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

: 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 ( B ) . 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


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 integrodifferential 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: where ∂ d+1 W/∂t∂x is the space-time white noise corresponding to the real-valued Brownian sheet W on R + × R d (d = 1, 2, 3); ∆ is the d-dimensional Laplacian operator; the time-fractional derivative of order β, C , is the Caputo fractional operator and the time-fractional integral of order α; I α t , is the Riemann-Liouville fractional integral α dτ, for t > 0 and α > 0, and I 0 t = 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 < γ ≤ 1 such that where C α,γ b (R d ; R) is the set of α-continuously differentiable functions on R d , whose αderivative is locally Hölder continuous with the exponent γ.
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 ≥ 0, as well as the laws of iterated logarithm (LILs) for U β (•, x) and ∂ x U β (•, 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 ∈ N + , we define K β,d and K β,0 by and In this article, we obtain the following exact global temporal continuity moduli and temporal LILs for the TFSPIDE U β (t, x) and the gradient process ∂ x U β (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].
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)  > 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 (β,d) 8 = 2K β,d and k 12 = 2K β,0 , as was obtained in [13].In this sense, the results of this paper reinforce those in [13] log log(1/h) taking the place of φβ,h were established in [13], where k > 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 (β,d) 9 = 2K β,d and k 13 = 2K β,0 , as was obtained in [13].In this sense, the results of this paper reinforce those in [13] which has a Lebesgue measurement of zero with a probability of one.Nevertheless, S β,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 ∂ x U β (•, x).
Fix x ∈ R d .For every λ ∈ (0, 1], the set of temporal λ-fast points for the fourth-order TFSPIDE are defined by where φ β,d,h is given in (5).For every χ ∈ (0, 1], the set of the temporal χ-fast points for the fourth-order TFSPIDE gradients are defined by where ϕ β,h is given in (7).The S β,d,x (λ) are the sets of t, where the temporal LIL of TFSPIDEs fail, and the S β,x (χ) 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 β,d,x (λ) and S β,x (χ).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 β (t, x) and the gradient process ∂ x U β (t, x).
The Hausdorff g-measure of a subset B of a real line for any continuous increasing function g : [0, 1] → [0, +∞] with g(0) = 0 is defined as follows: where the infimum in (11) extends over all countable covers of B by sets C i of diameter d(C i ) < δ.Keep in mind that, while µ g (B) simplifies to an Lebesgue outer measure if g(s) = s, using a distinct g creates a hierarchy of measures.By being familiar with the class of measure functions g for which µ g (B) = 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 β,d,x (λ) and S β,x (χ) 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 (B) rather than its Hausdorff dimension dim(B).For a definition of packing dimension, see [22].
Theorem 2. (Fractal nature for the sets of the temporal fast points.)Let β ∈ (0, ) and u 0 ≡ 0 in (1) be fixed.(a) Suppose d ∈ {1, 2, 3}.For every λ ∈ [0, 1] with a probability of one, we have with a probability of one, we have The following theorem demonstrates that the appropriate index through which to determine whether sets overlap S β,d,x (λ) and S β,x (χ) is the packing dimension.Theorem 3. (Hitting probabilities for the sets of temporal fast points.)Let β ∈ (0, ) and u 0 ≡ 0 in (1) be fixed.
(a) Suppose d ∈ {1, 2, 3}.For every λ ∈ [0, 1] and every analytic set B ⊂ R + , we have For every χ ∈ [0, 1] and every analytic set B ⊂ R + , we have Remark 2. It is easy to see that Equations ( 14) and ( 15) are respectively equivalent for every analytic set B ⊂ R + .As such, we have and 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 ⊂ R + .
Remark 3. We obtain the following probabilistic interpretations of the upper and lower Minkowski dimensions of B, which are denoted by dim M (B) and dim M (B), respectively.This was achieved by reversing the order of sup and lim sup in Equation ( 16); these definitions are provided in [22].
P lim inf 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 , ... were found to be more particularly positive and finite constants (independent of x), as shown in Section i.
The remainder of the article is organized as follows.In Section 2, using the timefractional 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.

