Temporal Fractal Nature of the Time-Fractional SPIDEs and Their Gradient

Abstract

Fractional and high-order PDEs have become prominent in theory and in the modeling of many phenomena. In this article, we study the temporal fractal nature for fourth-order time-fractional stochastic partial integro-differential equations (TFSPIDEs) and their gradients, which are driven in one-to-three dimensional spaces by space–time white noise. By using the underlying explicit kernels, we prove the exact global temporal continuity moduli and temporal laws of the iterated logarithm for the TFSPIDEs and their gradients, as well as prove that the sets of temporal fast points (where the remarkable oscillation of the TFSPIDEs and their gradients happen infinitely often) are random fractals. In addition, we evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of the TFSPIDEs and their gradients, in time, are most likely one everywhere, and are dense with the power of the continuum. Moreover, their hitting probabilities are determined by the target set B’s packing dimension ${dim}_{{}_p}\left(B\right)$. On the one hand, this work reinforces the temporal moduli of the continuity and temporal LILs obtained in relevant literature, which were achieved by obtaining the exact values of their normalized constants; on the other hand, this work obtains the size of the set of fast points, as well as a potential theory of TFSPIDEs and their gradients.

1. Introduction

Fractional and higher-order evolution equations have been used as (stochastic) models in mathematical finance, fluid dynamics, turbulence, and mathematical physics by numerous authors in recent years (see, e.g., [1,2,3]). Time-fractional stochastic partial integro-differential equations (TFSPIDEs) are related to diffusion or slow diffusion in materials with memory. (For connected deterministic PDEs, see [4,5,6]; for connected stochastic PDEs, see [7,8]; and, for the associated stochastic integral equations (SIEs), see [9,10,11].)

Expanded upon by [11], Brownian-time processes (BTP) provide the foundation for the deterministic version of the TFSPIDEs. The precise dimensions and hitting probabilities for the sets of fast points, in time, for these important class of stochastic equations are obtained in this article as follows:

$$\left\{\begin{array}{cc}{\displaystyle C_{{\partial }_t^{\beta }{U}_{\beta }}=\frac{1}{2}\Delta U_{\beta }+I_t^{1-\beta }\left(\frac{{\partial }^{d+1}W}{\partial t\partial x}\right),}\hfill & (t,x)\in {\mathring{\mathbb{R}}}_{+}\times {\mathbb{R}}^d;\hfill \\ U_{\beta }(0,x)=u_0\left(x\right),\hfill & x\in {\mathbb{R}}^d,\hfill \end{array}\right.$$

(1)

where ${\partial }^{d+1}W/\partial t\partial x$ is the space–time white noise corresponding to the real-valued Brownian sheet W on ${\mathbb{R}}_{+}\times {\mathbb{R}}^d$$(d=1,2,3)$; $\Delta$ is the d-dimensional Laplacian operator; the time-fractional derivative of order $\beta$, $C_{{\partial }_t^{\beta }}$, is the Caputo fractional operator

$$C_{{\partial }_t^{\beta }f\left(t\right)}:=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (1-\beta )}{\int }_0^t\frac{f^{\prime }\left(\tau \right)}{{(t-\tau )}^{\beta }}d\tau ,}\hfill & \mathrm{if}0<\beta <1;\hfill \\ {\displaystyle \frac{d}{dt}f\left(t\right),}\hfill & \mathrm{if}\beta =1,\hfill \end{array}\right.$$

and the time-fractional integral of order $\alpha$; $I_t^{\alpha }$, is the Riemann-Liouville fractional integral

$${\displaystyle I_t^{\alpha }f:=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t\frac{f\left(\tau \right)}{{(t-\tau )}^{1-\alpha }}d\tau ,\mathrm{for}t>0\mathrm{and}\alpha >0,}$$

and $I_t^0=\mathrm{I}$ is the identity operator. Here, it was assumed that the initial data $u_0$ are deterministic and the Borel measurable, and that there exists a constant $0<\gamma \le 1$ such that

$$u_0\in C_b^{2^{k+1}-2,\gamma }({\mathbb{R}}^d;\mathbb{R}),\mathrm{for}{2}^{k-1}<{\beta }^{-1}\le 2^k,k\in \mathbb{N},$$

(2)

where $C_b^{\alpha ,\gamma }({\mathbb{R}}^d;\mathbb{R})$ is the set of $\alpha$-continuously differentiable functions on ${\mathbb{R}}^d$, whose $\alpha$-derivative is locally Hölder continuous with the exponent $\gamma$.

It is clear that the formal (and non-rigorous) equation is Equation (1). In this article, we work with its rigorous formulation, which is the mild form kernel SIE. Refs. [10,11,12] presented and addressed this SIE for the first time. We include them in Section 2 below, along with some other pertinent information.

Refs. [10,12] obtained the existence, uniqueness, sharp dimension-dependent $L^p$, and the Hölder regularity of the linear and non-linear noise versions of (1). The exact uniform and local continuity moduli for the TFSPIDEs in the time variable t and space variable x were separately obtained in [13]. Specifically, it was shown, in [13], that the fourth-order TFSPIDEs and their gradients have exact, spatio-temporal, dimension-dependent, uniform, and local continuity moduli. In addition to obtaining temporal central limit theorems for modifications of the quadratic variation of the solution to Equation (1) in time, it was also investigated in [14] that the solution to Equation (1) in time has infinite quadratic variation and is not a semimartingale. Ref. [15] obtained the precise, dimension-dependent, non-differentiability moduli for the TFSPIDEs and their gradients in the time variable t.

Here, we would like to mention the global temporal continuity moduli and the local temporal continuity moduli at a prescribed time $t_0\ge 0$, as well as the laws of iterated logarithm (LILs) for $U_{\beta }(·,x)$ and ${\partial }_x{U}_{\beta }(·,x)$, which were obtained in [13]. These phenomena showed the existence of normalized constants for the global temporal continuity moduli and temporal LILs. But their exact values remain unknown. In this paper, we give the exact values of these normalized constants by obtaining precise estimations of the second-order increment moments. For any $d\in {\mathbb{N}}_{+}$, we define $K_{\beta ,d}$ and $K_{\beta ,0}$ by

$${\displaystyle K_{\beta ,d}=\frac{4}{{\left(2\pi \right)}^d}{\int }_{\mathbb{R}}\frac{1-cosu}{u^{2-\left(d\beta \right)/2}}du{\int }_{{\mathbb{R}}^d}\frac{1}{4+{4\left|y\right|}^{2}cos\left(\frac{\beta \pi }{2}\right)+{\left|y\right|}^4}dy,}$$

(3)

and

$${\displaystyle K_{\beta ,0}=\frac{2}{\pi }{\int }_{\mathbb{R}}\frac{1-cosu}{u^{2-3\beta /2}}du{\int }_{\mathbb{R}}\frac{y^2}{4+4y^{2}cos\left(\frac{\beta \pi }{2}\right)+y^4}dy.}$$

(4)

In this article, we obtain the following exact global temporal continuity moduli and temporal LILs for the TFSPIDE $U_{\beta }(t,x)$ and the gradient process ${\partial }_x{U}_{\beta }(t,x)$. Equations (5) and (7) below are other forms of the global temporal continuity moduli of the TFSPIDEs and their gradients, which are slightly different from those obtained in [13].

Theorem 1. 

(Temporal continuity moduli) Let $\beta \in (0,1/2]$, $x\in {\mathbb{R}}^d$ ($d=1,2,3$), and $u_0\equiv 0$ in (1) be fixed.

(a) (Global temporal continuity modulus and temporal LIL for the TFSPIDEs) for every compact interval $I_{\mathrm{time}}\subset {\mathbb{R}}_{+}$,

$$\mathbb{P}\left\{\underset{h\to 0+}{lim}\underset{s,t\in I_{\mathrm{time}},|t-s|<h}{sup}{\varphi }_{\beta ,d,h}^{-1}|U_{\beta }(t,x)-U_{\beta }(s,x)|=1\right\}=1,$$

(5)

where ${\varphi }_{\beta ,d,h}=h^{\frac{2-d\beta }{4}}\sqrt{2K_{\beta ,d}log(1/h)}$, and for every fixed $t\ge 0$

$$\mathbb{P}\left\{\underset{h\to 0+}{lim\; sup}\underset{s,t\in I_{\mathrm{time}},|t-s|<h}{sup}{\widehat{\varphi }}_{\beta ,d,h}^{-1}|U_{\beta }(t,x)-U_{\beta }(s,x)|=1\right\}=1,$$

(6)

where ${\widehat{\varphi }}_{\beta ,d,h}=h^{\frac{2-d\beta }{4}}\sqrt{2K_{\beta ,d}loglog(1/h)}$. Here, $K_{\beta ,d}$ is given in (3).

(b) (Global temporal continuity modulus and temporal LIL for the TFSPIDE gradients.) Let $d=1$. For every compact interval $I_{\mathrm{time}}\subset {\mathbb{R}}_{+}$,

$$\mathbb{P}\left\{\underset{h\to 0+}{lim}\underset{s,t\in I_{\mathrm{time}},|t-s|<h}{sup}{\phi }_{\beta ,h}^{-1}|{\partial }_x{U}_{\beta }(t,x)-{\partial }_x{U}_{\beta }(s,x)|=1\right\}=1,$$

(7)

where ${\phi }_{\beta ,h}=h^{\frac{2-3\beta }{4}}\sqrt{2K_{\beta ,0}log(1/h)}$, and, for every fixed $t\in {\mathbb{R}}_{+}$,

$$\mathbb{P}\left\{\underset{h\to 0+}{lim\; sup}\underset{s,t\in I_{\mathrm{time}},|t-s|<h}{sup}{\widehat{\phi }}_{\beta ,h}^{-1}|{\partial }_x{U}_{\beta }(t,x)-{\partial }_x{U}_{\beta }(s,x)|=1\right\}=1,$$

(8)

where ${\widehat{\phi }}_{\beta ,h}=h^{\frac{2-3\beta }{4}}\sqrt{2K_{\beta ,0}loglog(1/h)}$. Here, $K_{\beta ,0}$ is given in (4).

Remark 1. 

We can infer the following from the aforementioned theorem:

• Equations (5) and (7) are other forms of the global temporal continuity moduli of the TFSPIDEs and the TFSPIDE gradients, respectively, which are slightly different from those obtained in [13]. Equation (5) with $k_8^{(\beta ,d)}{|t-s|}^{\frac{2-d\beta }{4}}\sqrt{2log(1/|t-s\left|\right)}$ taking the place of ${\varphi }_{\beta ,d,h}$, and Equation (7) with $k_{12}{|t-s|}^{\frac{2-3\beta }{4}}$$\sqrt{log(1/|t-s\left|\right)}$ taking the place of ${\phi }_{\beta ,h}$ were established in [13], where $k_8^{(\beta ,d)}>0$ and $k_{12}>0$ were understood as dimension-dependent constants, i.e., independent of x (whose exact values were unknown). Here, in Equations (5) and (7), we give the exact constants for the global temporal continuity moduli of the TFSPIDEs and the TFSPIDE gradients. Moreover, by using Lemma 5 below, we can obtain $k_8^{(\beta ,d)}=\sqrt{2K_{\beta ,d}}$ and $k_{12}=\sqrt{2K_{\beta ,0}}$, as was obtained in [13]. In this sense, the results of this paper reinforce those in [13].
• Equation (6) with $k_9^{(\beta ,d)}{h}^{\frac{2-d\beta }{4}}\sqrt{loglog(1/h)}$ taking the place of ${\widehat{\varphi }}_{\beta ,d,h}$, and Equation (8) with $k_{13}{h}^{\frac{2-3\beta }{4}}\sqrt{loglog(1/h)}$ taking the place of ${\widehat{\phi }}_{\beta ,h}$ were established in [13], where $k_9^{(\beta ,d)}>0$ and $k_{13}>0$ were understood as dimension-dependent constants, i.e., independent of x (whose exact values were unknown). Here, in Equations (6) and (8), we give the exact constants for the temporal LILs of the TFSPIDEs and the TFSPIDE gradients. Moreover, by using Lemma 5 below, we can obtain $k_9^{(\beta ,d)}=\sqrt{2K_{\beta ,d}}$ and $k_{13}=\sqrt{2K_{\beta ,0}}$, as was obtained in [13]. In this sense, the results of this paper reinforce those in [13].
• Equation (5) gives the magnitude of the global maximal oscillation of the TFSPIDE solution $U_{\beta }(·,x)$ over the compact rectangle $I_{\mathrm{time}}$, which is ${\varphi }_{\beta ,d,h}$. Equation (7) gives the magnitude of the global maximal oscillation of the TFSPIDE gradient solution ${\partial }_x{U}_{\beta }(·,x)$ over the compact rectangle $I_{\mathrm{time}}$, which is ${\phi }_{\beta ,h}$.
• Equation (6) gives the magnitude of the local oscillation of the TFSPIDE solution $U_{\beta }(·,x)$ at a prescribed time $t_0\ge 0$ is ${\widehat{\varphi }}_{\beta ,d,h}$. Equation (8) gives the magnitude of the local oscillation of the TFSPIDE gradient solution ${\partial }_x{U}_{\beta }(·,x)$ at a prescribed time $t_0\ge 0$ is ${\widehat{\phi }}_{\beta ,h}$.
• It is interesting to compare Equations (5) and (6). The latter one states that, at some given point, the LIL of $U_{\beta }(·,x)$ for any fixed x is not more than ${\widehat{\varphi }}_{\beta ,d,h}$. On the other hand, the former tells us that the global continuity modulus of $U_{\beta }(·,x)$ can be much larger, namely ${\varphi }_{\beta ,d,h}$. Similarly, by Equations (7) and (8), the LIL of ${\partial }_x{U}_{\beta }(·,x)$ for every fixed x is less than ${\widehat{\phi }}_{\beta ,h}$. On the other hand, the continuity modulus of ${\partial }_x{U}_{\beta }(·,x)$ can be much larger, namely ${\phi }_{\beta ,h}$.
• With Equation (6) and Fubini’s theorem, we have the random time set at

