times Q_2)={\displaystyle \prod _{n=1}^2\prod _{i=1}^N}(m{s}_{n,i},m{t}_{n,i}]\in A_2$.

Remark 13. 

If we take $Q_1\times Q_2\in A_2$ with $\delta {(}_{Q_1\times Q_2}\mathrm{t})\ne 0$ and $d<4N$, the equality (17) in Proposition 3 above along with Corollary 3 yields that there is a similar result on the convergence of ergodic averages to the expectation in the stationary distribution

$$\begin{array}{c}\hfill \underset{m\to \infty }{lim}{m}^{dN/2-2N}\mathbb{E}{\int }_{m(Q_1\times Q_2)}f(W_1\left(t_1\right)-W_2\left(t_2\right))\mathrm{d}{t}_1\mathrm{d}{t}_2={\int }_{{\mathbb{R}}^d}f\left(x\right){\mu }^{*}\left(\mathrm{d}x\right),\end{array}$$

where ${\mu }^{*}\left(\mathrm{d}x\right)={\left(2\pi \right)}^{-d/2}\left({\int }_{Q_1\times Q_2}{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^N}{t}_{n,i}\right)}^{-d/2}\mathrm{d}{t}_1\mathrm{d}{t}_2\right)\mathrm{d}x$.

The following result on the law of the iterated logarithm of the intersection local times for two independent Brownian sheets is the analogy to Theorem 3, and it reveals more delicate behaviors and properties of the intersection for two independent Brownian sheets.

Corollary 4. 

Let $W_1=\left\{W_1\left(t_1\right)\in {\mathbb{R}}^d;t_1\in {\mathbb{R}}_{+}^{N_1}\right\}$ and $W_2=\left\{W_2\left(t_2\right)\in {\mathbb{R}}^d;t_2\in {\mathbb{R}}_{+}^{N_2}\right\}$ be two independent Brownian sheets, and let $I\left(W_1,W_2,Q_1\times Q_2\right)$ be their continuous intersection local time for any $Q_1\times Q_2\in A_2$. If $d<2(N_1+N_2)$; then, there exist finite constants $c_1^{\prime },c_2^{\prime }>0$ such that, for every $s_1\times s_2\in {(0,\infty )}^{N_1+N_2}$ and every $T\in A_2$,

$$\begin{array}{cc}\hfill \underset{r\to 0}{lim\; sup}& {\displaystyle \frac{I\left(W_1,W_2,\left(s_1-\langle r\rangle ,s_1+\langle r\rangle \right]\times \left(s_2-\langle r\rangle ,s_2+\langle r\rangle \right]\right)}{r^{N_1+N_2-d/2}{\left(loglog{r}^{-1}\right)}^{d/2}}}\\ & ⩽c_1^{\prime }\delta {(s_1\times s_2)}^{-(1-1/(N_1+N_2))d/2}a.s.\end{array}$$

$$\begin{array}{cc}\hfill \underset{\epsilon \to 0}{lim}& \underset{\begin{array}{c}{Q}_1\times Q_2\in A_2,\\ Q_1\times Q_2\subset T,\\ \lambda (Q_1\times Q_2)<\epsilon \end{array}}{sup}{\displaystyle \frac{I\left(W_1,W_2,Q_1\times Q_2\right)}{\lambda {(Q_1\times Q_2)}^{1-d/\left(2(N_1+N_2)\right)}{(log\lambda {(Q_1\times Q_2)}^{-1})}^{d/2}}}\\ & ⩽c_2^{\prime }\delta {\left({}_T\mathrm{t}\right)}^{-(1-1/(N_1+N_2))d/2}a.s.\end{array}$$

Furthermore, the basic quantity in the study of the intersection local times for two independent Brownian sheets is associated with the energy function in a Hamiltonian system. The upper tail behavior $\mathbb{P}\left(I\left(W_1,W_2,Q_1\times Q_2\right)>a\right)$ is significant and appears in some certain self-attracting random models with weight such as $\mathbb{E}exp\left\{\theta I\left(W_1,W_2,Q_1\times Q_2\right)\right\}$ for $\theta >0$ and $Q_1\times Q_2\in A_2$. In the light of Corollary 5, we establish the upper bound for the large deviation of the intersection local times for two independent Brownian sheets.

Proposition 4. 

Let $W_1=\left\{W_1\left(t_1\right)\in {\mathbb{R}}^d;t_1\in {\mathbb{R}}_{+}^{N_1}\right\}$ and $W_2=\left\{W_2\left(t_2\right)\in {\mathbb{R}}^d;t_2\in {\mathbb{R}}_{+}^{N_2}\right\}$ be two independent Brownian sheets, and let $I\left(W_1,W_2,Q_1\times Q_2\right)$ be their intersection local time for any $Q_1\times Q_2\in A_2$. If $d<2(N_1+N_2)$, then there exists finite constant C such that

$$\begin{array}{ccc}\hfill \underset{a\to \infty }{lim\; sup}{\displaystyle \frac{log\mathbb{P}\left(I\left(W_1,W_2,Q_1\times Q_2\right)>a\right)}{a^{2/d}}}& ⩽& -C\delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{1-1/(N_1+N_2)}\hfill \\ & & \times \lambda {(Q_1\times Q_2)}^{1/(N_1+N_2)-2/d},\hfill \end{array}$$

(25)

where

$$C=2d{\pi }^2{M}^{-1}{3}^{-2(N_1+N_2)/d}{\left({\int }_{{\mathbb{R}}^d}exp\{-{\displaystyle \frac{N_1+N_2}{2}}{\parallel \gamma \parallel }^2\}\mathrm{d}\gamma \right)}^{-2/d}$$

and

$$M=\underset{m\in \mathbb{N}}{sup}\left\{{(\left(2m\right)!)}^{(2(N_1+N_2)-d)/\left(2md\right)}{\left(\Gamma [1+2m(1-d/\left(2(N_1+N_2)\right))]\right)}^{-(N_1+N_2)/\left(md\right)}\right\}.$$

Remark 14. 

(1) 

We should point out that the above conclusion is pointless without the condition $\delta {(}_{Q_1\times Q_2}\mathrm{t})\ne 0$, and ${lim\; sup}_{a\to \infty }{\displaystyle \frac{log\mathbb{P}\left(I\left(W_1,W_2,Q_1\times Q_2\right)>a\right)}{a^{2/d}}}⩽0$ is trivial. Unfortunately, from this, we do not gain an accurate large deviation conclusion about the intersection local times for two independent Brownian sheets. It is even difficult to obtain the lower bound of it. It seems to be beyond the reach of the techniques that are known to us. Nevertheless, the result of Proposition 4 provides the necessary assurance for finally obtaining the refined large deviation of the intersection local times for two independent Brownian sheets. In particular, by Fourier analysis, moment computation, time exponentiation and the approach given by Chen [54], we can give the following statement in the case when $N_1=N_2=1$, namely the large deviation of the intersection local times for two independent Brownian motions. Setting $Q_1\times Q_2=[0,1]\times [0,1]$, for $d<4$,

$$\begin{array}{c}\hfill \underset{a\to \infty }{lim}{\displaystyle \frac{log\mathbb{P}\left(I\left(W_1,W_2,[0,1]\times [0,1]\right)>a\right)}{a^{2/d}}}=-2d{\pi }^2{\rho }_2^{-2/d}{\left(1-{\displaystyle \frac{d}{4}}\right)}^{4/d-1},\end{array}$$

where the positive constant ${\rho }_2$ is given by terms of (15) in Remark 8. More specifically, we have ${\rho }_2={\displaystyle \frac{3\pi }{4\sqrt{2}}}$ by simple calculation. Obviously, we can straightforwardly obtain an interesting result, which is shown in [52] by Chen and Li. That is

$$\begin{array}{c}\hfill \underset{a\to \infty }{lim}{\displaystyle \frac{log\mathbb{P}\left(I\left(W_1,W_2,[0,1]\times [0,1]\right)>a\right)}{a^2}}=-3\end{array}$$

and equivalently

$$\begin{array}{c}\hfill \underset{a\to \infty }{lim}{\displaystyle \frac{log\mathbb{E}exp\left\{a{\left(I\left(W_1,W_2,[0,1]\times [0,1]\right)\right)}^{1/2}\right\}}{a^{4/3}}}={\left({\displaystyle \frac{9}{16}}\right)}^{1/3}.\end{array}$$

(2) 

It is easy to see that $\mathbb{E}exp\left\{\theta I\left(W_1,W_2,Q_1\times Q_2\right)\right\}<\infty$ when $\theta ⩽0$. Then, it is interesting to investigate if there is a critical exponent ${\theta }_I>0$ such that

$$\begin{array}{c}\hfill \mathbb{E}exp\left\{\theta I\left(W_1,W_2,Q_1\times Q_2\right)\right\}\left\{\begin{array}{cc}\hfill <\infty ,& \mathrm{if}\theta <{\theta }_I,\hfill \\ \hfill =\infty ,& \mathrm{if}\theta >{\theta }_I.\hfill \end{array}\right.\end{array}$$

Concretely speaking, when $d=1$, for any $Q_1\times Q_2\in A_2$ with $\delta {(}_{Q_1\times Q_2}\mathrm{t})\ne 0$, if $0<\theta <{\tilde{\theta }}_I$, according to (25), we can show that

$$\begin{array}{ccc}\hfill \mathbb{E}exp\left\{\theta I\left(W_1,W_2,Q_1\times Q_2\right)\right\}& =& {\int }_0^{\infty }\mathbb{P}\left(exp\left\{\theta I\left(W_1,W_2,Q_1\times Q_2\right)\right\}⩾y\right)\mathrm{d}y\hfill \\ & ⩽& e+{\int }_e^{\infty }\mathbb{P}\left(I\left(W_1,W_2,Q_1\times Q_2\right)⩾{\displaystyle \frac{logy}{\theta }}\right)\mathrm{d}y\hfill \\ & ⩽& e+{\int }_e^{\infty }exp\left\{-{\left({\displaystyle \frac{{\tilde{\theta }}_I}{\theta }}\right)}^2{(logy)}^2\right\}\mathrm{d}y\hfill \\ & ⩽& e+{\int }_e^{\infty }exp\left\{-{\left({\displaystyle \frac{{\tilde{\theta }}_I}{\theta }}\right)}^{2}logy\right\}\mathrm{d}y\hfill \\ & =& e+exp\left\{1-{\left({\displaystyle \frac{{\tilde{\theta }}_I}{\theta }}\right)}^2\right\}/\left({\left({\displaystyle \frac{{\tilde{\theta }}_I}{\theta }}\right)}^2-1\right)\hfill \\ & <& \infty \hfill \end{array}$$

holds for any $N_1,N_2\in \mathbb{N}$, where

$$\begin{array}{ccc}\hfill {\tilde{\theta }}_I:& =& 2^{1/2}{3}^{-(N_1+N_2)}\pi M^{-1/2}{\left({\int }_{\mathbb{R}}exp\{-{\displaystyle \frac{N_1+N_2}{2}}{\parallel \gamma \parallel }^2\}\mathrm{d}\gamma \right)}^{-1}\hfill \\ & & \times \delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{1/2-1/\left(2(N_1+N_2)\right)}\lambda {(Q_1\times Q_2)}^{1/\left(2(N_1+N_2)\right)-1}.\hfill \end{array}$$

