Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

: High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient.

The fundamental kernel associated with the deterministic version of the time-fractional SPIDE is built on the BTP [20,31,35] and extensions thereof. In this article, we give exact, dimension-dependent, spatial moduli of non-differentiability for the important class of stochastic equation: whereR + = (0, ∞), the noise term ∂ d+1 W/∂t∂x is the space-time white noise corresponding to the real-valued Brownian sheet W onR + × R d , d = 1, 2, 3; the time fractional derivative of order β, C and the time fractional integral of order α, I α t , is the Riemann-Liouville fractional integral of order α: 1−α dτ, for t > 0 and α > 0, and I 0 t = I, the identity operator. Here Γ(s) = ∞ 0 x s−1 e −x dx, s > 0, is the Gamma function. The initial data u 0 here is assumed Borel measurable, deterministic, and that there is a constant 0 < γ ≤ 1 such that where C α,γ b (R d ; R) denotes the set of α-continuously differentiable functions on R d whose α-derivative is locally Hölder continuous with exponent γ.
Of course, Equation (1) is the formal (and nonrigorous) equation. Its rigorous formulation, which we work with in this paper, is given in mild form as kernel stochastic integral equation (SIE). This SIE was first introduced and treated by [21,30,31,[35][36][37][38]. We give them below in Section 2, along with some relevant details.
The existence/uniqueness as well as sharp dimension-dependent L p and Hölder regularity of the linear and nonlinear noise version of Equation (1) were investigated in [30,[36][37][38]. These results naturally lead to the following list of motivating questions:

•
Are the solutions to Equation (1)  It was studied in [39] that the exact uniform and local moduli of continuity for the time fractional SPIDE in the time variable t and space variable x, separately. In fact, it was established in [39] that the exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for the fourth order time fractional SPIDEs and their gradient. These results give the answers to spatially continuity and exact moduli of continuity of the solutions to Equation (1), and give partial answers to above questions. In this paper, we are concerned with spatially differentiability of the solutions to Equation (1). We delve into the exact moduli of non-differentiability of the process U β and its gradient in space. It builds on and complements works in [39], and together answers all of the above questions.
The rest of the paper is organized as follows. In Section 2, we discuss the rigorous timefractional SPIDE kernel SIE (mild) formulation, spatial spectral density and spatial zero-one laws for time fractional SPIDEs and their gradient by using the time-fractional SPIDE kernel SIE formulation and spectral/harmonic analysis. In Section 3, we investigate the exact spatial moduli of non-differentiability for time fractional SPIDEs and their gradient by making use of the theory on limsup random fractals in [40]. In order to apply their results, the Gaussian correlation inequality in [41] will also play an important role. In the final section, the results are summarized and discussed.
Then, the rigorous time-fractional SPIDE kernel SIE (mild) formulation is the stochastic integral equation (see p. 530 in [32], and Definition 1.1 and Equation (1.11) in [36]). Of course, the mild formulation of (1.1) is then obtained by setting a ≡ 1 and b ≡ 0 in Equation (5).

Notation 1.
Positive and finite constants (independent of x) in Section i are numbered as c i,1 , c i,2 , .... We conclude this section by citing the following spatial Fourier transform of the β-time-fractional (including the β = 1/2 BTBM case) kernels from Lemma 2.1 in [39].
t;x,y be the β-time-fractional kernel, and let 0 < β < 1. The spatial Fourier transform of the β-time-fractional kernel is given bŷ where is the well known Mittag-Leffler function. Here, the following symmetric form of the spatial Fourier transform has been used:

Spatial Spectral Density for Time Fractional SPIDEs and Their Gradient
Our spatial results are crucially depend on the following Lemma. In this lemma, (a) is Lemma 4.1 in [39], and (b) follows from (4.27) in [39].

Spatial Zero-One Laws for Time Fractional SPIDEs and Their Gradient
We establish spatial zero-one laws for time fractional SPIDEs and their gradient to have moduli of non-differentiability, which may be of independent interest. Fix t ∈R + . For
(a) Suppose d = 1, 2, 3. For any compact rectangle where where Remark 1. Equation (11) establishes zero-one law for the minimum oscillation inf x∈I space M β (x, h) of the sample function x → U β (t, x) over the compact rectangle I space . Equation (13) establishes zero-one law for the minimum oscillation inf x∈I space V β (x, h) of the sample function x → ∂ x U β (t, x) over the compact rectangle I space .
Proof of Proposition 1. Since the proof of Equation (13) is similar to Equation (11), we only prove Equation (11). Let .., are mutually disjoint, where the following notation is used: Then ξ n = {ξ n (t, x), (t, x) ∈ R + × R d }, n = 1, 2, ..., are independent Gaussian fields. By Equation (6), we express and denote by N(d ξ n , I space , δ) the smallest number of d ξ n -balls of radius δ > 0 needed to cover I space . Since {U β (t, x), x ∈ R d } is a stationary Gaussian random field with spectral density S β (t, ξ), by the definition of spectral density, one has To obtain the last inequality, in the integral we bound 1 − cos( u, v ) by |u| 2 |v| 2 /2 for u, v ∈ R d . Then, by Theorem 4.1 in Meerschaert et al. [44], one has sup x,y∈I space :|y|≤h where Then, by Equation (17), one has lim h→0+ sup x,y∈I space :|y|≤h Therefore, the random variable lim inf is measurable with respect to the tail field of {ξ n } ∞ n=1 and hence is constant almost surely. This implies Equation (11).