  $$S_{\beta ,d,x,+}:=\left\{t\in [0,1]:\underset{h\to 0+}{lim\; sup}{\widehat{\varphi }}_{\beta ,d,h}^{-1}|U_{\beta }(t+h,x)-U_{\beta }(t,x)|>1\right\},$$

  which has a Lebesgue measurement of zero with a probability of one. Nevertheless, $S_{\beta ,d,x,+}$ is not null. It is almost certain that the set of t that satisfies the stronger growth criterion (9) below is dense everywhere with the power of the continuum. There are similar properties for the TFSPIDE gradient ${\partial }_x{U}_{\beta }(·,x)$.

Fix $x\in {\mathbb{R}}^d$. For every $\lambda \in (0,1]$, the set of temporal $\lambda$-fast points for the fourth-order TFSPIDE are defined by

$$S_{\beta ,d,x}\left(\lambda \right):=\left\{t\in [0,1]:\underset{h\to 0+}{lim\; sup}{\varphi }_{\beta ,d,h}|U_{\beta }(t+h,x)-U_{\beta }(t,x)|\ge \lambda \right\},$$

(9)

where ${\varphi }_{\beta ,d,h}$ is given in (5). For every $\chi \in (0,1]$, the set of the temporal $\chi$-fast points for the fourth-order TFSPIDE gradients are defined by

$${\mathcal{S}}_{\beta ,x}\left(\chi \right):=\left\{t\in [0,1]:\underset{h\to 0+}{lim\; sup}{\phi }_{\beta ,h}|{\partial }_x{U}_{\beta }(t+h,x)-{\partial }_x{U}_{\beta }(t,x)|\ge \chi \right\},$$

(10)

where ${\phi }_{\beta ,h}$ is given in (7).

The $S_{\beta ,d,x}\left(\lambda \right)$ are the sets of t, where the temporal LIL of TFSPIDEs fail, and the ${\mathcal{S}}_{\beta ,x}\left(\chi \right)$ are the sets of t, where the temporal LIL of TFSPIDE gradients fail. This kind of set is usually called the fast point set or exceptional time set. It is interesting to obtain information about the sizes of $S_{\beta ,d,x}\left(\lambda \right)$ and ${\mathcal{S}}_{\beta ,x}\left(\chi \right)$. We usually do this by considering their Hausdorff measures. This problem was first introduced in Orey and Taylor [16] on the fast set for Brownian motion. After this famous paper, there were several papers that studied this problem for general Gaussian processes. Among other things, the fractal nature of the fast set of empirical processes with independent increments was studied in [17]. The fractal nature of the fast point set of $L^p$-valued Gaussian processes was studied in [18]. The limsup fractal nature of the fast point sets of Gaussian processes was studied in [19]. The solutions and gradient solutions for TFSPIDEs are spatio-temporal Gaussian random fields. It is, therefore, natural to study this type of fractal nature (in the sense of [16,19]). This paper is devoted to establishing the fractal nature and hitting probabilities for the sets of temporal fast points for TFSPIDE $U_{\beta }(t,x)$ and the gradient process ${\partial }_x{U}_{\beta }(t,x)$.

Recall (see, e.g., [20,21]) that the Hausdorff dimension $dimB$ of a subset B of $[0,1]$ is defined by

$$dim\left(B\right)=inf\{\alpha >0:{\mu }_g\left(B\right)=0\mathrm{for}g\left(s\right)=s^{\alpha }\}.$$

The Hausdorff g-measure of a subset B of a real line for any continuous increasing function $g:[0,1]\to [0,+\infty ]$ with $g\left(0\right)=0$ is defined as follows:

$${\mu }_g\left(B\right)=\underset{\delta \to 0}{lim}\left[\underset{\genfrac{}{}{0pt}{}{B\subseteq \cup C_i}{d\left(C_i\right)<\delta }}{inf}\sum g\left(d\left(C_i\right)\right)\right],$$

(11)

where the infimum in (11) extends over all countable covers of B by sets $C_i$ of diameter $d\left(C_i\right)<\delta$. Keep in mind that, while ${\mu }_g\left(B\right)$ simplifies to an Lebesgue outer measure if $g\left(s\right)=s$, using a distinct g creates a hierarchy of measures. By being familiar with the class of measure functions g for which ${\mu }_g\left(B\right)=0$, one may determine the metric features of B. The purpose of this article is to show the following two theorems. In the first one, we show that $S_{\beta ,d,x}\left(\lambda \right)$ and ${\mathcal{S}}_{\beta ,x}\left(\chi \right)$ are random fractals, and we also evaluate their Hausdorff dimensions. In the second one, we show that hitting probabilities are determined by the the target set B’s packing dimension ${dim}_{{}_P}\left(B\right)$ rather than its Hausdorff dimension $dim\left(B\right)$. For a definition of packing dimension, see [22].

Theorem 2. 

(Fractal nature for the sets of the temporal fast points.) Let $\beta \in (0,1/2]$, $x\in {\mathbb{R}}^d$ ($d=1,2,3$) and $u_0\equiv 0$ in (1) be fixed.

(a) Suppose $d\in \{1,2,3\}$. For every $\lambda \in [0,1]$ with a probability of one, we have

$$dim\left(S_{\beta ,d,x}\left(\lambda \right)\right)=1-{\lambda }^2.$$

(12)

(b) Suppose $d=1$. For every $\chi \in [0,1]$ with a probability of one, we have

$$dim\left({\mathcal{S}}_{\beta ,x}\left(\chi \right)\right)=1-{\chi }^2.$$

(13)

The following theorem demonstrates that the appropriate index through which to determine whether sets overlap $S_{\beta ,d,x}\left(\lambda \right)$ and ${\mathcal{S}}_{\beta ,x}\left(\chi \right)$ is the packing dimension.

Theorem 3. 

(Hitting probabilities for the sets of temporal fast points.) Let $\beta \in (0,1/2]$, $x\in {\mathbb{R}}^d$ ($d=1,2,3$) and $u_0\equiv 0$ in (1) be fixed.

(a) Suppose $d\in \{1,2,3\}$. For every $\lambda \in [0,1]$ and every analytic set $B\subset {\mathbb{R}}_{+}$, we have

$$P\{S_{\beta ,d,x}\left(\lambda \right)\cap B\ne \varnothing \}=\left\{\begin{array}{cc}1,\hfill & if{dim}_{{}_P}\left(B\right)>{\lambda }^2,\hfill \\ 0,\hfill & if{dim}_{{}_P}\left(B\right)<{\lambda }^2.\hfill \end{array}\right.$$

(14)

(b) Suppose $d=1$. For every $\chi \in [0,1]$ and every analytic set $B\subset {\mathbb{R}}_{+}$, we have

$$P\{{\mathcal{S}}_{\beta ,x}\left(\chi \right)\cap B\ne \varnothing \}=\left\{\begin{array}{cc}1,\hfill & if{dim}_{{}_P}\left(B\right)>{\chi }^2,\hfill \\ 0,\hfill & if{dim}_{{}_P}\left(B\right)<{\chi }^2.\hfill \end{array}\right.$$

(15)

Remark 2. 

It is easy to see that Equations (14) and (15) are respectively equivalent for every analytic set $B\subset {\mathbb{R}}_{+}$. As such, we have

$$\mathbb{P}\left\{\underset{t\in B}{sup}\underset{h\to 0+}{lim\; sup}{\varphi }_{\beta ,d,h}|U_{\beta }(t+h,x)-U_{\beta }(t,x)|={\left({dim}_{{}_P}\left(B\right)\right)}^{1/2}\right\}=1,$$

(16)

and

$$\mathbb{P}\left\{\underset{t\in B}{sup}\underset{h\to 0+}{lim\; sup}{\phi }_{\beta ,h}|{\partial }_x{U}_{\beta }(t+h,x)-{\partial }_x{U}_{\beta }(t,x)|={\left({dim}_{{}_P}\left(B\right)\right)}^{1/2}\right\}=1.$$

(17)

Thus, in the context of TFSPIDEs and their gradients, Equations (16) and (17) can be understood as two probabilistic interpretations of the packing dimension of an analytic set $B\subset {\mathbb{R}}_{+}$.

Remark 3. 

We obtain the following probabilistic interpretations of the upper and lower Minkowski dimensions of B, which are denoted by ${\overline{dim}}_{{}_M}\left(B\right)$ and ${\underline{dim}}_{{}_M}\left(B\right)$, respectively. This was achieved by reversing the order of sup and lim sup in Equation (16); these definitions are provided in [22].

$$\mathbb{P}\left\{\underset{h\to 0+}{lim\; sup}\underset{t\in B}{sup}{\varphi }_{\beta ,d,h}|U_{\beta }(t+h,x)-U_{\beta }(t,x)|={\left({\overline{dim}}_{{}_M}\left(B\right)\right)}^{1/2}\right\}=1,$$

(18)

$$\mathbb{P}\left\{\underset{h\to 0+}{lim\; inf}\underset{t\in B}{sup}{\varphi }_{\beta ,d,h}|U_{\beta }(t+h,x)-U_{\beta }(t,x)|={\left({\underline{dim}}_{{}_M}\left(B\right)\right)}^{1/2}\right\}=1.$$

(19)

According to Equations (18) and (19), there are also probabilistic interpretations of the upper and lower Minkowski dimensions of B.

An undefined positive, finite constant, c, will be used throughout this work; however, it might not always be the same. $c_{i,1},c_{i,2}\dots$ were found to be more particularly positive and finite constants (independent of x), as shown in Section 1.

The remainder of the article is organized as follows. In Section 2, using the time-fractional SPIDEs kernel SIE formulation, the rigorous TFSPIDE kernel SIE (mild) formulation and temporal spectral density for TFSPIDEs and their gradients are discussed. Estimations on the second-order moments of temporal increments of the fourth-order TFSPIDEs and their gradients are also obtained. In Section 3, we prove Theorem 1 and thereby establish the exact temporal continuity moduli for the TFSPIDEs and their gradients; in addition, we prove Theorem 2 and thereby obtain Hausdorff dimensions of the sets of temporal fast points for the TFSPIDEs and their gradients. Furthermore, we prove Theorem 3 and thereby obtain the hitting probabilities of the sets of temporal fast points for the TFSPIDEs and their gradients. In Section 4, the results are summarized and discussed.