However, for any $Q_1\times Q_2\in A_2$ with $\delta {(}_{Q_1\times Q_2}\mathrm{t})\ne 0$ and any $N_1,N_2\in \mathbb{N}$, there is no certainty that $\theta >{\tilde{\theta }}_I$ would lead to

$$\mathbb{E}exp\left\{\theta I\left(W_1,W_2,Q_1\times Q_2\right)\right\}=\infty$$

as $d=1$. It is similar to the case $d=2$. Also by (25) and simple calculates, for any $Q_1\times Q_2\in A_2$, we merely know that

$$\mathbb{E}exp\left\{\theta I\left(W_1,W_2,Q_1\times Q_2\right)\right\}<\infty ,0<\theta <{\widehat{\theta }}_I,$$

where $N_1,N_2\in \mathbb{N}$, $\delta {(}_{Q_1\times Q_2}\mathrm{t})\ne 0$, and

$$\begin{array}{ccc}\hfill {\widehat{\theta }}_I:& =& 2^2{3}^{-(N_1+N_2)}\pi M^{-1}{\left({\int }_{{\mathbb{R}}^2}exp\{-{\displaystyle \frac{N_1+N_2}{2}}{\parallel \gamma \parallel }^2\}\mathrm{d}\gamma \right)}^{-1}\hfill \\ & & \times \delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{1-1/(N_1+N_2)}\lambda {(Q_1\times Q_2)}^{1/(N_1+N_2)-1}.\hfill \end{array}$$

In fact, it is more difficult to find critical exponent ${\theta }_I$, which can determine if

$$\mathbb{E}exp\left\{\theta I\left(W_1,W_2,Q_1\times Q_2\right)\right\}$$

is finite or infinite for $d⩾3$. It is worthwhile to note that the large deviations principle for intersection local times generated by independent stochastic processes remains open in general. The value of the critical exponent is closely connected with the best constants for Gagliardo–Nirenberg type inequalities (see [108,109]). It has been attracting considerable attention partially due to its connection to some problems in physics; we refer the reader to [110,111,112] for further reference.

There are a large number of studies concerning the intersection local times for independent one-parameter processes by mathematicians and physicists, and relevant research literature is also around (cf. [52,93,113]). It is reasonable to suppose that the properties of the intersections for independent Brownian sheets are analogous to those of a number of more complicated models in equilibrium statistical physics, just as intersections of one-parameter process (e.g., Brownian motion and random walk) paths have been studied for quite a long time in probability theory and statistical mechanics. We conclude this section with an interesting result that links the intersection local times for independent random walks and the intersection local times for independent Brownian motions. Let $S\left(n\right)={\sum }_{k=1}^n{X}_k\in \mathbb{Z}$ be a random walk, where ${\left\{X_n\right\}}_{n⩾1}$ is an integer-valued symmetric i.i.d. sequence with $\mathbb{E}{X}_1^2<\infty$. There are two independent copies of $\left\{S\left(n\right)\in \mathbb{Z};n\in \mathbb{N}\right\}$, which we denote by $S_1=\left\{S_1\left(n\right)\in \mathbb{Z};n\in \mathbb{N}\right\}$ and $S_2=\left\{S_2\left(n\right)\in \mathbb{Z};n\in \mathbb{N}\right\}$. Let $W_1=\left\{W_1\left(t_1\right)\in \mathbb{R};t_1\in {\mathbb{R}}_{+}\right\}$ and $W_2=\left\{W_2\left(t_2\right)\in \mathbb{R};t_2\in {\mathbb{R}}_{+}\right\}$ be independent standard Brownian motions. Then the following weak law holds.

$$\begin{array}{c}\hfill n^{-3/2}\sum _{x\in \mathbb{Z}}\left(l_1(n,x)l_2(n,x)\right)\stackrel{\mathrm{law}}{⟶}{\left(\mathbb{E}{X}_1^2\right)}^{-1}I\left(W_1,W_2,[0,1]\times [0,1]\right),\end{array}$$

where $l_1(n,x)={\displaystyle \sum _{k=1}^n}{\mathbf{1}}_x\left(S_1\left(k\right)\right)$ and $l_2(n,x)={\displaystyle \sum _{k=1}^n}{\mathbf{1}}_x\left(S_2\left(k\right)\right)$ are the local times of $S_1$ and $S_2$, respectively. Actually, the random quantity ${\displaystyle \sum _{x\in \mathbb{Z}}}\left(l_1(n,x)l_2(n,x)\right)$ is the intersection local times for $S_1$ and $S_2$. More precisely, ${\displaystyle \sum _{x\in \mathbb{Z}}}\left(l_1(n,x)l_2(n,x)\right)$ can be formally written as

$$\begin{array}{c}\hfill \sum _{x\in \mathbb{Z}}\left(l_1(n,x)l_2(n,x)\right)={\displaystyle \sum _{k_1,k_2=1}^n}{\mathbf{1}}_0\left(S_1\left(k_1\right)-S_2\left(k_2\right)\right).\end{array}$$

In view of the above representation, we can take $\sum _{x\in \mathbb{Z}}\left(l_1(n,x)l_2(n,x)\right)$ as a means of counting the number of times that up to the time n, the trajectories of $S_1$ and $S_2$ meet together. Here, we refer the interested reader to [52] by Chen and Li for a relevant study on the above result. Based on the progress made in this paper, it seems natural to investigate the extension of the multi-parameter case for the above result. More exactly, we could ask an analogous question about the intersection local times for independent multi-parameter random walks that converge to the intersection local times for independent Brownian sheets. We leave this to the follow-up research. The examples of one-parameter processes and the special cases indicate that the intersection local times for independent multi-parameter processes (e.g., Brownian sheets) may be revealing something interesting and have some challenges to explore and analyze thoroughly.

3. Proofs of the Main Results

This section is devoted to proving the results of the local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ as stated in the previous section. We begin with a technical remark for simplicity in the proofs. Throughout this section, we only consider the case $k=2$ in some complicated proofs without loss of much generality, and the situation $k⩾3$ follows along similar lines.

3.1. Proof of Theorem 1

In this subsection, we prove a sufficient condition for the existence of the local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$.

Proof of Theorem 1. 

By using (21.10) in Geman and Horowize [114], it is sufficient to prove that

$$\begin{array}{c}\hfill \mathbb{E}{\int }_{{\mathbb{R}}^d}{\int }_Q{\int }_{Q}exp\left\{\mathbf{i}\gamma ·(\mathbb{X}\left({\mathrm{t}}^1\right)-\mathbb{X}\left({\mathrm{t}}^2\right))\right\}\mathrm{d}{\mathrm{t}}^1\mathrm{d}{\mathrm{t}}^2\mathrm{d}\gamma <\infty ,\end{array}$$

(26)

where $Q=(u_1,v_1]\times (u_2,v_2]\in A_2$.

By the additive composition of $\mathbb{X}$, we notice that

$$\begin{array}{cc}\hfill & \gamma ·(\mathbb{X}\left({\mathrm{t}}^1\right)-\mathbb{X}\left({\mathrm{t}}^2\right))\hfill \\ \hfill =& \gamma ·(W_1\left(t_1^1\right)+W_2\left(t_2^1\right)-W_1\left(t_1^2\right)-W_2\left(t_2^2\right))\hfill \\ \hfill =& \gamma ·(W_1\left(t_1^1\right)-W_1\left(t_1^2\right))+\gamma ·(W_2\left(t_2^1\right)-W_2\left(t_2^2\right)).\hfill \end{array}$$

(27)

Let ${▵}_i=\left\{(t_i^1,t_i^2)\right|t_{i,p_i}^{{\pi }_{p_i}^i\left(1\right)}<t_{i,p_i}^{{\pi }_{p_i}^i\left(2\right)},p_i=1,\cdots ,N_i\}$ for $i=1,2,$ where ${\pi }_{p_i}^i,p_i=1,\cdots ,N_i$ denote a permutation of $\{1,2\}$. Using the path decomposition of Brownian sheet ([7] Lemma 2.4), we have

$$\begin{array}{c}\hfill W_i\left(t_i^1\right)=W_i\left(u_i\right)+{\displaystyle \sum _{p_i=1}^{N_i}}(W_i(u_{i,1},\cdots ,u_{i,p_i-1},t_{i,p_i}^1,u_{i,p_i+1},\cdots ,u_{i,N_i})-W_i\left(u_i\right))+{\eta }^{i1}\end{array}$$

and

$$\begin{array}{c}\hfill W_i\left(t_i^2\right)=W_i\left(u_i\right)+{\displaystyle \sum _{p_i=1}^{N_i}}(W_i(u_{i,1},\cdots ,u_{i,p_i-1},t_{i,p_i}^2,u_{i,p_i+1},\cdots ,u_{i,N_i})-W_i\left(u_i\right))+{\eta }^{i2},\end{array}$$

it holds that

$$\begin{array}{c}\hfill W_i\left(t_i^1\right)-W_i\left(t_i^2\right)={\displaystyle \sum _{p_i=1}^{N_i}}{\xi }_{p_i}^i+{\zeta }^i,i=1,2,\end{array}$$

where

$${\xi }_{p_i}^i=\left\{\begin{array}{c}{W}_i(u_{i,1},\cdots ,u_{i,p_i-1},t_{i,p_i}^{{\pi }_{p_i}^i\left(1\right)},u_{i,p_i+1},\cdots ,u_{i,N_i})\hfill \\ -W_i(u_{i,1},\cdots ,u_{i,p_i-1},t_{i,p_i}^{{\pi }_{p_i}^i\left(2\right)},u_{i,p_i+1},\cdots ,u_{i,N_i}),\mathrm{if}{\pi }_{p_i}^i\left(1\right)>{\pi }_{p_i}^i\left(2\right);\hfill \\ W_i(u_{i,1},\cdots ,u_{i,p_i-1},t_{i,p_i}^{{\pi }_{p_i}^i\left(2\right)},u_{i,p_i+1},\cdots ,u_{i,N_i})\hfill \\ -W_i(u_{i,1},\cdots ,u_{i,p_i-1},t_{i,p_i}^{{\pi }_{p_i}^i\left(1\right)},u_{i,p_i+1},\cdots ,u_{i,N_i}),\mathrm{if}{\pi }_{p_i}^i\left(1\right)\le {\pi }_{p_i}^i\left(2\right).\hfill \end{array}\right.$$

It is trivial to see that ${\xi }_{p_i}^i,p_i=1,\cdots ,N_i,i=1,2$ are independent Brownian motions in ${\mathbb{R}}^d$. Furthermore, ${\zeta }^i,{\xi }_{p_i}^i,p_i=1,\cdots ,N_i$ are independent as $(t_i^1,t_i^2)\in {▵}_i$ for $i=1,2$ by the independent increments property of the Brownian sheet.