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: where δ(x) is the Dirac function.This solution is the transition density of a d-dimensional , where the inverse stable Lévy motion A β of index β ∈ (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: where Here, the density of a stable subordinator is denoted by g β (u), and its Laplace transform is e −s β .When β = 1/2, the density of the Brownian-time Brownian motion (BTBM) is represented by the kernel H (β,d) t;x , as described in [9]; for β ∈ {1/2 k ; k ∈ N}, the density of the k-iterated BTBM is represented by the kernel H (β,d) t;x , as explained in [10,11].Let b : R → R be Borel measurable.The non-linear drift diffusion TFSPIDE is thus 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: Naturally, this yields the mild formulation of (1.1), which is when a ≡ 1 and b ≡ 0 are set in (22).The spatial Fourier transform of the β-time-fractional (including the β = 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 < β < 1 and H (β,d) t;x,y be the β-timefractional kernel.The β-time-fractional kernel's spatial Fourier transform is provided by where is the well-known function of Mittag-Leffler.The spatial Fourier transform in its symmetric form is applied here as follows:

Estimations on the Variances of Temporal Increments of TFSPIDEs and Their Gradients
For the purposes of this subsection, let x ∈ R d be an arbitrary, fixed variable.The auxiliary Gaussian random field {X β (t, x), t ∈ R + , x ∈ R d } is defined by the following: where z + = max{z, 0} for any z ∈ R.Then, the TFSPIDE solution U β has a decomposition as where 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 β , we first calculated the exact variance for the temporal increments of the auxiliary process X β .Then, we transferred these to our TFSPIDE solution U β in terms of X β and a smooth process of Y β .The outcome that followed was crucial.
) and u 0 ≡ 0 in (1) be fixed.Then, for any s, t ∈ (0, T] such that t/s is sufficiently close to 1, we have where K β,d is given in (3).
We also need the following estimation on the variances of temporal increments of the TFSPIDE gradient process ) and u 0 ≡ 0 in (1) be fixed.Then, for all s, t ∈ (0, T], such that t/s is sufficiently close to 1, we have where K β,0 is given in (4).
Proof.Through (4.40) in [13], we have where Via a change in the variables to the integral in τ, (42) yields Let x ∈ R d be fixed.For each 0 < s < t, we obtain the following by using Parseval's identity to the integral in y: Following the same route as the proof of (38), via (44), we have Thus, with (43) and (45), similar to the proof of (40), we obtain (41).This completes the proof.

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.
(a) Suppose d ∈ {1, 2, 3}.Then, for any t, h ∈ R+ , such that h/t is sufficiently close to 0, we have Then, for any t, h ∈ R+ , such that h/t is sufficiently close to 0, we have 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 E[(U Thus, via a well-known estimation (cf., e.g., [28] (p.23)), we have in which we obtain (46) immediately.The proof is thus completed.
(a) Suppose d ∈ {1, 2, 3}.Then, for any > 0, there exist positive and finite constants, i.e., independent of x, and h 0 = h 0 ( ) and c = c( ) are such that, for any compact interval I time ⊂ R + , 0 < h < h 0 and u > 0, we have Then, for any > 0, there exist positive and finite constants, i.e., independent of x, and h 0 = h 0 ( ) and c = c( ) are such that, for any compact interval I time ⊂ R + , 0 < h < h 0 and u > 0, we have 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.