2. Preliminaries

2.1. Rigorous Kernel SIE Formulations

We define the rigorous mild SIE formulations of the TFSPIDEs, as in [13], using the density of an inverse stable Lévy time Brownian motion. According to [10,11,12], this density is the time-fractional PDE’s solution as follows:

$$\left\{\begin{array}{cc}{\displaystyle C_{{\partial }_t^{\beta }{U}_{\beta }}=\frac{1}{2}\Delta U_{\beta },}\hfill & (t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^d;\hfill \\ U_{\beta }(0,x)=\delta \left(x\right),\hfill & x\in {\mathbb{R}}^d,\hfill \end{array}\right.$$

(20)

where $\delta \left(x\right)$ is the Dirac function. This solution is the transition density of a d-dimensional $\beta$-inverse-stable-Lévy-time Brownian motion ($\beta$-ISLTBM). It starts from $x\in {\mathbb{R}}^d$, ${\mathbb{B}}_{A_{\beta }}^x:=\{B^x\left(A_{\beta }\left(t\right)\right),t\ge 0\}$, where the inverse stable Lévy motion $A_{\beta }$ of index $\beta \in (0,1/2]$ serves as the time clock for an independent d-dimensional Brownian motion $B^x$ (see [10,23]), which is given by the following:

$${\displaystyle {\mathbb{H}}_{t;x,y}^{(\beta ,d)}={\int }_0^{+\infty }{H}_{s;x,y}^{{\mathrm{BM}}^d}{H}_{t;0,s}^{A_{\beta }}ds,}$$

(21)

where $H_{s;x,y}^{{\mathrm{BM}}^d}=\frac{{-|x-y|}^2/2s}{{\left(2\pi s\right)}^{d/2}}$ and $H_{t;0,s}^{A_{\beta }}=t{\beta }^{-1}{s}^{-1-1/\beta }{g}_{\beta }\left(t{s}^{-1/\beta }\right)$. Here, the density of a stable subordinator is denoted by $g_{\beta }\left(u\right)$, and its Laplace transform is $e^{-s^{\beta }}$. When $\beta =1/2$, the density of the Brownian-time Brownian motion (BTBM) is represented by the kernel ${\mathbb{H}}_{t;x}^{(\beta ,d)}$, as described in [9]; for $\beta \in \{1/2^k;k\in \mathbb{N}\}$, the density of the k-iterated BTBM is represented by the kernel ${\mathbb{H}}_{t;x}^{(\beta ,d)}$, as explained in [10,11].

Let $b:\mathbb{R}\to \mathbb{R}$ be Borel measurable. The non-linear drift diffusion TFSPIDE is thus

$$\left\{\begin{array}{cc}{\displaystyle C_{{\partial }_t^{\beta }{U}_{\beta }}=\frac{1}{2}\Delta U_{\beta }+I_t^{1-\beta }\left[b\left(U_{\beta }\right)+a\left(U_{\beta }\right)\frac{{\partial }^{d+1}W}{\partial t\partial x}\right],}\hfill & (t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^d;\hfill \\ U_{\beta }(0,x)=u_0\left(x\right),\hfill & x\in {\mathbb{R}}^d.\hfill \end{array}\right.$$

(22)

Then, the rigorous TFSPIDE kernel SIE formulation is the SIE (see Equation (1.11), and Definition 1.1 in [12], as well as p. 530 in [9]), is as follows:

$$\begin{array}{cc}U(t,x)=\hfill & {\displaystyle {\int }_{{\mathbb{R}}^d}{\mathbb{H}}_{t;x,y}^{(\beta ,d)}{u}_0\left(y\right)dy}\hfill \\ \hfill & {\displaystyle {\int }_{{\mathbb{R}}^d}{\int }_0^t{\mathbb{H}}_{t-s;x,y}^{(\beta ,d)}[b\left(U(s,y)\right)dsdy+a\left(U(s,y)\right)W(ds\times dy)].}\hfill \end{array}$$

(23)

Naturally, this yields the mild formulation of (1.1), which is when $a\equiv 1$ and $b\equiv 0$ are set in (22).

The spatial Fourier transform of the $\beta$-time-fractional (including the $\beta =1/2$ BTBM example) kernels from Lemma 2.1 in [13] is cited to conclude this section.

Lemma 1 

(Transforms of a spatial Fourier type). Let $0<\beta <1$ and ${\mathbb{H}}_{t;x,y}^{(\beta ,d)}$ be the β-time-fractional kernel. The β-time-fractional kernel’s spatial Fourier transform is provided by

$${\displaystyle {\widehat{\mathbb{H}}}_{t;x,\xi }^{(\beta ,d)}={\left(2\pi \right)}^{-\frac{d}{2}}{E}_{\beta }\left(-\frac{{\left|\xi \right|}^2}{2}{t}^{\beta }\right),}$$

(24)

where

$${\displaystyle E_{\beta }\left(u\right)=\sum _{k=0}^{+\infty }\frac{u^k}{\Gamma (1+\beta k)},}$$

(25)

is the well-known function of Mittag–Leffler. The spatial Fourier transform in its symmetric form is applied here as follows: $\widehat{f}\left(\xi \right)={\left(2\pi \right)}^{-d/2}{\int }_{{\mathbb{R}}^d}f\left(u\right)$$e^{-\mathbf{i}\xi ·u}du$.

2.2. Estimations on the Variances of Temporal Increments of TFSPIDEs and Their Gradients

For the purposes of this subsection, let $x\in {\mathbb{R}}^d$ be an arbitrary, fixed variable. The auxiliary Gaussian random field $\{X_{\beta }(t,x),t\in {\mathbb{R}}_{+},x\in {\mathbb{R}}^d\}$ is defined by the following:

$${\displaystyle X_{\beta }(t,x)={\int }_{{\mathbb{R}}^d}{\int }_{\mathbb{R}}\left({\mathbb{H}}_{{(t-r)}_{+};x,y}^{(\beta ,d)}-{\mathbb{H}}_{{(-r)}_{+};x,y}^{(\beta ,d)}\right)W(dr\times dy),}$$

(26)

where $z_{+}=max\{z,0\}$ for any $z\in \mathbb{R}$. Then, the TFSPIDE solution $U_{\beta }$ has a decomposition as $U_{\beta }(t,x)=X_{\beta }(t,x)-Y_{\beta }(t,x)$, where

$${\displaystyle Y_{\beta }(t,x)={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}_{-}}\left({\mathbb{H}}_{{(t-r)}_{+};x,y}^{(\beta ,d)}-{\mathbb{H}}_{{(-r)}_{+};x,y}^{(\beta ,d)}\right)W(dr\times dy).}$$

(27)

This decomposition idea was first introduced in the second-order SPDE setting in [23]. It has since been implemented in the second-order heat SPDE setting in [24,25].

Using the previously mentioned decomposition of $U_{\beta }$, we first calculated the exact variance for the temporal increments of the auxiliary process $X_{\beta }$. Then, we transferred these to our TFSPIDE solution $U_{\beta }$ in terms of $X_{\beta }$ and a smooth process of $Y_{\beta }$. The outcome that followed was crucial.

Lemma 2. 

Let $\beta \in (0,1/2]$, $x\in {\mathbb{R}}^d$ ($d=1,2,3$) and $u_0\equiv 0$ in (1) be fixed. Then, for any $s,t\in (0,T]$ such that $t/s$ is sufficiently close to 1, we have

$$\mathbb{E}\left[{(U_{\beta }(t,x)-U_{\beta }(s,x))}^2\right]=(K_{\beta ,d}+o\left(1\right)){|t-s|}^{\frac{2-d\beta }{2}},$$

(28)

where $K_{\beta ,d}$ is given in (3).

Proof. 

With Theorem 4.1 in [13], we have

$${\displaystyle \mathbb{E}\left[\right|X_{\beta }(t,x)-X_{\beta }(s,x){|}^2]=2{\int }_{\mathbb{R}}(1-cos\left((t-s)\tau \right))f_{\beta }\left(\tau \right)d\tau ,}$$

(29)

where

$${\displaystyle f_{\beta }\left(\tau \right)={\left(2\pi \right)}^{-d}\frac{1}{{\left|\tau \right|}^{2-\left(\right(d\beta \left)\right)/2}}{\int }_{{\mathbb{R}}^d}\frac{1}{1+{\left|\xi \right|}^{2}cos\left(\frac{\pi \beta }{2}\right)+\frac{1}{4}{\left|\xi \right|}^4}d\xi .}$$

With the change in variable $\tau \mapsto u:u=(t-s)\tau$, (29) yields

$$\mathbb{E}\left[\right|X_{\beta }(t,x)-X_{\beta }{(s,x)|}^2]=K_{\beta ,d}{|t-s|}^{1-\frac{d\beta }{2}}.$$

(30)

Let $x\in {\mathbb{R}}^d$ be fixed. For each $0<s<t$, we can obtain the following by using Parseval’s identity to the integral in y:

$$\begin{array}{cc}\mathbb{E}\left[\right|Y_{\beta }(t,x)-Y_{\beta }(s,x){|}^2]\hfill & {\displaystyle ={\int }_{{\mathbb{R}}^d}{\int }_{\mathbb{R}}|{\mathbb{H}}_{t-r;x,y}^{(\beta ,d)}{\mathbb{I}}_{\{0>r\}}-{\mathbb{H}}_{s-r;x,y}^{(\beta ,d)}{\mathbb{I}}_{\{0>r\}}{|}^{2}drdy}\hfill \\ \hfill & {\displaystyle ={\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^d}|{\widehat{\mathbb{H}}}_{t-r;x,\xi }^{(\beta ,d)}{\mathbb{I}}_{\{0>r\}}-{\widehat{\mathbb{H}}}_{s-r;x,\xi }^{(\beta ,d)}{\mathbb{I}}_{\{0>r\}}{|}^{2}d\xi dr.}\hfill \end{array}$$

(31)

Note that

$${\displaystyle {\widehat{\mathbb{H}}}_{t-r;x,\xi }^{(\beta ,d)}={\left(2\pi \right)}^{-\frac{d}{2}}{E}_{\beta }\left(-\frac{{\left|\xi \right|}^2{(t-r)}^{\beta }}{2}\right).}$$

(32)

Through the corollary on page 23 in [26], we have

$${\displaystyle {\int }_{{\mathbb{R}}^d}f\left(\sum _{i=1}^d{x}_i^2\right)d{x}_1\cdots d{x}_d=\frac{{\pi }^{d/2}}{\Gamma (d/2)}{\int }_0^{+\infty }{y}^{d/2-1}f\left(y\right)dy,}$$

(33)

Through Equations (32) and (33), Equation (31) becomes

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\right|Y_{\beta }(t,x)-Y_{\beta }(s,x){|}^2]\hfill \\ \hfill & {\displaystyle ={\int }_{{\mathbb{R}}^d}{\int }_{\mathbb{R}}\frac{|E_{\beta }\left(-\frac{{\left|\xi \right|}^2}{2}{(t-r)}^{\beta }\right){\mathbb{I}}_{\{0>r\}}-E_{\beta }\left(-\frac{{\left|\xi \right|}^2}{2}{(s-r)}^{\beta }\right){\mathbb{I}}_{\{0>r\}}{|}^2}{{\left(2\pi \right)}^d}drd\xi }\hfill \\ \hfill & {\displaystyle ={\left(2\pi \right)}^{-d}\frac{{\pi }^{d/2}}{\Gamma (d/2)}{\int }_0^{+\infty }{y}^{d/2-1}{\int }_{-\infty }^0|E_{\beta }\left(-\frac{y}{2}{(t-r)}^{\beta }\right)-E_{\beta }\left(-\frac{y}{2}{(s-r)}^{\beta }\right){|}^{2}drdy.}\hfill \end{array}$$

(34)

It follows from (7.7) in [27] that

$${\displaystyle E_{\beta }(-x)=\frac{sin\left(\beta \pi \right)}{\pi }{\int }_0^{+\infty }{a}_{\beta }\left(\zeta \right)e^{-\zeta x^{1/\beta }}d\zeta ,}$$

(35)

where

$${\displaystyle a_{\beta }\left(\zeta \right)=\frac{{\zeta }^{\beta }}{1+2{\zeta }^{\beta }cos\left(\beta \pi \right)+{\zeta }^{2\beta }}.}$$