Combining (27) and the above arguments, we have

$$\begin{array}{cc}\hfill & \left|\mathbb{E}\left(exp\left\{\mathbf{i}\gamma ·(\mathbb{X}\left({\mathrm{t}}^1\right)-\mathbb{X}\left({\mathrm{t}}^2\right))\right\}\right)\right|\hfill \\ \hfill ⩽& \left|\mathbb{E}(exp\{\mathbf{i}{\displaystyle \sum _{p_1=1}^{N_1}}\gamma ·{\xi }_{p_1}^1\})\mathbb{E}(exp\{\mathbf{i}{\displaystyle \sum _{p_2=1}^{N_2}}\gamma ·{\xi }_{p_2}^2\})\right|\hfill \\ \hfill =& \left(\prod _{p_1=1}^{N_1}\mathbb{E}(exp\{\mathbf{i}\gamma ·{\xi }_{p_1}^1\})\right)\left(\prod _{p_2=1}^{N_2}\mathbb{E}(exp\{\mathbf{i}\gamma ·{\xi }_{p_2}^2\})\right)\hfill \\ \hfill =& \left(\prod _{p_1=1}^{N_1}exp\{-{\displaystyle \frac{\delta \left(u_1\right)}{2u_{1,p_1}}}(t_{1,p_1}^{{\pi }_{p_1}^1\left(2\right)}-t_{1,p_1}^{{\pi }_{p_1}^1\left(1\right)}){\parallel \gamma \parallel }^2\}\right)\hfill \\ & \times \left(\prod _{p_2=1}^{N_2}exp\{-{\displaystyle \frac{\delta \left(u_2\right)}{2u_{2,p_2}}}(t_{2,p_2}^{{\pi }_{p_2}^2\left(2\right)}-t_{2,p_2}^{{\pi }_{p_2}^2\left(1\right)}){\parallel \gamma \parallel }^2\}\right)\hfill \\ \hfill =& \left(\prod _{p_1=1}^{N_1}exp\{-k_{1,p_1}({\tilde{t}}_{1,p_1}^{\left(2\right)}-{\tilde{t}}_{1,p_1}^{\left(1\right)}){\parallel \gamma \parallel }^2\}\right)\hfill \\ & \times \left(\prod _{p_2=1}^{N_2}exp\{-k_{2,p_2}({\tilde{t}}_{2,p_2}^{\left(2\right)}-{\tilde{t}}_{2,p_2}^{\left(1\right)}){\parallel \gamma \parallel }^2\}\right),\hfill \end{array}$$

(28)

where $k_{i,p_i}={\displaystyle \frac{\delta \left(u_i\right)}{2u_{i,p_i}}},{\tilde{t}}_{i,p_i}^{\left(1\right)}=t_{i,p_i}^{{\pi }_{p_i}^i\left(1\right)},{\tilde{t}}_{i,p_i}^{\left(2\right)}=t_{i,p_i}^{{\pi }_{p_i}^i\left(2\right)},p_i=1,\cdots ,N_i,i=1,2.$

Therefore, by (28) and Fubini’s theorem, we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_{{\mathbb{R}}^d}{\int }_{Q\cap ({▵}_1\times {▵}_2)}{\int }_{Q\cap ({▵}_1\times {▵}_2)}exp\left\{\mathbf{i}\gamma ·(\mathbb{X}\left({\mathrm{t}}^1\right)-\mathbb{X}\left({\mathrm{t}}^2\right))\right\}\mathrm{d}{\mathrm{t}}^1\mathrm{d}{\mathrm{t}}^2\mathrm{d}\gamma \hfill \\ \hfill =& {\int }_{Q\cap ({▵}_1\times {▵}_2)}{\int }_{Q\cap ({▵}_1\times {▵}_2)}{\int }_{{\mathbb{R}}^d}\mathbb{E}exp\left\{\mathbf{i}\gamma ·(\mathbb{X}\left({\mathrm{t}}^1\right)-\mathbb{X}\left({\mathrm{t}}^2\right))\right\}\mathrm{d}\gamma \mathrm{d}{\mathrm{t}}^1\mathrm{d}{\mathrm{t}}^2\hfill \\ \hfill ⩽& {\int }_{{(Q\cap ({▵}_1\times {▵}_2))}^2}{\int }_{{\mathbb{R}}^d}\left(\prod _{p_1=1}^{N_1}exp\{-k_{1,p_1}({\tilde{t}}_{1,p_1}^{\left(2\right)}-{\tilde{t}}_{1,p_1}^{\left(1\right)}){\parallel \gamma \parallel }^2\}\right)\hfill \\ & \times \left(\prod _{p_2=1}^{N_2}exp\{-k_{2,p_2}({\tilde{t}}_{2,p_2}^{\left(2\right)}-{\tilde{t}}_{2,p_2}^{\left(1\right)}){\parallel \gamma \parallel }^2\}\right)\mathrm{d}\gamma \mathrm{d}{\mathrm{t}}^1\mathrm{d}{\mathrm{t}}^2\hfill \\ \hfill ⩽& {\int }_{{(Q\cap ({▵}_1\times {▵}_2))}^2}\prod _{p_1=1}^{N_1}{\left({\int }_{{\mathbb{R}}^d}exp\{-(N_1+N_2)k_{1,p_1}({\tilde{t}}_{1,p_1}^{\left(2\right)}-{\tilde{t}}_{1p_1}^{\left(1\right)}){\parallel \gamma \parallel }^2\}\mathrm{d}\gamma \right)}^{{\displaystyle \frac{1}{N_1+N_2}}}\hfill \\ & \times \prod _{p_2=1}^{N_2}{\left({\int }_{{\mathbb{R}}^d}exp\{-(N_1+N_2)k_{2,p_2}({\tilde{t}}_{2,p_2}^{\left(2\right)}-{\tilde{t}}_{2,p_2}^{\left(1\right)}){\parallel \gamma \parallel }^2\}\mathrm{d}\gamma \right)}^{{\displaystyle \frac{1}{N_1+N_2}}}\mathrm{d}{\mathrm{t}}^1\mathrm{d}{\mathrm{t}}^2,\hfill \end{array}$$

(29)

where the last inequality follows by General Hölder inequality.

Then, for $p_i=1,\cdots ,N_i,i=1,2,$ we have

$$\begin{array}{cc}\hfill & {\left({\int }_{{\mathbb{R}}^d}exp\{-(N_1+N_2)k_{1,p_1}({\tilde{t}}_{1,p_1}^{\left(2\right)}-{\tilde{t}}_{1,p_1}^{\left(1\right)}){\parallel \gamma \parallel }^2\}\mathrm{d}\gamma \right)}^{{\displaystyle \frac{1}{N_1+N_2}}}\hfill \\ \hfill =& ({\int }_{{\mathbb{R}}^d}{\left(2\pi \right)}^{d/2}{\left(2(N_1+N_2)k_{i,p_i}({\tilde{t}}_{i,p_i}^{\left(2\right)}-{\tilde{t}}_{i,p_i}^{\left(1\right)})\right)}^{-d/2}\hfill \\ & \times {\displaystyle \frac{exp\{-(N_1+N_2)k_{i,p_i}({\tilde{t}}_{i,p_i}^{\left(2\right)}-{\tilde{t}}_{i,p_i}^{\left(1\right)})\parallel \gamma {\parallel }^2\}}{{\left(2\pi \right)}^{d/2}{\left(2(N_1+N_2)k_{i,p_i}({\tilde{t}}_{i,p_i}^{\left(2\right)}-{\tilde{t}}_{i,p_i}^{\left(1\right)})\right)}^{-d/2}}}\mathrm{d}\gamma {)}^{{\displaystyle \frac{1}{N_1+N_2}}}\hfill \\ \hfill =& {\left({\left(2\pi \right)}^{d/2}{\left(2(N_1+N_2)k_{i,p_i}({\tilde{t}}_{i,p_i}^{\left(2\right)}-{\tilde{t}}_{i,p_i}^{\left(1\right)})\right)}^{-d/2}\right)}^{{\displaystyle \frac{1}{N_1+N_2}}}\hfill \\ \hfill =& C_{i,p_i}{({\tilde{t}}_{i,p_i}^{\left(2\right)}-{\tilde{t}}_{i,p_i}^{\left(1\right)})}^{-d/\left(2(N_1+N_2)\right)},\hfill \end{array}$$

where $C_{i,p_i}={\left(\pi /((N_1+N_2)k_{i,p_i})\right)}^{{\displaystyle \frac{d}{\left(2(N_1+N_2)\right)}}}$.

Thus, (29) is finite for $d<2(N_1+N_2)$, and (26) holds by the fact that there are only a finite number of permutations. □

3.2. Proof of Theorem 2

Using the argument as Berman [27,58] and the Fourier analytic approach, the key ingredient in proving the jointly continuous of local times for ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ is the estimate for the following multiple integral

$$\begin{array}{c}\hfill J(Q,m,\eta )={\int }_{Q^{2m}}{\int }_{{\left({\mathbb{R}}^d\right)}^{2m}}\left|Eexp\{\mathbf{i}\sum _{j=1}^{2m}{\gamma }^j·\mathbb{X}\left({\mathrm{t}}^j\right)\}\right|\prod _{j=1}^{2m}{\parallel {\gamma }^j\parallel }^{\eta }\mathrm{d}{\gamma }^j\mathrm{d}{\mathrm{t}}^j,\end{array}$$

where $Q\in A_k,m\in \mathbb{N},\eta ⩾0.$ In order to prove Theorem 2 conveniently, we present the below Lemma at first, which gives the suitable upper bound for $J(Q,m,\eta )$.

Lemma 1. 

Let $d<2{\displaystyle \sum _{n=1}^k}{N}_n$ and $\eta \in [0,(2{\displaystyle \sum _{n=1}^k}{N}_n-d)/2)$, and there is a finite constant $K_1$ such that for any $Q=(\mathrm{u},\mathrm{v}]\in A_k$ and $m\in \mathbb{N}$

$$\begin{array}{ccc}\hfill J(Q,m,\eta )& ⩽& K_1^{2m}{\left(\left(2m\right)!/\Gamma [1+2m(1-(d+\eta )/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right))]\right)}^{{\displaystyle \sum _{n=1}^k}{N}_n}\hfill \\ & & \times \delta {\left(\mathrm{u}\right)}^{-m(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)(d+\eta )}\lambda {\left(Q\right)}^{2m(1-(d+\eta )/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right))}.\hfill \end{array}$$

(30)

Proof. 