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, 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, ∀λ ∈ We follow Theorem 1.1 of [18].Without a loss of generality, we can assume 0 < λ < 1.For every fixed 0 < λ 0 < λ < 1, we show that S β,d,x (λ) contains a Cantor-like subset of dimension of at least η − 2 , where 0 < < η/2 < 1 and η = 1 − λ 2 0 .A sequence of values for λ 0 converging to λ, as well as 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]).
Then, if there exist two constants δ > 0 and C > 0, such that, for every interval I ⊂ [0, 1] with |I| ≤ δ, there is a constant m(I), such that, for all m ≥ m(I), we have we have µ g (F) > 0.
Let T be the collection of intervals [s, t] ⊂ [0, 1] such that The modulus of continuity ( 5) tells us that for all s, t ∈ [0, 1] that have a |s − t| that is small enough.Thus, there is b > 0, which depends only on λ and λ 0 such that, for every small value, we have 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 < br m , and br m /r m+1 is an integer for m = 1, 2, ... Let δ be a positive number such that δ < /16.For every m ≥ 1, define For every m ≥ 1 and where 4 .Moreover, we define where 0 < η(m) → η := 1 − λ 2 0 as m → +∞.From (4), we derive that, for any m large enough, ) and u 0 ≡ 0 in (1) be fixed.Then, there exists a positive, independent of x, constant c = c(d) > 0, such that, for all I time = [t m,i , t m,i + r m ] ∈ J m and J time = [t m,j , t m,j + r m ] ∈ J m with I time ∩ J time = ∅, as well as all m ≥ m 0 with some m 0 > 0, we have Proof.For convenience, we assume that j > i > 0. For brevity, we define Z ξ,x (•, •) with the increments of the process ξ(•, •) as follows: It follows from ( 28) that, for j > i > 0 and large m, we have where . This yields the following for j > i > 0: where As such, together with (57), we have Similarly to (57), where Y β (•, •) is given in (27).It follows from (36) that, for any t, s ∈ R + , we have where the following notation is used: By some element calculations, we can conclude that, for j > i > 0, we have Proof.We follow Lemma 2.3 of [18].For brevity, we denote Let {ξ m , Y m,i , i = 1, ..., m } be independent mean zero Gaussian random variables with Let f (z) = e z if 0 ≤ z ≤ q m , and = e q m (z ∨ m e q m .Via the well-known comparison property (cf.Theorem 3.11 of [30] (p.74)), we have Thus, we conclude that Via the fact that {Y m,i , i = 1, ..., m } are independent, it is easy to see that .
Lemma 10.Given η < η = 1 − λ 2 , there is an absolute constant c such that, with a probability of one, there exists m 1 such that Proof.It follows from Lemma 9 that it is enough to show that We deduce from Lemma 8 that for any m large enough, we have This implies that, with a probability of one, there is a m 1 = m 1 (η ) > 0 such that (72) holds.The proof is thus completed.
Next, we shall show that the existence of a sequence of sets F 1 ⊃ F 2 ⊃ • • • are such that they satisfy Lemma 2.1's presumptions and that F = ∩ +∞ m=1 F m ⊂ S β,d,x (λ).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 η = η − 1 4 and define m 1 =: m 1 (η ) such that m ≥ m 1 and (71) hold.Assume that the sequence of positive numbers ( k ) satisfies ∑ k < +∞.In the first step, when using Lemma 9, we determine an integer n 1 ≥ m 1 such that .
And then we shall define an increasing sequence n 1 , n 2 , ... inductively, as well as define for For each k ≥ 2, suppose that n k−1 has been defined; as such, we can define an n k large enough to ensure the following: where m 0 ( , δ) is the integer determined in Lemma 9 to invalidate (70) and Then, we have for all ⊆ [0, 1], such that |J time | ≥ R k−1 and all m ≥ 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 , we can conclude that Thus, it follows from Lemma 6, as well as from the fact that r −2δ n k → 0 (k → +∞), with a probability of one, we have Hence, we have proved (52).The proof is thus completed.