Thus, Equation (34) yields

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\right|Y_{\beta }(t,x)-Y_{\beta }(s,x){|}^2]\hfill \\ \hfill & {\displaystyle =\frac{1}{2^d\Gamma (d/2)}\frac{{sin}^2\left(\beta \pi \right)}{{\pi }^{2+d/2}}{\int }_0^{+\infty }{y}^{d/2-1}{\int }_{-\infty }^0|{\int }_0^{+\infty }{a}_{\beta }\left(\zeta \right)(e^{-\zeta {\left(\frac{y}{2}\right)}^{1/\beta }(t-r)}-e^{-\zeta {\left(\frac{y}{2}\right)}^{1/\beta }(s-r)})d\zeta {|}^{2}drdy}\hfill \\ \hfill & {\displaystyle \le \frac{1}{2^d\Gamma (d/2)}\frac{{sin}^2\left(\beta \pi \right)}{{\pi }^{2+d/2}}{\int }_0^{+\infty }{y}^{d/2-1}{\int }_{-\infty }^0\left|{\int }_0^{+\infty }{a}_{\beta }\left(\zeta \right)e^{-\zeta {\left(\frac{y}{2}\right)}^{1/\beta }(s-r)}\right|e^{-\zeta {\left(\frac{y}{2}\right)}^{1/\beta }(t-s)}-1|d\zeta {|}^{2}drdy.}\hfill \end{array}$$

(36)

Since, for any $u\ge 0$, $|1-e^{-u}|\le 2u$, Equation (36) yields

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\right|Y_{\beta }(t,x)-Y_{\beta }(s,x){|}^2]\hfill \\ \hfill & {\displaystyle \le \frac{1}{2^{d+2/\beta -2}\Gamma (d/2)}\frac{{sin}^2\left(\beta \pi \right)}{{\pi }^{d/2}}{(t-s)}^2{\int }_0^{+\infty }{y}^{d/2+2/\beta -1}{\int }_{-\infty }^0|{\int }_0^{+\infty }{a}_{\beta }\left(\zeta \right)e^{-\zeta {\left(\frac{y}{2}\right)}^{1/\beta }(s-r)}d\zeta {|}^{2}drdy.}\hfill \end{array}$$

(37)

By changing the variables $r\mapsto u:u={\left(\frac{y}{2}\right)}^{1/\beta }r$ and $y\mapsto v:v={\left(\frac{y}{2}\right)}^{1/\beta }s$, (37) yields

$$\begin{array}{cc}\hfill & \mathbb{E}\left[\right|Y_{\beta }(t,x)-Y_{\beta }(s,x){|}^2]\hfill \\ \hfill & {\displaystyle \le \frac{2^{d/2+2}\beta }{{\left(2\pi \right)}^d}\frac{{\pi }^{d/2}}{\Gamma (d/2)}\frac{{sin}^2\left(\beta \pi \right)}{{\pi }^2}\frac{{(t-s)}^2}{s^{\left(d\beta \right)/2+1}}{\int }_0^{+\infty }{v}^{\left(d\beta \right)/2}{\int }_0^{+\infty }|{\int }_0^{+\infty }\zeta a_{\beta }\left(\zeta \right)e^{-\zeta v}{e}^{-\zeta u}d\zeta {|}^{2}dudv}\hfill \\ \hfill & {\displaystyle \le \frac{c_{{}_{2,1}}}{s^{\left(d\beta \right)/2+1}}{(t-s)}^2,}\hfill \end{array}$$

(38)

since the integral above is finite for $0<\beta \le 1/2$. Furthermore, as $U_{\beta }$ and $Y_{\beta }$ are independent, we have

$$\mathbb{E}\left[\right|X_{\beta }(t,x)-X_{\beta }{(s,x)|}^2]=\mathbb{E}[|U_{\beta }(t,x)-U_{\beta }{(s,x)|}^2]+\mathbb{E}[|Y_{\beta }(t,x)-Y_{\beta }(s,x){|}^2].$$

(39)

Note that $s^{-\left(\right(d\beta )/2+1)}{(t-s)}^2={(t-s)}^{1-\left(d\beta \right)/2}{(t/s-1)}^{1+\left(d\beta \right)/2}$. Combining (30), (38) and (39), we also obtain the following for any $0<s<t$:

$$|\mathbb{E}[{(U_{\beta }(t,x)-U_{\beta }(s,x))}^2]-K_{\beta ,d}{(t-s)}^{1-\left(d\beta \right)/2}|\le c_{{}_{2,2}}{(t-s)}^{1-\left(d\beta \right)/2}{(t/s-1)}^{1+\left(d\beta \right)/2}.$$

(40)

This yields (28) and completes the proof. □

We also need the following estimation on the variances of temporal increments of the TFSPIDE gradient process ${\partial }_x{U}_{\beta }(·,x)$.

Lemma 3. 

Let $\beta \in (0,1/2]$, $x\in {\mathbb{R}}^d$ ($d=1,2,3$) and $u_0\equiv 0$ in (1) be fixed. Then, for all $s,t\in (0,T]$, such that $t/s$ is sufficiently close to 1, we have

$$\mathbb{E}[{({\partial }_x{U}_{\beta }(t,x)-{\partial }_x{U}_{\beta }(s,x))}^2]=(K_{\beta ,0}+o\left(1\right)){|t-s|}^{\frac{2-3\beta }{2}},$$

(41)

where $K_{\beta ,0}$ is given in (4).

Proof. 

Through (4.40) in [13], we have

$${\displaystyle \mathbb{E}\left[\right|{\partial }_x{X}_{\beta }(t,x)-{\partial }_x{X}_{\beta }(s,x){|}^2]=2{\int }_{\mathbb{R}}(1-cos\left((t-s)\tau \right))f_{\beta }\left(\tau \right)d\tau ,}$$

(42)

where

$${\displaystyle f_{\beta }\left(\tau \right)={\left(2\pi \right)}^{-1}{\left|\tau \right|}^{-2+\frac{3\beta }{2}}{\int }_{\mathbb{R}}\frac{{\xi }^2}{1+{\xi }^{2}cos\left(\frac{\pi \beta }{2}\right)+\frac{1}{4}{\xi }^4}d\xi .}$$

Via a change in the variables to the integral in $\tau$, (42) yields

$$\mathbb{E}\left[\right|{\partial }_x{X}_{\beta }(t,x)-{\partial }_x{X}_{\beta }{(s,x)|}^2]=K_{\beta ,0}{|t-s|}^{1-\frac{3\beta }{2}}.$$

(43)

Let $x\in {\mathbb{R}}^d$ be fixed. For each $0<s<t$, we obtain the following by using Parseval’s identity to the integral in y:

$$\begin{array}{cc}\mathbb{E}\left[\right|{\partial }_x{Y}_{\beta }(t,x)-{\partial }_x{Y}_{\beta }(s,x){|}^2]\hfill & {\displaystyle ={\int }_{{\mathbb{R}}^d}{\int }_{\mathbb{R}}|{\partial }_x{\mathbb{H}}_{t-r;x,y}^{(\beta ,1)}{\mathbb{I}}_{\{0>r\}}-{\partial }_x{\mathbb{H}}_{s-r;x,y}^{(\beta ,1)}{\mathbb{I}}_{\{0>r\}}{|}^{2}drdy}\hfill \\ \hfill & {\displaystyle ={\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^d}{\xi }^2|{\widehat{\mathbb{H}}}_{t-r;x,\xi }^{(\beta ,1)}{\mathbb{I}}_{\{0>r\}}-{\widehat{\mathbb{H}}}_{s-r;x,\xi }^{(\beta ,1)}{\mathbb{I}}_{\{0>r\}}{|}^{2}d\xi dr.}\hfill \end{array}$$

(44)

Following the same route as the proof of (38), via (44), we have

$${\displaystyle \mathbb{E}\left[\right|{\partial }_x{Y}_{\beta }(t,x)-{\partial }_x{Y}_{\beta }(s,x){|}^2]\le \frac{c_{{}_{2,3}}}{s^{3\beta /2+1}}{(t-s)}^2.}$$

(45)

Thus, with (43) and (45), similar to the proof of (40), we obtain (41). This completes the proof. □

3. Results

3.1. Temporal Moduli of Continuity

We prove Theorem 1 in this subsection, thus establishing the temporal moduli of continuity for the TFSPIDEs, as well as their gradients, in the process. The following precise large deviation estimates for the TFSPIDEs and their gradients are necessary for our results.

Lemma 4. 

Let $\beta \in (0,1/2]$, $x\in {\mathbb{R}}^d$ ($d=1,2,3$) and $u_0\equiv 0$ in (1) be fixed.

(a) Suppose $d\in \{1,2,3\}$. Then, for any $t,h\in {\mathring{\mathbb{R}}}_{+}$, such that $h/t$ is sufficiently close to 0, we have

$${\displaystyle \underset{u\to +\infty }{lim}{u}^{-2}log\mathbb{P}\left(|U_{\beta }(t+h,x)-U_{\beta }(t,x)|\ge u{K}_{\beta ,d}^{1/2}{h}^{\frac{2-d\beta }{4}}\right)=-\frac{1}{2}.}$$

(46)

(b) Suppose $d=1$. Then, for any $t,h\in {\mathring{\mathbb{R}}}_{+}$, such that $h/t$ is sufficiently close to 0, we have

$${\displaystyle \underset{u\to +\infty }{lim}{u}^{-2}log\mathbb{P}\left(|{\partial }_x{U}_{\beta }(t+h,x)-{\partial }_x{U}_{\beta }(t,x)|\ge u{K}_{\beta ,0}^{1/2}{h}^{\frac{2-3\beta }{4}}\right)=-\frac{1}{2}.}$$

(47)

Proof. 

We only show (46) because the proof of (47), which is similar to that of (46). Since $h/t$ is sufficiently close to 0, via Lemma 2, we have $\mathbb{E}\left[{(U_{\beta }(t+h,x)-U_{\beta }(t,x))}^2\right]=(K_{\beta ,d}+o\left(1\right))h^{1-\frac{d\beta }{2}}$. Thus, via a well-known estimation (cf., e.g., [28] (p. 23)), we have

$${\displaystyle \frac{1}{\sqrt{2\pi }}\left(\frac{1}{x}-\frac{1}{x^3}\right)e^{-x^2/2}\le 1-\Phi \left(x\right)\le \frac{1}{\sqrt{2\pi }x}{e}^{-x^2/2},\forall x>0,}$$

(48)

in which we obtain (46) immediately. The proof is thus completed. □

We needed the following Fernique-type inequality for the TFSPIDEs and their gradients as it is required in the proof.

Lemma 5. 

Let $\beta \in (0,1/2]$, $x\in {\mathbb{R}}^d$ ($d=1,2,3$) and $u_0\equiv 0$ in (1) be fixed.

(a) Suppose $d\in \{1,2,3\}$. Then, for any $ϵ>0$, there exist positive and finite constants, i.e., independent of x, and $h_0=h_0\left(ϵ\right)$ and $c=c\left(ϵ\right)$ are such that, for any compact interval $I_{\mathrm{time}}\subset {\mathbb{R}}_{+}$, $0<h<h_0$ and $u>0$, we have

$${\displaystyle \mathbb{P}\left(\underset{s,t\in I_{\mathrm{time}},|t-s|<h}{sup}|U_{\beta }(t,x)-U_{\beta }(s,x)|\ge u{K}_{\beta ,d}^{1/2}{h}^{\frac{2-d\beta }{4}}\right)\le \frac{c}{h}{e}^{-\frac{u^2}{2+ϵ}}.}$$

(49)

(b) Suppose $d=1$. Then, for any $ϵ>0$, there exist positive and finite constants, i.e., independent of x, and $h_0=h_0\left(ϵ\right)$ and $c=c\left(ϵ\right)$ are such that, for any compact interval $I_{\mathrm{time}}\subset {\mathbb{R}}_{+}$, $0<h<h_0$ and $u>0$, we have

$${\displaystyle \mathbb{P}\left(\underset{s,t\in I_{\mathrm{time}},|t-s|<h}{sup}|{\partial }_x{U}_{\beta }(t,x)-{\partial }_x{U}_{\beta }(s,x)|\ge u{K}_{\beta ,0}^{1/2}{h}^{\frac{2-3\beta }{4}}\right)\le \frac{c}{h}{e}^{-\frac{u^2}{2+ϵ}}.}$$

(50)

Proof. 

By using (46) and (47), as well as by following the same route as the proof of Proposition 3.3 in [29], we obtain (49) and (50), respectively. This completes the proof. □

Now, we can complete the Proof of Theorem 1.

Proof of Theorem 1. 