Let $Q=(\mathrm{u},\mathrm{v}]\in A_2$ be fixed. For simplicity, we set $(\mathrm{u},\mathrm{v}]=(u_1,v_1]\times (u_2,v_2]$, then applying the argument of [24], there is a following decomposition:

$$\mathbb{X}\left((u_1,u_2)+(s_1,s_2)\right)={\displaystyle \sum _{i=1}^2}\left(W_i\left(u_i\right)+{\displaystyle \sum _{n=1}^{N_i}}{\zeta }_n^i\left(s_{i,n}\right|u_i)+R^i\left(s_i\right|u_i)\right),$$

where ${\zeta }_n^i\left(s_{i,n}\right|u_i)=W_i(u_{i,1},\cdots ,u_{i,n-1},u_{i,n}+s_{i,n},u_{i,n+1},\cdots ,u_{i,N_i})-W_i\left(u_i\right),s_i\in {\mathbb{R}}_{+}^{N_i},n=1,\cdots ,N_i,i=1,2.$ By the independent increments property of the Brownian sheet and the assumption that independent of $W_1$ and $W_2$, we learn that ${\zeta }_n^i(·|u_i),n=1,\cdots ,N_i,i=1,2,R^1(·|u_1),R^2(·|u_2),W_1\left(u_1\right)$ and $W_2\left(u_2\right)$ are mutually independent. In this case, for every $m\in \mathbb{N},$ we have

$$\begin{array}{cc}\hfill |\mathbb{E}exp\{\mathbf{i}{\displaystyle \sum _{j=1}^{2m}}{\gamma }^j·\mathbb{X}((u_1,u_2)+(s_1^j,s_2^j))\left\}\right|⩽& {\displaystyle \prod _{i=1}^2}{\displaystyle \prod _{n=1}^{N_i}}|\mathbb{E}exp\{\mathbf{i}{\displaystyle \sum _{j=1}^{2m}}{\gamma }^j·{\zeta }_n^i\left(s_{i,n}^j\right|u_i)\}|,\end{array}$$

where $(s_1^j,s_2^j)\in {\mathbb{R}}_{+}^{N_1+N_2},{\gamma }^j\in {\mathbb{R}}^d$.

Then using Hölder’s inequality, it yields,

$$\begin{array}{cc}\hfill {\int }_{{\left({\mathbb{R}}^d\right)}^{2m}}& |\mathbb{E}exp\{\mathbf{i}{\displaystyle \sum _{j=1}^{2m}}{\gamma }^j·\mathbb{X}((u_1,u_2)+(s_1^j,s_2^j))\}|{\displaystyle \prod _{j=1}^{2m}}{\parallel {\gamma }^j\parallel }^{\eta }\mathrm{d}{\gamma }^j\hfill \\ \hfill & ⩽{\displaystyle \prod _{i=1}^2\prod _{n=1}^{N_i}}{\left({\int }_{{\left({\mathbb{R}}^d\right)}^{2m}}|\mathbb{E}exp\{\mathbf{i}\sum _{j=1}^{2m}{\gamma }^j·{\zeta }_n^i\left(s_{i,n}^j\right|u_i)\}{|}^{N_1+N_2}\prod _{j=1}^{2m}{\parallel {\gamma }^j\parallel }^{\eta }\mathrm{d}{\gamma }^j\right)}^{1/(N_1+N_2)}.\hfill \end{array}$$

(31)

Set ${▵}_{i,n}=\left\{(s_{i,n}^1,\cdots ,s_{i,n}^{2m})|s_{i,n}^{{\pi }_{i,n}\left(j\right)}<s_{i,n}^{{\pi }_{i,n}(j+1)},j=1,\cdots ,2m-1\right\}$, where ${\pi }_{i,n}$ denote the permutation of $\{1,\cdots ,2m\}$, $n=1,\cdots ,N_i,i=1,2$. Notice that ${\zeta }_n^i(·|u_i),n=1,\cdots ,N_i,i=1,2$ are independent Brownian motions in ${\mathbb{R}}^d$ with fixing $u_i$. Hence taking advantage of the independent increments property of Brownian motion, let $(s_{i,n}^1,\cdots ,s_{i,n}^{2m})\in {▵}_{i,n}$ and $\eta \in [0,(2{\displaystyle \sum _{n=1}^k}{N}_n-d)/2)$, we obtain

$$\begin{array}{cc}\hfill & {\int }_{{\left({\mathbb{R}}^d\right)}^{2m}}|\mathbb{E}exp\{\mathbf{i}\sum _{j=1}^{2m}{\gamma }^j·{\zeta }_n^i\left(s_{i,n}^j\right|u_i)\}{|}^{N_1+N_2}\prod _{j=1}^{2m}{\parallel {\gamma }^j\parallel }^{\eta }\mathrm{d}{\gamma }^j\hfill \\ \hfill =& {\int }_{{\left({\mathbb{R}}^d\right)}^{2m}}\prod _{j=1}^{2m}exp\left\{-{\displaystyle \frac{(N_1+N_2)\delta \left(u_i\right)}{2u_{i,n}}}(s_{i,n}^{{\pi }_{i,n}\left(j\right)}-s_{i,n}^{{\pi }_{i,n}(j-1)}){\parallel \sum _{l=j}^{2m}{\gamma }^l\parallel }^2\right\}{\parallel {\gamma }^j\parallel }^{\eta }\mathrm{d}{\gamma }^j\hfill \\ \hfill =& {\int }_{{\left({\mathbb{R}}^d\right)}^{2m}}\prod _{j=1}^{2m}exp\left\{-{\displaystyle \frac{(N_1+N_2)\delta \left(u_i\right)}{2u_{i,n}}}(r_i^j-r_i^{j-1}){\parallel {\beta }^j\parallel }^2\right\}{\parallel {\beta }^j-{\beta }^{j+1}\parallel }^{\eta }\mathrm{d}{\beta }^j\hfill \\ \hfill ⩽& K^{2m}{\sum }^{\prime }\prod _{j=1}^{2m}{\left((r_i^j-r_i^{j-1})\delta \left(u_i\right)/u_{i,n}\right)}^{-(d+q_j\eta )/2}\hfill \\ \hfill ⩽& K^{2m}{\left(\delta \left(u_i\right)/u_{i,k}\right)}^{-m(d+\eta )}{\sum }^{\prime }\prod _{j=1}^{2m}{(r_i^j-r_i^{j-1})}^{-(d+q_j\eta )/2},\hfill \end{array}$$

(32)

where we used the substitution $r_i^j=s_{i,n}^{{\pi }_{i,n}\left(j\right)},{\beta }^j={\displaystyle \sum _{l=j}^{2m}}{\gamma }^l,j=1,\cdots ,2m,r_i^0={\beta }^{2m+1}=0$, and $K=2^{\eta }\underset{q\in \{0,1,2\}}{max}{\int }_{{\mathbb{R}}^d}exp\{-{\displaystyle \frac{N_1+N_2}{2}}{\parallel \gamma \parallel }^2{\}\parallel \gamma \parallel }^{q\eta }\mathrm{d}\gamma$. Notice that in obtaining the first inequality in (32), it follows from

$$\begin{array}{c}\hfill {\displaystyle \prod _{j=1}^{2m}}\parallel {\beta }^j-{\beta }^{j+1}{\parallel }^{\eta }⩽2^{2m\eta }{\sum }^{\prime }\prod _{j=1}^{2m}{\parallel {\beta }^j\parallel }^{q_j\eta },\end{array}$$

in which ${\sum }^{\prime }=♯\left\{(q_1,\cdots ,q_{2m}):(q_1,\cdots ,q_{2m})\in {\{0,1,2\}}^{2m},{\displaystyle \sum _{j=1}^{2m}}{q}_j=2m\right\}$.

By (31) and (32), let $Q_{-}=[0,v_1-u_1]\times [0,v_2-u_2]$,

$$\begin{array}{cc}\hfill & J(Q,m,\eta )\hfill \\ \hfill =& {\int }_{Q_{-}^{2m}}{\int }_{{\left({\mathbb{R}}^d\right)}^{2m}}|\mathbb{E}exp\{\mathbf{i}\sum _{j=1}^{2m}{\gamma }^j·\mathbb{X}((u_1,u_2)+(s_1^j,s_2^j))\}|\prod _{j=1}^{2m}{\parallel {\gamma }^j\parallel }^{\eta }\mathrm{d}{\gamma }^j\mathrm{d}{s}^j\hfill \\ \hfill ⩽& \prod _{i=1}^2\prod _{n=1}^{N_i}[\left(2m\right)!\underset{0=r_i^0⩽\cdots ⩽r_i^{2m}⩽v_{i,n}-u_{i,n}}{\int }(K_1^{2m}{\left({\displaystyle \frac{\delta \left(u_i\right)}{u_{i,n}}}\right)}^{-m(d+\eta )}\hfill \\ & \times {\sum }^{\prime }\prod _{j=1}^{2m}{(r_i^j-r_i^{j-1})}^{-(d+q_j\eta )/2}{)}^{1/(N_1+N_2)}\mathrm{d}{r}_i^1\cdots \mathrm{d}{r}_i^{2m}]\hfill \\ \hfill ⩽& K^{2m}{(\left(2m\right)!)}^{N_1+N_2}\prod _{i=1}^2\prod _{n=1}^{N_i}[{\left({\displaystyle \frac{\delta \left(u_i\right)}{u_{i,n}}}\right)}^{-m(d+\eta )/(N_1+N_2)}\hfill \\ & \times \left({\sum }^{\prime }\underset{0=r_i^0⩽\cdots ⩽r_i^{2m}⩽v_{i,n}-u_{i,n}}{\int }\prod _{j=1}^{2m}{(r_i^j-r_i^{j-1})}^{-(d+q_j\eta )/\left(2(N_1+N_2)\right)}\mathrm{d}{r}_i^j\right)].\hfill \end{array}$$