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 ⊂ R + , we have 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 We now turn to the proof of the opposite inequality.That is, it is enough to show that Fix such that dim P (B) > .For each integer n ≥ 1, which are denoted by Q n , the set of all intervals of the form [m2 −n , (m + 1)2 ) is obtained, where the following notation is used: In other words, ω n (I time ) is a Bernoulli random variable whose values take 1 or 0 according as to whether we have Define via D := lim sup n D(n) a discrete limsup random fractal, where and where I 0 time denotes the interior of I time .We can claim that, whenever dim P (B) > , then P(D We postpone the verification of (81) and prove (79) first, which thereby completes the proof.Since dim P (B) > , (81), implies that there exists t ∈ B such that there is Θ β,d,x (2 −n [t2 n ], 2 −n (log) −1 ) ≥ for infinitely many instances of n, then, we have, in particular, Thus, if dim P (F) > , then (79) holds; as such, (77) also holds.
(81) remains to be verified.Fix a small η > 0 such that dim P (B) > + η.By [31], there is a closed B * ⊂ B, such that, for all open sets F, (whenever B * ∩ F = ∅), then dim M (B * ∩ F) > + η (see [22] for the definition of an upper Minkowski dimension).It is enough to show that D ∩ B * = ∅ when fixing an open set F such that F ∩ B * = ∅.We can claim that, with a probability of one, D(n) ∩ F ∩ B * = ∅ is such for infinitely many n.When defined via V(n) := ∪ +∞ k=n D(k), n ≥ 1, the open sets are obtained.As such, this claim implies that, with a probability of one, V(n) ∩ F ∩ B * = ∅ is such for all n.Furthermore, via letting F run over a countable base for the open sets, we can obtain a V(n) ∩ B * that is as dense as in (the complete metric space) B * .Via Baire's category theorem (see [32]), we have a B * ∩ ∩ +∞ n=1 V(n) that is dense in B * and, in particular, non-empty.Since D = ∩ +∞ n=1 V(n), we can conclude that D ∩ B * = ∅, which, in turn, means that (81) holds and its results follow.
Fix an open set F by satisfying F ∩ B * = ∅.This is denoted by N n , which are the total number of intervals I time ∈ Q n that satisfy I time ∩ F ∩ B * = ∅.Since dim M (F ∩ B * ) > + η, via the definition of an upper Minkowski dimension, there exists 1 > + η such that N n ≥ 2 n 1 is the case for the infinitely many integers of n.Thus, #(ℵ) = +∞, where As denote by Ω n := ∑ ω n (I time ), the total number of intervals I time ∈ Q n is such that I time ∩ F ∩ B * ∩ D(n) = ∅, where the sum is taken over for all I time ∈ Q n such that In order to show that, with a probability of one, D(n) ∩ F ∩ B * = ∅ applies for the infinitely many instances of n, it suffices to show that Ω n > 0 applies for the infinitely many instances of n.That is, it is enough to show that P(Ω n > 0 i.o.) = 1. (83) It follows from Lemma 4 that p n = 2 −n( +a n ) , where a n → 0 is to n → +∞.Hence, E[Ω n ] = N n p n ≥ 2 n( 1 − −a n ) .Thus, it follows from Lemma 9 that, with a probability of one, Ω n ≥ c2 n( 1 − −a n ) applies, which implies that P(Ω n = 0) → 0 as is to n → +∞.Via Fatou's lemma, one can obtain P(Ω n > 0 i.o.) ≥ lim sup n→+∞ P(Ω n > 0) = 1.

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 fourthorder 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 (B).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.
. • Equation (5) gives the magnitude of the global maximal oscillation of the TFSPIDE solution U β (•, x) over the compact rectangle I time , which is φ β,d,h .Equation (7) gives the magnitude of the global maximal oscillation of the TFSPIDE gradient solution ∂ x U β (•, x) over the compact rectangle I time , which is ϕ β,h .
−n ], m ∈ Z + are obtained.In words, Q n denotes the totality of all intervals.For all I time ∈ Q n , define π n (I time ) = m2 −n to be the smallest element in I time .For I time ∈ Q n , which is denote by ω n (I time ), the indicator function of the event