By making use of (49) and (50), as well as by following the same route as the proof of Theorems 1.4 and 1.7 in [13], we obtain (5)–(8). This completes the proof. □

3.2. Hausdorff Dimensions for the Sets of Temporal Fast Points

We prove Theorem 2 in this subsection, thus obtaining Hausdorff dimensions for the sets of temporal fast points of the TFSPIDEs, as well as their gradients, in the process.

Proof of Theorem 2. 

We only show Equation (11) because Equation (12) can be proved similarly. Equation (11). Via Lemma 5 and the following, i.e., the same lines in the proof of Theorem 2 of [16] (p. 180), we can show that, with a probability of one,

$$\forall \lambda \in [0,1],dim\left(S_{\beta ,d,x}\left(\lambda \right)\right)\le 1-{\lambda }^2.$$

(51)

That is, the upper bound of Equation (11) is validated. □

We now turn to the proof of the opposite inequality. It suffices to show that, with a probability of one,

$$\forall \lambda \in [0,1],dim\left(S_{\beta ,d,x}\left(\lambda \right)\right)\ge 1-{\lambda }^2.$$

(52)

We follow Theorem 1.1 of [18]. Without a loss of generality, we can assume $0<\lambda <1$. For every fixed $0<{\lambda }_0<\lambda <1$, we show that $S_{\beta ,d,x}\left(\lambda \right)$ contains a Cantor-like subset of dimension of at least $\eta -2ϵ$, where $0<ϵ<\eta /2<1$ and $\eta =1-{\lambda }_0^2$. A sequence of values for ${\lambda }_0$ converging to $\lambda$, as well as $ϵ$ converging to 0, was then used to determine the outcome. The focus of the proof was on creating this Cantor-like subset, which was essentially a generalized version of the reasoning presented in the proofs of [16,18].

We state the following lemma that is required in the proof (see [18]).

Lemma 6. 

Suppose $g:[0,1]\to [0,+\infty )$ is a continuous function with $g\left(0\right)=0$. Let $F\subset [0,1]$ be such that $F={\cap }_{m=1}^{+\infty }{F}_m$, where $F_1\supset \cdots \supset F_m\cdots$ for $m=1,2,...$, and $F_m={\cup }_{k=1}^{N_m}{I}_{m,i}$ with $\{I_{m,i}:1\le i\le N_m\}$ being, for each $m\ge 1$, a collection of disjoint closed subintervals of $[0,1]$. Then, if there exist two constants $\delta >0$ and $C>0$, such that, for every interval $I\subset [0,1]$ with $\left|I\right|\le \delta$, there is a constant $m\left(I\right)$, such that, for all $m\ge m\left(I\right)$, we have

$$N_m\left(I\right)=:\#\{I_{m,i}\subset I;1\le i\le N_m\}\le Cg\left(\right|I\left|\right)N_m,$$

(53)

we have ${\mu }_g\left(F\right)>0$.

Let $\mathcal{T}$ be the collection of intervals $[s,t]\subset [0,1]$ such that

$$U_{\beta }(t,x)-U_{\beta }(s,x)\ge \lambda {\varphi }_{|t-s|}.$$

The modulus of continuity (5) tells us that

$$|U_{\beta }(t,x)-U_{\beta }(s,x)|\le \sqrt{2}{\varphi }_{|t-s|},$$

(54)

for all $s,t\in [0,1]$ that have a $|s-t|$ that is small enough. Thus, there is $b>0$, which depends only on $\lambda$ and ${\lambda }_0$ such that, for every small value, we have $I_{\mathrm{time}}=[s,t]\subset [0,1]$,

$$U_{\beta }(t,x)-U_{\beta }(s,x)\ge {\lambda }_0{\varphi }_{|t-s|},$$

(55)

which implies that $[v,t]\in \mathcal{T}$ for all $v\in I_{\mathrm{time}}\left(b\right)=[s,s+b(t-s)]$. For convenience, we assume that b is the reciprocal of an integer.

Suppose that $r_m$ is the reciprocal of an integer, $r_{m+1}<b{r}_m$, and $b{r}_m/r_{m+1}$ is an integer for $m=1,2,...$ Let $\delta$ be a positive number such that $\delta <ϵ/16$. For every $m\ge 1$, define ${\nu }_m=\lfloor r_m^{-\delta }\rfloor$, ${\varrho }_m=\lfloor (r_m^{-1}-1)/{\nu }_m\rfloor +1$ and

$$t_{m,i}=i{\nu }_m{r}_m,i=0,1,...,{\varrho }_m-1,$$

$${\mathcal{J}}_m=\{[t_{m,i},t_{m,i}+r_m];i=0,1,...,{\varrho }_m-1\}.$$

For every $m\ge 1$ and $I_{\mathrm{time}}=[t_{m,i},t_{m,i}+r_m]\in {\mathcal{J}}_m$, define

$${\Lambda }_{\beta ,d,x}(m,I_{\mathrm{time}})={\gamma }_{r_m}^{-1}(U_{\beta }(t_{m,i}+r_m,x)-U_{\beta }(t_{m,i},x)),$$

where ${\gamma }_h=K_{\beta ,d}^{1/2}{h}^{\frac{2-d\beta }{4}}$. Moreover, we define

$$\begin{array}{cc}& {\mathcal{J}}_{m,+}=\{I_{\mathrm{time}}\in {\mathcal{F}}_m;{\Lambda }_{\beta ,x}(m,I_{\mathrm{time}})>\lambda {(2log(1/r_m))}^{1/2}\},\hfill \\ & {\mathcal{J}}_{m,+}\left(b\right)=\{I_{\mathrm{time}}\left(b\right)=[s,s+b(t-s)],I_{\mathrm{time}}=[s,t]\in {\mathcal{J}}_{m,+}\},\hfill \\ & {\rho }_m\left(J_{\mathrm{time}}\right)=\#\{I_{\mathrm{time}}\in {\mathcal{J}}_{m,+},I_{\mathrm{time}}\subset J_{\mathrm{time}}\},{\rho }_m={\rho }_m\left([0,1]\right),\hfill \\ & {\varrho }_m\left(J_{\mathrm{time}}\right)=\#\{I_{\mathrm{time}}\in {\mathcal{J}}_m,I_{\mathrm{time}}\subset J_{\mathrm{time}}\},{\varrho }_m={\varrho }_m\left([0,1]\right),\hfill \\ & r_m^{1-\eta \left(m\right)}=P(N(0,1)>\lambda {(2log(1/r_m))}^{1/2}),\hfill \end{array}$$

where $0<\eta \left(m\right)\to \eta :=1-{\lambda }_0^2$ as $m\to +\infty$.

From (4), we derive that, for any m large enough, $I_{\mathrm{time}}=[s,t]\in {\mathcal{J}}_{m,+}$ is implied (55). Then, we have $[v,t]\in \mathcal{T}$ for any $s\in I_{\mathrm{time}}\left(b\right)\in {\mathcal{J}}_{m,+}\left(b\right)$.

Lemma 7. 

Let $\beta \in (0,1/2]$, $x\in {\mathbb{R}}^d$ ($d=1,2,3$) and $u_0\equiv 0$ in (1) be fixed. Then, there exists a positive, independent of x, constant $c=c\left(d\right)>0$, such that, for all $I_{\mathrm{time}}=[t_{m,i},t_{m,i}+r_m]\in {\mathcal{J}}_m$ and $J_{\mathrm{time}}=[t_{m,j},t_{m,j}+r_m]\in {\mathcal{J}}_m$ with $I_{\mathrm{time}}\cap J_{\mathrm{time}}=\varnothing$, as well as all $m\ge m_0$ with some $m_0>0$, we have

$$\mathbb{E}\left[{\Lambda }_{\beta ,d,x}(m,I_{\mathrm{time}}){\Lambda }_{\beta ,d,x}(m,J_{\mathrm{time}})\right]\le c{\nu }_m^{-1}.$$

(56)

Proof. 

For convenience, we assume that $j>i>0$. For brevity, we define $Z_{\xi ,x}(·,·)$ with the increments of the process $\xi (·,·)$ as follows:

$$Z_{\xi ,x}(s,t)=\xi (t,x)-\xi (s,x),t,s\in {\mathbb{R}}_{+},x\in {\mathbb{R}}^d.$$

It follows from (28) that, for $j>i>0$ and large m, we have

$$\begin{array}{cc}\hfill & \mathbb{E}\left[Z_{X_{\beta },x}(i{\nu }_m,i{\nu }_m+1)Z_{X_{\beta },x}(j{\nu }_m,j{\nu }_m+1)\right]\hfill \\ \hfill & =\mathbb{E}\left[{\left(Z_{X_{\beta },x}(j{\nu }_m,i{\nu }_m+1)\right)}^2\right]-\mathbb{E}\left[{\left(Z_{X_{\beta },x}(j{\nu }_m,i{\nu }_m)\right)}^2\right]\hfill \\ \hfill & -\mathbb{E}\left[{\left(Z_{X_{\beta },x}(j{\nu }_m+1,i{\nu }_m+1)\right)}^2\right]+\mathbb{E}\left[{\left(Z_{X_{\beta },x}(j{\nu }_m+1,i{\nu }_m)\right)}^2\right]\hfill \\ \hfill & =K_{\beta ,d}(1+o\left(1\right))[{(j-i+1)}^{1-\frac{d\beta }{2}}-2{(j-i)}^{1-\frac{d\beta }{2}}+{(j-i-1)}^{1-\frac{d\beta }{2}}],\hfill \end{array}$$

(57)

where $X_{\beta }(·,·)$ is given in (26). Let $f\left(x\right)=x^{1-\frac{d\beta }{2}}$ for $x>0$. Then, $f^{″}\left(x\right)<0$ for $x>0$. Note that $f(j-i+1)-f(j-i)={\nu }_m^{-1}{f}^{\prime }(j-i+{\theta }_1)$ and $f(j-i)-f(j-i-1)=f^{\prime }(j-i-{\theta }_2)$, where ${\theta }_1,{\theta }_2\in [0,1]$. This yields the following for $j>i>0$:

$$\begin{array}{cc}& {(j-i+1)}^{1-\frac{d\beta }{2}}-2{(j-i)}^{1-\frac{d\beta }{2}}+{(j-i-1)}^{1-\frac{d\beta }{2}}\hfill \\ & =f^{\prime }(j-i+{\theta }_1)-f^{\prime }(j-i-{\theta }_2)\hfill \\ & =({\theta }_1+{\theta }_2)f^{″}(j-i+{\theta }_3)\hfill \\ & <0,\hfill \end{array}$$

where ${\theta }_3\in [-1,1]$. As such, together with (57), we have

$$\mathbb{E}\left[Z_{X_{\beta },x}(i{\nu }_m,i{\nu }_m+1)Z_{X_{\beta },x}(j{\nu }_m,j{\nu }_m+1)\right]<0.$$

(58)

Similarly to (57),

$$\begin{array}{cc}\hfill & \mathbb{E}\left[Z_{Y_{\beta },x}(i{\nu }_m,i{\nu }_m+1)Z_{Y_{\beta },x}(j{\nu }_m,j{\nu }_m+1)\right]\hfill \\ \hfill & =\mathbb{E}\left[{\left(Z_{Y_{\beta },x}(j{\nu }_m,i{\nu }_m+1)\right)}^2\right]-\mathbb{E}\left[{\left(Z_{Y_{\beta },x}(j{\nu }_m,i{\nu }_m)\right)}^2\right]\hfill \\ \hfill & -\mathbb{E}\left[{\left(Z_{Y_{\beta },x}(j{\nu }_m+1,i{\nu }_m+1)\right)}^2\right]+\mathbb{E}\left[{\left(Z_{Y_{\beta },x}(j{\nu }_m+1,i{\nu }_m)\right)}^2\right],\hfill \end{array}$$