Therefore, with the similar argument as that of Ehm [28], we obtain

$$\begin{array}{cc}\hfill & J(Q,m,\eta )\hfill \\ \hfill ⩽& K^{2m}{(\left(2m\right)!)}^{N_1+N_2}{\displaystyle \prod _{i=1}^2\prod _{n=1}^{N_i}}(\delta {\left(u_i\right)}^{-m(1-1/(N_1+N_2))(d+\eta )}\hfill \\ & \times \left({\displaystyle \frac{{\sum }^{\prime }{(v_{i,n}-u_{i,n})}^{2m-{\displaystyle \sum _{j=1}^{2m}}(d+q_j\eta )/\left(2(N_1+N_2)\right)}}{\Gamma [1+2m-{\displaystyle \sum _{j=1}^{2m}}(d+q_j\eta )/\left(2(N_1+N_2)\right)]}}\right))\hfill \\ \hfill ⩽& K_1^{2m}{\left(\left(2m\right)!/\Gamma [1+2m(1-(d+\eta )/\left(2(N_1+N_2)\right))]\right)}^{N_1+N_2}\hfill \\ & \times \delta {\left(u\right)}^{-m(1-1/(N_1+N_2))(d+\eta )}\lambda {\left(Q\right)}^{2m(1-(d+\eta )/\left(2(N_1+N_2)\right))},\hfill \end{array}$$

where $K_1=3^{(N_1+N_2)}K$, the first inequality follows from the fact that $d/2(N_1+N_2)<1$ and the following straightforward estimate of integral

$$\underset{0=r_i^0⩽\cdots ⩽r_i^{2m}⩽a_i}{\int }{\displaystyle \prod _{j=1}^{2m}}{(r_i^j-r_i^{j-1})}^{-b_j}\mathrm{d}{r}_i^j⩽a_i^{2m-{\displaystyle \sum _{j=1}^{2m}}{b}_j}/\Gamma [1+2m-{\displaystyle \sum _{j=1}^{2m}}{b}_j].$$

This completes the proof of (30). □

Having proved the above Lemma by (30) and using the argument similar to the proof of the sufficiency of the jointly continuous local times given by Berman [58], it is easy to see that under $d<2{\displaystyle \sum _{n=1}^k}{N}_n$, the ABS $\mathbb{X}={⨁}_{n=1}^k{W}_n$ have jointly continuous local times $\left\{L(Q,x),x\in {\mathbb{R}}^d\right\}$ on $Q\in A_{k}a.s.$ In the following, we present a spontaneous Corollary of the Lemma 1, which contain the even-order moments estimates of $L(Q,\mathbb{X}(\mathrm{t})+x)$ and $L(Q,\mathbb{X}(\mathrm{t})+x)-L(Q,\mathbb{X}(\mathrm{t})+y)$ for any $Q\in A_k$, $x,y\in {\mathbb{R}}^d$ with $\mathrm{t}=0$ or ${}_Q\mathrm{t}$. It should be noted that the result below is not only usefully stated in terms of the proving of Theorem 3 and Propositions 3 and 4 but also is of interest itself.

Corollary 5. 

If $\eta \in [0,1\wedge (2{\displaystyle \sum _{n=1}^k}{N}_n-d)/2)$, there are constants $K_2,K_3$ such that for any $Q\in A_k$, $x,y\in {\mathbb{R}}^d$ and $m\in \mathbb{N}$

$$\begin{array}{cc}\hfill & \mathbb{E}{\left[{\displaystyle \frac{L(Q,\mathbb{X}(\mathrm{t})+x)}{\delta {\left({}_Q\mathrm{t}\right)}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}}}\right]}^{2m}\hfill \\ & ⩽K_2^{2m}{(\left(2m\right)!/\Gamma [1+2m(1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right))])}^{{\displaystyle \sum _{n=1}^k}{N}_n}\hfill \end{array}$$

(33)

and

$$\begin{array}{cc}\hfill & \mathbb{E}{\left[{\displaystyle \frac{\left(L(Q,\mathbb{X}(\mathrm{t})+x)-L(Q,\mathbb{X}(\mathrm{t})+y)\right)}{\delta {\left({}_Q\mathrm{t}\right)}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)(d+\eta )/2}\lambda {\left(Q\right)}^{1-(d+\eta )/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}{\parallel x-y\parallel }^{\eta }}}\right]}^{2m}\hfill \\ & ⩽K_3^{2m}{(\left(2m\right)!/\Gamma [1+2m(1-(d+\eta )/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right))])}^{{\displaystyle \sum _{n=1}^k}{N}_n},\hfill \end{array}$$

(34)

where $\mathrm{t}$ may be either 0 or ${}_Q\mathrm{t}$.

Proof. 

If $\mathrm{t}=0$, i.e., $\mathbb{X}\left(\mathrm{t}\right)=0$, then Berman [58] has shown that for any $Q\in A_k$, $x,y\in {\mathbb{R}}^d$, $m\in \mathbb{N}$ and $\eta \in (0,1]$,

$${\mathbb{E}\left|L(Q,x)\right|}^{2m}⩽{\left(2\pi \right)}^{-2md}J(Q,m,0)$$

(35)

and

$${\mathbb{E}|L(Q,x)-L(Q,y)|}^{2m}⩽{\left(2\pi \right)}^{-2md}{\parallel x-y\parallel }^{2m\eta }J(Q,m,\eta ).$$

(36)

Hence for any $\eta \in [0,1\wedge (2{\displaystyle \sum _{n=1}^k}{N}_n-d)/2)$, puting (30) into (35) and (36), respectively, we see that

$$\begin{array}{cc}\hfill & \mathbb{E}{\left[{\displaystyle \frac{L(Q,x)}{\delta {{(}_{Q}t)}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}}}\right]}^{2m}\hfill \\ & ⩽K_2^{2m}{(\left(2m\right)!/\Gamma [1+2m(1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right))])}^{{\displaystyle \sum _{n=1}^k}{N}_n}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \mathbb{E}{\left[{\displaystyle \frac{\left(L(Q,x)-L(Q,y)\right)}{\delta {{(}_{Q}t)}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)(d+\eta )/2}\lambda {\left(Q\right)}^{1-(d+\eta )/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}{\parallel x-y\parallel }^{\eta }}}\right]}^{2m}\hfill \\ & ⩽K_3^{2m}{(\left(2m\right)!/\Gamma [1+2m(1-(d+\eta )/\left(2\sum _{n=1}^k{N}_n\right))])}^{{\displaystyle \sum _{n=1}^k}{N}_n},\hfill \end{array}$$

where

$$K_2=3^{\left({\displaystyle \sum _{n=1}^k}{N}_n\right)}{\left(2\pi \right)}^{-d}{\int }_{{\mathbb{R}}^d}exp\{-{\displaystyle \frac{{\displaystyle \sum _{n=1}^k}{N}_n}{2}}{\parallel \gamma \parallel }^2\}\mathrm{d}\gamma$$

and

$$K_3=3^{\left({\displaystyle \sum _{n=1}^k}{N}_n\right)}{\left(2\pi \right)}^{-d}{2}^{\eta }\underset{q\in \{0,1,2\}}{max}{\int }_{{\mathbb{R}}^d}exp\{-{\displaystyle \frac{{\displaystyle \sum _{n=1}^k}{N}_n}{2}}{\parallel \gamma \parallel }^2{\}\parallel \gamma \parallel }^{q\eta }\mathrm{d}\gamma .$$

On the other hand, for $\mathrm{t}={}_Q\mathrm{t}$, taking $\mathbb{X}\left({\mathrm{t}}^j\right)-X\left({}_Q\mathrm{t}\right)$ to replace $\mathbb{X}\left({\mathrm{t}}^j\right)$ in the definition of multiple integral $J(Q,m,\eta )$, then one can obtain the upper bound of $J(Q,m,\eta )$ in common with that given in Lemma 1, and we omit the details to save pages. Hence, combining (35) and (36) together with the above argument, we can easily obtain (33) and (34). □

3.3. Proof of Theorem 3

Let us begin by formulating some crucial technical lemmas before we proceed to prove the Theorem 3.

Lemma 2. 

For any $\eta \in (0,1\wedge (2{\displaystyle \sum _{n=1}^k}{N}_n-d)/2)$, there are constants $b_1,b_2>0$ and $K_4,K_5<\infty$ such that, for any $Q\in A_k$, $x,y\in {\mathbb{R}}^d$ and $a>0$

$$\mathbb{P}\left(L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)⩾\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}{a}^{d/2}\right)⩽K_4{e}^{-b_{1}a}$$

(37)

and

$$\begin{array}{c}\mathbb{P}\left(|L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)-L(Q,\mathbb{X}\left(\mathrm{t}\right)+y)|⩾(\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)(d+\eta )/2}\right.\hfill \\ \times \left.\lambda {\left(Q\right)}^{1-(d+\eta )/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}{\parallel x-y\parallel }^{\eta }{a}^{(d+\eta )/2})\right)⩽K_5{e}^{-b_{2}a},\hfill \end{array}$$

(38)

where $\mathrm{t}$ may be either 0 or ${}_Q\mathrm{t}$.

Proof. 

For any $d⩾2$, i.e., $2/d⩽1$, $Q\in A_k$, $x\in {\mathbb{R}}^d$ and $m\in \mathbb{N}$, it follows from the Jensen’s inequality that

$$\begin{array}{ccc}\hfill & & \mathbb{E}{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2m(2/d)}\hfill \\ & ⩽& {\left(\mathbb{E}{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2m}\right)}^{2/d}\hfill \\ & ⩽& K_2^{4m/d}{M}_1^{2m}\left(2m\right)!\hfill \\ & ⩽& {\left(K_2^{2/d}{M}_1\right)}^{2m}(2m+1)!\hfill \end{array}$$

(39)

by (33) and the following fact

$${\left(\left(2m\right)!/\Gamma [1+2m(1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right))]\right)}^{2{\displaystyle \sum _{n=1}^k}{N}_n/d}⩽M_1^{2m}\left(2m\right)!,$$

where $M_1$ is a suitable finite constant by Stirling’s formula.

Then, by the Jensen’s inequality and (39),

$$\begin{array}{cc}\hfill & \mathbb{E}{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{(2m-1)(2/d)}\hfill \\ \hfill =& \mathbb{E}{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2m(2/d)\left(\right(2m-1)/(2m\left)\right)}\hfill \\ \hfill ⩽& {\left(\mathbb{E}{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2m(2/d)}\right)}^{(2m-1)/\left(2m\right)}\hfill \\ \hfill ⩽& {\left(K_2^{2/d}{M}_1\right)}^{2m-1}\left(2m\right)!.\hfill \end{array}$$

(40)

Consequently, combining (39) and (40), for any $n\in \mathbb{N}$, we obtain

$$\begin{array}{c}\hfill \mathbb{E}{\left[{\displaystyle \frac{L(Q,\mathbb{X}(\mathrm{t})+x)}{\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}}}\right]}^{n(2/d)}⩽{\left(K_2^{2/d}{M}_1\right)}^n(n+1)!.\end{array}$$

(41)

Hence, applying the Taylor expansions and (41),

$$\begin{array}{cc}\hfill & \mathbb{E}exp\left\{b_1{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2/d}\right\}\hfill \\ \hfill =& {\displaystyle \sum _{n=0}^{\infty }}{b}_1^n\mathbb{E}{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{n(2/d)}/n!\hfill \\ \hfill ⩽& {\displaystyle \sum _{n=0}^{\infty }}{\left(b_1{K}_2^{2/d}{M}_1\right)}^n(n+1),\hfill \end{array}$$

where positive constant $b_1$ is determined later.