(59)

where $Y_{\beta }(·,·)$ is given in (27). It follows from (36) that, for any $t,s\in {\mathbb{R}}_{+}$, we have

$$\begin{array}{cc}\hfill & \mathbb{E}\left[{\left(Z_{Y_{\beta },x}(t,s)\right)}^2\right]\hfill \\ \hfill & {\displaystyle =\frac{1}{2^d\Gamma (d/2)}\frac{{sin}^2\left(\beta \pi \right)}{{\pi }^{2+d/2}}{\int }_0^{+\infty }{y}^{d/2-1}{\int }_{-\infty }^0|{\int }_0^{+\infty }{a}_{\beta }\left(\zeta \right){\mathbb{K}}_{\beta ;r,y}(\zeta ,t,s)d\zeta {|}^{2}drdy}\hfill \\ \hfill & {\displaystyle =\frac{1}{2^d\Gamma (d/2)}\frac{{sin}^2\left(\beta \pi \right)}{{\pi }^{2+d/2}}{\int }_0^{+\infty }{y}^{d/2-1}{\int }_{-\infty }^0{\int }_0^{+\infty }{\int }_0^{+\infty }{a}_{\beta }\left({\zeta }_1\right)a_{\beta }\left({\zeta }_2\right)}\hfill \\ \hfill & \times {\mathbb{K}}_{\beta ;r,y}({\zeta }_1,t,s){\mathbb{K}}_{\beta ;r,y}({\zeta }_2,t,s)d{\zeta }_{1}d{\zeta }_{2}drdy,\hfill \end{array}$$

(60)

where the following notation is used:

$${\mathbb{K}}_{\beta ;r,y}(\zeta ,t,s)=e^{-\zeta {\left(\frac{y}{2}\right)}^{1/\beta }(s-r)}\left(e^{-\zeta {\left(\frac{y}{2}\right)}^{1/\beta }(t-s)}-1\right).$$

By some element calculations, we can conclude that, for $j>i>0$, we have

$$\begin{array}{cc}\hfill & {\mathbb{K}}_{\beta ;r,y}({\zeta }_1,j{\nu }_m,i{\nu }_m+1){\mathbb{K}}_{\beta ;r,y}({\zeta }_2,j{\nu }_m,i{\nu }_m+1)\hfill \\ \hfill & -{\mathbb{K}}_{\beta ;r,y}({\zeta }_1,j{\nu }_m,i{\nu }_m){\mathbb{K}}_{\beta ;r,y}({\zeta }_2,j{\nu }_m,i{\nu }_m)\hfill \\ \hfill & -{\mathbb{K}}_{\beta ;r,y}({\zeta }_1,j{\nu }_m+1,i{\nu }_m+1){\mathbb{K}}_{\beta ;r,y}({\zeta }_2,j{\nu }_m+1,i{\nu }_m+1)\hfill \\ \hfill & +{\mathbb{K}}_{\beta ;r,y}({\zeta }_1,j{\nu }_m+1,i{\nu }_m){\mathbb{K}}_{\beta ;r,y}({\zeta }_2,j{\nu }_m+1,i{\nu }_m)\hfill \end{array}$$

(61)

$$\begin{array}{cc}& =e^{r({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}(e^{-({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}-1)\hfill \\ & \times [e^{-({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }(j{\nu }_m-1)}(1-e^{-{\zeta }_1{\left(\frac{y}{2}\right)}^{1/\beta }})(1-e^{-{\zeta }_2{\left(\frac{y}{2}\right)}^{1/\beta }})\hfill \\ & +e^{-((j{\nu }_m-1){\zeta }_1+i{\nu }_m{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}(1-e^{-{\zeta }_1{\left(\frac{y}{2}\right)}^{1/\beta }})(e^{-{\zeta }_2{\left(\frac{y}{2}\right)}^{1/\beta }(j-i){\nu }_m}-1)\hfill \\ & +e^{-(i{\nu }_m{\zeta }_1+(j{\nu }_m-1){\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}(1-e^{-{\zeta }_2{\left(\frac{y}{2}\right)}^{1/\beta }})(e^{-{\zeta }_1{\left(\frac{y}{2}\right)}^{1/\beta }(j-i){\nu }_m}-1)]\hfill \end{array}$$

$$\begin{array}{cc}& +e^{r({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}\{e^{-({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }(j{\nu }_m-1)}(1-e^{-{\zeta }_1{\left(\frac{y}{2}\right)}^{1/\beta }})(1-e^{-{\zeta }_2{\left(\frac{y}{2}\right)}^{1/\beta }})\hfill \\ & -e^{-j({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }{\nu }_m}(1-e^{-{\zeta }_1{\left(\frac{y}{2}\right)}^{1/\beta }})(1-e^{-{\zeta }_2{\left(\frac{y}{2}\right)}^{1/\beta }})\hfill \\ & +e^{-((j{\nu }_m-1){\zeta }_1+i{\nu }_m{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}(1-e^{-{\zeta }_1{\left(\frac{y}{2}\right)}^{1/\beta }})\hfill \\ & \times [e^{-{\zeta }_2{\left(\frac{y}{2}\right)}^{1/\beta }(j-i){\nu }_m}(1-e^{-({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }})+(e^{-{\zeta }_1{\left(\frac{y}{2}\right)}^{1/\beta }}-1)]\hfill \\ & +e^{-(i{\nu }_m{\zeta }_1+(j{\nu }_m-1){\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}(1-e^{-{\zeta }_2{\left(\frac{y}{2}\right)}^{1/\beta }})\hfill \\ & \times [e^{-{\zeta }_1{\left(\frac{y}{2}\right)}^{1/\beta }(j-i){\nu }_m}(1-e^{-({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }})+(e^{-{\zeta }_2{\left(\frac{y}{2}\right)}^{1/\beta }}-1)]\}.\hfill \end{array}$$

Since for any $u\ge 0$, $|1-e^{-u}|\le 2u$, the absolute value of the above equation is less than the following quantity, for any $j>i>0$, we have

$$\begin{array}{cc}\hfill & 48e^{r({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}{e}^{-({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }(i{\nu }_m-1)}.\hfill \end{array}$$

(62)

Integrating this first in r, we have ${\int }_{-\infty }^0{e}^{r({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}dr=\frac{1}{({\zeta }_1+{\zeta }_2){\left(\frac{y}{2}\right)}^{1/\beta }}$. Then, noting that $a_{\beta }\left({\zeta }_i\right)\le \frac{1}{2cos\left(\beta \pi \right)}$ for all ${\zeta }_i\in {\mathbb{R}}_{+}$ and $i=1,2$, via the change in variables ${\zeta }_1\mapsto u_1:u_1={\left(\frac{y}{2}\right)}^{1/\beta }(i{\nu }_m-1){\zeta }_1$ and ${\zeta }_2\mapsto u_2:u_2={\left(\frac{y}{2}\right)}^{1/\beta }(i{\nu }_m-1){\zeta }_2$—as well as by integrating r, ${\zeta }_1$ and ${\zeta }_2$ in (62) separately—we can conclude this integration is less that $\frac{c}{{\nu }_m{\left(\frac{y}{2}\right)}^{1/\beta }}$. Thus, together with (59)–(62), we obtain

$$\begin{array}{cc}\hfill & |\mathbb{E}\left[Z_{Y_{\beta },x}(i{\nu }_m,i{\nu }_m+1)Z_{Y_{\beta },x}(j{\nu }_m,j{\nu }_m+1)\right]|\le c{\nu }_m^{-1}.\hfill \end{array}$$

(63)

Since the Gaussian process $\{U_{\beta }(t,x),t\ge 0\}$ is self-similar with the index $(2-\beta d)/4$ (see [13] (p. 1591)), we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\left[{\Lambda }_{\beta ,d,x}(m,I_{\mathrm{time}}){\Lambda }_{\beta ,d,x}(m,J_{\mathrm{time}})\right]=\mathbb{E}\left[Z_{U_{\beta },x}(i{\nu }_m,i{\nu }_m+1)Z_{U_{\beta },x}(j{\nu }_m,j{\nu }_m+1)\right].\hfill \end{array}$$

(64)

Since $U_{\beta }(t,x)=X_{\beta }(t,x)-Y_{\beta }(t,x)$, $U_{\beta }(t,x)$ and $Y_{\beta }(t,x)$ are independent for $(t,x)\in {\mathring{\mathbb{R}}}_{+}\times {\mathbb{R}}^d$, Equation (64) becomes

$$\begin{array}{cc}\hfill & \mathbb{E}\left[{\Lambda }_{\beta ,d,x}(m,I_{\mathrm{time}}){\Lambda }_{\beta ,d,x}(m,J_{\mathrm{time}})\right]\hfill \\ \hfill & =\mathbb{E}\left[Z_{X_{\beta },x}(i{\nu }_m,i{\nu }_m+1)Z_{X_{\beta },x}(j{\nu }_m,j{\nu }_m+1)\right]-\mathbb{E}\left[Z_{Y_{\beta },x}(i{\nu }_m,i{\nu }_m+1)Z_{Y_{\beta },x}(j{\nu }_m,j{\nu }_m+1)\right].\hfill \end{array}$$

(65)

With (58), (63) and (65), we obtain (56). The proof is thus completed. □

We also need the following three lemmas.

Lemma 8. 

For any $0<\zeta \le 1/2$, there exists an integer $m_0$, such that

$$\mathbb{P}\left(\right|{\rho }_m\left(J_{\mathrm{time}}\right)-\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right]|\ge \lambda \mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right])\le 2exp(-\zeta (\lambda -2\zeta )\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right])+r_m^5$$

(66)

for all $J_{\mathrm{time}}\subseteq [0,1]$, $m\ge m_0$ and $\lambda >0$.

Proof. 

We follow Lemma 2.3 of [18]. For brevity, we denote $Z_{m,i}=Z_{U_{\beta },x}(t_{m,i},t_{m,i}+r_m)$, $Y_{m,i}={\gamma }_{r_m}^{-1}{Z}_{m,i}$, ${\varrho }_m={\varrho }_m\left(J_{\mathrm{time}}\right)$, ${\mathsf{\ell }}_m={(2log(1/r_m))}^{1/2}$ and ${\delta }_m={\nu }_m^{-1}$. Note that

$${\rho }_m\left(J_{\mathrm{time}}\right)=\sum _{i=1}^{{\varrho }_m\left(J_{\mathrm{time}}\right)}\mathbb{I}({\gamma }_{r_m}^{-1}{Z}_{m,i}>\lambda {\mathsf{\ell }}_m).$$

Let $\{{\xi }_m,Y_{m,i}^{\prime },i=1,\dots ,{\varrho }_m\}$ be independent mean zero Gaussian random variables with $\mathbb{E}\left[{\xi }_m^2\right]={\delta }_m$ and $\mathbb{E}\left[{\left(Y_{m,i}^{\prime }\right)}^2\right]=1-{\delta }_m$. Then, $\mathbb{E}\left[{({\xi }_m+Y_{m,i}^{\prime })}^2\right]=\mathbb{E}\left[{\left(Y_{m,i}\right)}^2\right]=1$ and $\mathbb{E}\left[Y_{m,i}{Y}_{m,j}\right]\le \mathbb{E}\left[({\xi }_m+Y_{m,i}^{\prime })({\xi }_m+Y_{m,j}^{\prime })\right]=\mathbb{E}\left[{\xi }_m^2\right]={\delta }_m$$(i\ne j)$.

For any m large enough, define $p_{m,0}=p_m\left(\lambda \right)$, such that $q_m=\zeta (\lambda +1)\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right]=\zeta (\lambda +1){\varrho }_m{p}_{m,0}$, $p_{m,1}=p_m((\lambda -3{\delta }_m^{1/2})/{(1-{\delta }_m)}^{1/2})$ and $p_{m,2}=p_m((\lambda +3{\delta }_m^{1/2})/{(1-{\delta }_m)}^{1/2})$, where

$$p_m\left(z\right)=\mathbb{P}(N(0,1)>z{\mathsf{\ell }}_m),z>0.$$

Let $f\left(z\right)=e^z$ if $0\le z\le q_m$, and $=e^{q_m}(z-q_m+1)$ if $z\ge q_m$, and let $g(Y_{m,1},...,Y_{m,{\varrho }_m})=f\left(\zeta {\rho }_m\left(J_{\mathrm{time}}\right)\right)$. Then, $g(Y_{m,1},...,Y_{m,{\varrho }_m})\le e^{\zeta {\rho }_m\left(J_{\mathrm{time}}\right)}\vee {\varrho }_m{e}^{q_m}$. Via the well-known comparison property (cf. Theorem 3.11 of [30] (p. 74)), we have

$$\mathbb{E}\left[g(Y_{m,1},...,Y_{m,{\varrho }_m})\right]\le \mathbb{E}\left[g({\xi }_m+Y_{m,1}^{\prime },...,{\xi }_m+Y_{m,{\varrho }_m}^{\prime })\right].$$

Thus, we conclude that

$$\begin{array}{cc}& \mathbb{P}({\rho }_m\left(J_{\mathrm{time}}\right)-\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right]\ge \lambda \mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right])\hfill \\ & =\mathbb{P}(f\left(\zeta {\rho }_m\left(J_{\mathrm{time}}\right)\right)\ge f\left(q_m\right)))\hfill \\ & =\mathbb{P}(g(Y_{m,1},...,Y_{m,{\varrho }_m})\ge e^{q_m}))\hfill \\ & \le e^{-q_m}\mathbb{E}\left[g(Y_{m,1},...,Y_{m,{\varrho }_m})\right]\hfill \\ & \le e^{-q_m}\mathbb{E}\left[g({\xi }_m+Y_{m,1}^{\prime },...,{\xi }_m+Y_{m,{\varrho }_m}^{\prime })\right]\hfill \\ & \le e^{-q_m}\left\{\mathbb{E}\left[e^{\zeta {\sum }_{i=1}^{{\varrho }_m}\mathbb{I}\{{\xi }_m+Y_{m,i}^{\prime }>\lambda {\mathsf{\ell }}_m\}}\mathbb{I}({\xi }_m\le 3{\delta }_m^{1/2}{\mathsf{\ell }}_m)\right]+{\varrho }_m{e}^{q_m}\mathbb{P}({\xi }_m>3{\delta }_m^{1/2}{\mathsf{\ell }}_m)\right\}\hfill \\ & \le e^{-q_m}\mathbb{E}\left[e^{\zeta {\sum }_{i=1}^{{\varrho }_m}\mathbb{I}\{Y_{m,i}^{\prime }>(\lambda -3{\delta }_m^{1/2}){\mathsf{\ell }}_m\}}\right]+{\varrho }_m\mathbb{P}({\xi }_m>3{\delta }_m^{1/2}{\mathsf{\ell }}_m)\}.\hfill \end{array}$$

Via the fact that $\{Y_{m,i}^{\prime },i=1,...,{\varrho }_m\}$ are independent, it is easy to see that

$$\begin{array}{cc}& \mathbb{E}\left[e^{\zeta {\sum }_{i=1}^{{\varrho }_m}\mathbb{I}\{Y_{m,i}^{\prime }>(\lambda -3{\delta }_m^{1/2}){\mathsf{\ell }}_m\}}\right]\hfill \\ & =e^{\zeta {\varrho }_m{p}_{m,1}}{\left(\mathbb{E}\left[e^{\zeta (\mathbb{I}\{Y_{m,i}^{\prime }>(\lambda -3{\delta }_m^{1/2}){\mathsf{\ell }}_m\}-p_{m,1})}\right]\right)}^{{\varrho }_m}\hfill \\ & \le e^{\zeta {\varrho }_m{p}_{m,1}}{(1+p_{m,1}(1-p_{m,1}){\zeta }^2)}^{{\varrho }_m}\hfill \\ & \le e^{\zeta {\varrho }_m{p}_{m,1}+{\zeta }^2{\varrho }_m{p}_{m,1}(1-p_{m,1})}.\hfill \end{array}$$

Then, we have

$$\begin{array}{cc}\hfill & \mathbb{P}({\rho }_m\left(J_{\mathrm{time}}\right)-\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right]\ge \lambda \mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right])\hfill \\ \hfill & \le e^{-\zeta {\varrho }_m((\lambda +1)p_{m,0}-(1+\zeta )p_{m,1})}+{\varrho }_m\mathbb{P}({\xi }_m>3{\delta }_m^{1/2}{\mathsf{\ell }}_m).\hfill \end{array}$$

(67)

It follows from (48) that $p_{m,0}\sim p_{m,1}$ as $m\to +\infty$. This implies that $(1+\zeta )p_{m,1}\le (1+2\zeta )p_{m,0}$. Thus, (67) becomes

$$\begin{array}{cc}\hfill & \mathbb{P}({\rho }_m\left(J_{\mathrm{time}}\right)-\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right]\ge \lambda \mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right])\hfill \\ \hfill & \le e^{-\zeta {\varrho }_m(\lambda -2\zeta )p_{m,0}}+c{r}_m^{-1}{r}_m^9\hfill \\ \hfill & \le e^{-\zeta {\varrho }_m(\lambda -2\zeta )p_{m,0}}+r_m^5.\hfill \end{array}$$

(68)

Similarly to (68), by choosing $q_m=\zeta ((\lambda -1){\varrho }_m{p}_{m,0}+{\varrho }_m)$, we have

$$\begin{array}{cc}\hfill & \mathbb{P}(\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right]-{\rho }_m\left(J_{\mathrm{time}}\right)\ge \lambda \mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right])\hfill \\ \hfill & \le e^{-\zeta {\varrho }_m(\lambda -2\zeta )p_{m,0}}+r_m^5.\hfill \end{array}$$

(69)

Thus, together with (68), (66) is yielded. The proof is thus completed. □

Lemma 9. 

Given $ϵ>0$, $\delta >0$, with a probability of one, there exists an integer $m_0$ such that

$$|{\rho }_m\left(J_{\mathrm{time}}\right)-\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right]|\le ϵ\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right]$$

(70)

for all $J_{\mathrm{time}}\subseteq [0,1]$, such that $|J_{\mathrm{time}}|\ge \delta$ and all $m\ge m_0(ϵ,\delta )$.

Proof. 

It follows from (46) that $p_{m,0}=r_m^{{\lambda }^2(1+r_m)}$, where $r_m\to 0$ as $m\to +\infty$. This, together with Lemma 8 and the Borel–Cantelli argument, yields (70). The proof is thus completed. □

Lemma 10. 

Given ${\eta }^{\prime }<\eta =1-{\lambda }^2$, there is an absolute constant c such that, with a probability of one, there exists $m_1$ such that

$${\rho }_m\left(J_{\mathrm{time}}\right)\le c{\left|J_{\mathrm{time}}\right|}^{{\eta }^{\prime }}{\rho }_m\left([0,1]\right),$$

(71)

for all $J_{\mathrm{time}}\subseteq [0,1]$, $m\ge m_1$.

Proof. 

It follows from Lemma 9 that it is enough to show that

$${\rho }_m\left(J_{\mathrm{time}}\right)\le c|J_{\mathrm{time}}{|}^{{\eta }^{\prime }}\mathbb{E}\left[{\rho }_m\left([0,1]\right)\right]\le c{\left|J_{\mathrm{time}}\right|}^{{\eta }^{\prime }}{\varrho }_m{r}_m^{1-\eta \left(m\right)}$$

(72)

for $m\ge m_1$. Note that $|J_{\mathrm{time}}|<r_m$ implies ${\rho }_m\left(J_{\mathrm{time}}\right)=0$, $r_m\le \left|J_{\mathrm{time}}\right|<{\nu }_m{r}_m$, which implies ${\rho }_m\left(J_{\mathrm{time}}\right)\le 1$ and $|J_{\mathrm{time}}{|}^{{\eta }^{\prime }}{\varrho }_m{r}_m^{1-\eta \left(m\right)}\ge c{r}_m^{\delta +{\eta }^{\prime }-\eta \left(m\right)}\to +\infty$. Thus, we need only to consider the case of $|J_{\mathrm{time}}|\ge {\nu }_m{r}_m$. It is clearly sufficient to consider only the class ${\mathcal{D}}_m$ of intervals $[i{r}_m,j{r}_m]$, where $i,j$ are integers and $0\le i<j\le {\left({\nu }_m{r}_m\right)}^{-1}$. Note that ${\varrho }_m\sim {\nu }_m^{-1}{r}_m^{-1}\sim r_m^{\delta -1}$ and ${\varrho }_m\left(J_{\mathrm{time}}\right)=\left|J_{\mathrm{time}}\right|{\varrho }_m$. We deduce from Lemma 8 that for any m large enough, we have

$$\begin{array}{cc}& \mathbb{P}({\rho }_m\left(J_{\mathrm{time}}\right)>c|J_{\mathrm{time}}{|}^{{\eta }^{\prime }}{\varrho }_m{r}_m^{1-\eta \left(m\right)},J\in {\mathcal{D}}_m)\hfill \\ & \le r_m^{-2}exp(-c|r_m{|}^{{\eta }^{\prime }}{\varrho }_m{r}_m^{1-\eta \left(m\right)})+r_m^3\hfill \\ & \le r_m^{-2}exp(-c{r}_m^{\delta +{\eta }^{\prime }-\eta \left(m\right)})+r_m^3.\hfill \end{array}$$

Since, $\delta +{\eta }^{\prime }-\eta \left(m\right)\to \delta +{\eta }^{\prime }-\eta <0$, it follows that

$$\sum _{m=1}^{+\infty }\mathbb{P}({\rho }_m\left(J_{\mathrm{time}}\right)>c|J_{\mathrm{time}}{|}^{{\eta }^{\prime }}{\varrho }_m{r}_m^{1-\eta \left(m\right)},J\in {\mathcal{D}}_m)<+\infty .$$

This implies that, with a probability of one, there is a $m_1=m_1\left({\eta }^{\prime }\right)>0$ such that (72) holds. The proof is thus completed. □

Next, we shall show that the existence of a sequence of sets $F_1\supset F_2\supset \cdots$ are such that they satisfy Lemma 2.1’s presumptions and that $F={\cap }_{m=1}^{+\infty }{F}_m\subset S_{\beta ,d,x}\left(\lambda \right)$. We can assume that, for every stage of the construction that is completed in the same probability 1 set, there are only a countable number of steps required and that each step can be completed with a probability of 1. Select ${\eta }^{\prime }=\eta -\frac{1}{4}ϵ$ and define $m_1=:m_1\left({\eta }^{\prime }\right)$ such that $m\ge m_1$ and (71) hold. Assume that the sequence of positive numbers $\left({ϵ}_k\right)$ satisfies $\sum {ϵ}_k<+\infty$. In the first step, when using Lemma 9, we determine an integer $n_1\ge m_1$ such that

$$|{\rho }_m-\mathbb{E}\left[{\rho }_m\right]|<{ϵ}_1\mathbb{E}\left[{\rho }_m\right](m\ge n_1).$$

And then we shall define an increasing sequence $n_1,n_2,...$ inductively, as well as define for $k\ge 1$.

$$\begin{array}{cc}& \{I_{k.i},1\le i\le Q_k\}=\{I_{\mathrm{time}}\left(b\right)\in {\mathcal{J}}_{n_k,+},I_{\mathrm{time}}\left(b\right)\subset F_{k-1}\},\hfill \\ & F_0=[0,1],F_k={\cup }_{i=1}^{Q_k}{I}_{k,i},\hfill \\ & Q_k\left(J_{\mathrm{time}}\right)=\#\{i,I_{k,i}\subset J_{\mathrm{time}}\}\mathrm{for}{J}_{\mathrm{time}}\subset [0,1],Q_k=Q_k\left([0,1]\right),\hfill \\ & \varsigma \left(k\right)=\eta \left(n_k\right),\tau \left(k\right)=1-\varsigma \left(k\right),R_k=\left|I_{k,i}\right|=b{r}_{n_k}.\hfill \end{array}$$

For each $k\ge 2$, suppose that $n_{k-1}$ has been defined; as such, we can define an $n_k$ large enough to ensure the following:

$$\begin{array}{cc}& n_k\ge m_0(ϵ,R_{k-1}^{2\tau (k-1)/ϵ}),n_k=m_0({ϵ}_k,R_{k-1}),\hfill \\ & n_k\ge 2n_{k-1},r_{n_k}\le r_{n_{k-1}}^2,\hfill \end{array}$$

where $m_0(ϵ,\delta )$ is the integer determined in Lemma 9 to invalidate (70) and

$$R_k^{1/\left(2ϵ\right)}\le b^{2\eta }\prod _{i=1}^{k-1}{R}_i^{\tau \left(i\right)}{b}^{\varsigma \left(i\right)}.$$

(73)

Then, we have

$$|{\rho }_m\left(J_{\mathrm{time}}\right)-\mathbb{E}{\rho }_m\left(J_{\mathrm{time}}\right)|\le {ϵ}_k\mathbb{E}\left[{\rho }_m\left(J_{\mathrm{time}}\right)\right]$$

(74)

for all $\subseteq [0,1]$, such that $|J_{\mathrm{time}}|\ge R_{k-1}$ and all $m\ge n_k$.