The D’Alembert’s comparison test shows that by proper choice of $b_1$, the infinite series with non-negative terms ${\displaystyle \sum _{n=0}^{\infty }}{\left(b_1{K}_2^{2/d}{M}_1\right)}^n(n+1)$ converges. In fact, we fix $b_1$, which satisfies $0<b_1{K}_2^{2/d}{M}_1<1$ (i.e., $0<b_1<K_2^{-2/d}{M}_1^{-1}$), then there is a finite constant $K_4^{\prime }$, such that for any $d⩾2$,

$$\begin{array}{c}\hfill \mathbb{E}exp\left\{b_1{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2/d}\right\}⩽K_4^{\prime }.\end{array}$$

(42)

On the other hand, when $d=1$, by (33), we notice that for any $n\in \mathbb{N}$,

$$\begin{array}{cc}\hfill & \mathbb{E}{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{n(2/d)}\hfill \\ \hfill =& \mathbb{E}{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)/2}\lambda {\left(Q\right)}^{1-1/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2n}\hfill \\ \hfill ⩽& {\left(K_2^2{M}_2\right)}^n{(\left(2n\right)!)}^{1/2},\hfill \end{array}$$

(43)

where $M_2$ is a suitable finite constant in the following inequality by Stirling’s formula

$${\left(\left(2n\right)!/\Gamma [1+2n(1-1/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right))]\right)}^{{\displaystyle \sum _{n=1}^k}{N}_n}⩽M_2^n{(\left(2n\right)!)}^{1/2}.$$

Now by (43) and using the Taylor expansions again, we have

$$\begin{array}{cc}\hfill & \mathbb{E}exp\left\{b_1{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2/d}\right\}\hfill \\ \hfill =& \mathbb{E}exp\left\{b_1{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)/2}\lambda {\left(Q\right)}^{1-1/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^2\right\}\hfill \\ \hfill =& {\displaystyle \sum _{n=0}^{\infty }}{b}_1^n\mathbb{E}{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)/2}\lambda {\left(Q\right)}^{1-1/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2n}/n!\hfill \\ \hfill ⩽& {\displaystyle \sum _{n=0}^{\infty }}{\left(b_1{K}_2^2{M}_2\right)}^n{(\left(2n\right)!)}^{1/2}/n!\hfill \\ \hfill ⩽& {\displaystyle \sum _{n=0}^{\infty }}{\pi }^{-1/4}{\left(2b_1{K}_2^2{M}_2\right)}^n{n}^{-1/4},\hfill \end{array}$$

where the last inequality is verified by Stirling’s formula and positive constant $b_1$ is determined later.

Similar to the argument in deriving (42), we fix $b_1$ which satisfies $0<b_1<{\displaystyle \frac{1}{2}}{K}_2^{-2}{M}_2^{-1}$, and then there exists a finite constant $K_4^{″}$, such that

$$\begin{array}{c}\hfill \mathbb{E}exp\left\{b_1{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)/2}\lambda {\left(Q\right)}^{1-1/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^2\right\}⩽K_4^{″}.\end{array}$$

(44)

Therefore, combining (42) and (44), we have a choice of $b_1$ that satisfies $0<b_1<\left(K_2^{-2/d}{M}_1^{-1}\right)\wedge \left({\displaystyle \frac{1}{2}}{K}_2^{-2}{M}_2^{-1}\right)$, such that for any $d⩾1$,

$$\begin{array}{c}\hfill \mathbb{E}exp\left\{b_1{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2/d}\right\}⩽K_5,\end{array}$$

(45)

where $K_5=K_4^{\prime }\vee K_4^{″}$.

Finally, by Markov’s inequality and (45), for any $a>0$ we have

$$\begin{array}{ccc}& & \mathbb{P}\left(L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)⩾\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}{a}^{d/2}\right)\hfill \\ & =& \mathbb{P}\left({\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2/d}⩾a\right)\hfill \\ & =& \mathbb{P}\left(b_1{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2/d}⩾b_{1}a\right)\hfill \\ & ⩽& \mathbb{E}exp\left\{b_1{\left[L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)/\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)d/2}\lambda {\left(Q\right)}^{1-d/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}\right]}^{2/d}\right\}/e^{b_{1}a}\hfill \\ & ⩽& K_4{e}^{-b_{1}a}.\hfill \end{array}$$

This finishes the proof of (37). In order to prove (38), it is sufficient to hold that

$$\begin{array}{cc}\hfill & \mathbb{E}exp\left\{b_2\right[|L(Q,\mathbb{X}\left(\mathrm{t}\right)+x)-L(Q,\mathbb{X}\left(\mathrm{t}\right)+y)|/(\delta {{(}_Q\mathrm{t})}^{-(1-1/{\displaystyle \sum _{n=1}^k}{N}_n)(d+\eta )/2}\\ & \times \lambda {\left(Q\right)}^{1-(d+\eta )/\left(2{\displaystyle \sum _{n=1}^k}{N}_n\right)}{\parallel x-y\parallel }^{\eta }{a}^{(d+\eta )/2}\left){]}^{2/d}\right\}⩽K_5,\end{array}$$

for any $\eta \in (0,1\wedge (2{\displaystyle \sum _{n=1}^k}{N}_n-d)/2)$, $Q\in A_k$ and $x,y\in {\mathbb{R}}^d$, by proper choices of suitable constants $b_2$, $K_5$, which follow from the same lines of proving (42) and we omit the details. □

Lemma 3. 

There exist a finite constant $K_6$ such that for any $Q\in A_k$ and $\xi >0$

$$\begin{array}{c}\hfill \mathbb{P}\left(\underset{\mathrm{s}\in Q}{sup}\parallel \mathbb{X}\left(\mathrm{s}\right)-\mathbb{X}{(}_Q\mathrm{t})\parallel ⩾\xi \right)⩽K_6{(k/\xi )}^2{\displaystyle \sum _{n=1}^k}\delta ({}_{Q_n}\mathrm{t},{\mathrm{t}}^{Q_n}),\end{array}$$

(46)

where $Q={\displaystyle \prod _{n=1}^k}{Q}_n$.

Proof. 

For any $Q={\displaystyle \prod _{n=1}^2}{Q}_n\in A_2$ and $\xi >0$, it follows from ([28] Lemma 2.2) in the special case of $\alpha =2$ there exist finite constants $K_7,K_8>0$ such that

$$\begin{array}{c}\hfill \mathbb{P}\left(\underset{s_1\in Q_1}{sup}\parallel W_1\left(s_1\right)-W_1\left({}_{Q_1}\mathrm{t}\right)\parallel ⩾\xi \right)⩽K_7{\xi }^{-2}\delta ({}_{Q_1}\mathrm{t},{\mathrm{t}}^{Q_1})\end{array}$$

(47)

and

$$\begin{array}{c}\hfill \mathbb{P}\left(\underset{s_2\in Q_2}{sup}\parallel W_2\left(s_2\right)-W_2\left({}_{Q_2}\mathrm{t}\right)\parallel ⩾\xi \right)⩽K_8{\xi }^{-2}\delta ({}_{Q_2}\mathrm{t},{\mathrm{t}}^{Q_2}),\end{array}$$

(48)

where $\mathbb{X}=W_1+W_2$.

Therefore, by (47), (48) and applying triangle inequality, we have

$$\begin{array}{ccc}& & \mathbb{P}\left(\underset{\mathrm{s}\in Q}{sup}\parallel \mathbb{X}\left(\mathrm{s}\right)-\mathbb{X}\left({}_Q\mathrm{t}\right)\parallel ⩾\xi \right)\hfill \\ & =& \mathbb{P}\left(\underset{s_1\in Q_1,s_2\in Q_2}{sup}\parallel (W_1\left(s_1\right)-W_1\left({}_{Q_1}\mathrm{t}\right))+(W_2\left(s_2\right)-W_2\left({}_{Q_2}\mathrm{t}\right))\parallel ⩾\xi \right)\hfill \\ & ⩽& \mathbb{P}\left(\underset{s_1\in Q_1}{sup}\parallel W_1\left(s_1\right)-W_1\left({}_{Q_1}\mathrm{t}\right)\parallel +\underset{s_2\in Q_2}{sup}\parallel W_2\left(s_2\right)-W_2\left({}_{Q_2}\mathrm{t}\right)\parallel ⩾\xi \right)\hfill \\ & ⩽& \mathbb{P}\left(\underset{s_1\in Q_1}{sup}\parallel W_1\left(s_1\right)-W_1\left({}_{Q_1}\mathrm{t}\right)\parallel ⩾\xi /2\right)+\mathbb{P}\left(\underset{s_2\in Q_2}{sup}\parallel W_2\left(s_2\right)-W_2\left({}_{Q_2}\mathrm{t}\right)\parallel ⩾\xi /2\right)\hfill \\ & ⩽& K_7{(2/\xi )}^2\delta ({}_{Q_1}\mathrm{t},{\mathrm{t}}^{Q_1})+K_8{(2/\xi )}^2\delta ({}_{Q_2}\mathrm{t},{\mathrm{t}}^{Q_2})\hfill \\ & ⩽& K_6{(2/\xi )}^2\left(\delta ({}_{Q_1}\mathrm{t},{\mathrm{t}}^{Q_1})+\delta ({}_{Q_2}\mathrm{t},{\mathrm{t}}^{Q_2})\right),\hfill \end{array}$$

where $K_6=K_7\vee K_8$. □

Proof of Theorem 3. 

We present some notations firstly. For the fixed time point $\mathrm{s}=(s_1,s_2)\in {[0,\infty )}^{N_1+N_2}$, set a sequence of $N_1+N_2$ dimension time series ${\mathrm{s}}^n={(s_1^n,s_2^n)}_{n=1}^{\infty }$, which satisfies $\underset{n\to \infty }{lim}(s_1^n,s_2^n)=(s_1,s_2)$. Let $D_n=[s_1^n,s_1^n+\langle 2^{-n}\rangle ]\times [s_2^n,s_2^n+\langle 2^{-n}\rangle ],n\in \mathbb{N}$ denote a sequence of Hypercube and define $g\left(x\right)=x^{N_1+N_2-d/2}{(loglog{x}^{-1})}^{d/2}$ for small enough $x>0$. In order to establish the Hölder law for the increment of the local times for ABS $\mathbb{X}=W_1+W_2$ conveniently, we proceed in steps.

Step 1. Define the events

$$A_n=\left\{\omega :\underset{t\in D_n}{sup}\parallel \mathbb{X}\left(\mathrm{t}\right)-X\left({\mathrm{s}}^n\right)\parallel ⩾2^{-n/2}{n}^{\beta }\right\},$$

where the constant $\beta$ is determined later.

Clearly from (46) of Lemma 3

$$\begin{array}{cc}\hfill \mathbb{P}\left(A_n\right)⩽& 2^{n+2}{n}^{-2\beta }{K}_7\left({\left({\displaystyle \frac{1}{2^n}}\right)}^{N_1}+{\left({\displaystyle \frac{1}{2^n}}\right)}^{N_2}\right)\hfill \\ \hfill ⩽& 8K_7{n}^{-2\beta },\hfill \end{array}$$

In the following, we take $\beta >1/2$ such that ${\displaystyle \sum _{n=1}^{\infty }}\mathbb{P}\left(A_n\right)<\infty$. Then, we can conclude that $\mathbb{P}(A_n,i.o.)=0$ by applying the Borel–Cantelli lemma. That is, there exists an integer $n_1\left(\omega \right)$ such that for all $n>n_1\left(\omega \right)$

$$\begin{array}{c}\hfill \underset{t\in D_n}{sup}\parallel \mathbb{X}\left(\mathrm{t}\right)-\mathbb{X}\left({\mathrm{s}}^n\right)\parallel <2^{-n/2}{n}^{\beta }a.s.\end{array}$$

(49)

Step 2. Define

$$G_n=\left\{x\in {\mathbb{R}}^d:x={(2^{n}logn)}^{-1/2}p,p\in {\mathbb{Z}}^d,\parallel x\parallel ⩽2^{-n/2}{n}^{\beta }\right\}.$$

It follows from the definition of the $G_n$ that $\parallel x\parallel =\parallel {(2^{n}logn)}^{-1/2}p\parallel ⩽2^{-n/2}{n}^{\beta }$, thence $\parallel p\parallel ⩽n^{\beta }{(logn)}^{1/2}$. Moreover, since $p\in {\mathbb{Z}}^d$, we derive $♯G_n⩽{\left(n^{\beta }{(logn)}^{1/2}\right)}^d$.

Furthermore, define other events

$$B_n=\left\{\omega :\bigcup _{x\in G_n}\left\{x:L(D_n,\mathbb{X}\left({\mathrm{s}}^n\right)+x)⩾g\left(2^{-n}\right)\delta {\left({\mathrm{s}}^n\right)}^{-(1-1/(N_1+N_2))d/2}{a}_1^{d/2}\right\}\right\},$$

where the constant $a_1$ is selected below.

According to (37) of Lemma 2, we can easily see that

$$\begin{array}{cc}\hfill \mathbb{P}\left(B_n\right)⩽& K_4{\left(n^{\beta }{(logn)}^{1/2}\right)}^{d}exp\{-b_1{a}_{1}loglog{2}^n\}\hfill \\ \hfill ⩽& K_9{(logn)}^{d/2}{n}^{-(b_1{a}_1-d\beta )},\hfill \end{array}$$

where $K_9=K_4{(log2)}^{-b_1{a}_1}$.

Obviously, we have ${\displaystyle \sum _{n=1}^{\infty }}P\left(B_n\right)<\infty$, which can be due to $a_1$ being large enough. Therefore, applying the Borel–Cantelli lemma again, we have $\mathbb{P}(B_n,i.o.)=0$. This implies there is a integer $n_2\left(\omega \right)$ such that for all $n>n_2\left(\omega \right)$

$$\begin{array}{c}\hfill \underset{x\in G_n}{sup}L(D_n,\mathbb{X}\left({\mathrm{s}}^n\right)+x)<g\left(2^{-n}\right)\delta {\left({\mathrm{s}}^n\right)}^{-(1-1/(N_1+N_2))d/2}{a}_1^{d/2}a.s.\end{array}$$

(50)

Step 3. Define

$$H(n,m,x)=\left\{y\in {\mathbb{R}}^d:y=x+{(2^{n}logn)}^{-1/2}{\displaystyle \sum _{j=1}^m}{\epsilon }_j{2}^{-j},{\epsilon }_j\in {\{0,1\}}^d,1⩽j⩽m\right\},$$

which $n,m\in \mathbb{N}$ and $x\in G_n$.

Under the assumption $d<2(N_1+N_2)$, we choose $\gamma$ and $\eta$ such that $\gamma (d+\eta )/2<\eta <1\wedge (N_1+N_2-d/2)$. Then, we let $C_n$ denote events such that the element of that satisfies

$$\begin{array}{cc}\hfill \bigcup _{x\in G_n}{\displaystyle \bigcup _{m=1}^{\infty }}& \left\{\right|L(D_n,\mathbb{X}\left({\mathrm{s}}^n\right)+y_1)-L(D_n,\mathbb{X}\left({\mathrm{s}}^n\right)+y_2)|\hfill \\ & ⩾\lambda {\left(D_n\right)}^{1-(d+\eta )/2}{\parallel y_1-y_2\parallel }^{\eta }{\left(a_2\delta {\left({\mathrm{s}}^n\right)}^{-(1-1/(N_1+N_2))}{2}^{\gamma m}logn\right)}^{(d+\eta )/2}\},\hfill \end{array}$$

where $y_1,y_2\in H(n,m,x)$ with $y_1-y_2={(2^{n}logn)}^{-1/2}\epsilon 2^{-m}$ and $\epsilon \in {\{0,1\}}^d$.

By (38) of Lemma 2, we have

$$\begin{array}{cc}\hfill \mathbb{P}\left(C_n\right)⩽& K_5{\left(n^{\beta }{(logn)}^{1/2}\right)}^d{\displaystyle \sum _{m=1}^{\infty }}{2}^{(d+1)m}exp\{b_2{a}_2{2}^{\gamma m}logn\}\hfill \\ \hfill ⩽& K_5{(logn)}^{d/2}{n}^{-(b_2{a}_2/2-d\beta )},\hfill \end{array}$$

(51)

where the last inequality holds along with the following elementary fact

$${\displaystyle \sum _{m=1}^{\infty }}{2}^{(d+1)m}exp\{b_2{a}_2{2}^{\gamma m}logn\}⩽e^{-(b_2{a}_{2}logn)/2}$$

for selecting $a_2$ large enough.

Thus, (51) deduces to

$$\begin{array}{c}\hfill \mathbb{P}(C_n,i.o.)=0\end{array}$$

(52)

by Borel–Cantelli lemma within large enough $a_2$.

Step 4. Take $y\in {\mathbb{R}}^d$, which satisfies $\parallel y\parallel <2^{-n/2}{n}^{\beta }$ by fixing $n\in \mathbb{N}$. As already advertised, we can seek $x\in G_n$ such that $\underset{m\to \infty }{lim}{y}_m=y$ belong with

$$y_m=x+{(2^{n}logn)}^{-1/2}{\displaystyle \sum _{j=1}^m}{\epsilon }_j{2}^{-j},$$

where ${\epsilon }_j\in {\{0,1\}}^d$ and taking $y_0=x$ in the above.

As a partial consequence of Theorem 2, we notice that $L(·,x)$ is continuous in $x\in {\mathbb{R}}^d$. Consequently, on the complementary set of event $C_n$ perspective, we have

$$\begin{array}{ccc}\hfill & & |L(D_n,\mathbb{X}\left({\mathrm{s}}^n\right)+y)-L(D_n,\mathbb{X}\left({\mathrm{s}}^n\right)+x)|\hfill \\ & <& {\displaystyle \sum _{m=1}^{\infty }}\lambda {\left(D_n\right)}^{1-(d+\eta )/\left(2(N_1+N_2)\right)}{\parallel y_m-y_{m-1}\parallel }^{\eta }\hfill \\ & & \times {\left(a_2\delta {\left({\mathrm{s}}^n\right)}^{-(1-1/(N_1+N_2)}{2}^{\gamma m}logn\right)}^{(d+\eta )/2}\hfill \\ & ⩽& 2^{-n(N_1+N_2-(d+\eta )/2)}{\left(a_2\delta {\left({\mathrm{s}}^n\right)}^{-(1-1/(N_1+N_2)}logn\right)}^{(d+\eta )/2}\hfill \\ & & \times {\displaystyle \sum _{m=1}^{\infty }}{\left(\sqrt{d}{(2^{n}logn)}^{-1/2}\right)}^{\eta }{2}^{-m(\eta -\gamma (d+\eta )/2)}\hfill \\ & ⩽& K_{10}\delta {\left({\mathrm{s}}^n\right)}^{-(1-1/(N_1+N_2))(d+\eta )/2}g\left(2^{-n}\right),\hfill \end{array}$$

(53)

where $K_{10}$ is a finite constant depending on s and n.

Therefore, the proof of (11) is completed by summing up (49), (50), (52) and (53). □

3.4. Proof of Proposition 3 and 4

Proof of Proposition 3. 

On the one hand, by the view of

$$W_1\left(t_1\right)-W_2\left(t_2\right)\sim \mathcal{N}\left(0,\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right){\mathbb{I}}_d\right)$$

and applying Fubini’s theorem, for any $Q_1\times Q_2\in A_2$ with $\delta {(}_{Q_1\times Q_2}\mathrm{t})\ne 0$, we have

$$\begin{array}{ccc}\hfill & & \mathbb{E}I\left(W_1,W_2,Q_1\times Q_2\right)\hfill \\ & =& {\left(2\pi \right)}^{-d}{\int }_{{\mathbb{R}}^d}\mathrm{d}y{\int }_{Q_1\times Q_2}\mathbb{E}exp\left\{\mathbf{i}y·(W_1\left(t_1\right)-W_2\left(t_2\right))\right\}\mathrm{d}{t}_1\mathrm{d}{t}_2\hfill \\ & =& {\left(2\pi \right)}^{-d}{\int }_{{\mathbb{R}}^d}\mathrm{d}y{\int }_{Q_1\times Q_2}({\int }_{{\mathbb{R}}^d}exp\left\{\mathbf{i}y·x\right\}{\displaystyle \frac{1}{{\left(2\pi \right)}^{d/2}{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{d/2}}}\hfill \\ & & \times exp\left\{-{\displaystyle \frac{1}{2}}{x}^{\mathrm{T}}\left({\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{-1}{\mathbb{I}}_d\right)x\right\}\mathrm{d}x)\mathrm{d}{t}_1\mathrm{d}{t}_2\hfill \\ & =& {\left(2\pi \right)}^{-d}{\int }_{Q_1\times Q_2}{\displaystyle \prod _{j=1}^d}\left({\int }_{\mathbb{R}}\mathrm{d}{y}_j\right({\int }_{\mathbb{R}}exp\left\{\mathbf{i}{y}_j{x}_j\right\}\hfill \\ & & \times {\displaystyle \frac{1}{{\left(2\pi \right)}^{1/2}{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{1/2}}}exp\left\{-{\displaystyle \frac{x_j^2}{2{\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}}}\right\}\mathrm{d}{x}_j\left)\right)\mathrm{d}{t}_1\mathrm{d}{t}_2\hfill \\ & =& {\left(2\pi \right)}^{-d}{\int }_{Q_1\times Q_2}{\displaystyle \prod _{j=1}^d}\left({\int }_{\mathbb{R}}\mathrm{d}{y}_j\right({\int }_{\mathbb{R}}{\displaystyle \frac{1}{{\left(2\pi \right)}^{1/2}{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{1/2}}}\hfill \\ & & \times exp\left\{-{\displaystyle \frac{{\left(x_j-\mathbf{i}{y}_j{\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^2}{2{\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}}}\right\}\mathrm{d}{x}_j)exp\left\{{\displaystyle \frac{{\left(\mathbf{i}{y}_j{\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^2}{2{\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}}}\right\})\mathrm{d}{t}_1\mathrm{d}{t}_2\hfill \\ & =& {\left(2\pi \right)}^{-d}{\int }_{Q_1\times Q_2}{\displaystyle \prod _{j=1}^d}({\int }_{\mathbb{R}}{\displaystyle \frac{1}{\sqrt{2\pi }{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{-1/2}}}exp\left\{-{\displaystyle \frac{y_j^2}{2{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{-1}}}\right\}\mathrm{d}{y}_j\hfill \\ & & \times \sqrt{2\pi }{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{-1/2})\mathrm{d}{t}_1\mathrm{d}{t}_2\hfill \\ & =& {\left(2\pi \right)}^{-d/2}{\int }_{Q_1\times Q_2}{\left({\displaystyle \sum _{n=1}^2\prod _{i=1}^{N_n}}{t}_{n,i}\right)}^{-d/2}\mathrm{d}{t}_1\mathrm{d}{t}_2,\hfill \end{array}$$