By making use of (73), (74) and Lemmas 9 and 10, via following the same route as the proof of (2.23) in [18], we can obtain

$$Q_{k+j}\left(J_{\mathrm{time}}\right)\le c\left(\prod _{i=1}^k{\nu }_{n_i}\right)R_k^{ϵ}{\left|J_{\mathrm{time}}\right|}^{\eta -2ϵ}{Q}_{k+j}$$

(75)

for all $R_{k+1}<\left|J_{\mathrm{time}}\right|\le R_k$, $k\ge 1$, $j\ge 1$.

Noting that

$$r_{n_k}^2\le r_{n_k}^{1+\frac{1}{2}+\cdots +\frac{1}{2^{k-1}}}\le r_{n_k}{r}_{n_{k-1}}\cdots r_{n_1}$$

and

$$\prod _{i=1}^k{\nu }_{n_i}\le {\left(\prod _{i=1}^k{r}_{n_i}\right)}^{-\delta },$$

via (75), we can conclude that

$$Q_{k+j}\left(J_{\mathrm{time}}\right)\le c{b}^{ϵ}{r}_{n_k}^{ϵ-2\delta }{\left|J_{\mathrm{time}}\right|}^{\eta -2ϵ}{Q}_{k+j}$$

for all $R_{k+1}<\left|J_{\mathrm{time}}\right|\le R_k$, $k\ge 1$, $j\ge 1$. Thus, it follows from Lemma 6, as well as from the fact that $r_{n_k}^{ϵ-2\delta }\to 0$$(k\to +\infty )$, with a probability of one, we have

$${\mu }_{s^{\eta -2ϵ}}\left(F\right)>0.$$

(76)

Hence, we have proved (52). The proof is thus completed.

3.3. Hitting Probabilities for the Sets of Temporal Fast Points

We prove Theorem 3 in this subsection, thereby obtaining hitting probabilities for the sets of the temporal fast points of the TFSPIDEs, as well as their gradients, in the process.

Proof of Theorem 3. 

We only show Equation (13) because Equation (14) can be proved similarly. To prove Equation (13), via Remark 2, it is enough to show that, for every analytic set $B\subset {\mathbb{R}}_{+}$, we have

$$\mathbb{P}\left\{\underset{t\in B}{sup}\underset{h\to 0+}{lim\; sup}{\varphi }_{\beta ,d,h}|U_{\beta }(t+h,x)-U_{\beta }(t,x)|={\left({dim}_{{}_P}\left(B\right)\right)}^{1/2}\right\}=1.$$

(77)

□

By using (4) and Lemma 5, as well as by following the same route as the proof of the upper bound of Theorem 2.1 in [19], we obtain

$$\mathbb{P}\left\{\underset{t\in B}{sup}\underset{h\to 0+}{lim\; sup}{\varphi }_{\beta ,d,h}|U_{\beta }(t+h,x)-U_{\beta }(t,x)|\le {\left({dim}_{{}_P}\left(B\right)\right)}^{1/2}\right\}=1.$$

(78)

We now turn to the proof of the opposite inequality. That is, it is enough to show that

$$\mathbb{P}\left\{\underset{t\in B}{sup}\underset{h\to 0+}{lim\; sup}{\varphi }_{\beta ,d,h}|U_{\beta }(t+h,x)-U_{\beta }(t,x)|\ge {\left({dim}_{{}_P}\left(B\right)\right)}^{1/2}\right\}=1.$$

(79)

Fix $\varpi$ such that ${dim}_{{}_P}\left(B\right)>\varpi$. For each integer $n\ge 1$, which are denoted by ${\mathcal{Q}}_n$, the set of all intervals of the form $[m{2}^{-n},(m+1)2^{-n}]$, $m\in {\mathbb{Z}}_{+}$ are obtained. In words, ${\mathcal{Q}}_n$ denotes the totality of all intervals. For all $I_{\mathrm{time}}\in {\mathcal{Q}}_n$, define ${\pi }_n\left(I_{\mathrm{time}}\right)=m{2}^{-n}$ to be the smallest element in $I_{\mathrm{time}}$. For $I_{\mathrm{time}}\in {\mathcal{Q}}_n$, which is denote by ${\omega }_n\left(I_{\mathrm{time}}\right)$, the indicator function of the event $({\Theta }_{\beta ,d,x}({\pi }_n\left(I_{\mathrm{time}}\right),2^{-n}{(logn)}^{-1})>{\varpi }^{1/2})$ is obtained, where the following notation is used:

$${\Theta }_{\beta ,d,x}(t,h)={\varphi }_{\beta ,d,h}|U_{\beta }(t+h,x)-U_{\beta }(t,x)|.$$

(80)

In other words, ${\omega }_n\left(I_{\mathrm{time}}\right)$ is a Bernoulli random variable whose values take 1 or 0 according as to whether we have

$${\Theta }_{\beta ,d,x}({\pi }_n\left(I_{\mathrm{time}}\right),2^{-n}{(logn)}^{-1})>{\varpi }^{1/2}.$$

Define via $D:={lim\; sup}_{n}D\left(n\right)$ a discrete limsup random fractal, where

$$D\left(n\right)=\bigcup _{I_{\mathrm{time}}\in {\mathcal{Q}}_n:{\omega }_n\left(I_{\mathrm{time}}\right)=1}{I}_{\mathrm{time}}^0,$$

and where $I_{\mathrm{time}}^0$ denotes the interior of $I_{\mathrm{time}}$. We can claim that, whenever ${dim}_{{}_P}\left(B\right)>\varpi$, then

$$\mathbb{P}(D\cap B\ne \varnothing )=1.$$

(81)

We postpone the verification of (81) and prove (79) first, which thereby completes the proof.

Since ${dim}_{{}_P}\left(B\right)>\varpi$, (81), implies that there exists $t\in B$ such that there is ${\Theta }_{\beta ,d,x}(2^{-n}\left[t{2}^n\right]$, $2^{-n}{(log)}^{-1})\ge \varpi$ for infinitely many instances of n, then, we have, in particular,

$$\underset{t\in B}{sup}\underset{n\to +\infty }{lim\; sup}{\Theta }_{\beta ,d,x}(2^{-n}\left[t{2}^n\right],2^{-n}{(log)}^{-1})\ge \varpi \mathrm{a}.\mathrm{s}.$$

Via (4), we can obtain

$$\underset{n\to +\infty }{lim}\underset{t\in I_{\mathrm{time}}:I_{\mathrm{time}}\in {\mathcal{Q}}_n}{sup}|{\Theta }_{\beta ,d,x}(t,2^{-n}{(log)}^{-1})-{\Theta }_{\beta ,d,x}(2^{-n}\left[t{2}^n\right],2^{-n}{(log)}^{-1})|=0\mathrm{a}.\mathrm{s}.$$

Thus, if ${dim}_{\mathrm{P}}\left(F\right)>\varpi$, then (79) holds; as such, (77) also holds.

(81) remains to be verified. Fix a small $\eta >0$ such that ${dim}_{{}_P}\left(B\right)>\varpi +\eta$. By [31], there is a closed $B_{*}\subset B$, such that, for all open sets F, (whenever $B_{*}\cap F\ne \varnothing$), then ${\overline{dim}}_{{}_M}(B_{*}\cap F)>\varpi +\eta$ (see [22] for the definition of an upper Minkowski dimension). It is enough to show that $D\cap B_{*}\ne \varnothing$ when fixing an open set F such that $F\cap B_{*}\ne \varnothing$. We can claim that, with a probability of one, $D\left(n\right)\cap F\cap B_{*}\ne \varnothing$ is such for infinitely many n. When defined via $V\left(n\right):={\cup }_{k=n}^{+\infty }D\left(k\right)$, $n\ge 1$, the open sets are obtained. As such, this claim implies that, with a probability of one, $V\left(n\right)\cap F\cap B_{*}\ne \varnothing$ is such for all n. Furthermore, via letting F run over a countable base for the open sets, we can obtain a $V\left(n\right)\cap B_{*}$ that is as dense as in (the complete metric space) $B_{*}$. Via Baire’s category theorem (see [32]), we have a $B_{*}\cap {\cap }_{n=1}^{+\infty }V\left(n\right)$ that is dense in $B_{*}$ and, in particular, non-empty. Since $D={\cap }_{n=1}^{+\infty }V\left(n\right)$, we can conclude that $D\cap B_{*}\ne \varnothing$, which, in turn, means that (81) holds and its results follow.

Fix an open set F by satisfying $F\cap B_{*}\ne \varnothing$. This is denoted by ${\mathcal{N}}_n$, which are the total number of intervals $I_{\mathrm{time}}\in {\mathcal{Q}}_n$ that satisfy $I_{\mathrm{time}}\cap F\cap B_{*}\ne \varnothing$. Since $\overline{{dim}_{{}_M}}(F\cap B_{*})>\varpi +\eta$, via the definition of an upper Minkowski dimension, there exists ${\varpi }_1>\varpi +\eta$ such that ${\mathcal{N}}_n\ge 2^{n{\varpi }_1}$ is the case for the infinitely many integers of n. Thus, $\#(\aleph )=+\infty$, where

$$\aleph :=\left\{n\ge 1:{\mathcal{N}}_n\ge 2^{n{\varpi }_1}\right\}.$$

(82)

As denote by ${\Omega }_n:=\sum {\omega }_n\left(I_{\mathrm{time}}\right)$, the total number of intervals $I_{\mathrm{time}}\in {\mathcal{Q}}_n$ is such that $I_{\mathrm{time}}\cap F\cap B_{*}\cap D\left(n\right)\ne \varnothing$, where the sum is taken over for all $I_{\mathrm{time}}\in {\mathcal{Q}}_n$ such that $I_{\mathrm{time}}\cap B_{*}\cap F\ne \varnothing$;

$${\Omega }_n=\#\{I_{\mathrm{time}}\in {\mathcal{Q}}_n:I_{\mathrm{time}}\cap B_{*}\cap F\ne \varnothing ,{\Theta }_{\beta ,d,x}({\pi }_n\left(I_{\mathrm{time}}\right),2^{-n}{(logn)}^{-1})>{\varpi }^{1/2}\}.$$

In order to show that, with a probability of one, $D\left(n\right)\cap F\cap B_{*}\ne \varnothing$ applies for the infinitely many instances of n, it suffices to show that ${\Omega }_n>0$ applies for the infinitely many instances of n. That is, it is enough to show that

$$P({\Omega }_n>0\mathrm{i}.\mathrm{o}.)=1.$$

(83)

It follows from Lemma 4 that $p_n=2^{-n(\varpi +a_n)}$, where $a_n\to 0$ is to $n\to +\infty$. Hence, $\mathbb{E}\left[{\Omega }_n\right]={\mathcal{N}}_n{p}_n\ge 2^{n({\varpi }_1-\varpi -a_n)}$. Thus, it follows from Lemma 9 that, with a probability of one, ${\Omega }_n\ge c{2}^{n({\varpi }_1-\varpi -a_n)}$ applies, which implies that $\mathbb{P}({\Omega }_n=0)\to 0$ as is to $n\to +\infty$. Via Fatou’s lemma, one can obtain

$$\mathbb{P}({\Omega }_n>0\mathrm{i}.\mathrm{o}.)\ge \underset{n\to +\infty }{lim\; sup}\mathbb{P}({\Omega }_n>0)=1.$$

This yields (83). This thus completes the proof.

4. Conclusions

In this article, we established the exact, dimension-dependent temporal continuity moduli for fourth-order TFSPIDEs and their gradients. This was achieved by determining the precise values of the normalized constants, and these were supplemented by the prior efforts of Allouba and Xiao on the spatio-temporal Hölder regularity of the fourth-order TFSPIDEs and their gradients. We obtained Hausdorff dimensions and the hitting probabilities of the sets of the temporal fast points for the fourth-order TFSPIDEs and their gradients in a time variable t. It was confirmed that these points of the TFSPIDEs and their gradients, in time, have a probability of one everywhere, and that they are dense with the power of the continuum. In addition, their hitting probabilities were determined by the target set B’s packing dimension ${dim}_{{}_p}\left(B\right)$. On the one hand, this work has reinforced the temporal continuity moduli and temporal LILs obtained in [13] by obtaining the exact values of their normalized constants; on the other hand, this work has obtained the size of the set of fast points, as well as the potential theory of TFSPIDEs and their gradients.