(54)

where in the first equality we used (16).

On the other hand, by (30) and (35), for any $Q_1\times Q_2\in A_2$ with $\delta {(}_{Q_1\times Q_2}\mathrm{t})\ne 0$,

$$\begin{array}{cc}\hfill & \mathbb{E}|I\left(W_1,W_2,Q_1\times Q_2\right){|}^2\hfill \\ \hfill =& \mathbb{E}|L(Q_1\times Q_2,0){|}^2\hfill \\ \hfill ⩽& C\delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{(1-1/(N_1+N_2))d}\lambda {(Q_1\times Q_2)}^{2-d/(N_1+N_2)},\hfill \end{array}$$

(55)

where

$$\begin{array}{ccc}\hfill C& =& 3^{2(N_1+N_2)}{\left(2\pi \right)}^{-2d}{\left({\displaystyle \frac{2}{\Gamma (1+2(1-d/\left(2(N_1+N_2)\right)))}}\right)}^{(N_1+N_2)}\hfill \\ & & \times {\left({\int }_{{\mathbb{R}}^d}exp\{-{\displaystyle \frac{N_1+N_2}{2}}{\parallel \gamma \parallel }^2\}\mathrm{d}\gamma \right)}^2.\hfill \end{array}$$

Therefore, the proof of (18) is completed by (54) and (55). □

Proof of Proposition 4. 

Using some standard arguments of Berman [58] and Ehm [28], by (30) and (35), for every $m\in \mathbb{N}$ and $Q_1\times Q_2\in A_2$,

$$\begin{array}{cc}\hfill & \mathbb{E}|I\left(W_1,W_2,Q_1\times Q_2\right){|}^{2m}\hfill \\ \hfill =& \mathbb{E}|L(Q_1\times Q_2,0){|}^{2m}\hfill \\ \hfill ⩽& {\left(2\pi \right)}^{-2md}J(Q_1\times Q_2,2m,0)\hfill \\ \hfill ⩽& K_{11}^{2m}{(\left(2m\right)!/\Gamma (1+2m(1-d/\left(2(N_1+N_2)\right))))}^{(N_1+N_2)}\hfill \\ & \times \delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{-m(1-1/(N_1+N_2))d}\lambda {(Q_1\times Q_2)}^{2m(1-d/\left(2(N_1+N_2)\right))},\hfill \end{array}$$

(56)

where $K_{11}=3^{(N_1+N_2)}{\left(2\pi \right)}^{-d}{\int }_{{\mathbb{R}}^d}exp\{-{\displaystyle \frac{N_1+N_2}{2}}{\parallel \gamma \parallel }^2\}\mathrm{d}\gamma .$

Then, by means of Jensen’s inequality and (56), for every $m\in \mathbb{N}$,

$$\begin{array}{cc}\hfill & \mathbb{E}|I\left(W_1,W_2,Q_1\times Q_2\right){|}^{2m-1}\hfill \\ \hfill =& \mathbb{E}|L(Q_1\times Q_2,0){|}^{2m-1}\hfill \\ \hfill =& \mathbb{E}{\left(|L(Q_1\times Q_2,0){|}^{2m}\right)}^{{\displaystyle \frac{2m-1}{2m}}}\hfill \\ \hfill ⩽& {\left(\mathbb{E}|L(Q_1\times Q_2,0){|}^{2m}\right)}^{{\displaystyle \frac{2m-1}{2m}}}\hfill \\ \hfill ⩽& K_{11}^{2m-1}{(\left(2m\right)!/\Gamma (1+2m(1-d/\left(2(N_1+N_2)\right))))}^{(N_1+N_2)}\hfill \\ & \times \delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{-(2m-1)(1-1/(N_1+N_2))d/2}\lambda {(Q_1\times Q_2)}^{(2m-1)(1-d/\left(2(N_1+N_2)\right))}.\hfill \end{array}$$

(57)

It follows from (56) and (57) that

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}& {\displaystyle \frac{1}{n}}log\mathbb{E}\left(|I\left(W_1,W_2,Q_1\times Q_2\right){|}^n/{(n!)}^{d/2}\right)\hfill \\ & ⩽log\left(K_{11}{M}^{d/2}\delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{-(1-1/(N_1+N_2))d/2}\lambda {(Q_1\times Q_2)}^{1-d/\left(2(N_1+N_2)\right)}\right)\hfill \end{array}$$

(58)

by the fact

$${\left(\left(2m\right)!/\Gamma [1+2m(1-d/\left(2(N_1+N_2)\right))]\right)}^{(N_1+N_2)}⩽M^{md}{(\left(2m\right)!)}^{d/2},$$

where M is a suitable finite constant by Stirling’s formula, that is

$$M=\underset{m\in \mathbb{N}}{sup}\left\{{(\left(2m\right)!)}^{(2(N_1+N_2)-d)/\left(2md\right)}{\left(\Gamma [1+2m(1-d/\left(2(N_1+N_2)\right))]\right)}^{-(N_1+N_2)/\left(md\right)}\right\}.$$

Set $a_n=e^{\kappa }{n}^{d/2},$

where

$$\kappa =log\left(K_{11}{M}^{d/2}\delta {{(}_{Q_1\times Q_2}\mathrm{t})}^{-(1-1/(N_1+N_2))d/2}\lambda {(Q_1\times Q_2)}^{1-d/\left(2(N_1+N_2)\right)}\right),$$

by (58) and Markov’s inequality,

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{log\mathbb{P}\left(I\left(W_1,W_2,Q_1\times Q_2\right)>a_n\right)}{a_n^{2/d}}}\hfill \\ & =e^{-2\kappa /d}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n}}log\mathbb{P}\left({\left(I\left(W_1,W_2,Q_1\times Q_2\right)\right)}^n>e^{n\kappa }{n}^{nd/2}\right)\hfill \\ & ⩽e^{-2\kappa /d}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n}}log\mathbb{E}\left({\displaystyle \frac{{\left(I\left(W_1,W_2,Q_1\times Q_2\right)\right)}^n}{e^{n\kappa }{n}^{nd/2}}}\right)\hfill \\ & =e^{-2\kappa /d}\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n}}log\left[\mathbb{E}\left({\displaystyle \frac{{\left(I\left(W_1,W_2,Q_1\times Q_2\right)\right)}^n}{{\left(n!\right)}^{d/2}}}\right){\displaystyle \frac{{\left(n!\right)}^{d/2}}{e^{n\kappa }{n}^{nd/2}}}\right]\hfill \\ & =e^{-2\kappa /d}\underset{n\to \infty }{lim\; sup}\{{\displaystyle \frac{1}{n}}log\mathbb{E}\left({\displaystyle \frac{{\left(I\left(W_1,W_2,Q_1\times Q_2\right)\right)}^n}{{\left(n!\right)}^{d/2}}}\right)\hfill \\ & -\kappa +(d/2)log\left({\left(n!\right)}^{1/n}{n}^{-1}\right)\}\hfill \\ & =e^{-2\kappa /d}\underset{n\to \infty }{lim\; sup}\left\{{\displaystyle \frac{1}{n}}log\mathbb{E}\left({\displaystyle \frac{{\left(I\left(W_1,W_2,Q_1\times Q_2\right)\right)}^n}{{\left(n!\right)}^{d/2}}}\right)-\kappa -(d/2)\right\}\hfill \\ & ⩽-(d/2)e^{-2\kappa /d},\hfill \end{array}$$

(59)

where the last equality follows from the below fact that by Stirling’s formula

$$log\left({\left(n!\right)}^{1/n}{n}^{-1}\right)\sim -1(n\to \infty ).$$

Since $a_n^{2/d}/a_{n+1}^{2/d}\sim 1(n\to \infty )$, one can follow the proof of ([115] Theorem 3.4 (i)) to arrive at (25). □

4. Conclusions

We concentrate primarily on ABS in this paper, but the method also works for other similar random fields and, to some extent, additive multi-parameter α-stable processes and additive anisotropic Gaussian random fields. Additionally, in this paper, we show some properties of intersections for Brownian sheets by the intersection local times of two independent Brownian sheets in this paper. The study on intersections of Brownian motion or random walk paths in probability theory and statistical mechanics, whether it is theoretical exploration or practical demonstration, has entered a more mature stage, but the specialized research in the intersection local times of two independent Brownian sheets has only just begun. Theoretical physics predicts that intersection local times play a crucial role in the macroscopic behavior of a wide class of polymer physics models. In fact, the type of quantum field theory that was focused on the notion of random particles has moved to a base of random strings in the past decade, and the paths of particles have been substituted with the so-called “world sheet” of the string. The Brownian sheet is a important random surface as the natural generalization of the Brownian motion with the development of the string theory [116,117,118], its value will be more and more fully affirmed in practice. On the other hand, multi-parameter Gaussian random fields have become a core tool for complex system analysis due to their flexible multidimensional modeling capabilities [119,120,121,122]. With the improvement of computational efficiency and interdisciplinary integration, the ABS constructed in this article will unleash greater potential in fields such as metaverse simulation, financial mathematics, climate science, precision medicine, quantum physics, and string theory. In the future, we hope to develop efficient inference algorithms related to it and deeply integrate them with deep learning frameworks to meet the modeling needs of high-dimensional and multimodal data.