Extremes for Time Fractional SPIDEs and Their Gradient
Without loss of generality, we again assume that u 0 = 0, and the random field solution U β is given by Equation (1). Fix an arbitrary t > 0 throughout this subsection. Our spatial results are depend on the following the following small ball probability estimates for time fractional SPIDEs and their gradient. Lemma 3. Let t ∈R + be fixed and 0 < β ≤ 1/2, and assume that u 0 = 0 in Equation (1).
We also need the following lemma, which is Theorem 1.1 in [41]. Lemma 4. Let G = (G 1 , G 2 ) be an R n -valued normal random vector with mean vector 0, where G 1 = (X 1 , ..., X k ) , G 2 = (X k+1 , ..., X n ) and 1 ≤ k < n. Then ∀x > 0, where x ∞ denotes the maximum norm of a vector x and We also need the following lemma, which is Lemma 2.4 in [47].

Lemma 5.
Let A = (a ij , 1 ≤ i, j ≤ 2p) be a positive semidefinite matrix given by where A 11 and A 22 are two p × p matrices. Put S i = ∑ 2p j=p+1 |a ij | for 1 ≤ i ≤ p and = ∑ p j=1 |a ij | for p + 1 ≤ i ≤ 2p. Assume the following conditions are satisfied: (i) There is a constant δ such that for all 1 ≤ i ≤ 2p, (ii) There exists a finite constant B > 0 such that for all 1 ≤ i ≤ 2p, where A (i) is the submatrix of A obtained by deleting the ith row and ith column.

Spatial Moduli of Non-Differentiability for Time Fractional SPIDEs and Their Gradient
We investigate the exact spatial moduli of non-differentiability for time fractional SPIDE U β (t, x) and the gradient process ∂ x U β (t, x). Theorem 1. (Spatial moduli of non-differentiability) Let t ∈R + be fixed and 0 < β ≤ 1/2, and assume that u 0 = 0 in Equation (1).
(a) Suppose d = 3. For any compact rectangle I space ⊂ R 3 , Consequently, the sample paths of U β (t, x) are almost surely nowhere differentiable in all directions of x.
(b) Suppose d = 1. For any compact rectangle I space ⊂ R, Consequently, the sample paths of ∂ x U β (t, x) are almost surely nowhere differentiable in x.

Remark 3.
The following are some remarks on Theorem 1.
• Equations (30) and (31) describe the uniform minimal oscillations of the sample functions of U β (t, x) and ∂ x U β (t, x), respectively. To be precise, Equation (30) implies the size of the minimum oscillation inf x∈I space M β (x, h) of the sample function x → U β (t, x) over the compact rectangle I space is γ 1 (h) (up to a constant factor). Equation (31) implies the size of the minimum oscillation inf x∈I space V β (x, h) of the sample function x → ∂ x U β (t, x) over the compact rectangle I space is γ 2 (h) (up to a constant factor).

•
Together with the Khinchin-type law of the iterated logarithm and the uniform modulus of continuity in [39], they provide complete information on the regularity properties of U β (t, x) and ∂ x U β (t, x). (30), we only prove Equation (30). To prove Equation (30), we claim first the following two inequalities:
Fix an arbitrary θ > 1. For n ∈ Z + , let h n = θ −n and ρ n = θ 4n . For i = (i 1 , i 2 , i 3 ) ∈ Z 3 + and n ≥ 1, we define two sets A n and A n,i as follows: where 1 is a vector with elements 1. Observe that for all h ∈ (0, 1), there exists a set A n such that h ∈ A n , and for all x ∈ I, there exists a set A n,i such that x ∈ A n,i . Let as u → 0. Thus, by Equations (20) and (34), one has Hence, the sum of above probabilities with respect to n is finite and hence, by Borel-Cantelli lemma, one has Since the function x → γ 1 (x) is decreasing for x ∈ (0, 1), one has lim inf It follows from Equations (35)-(37) that Equation (32) holds. Next we prove Equation (33). For convenience, a typical parameter ("space point") Denote by u ≤ u < u + 1 the integer part of u ∈ R + . For each n ≥ 1, we denote h n = 2 −n , Θ = Θ n = h 2 n , ϑ = ϑ n = | log(h n )| −1 , D n = ϑ −1 h −2 n , S n = {1, ..., D n }, and T n = {i = (i 1 , i 2 , i 3 ) ∈ Z 3 + : i k ∈ {1, ..., ϑ −1 }, k = 1, 2, 3}. For each i ∈ S n , we define We define Y i,n , for i ∈ S n , to be 1 or 0 according as the random variable is or not. Define S n := ∑ i∈S n Y i,n . For each n ≥ 1, the mean p n := E[Y i,n ] is the same for all i ∈ S n , and that, by Equation (20), one has uniformly over i ∈ S n , as n → ∞, We need only show that S n > 0 for infinitely many n. We want to estimate Put Λ = Λ n = ϑ −12 . We make the following claim: ∀τ > 0, whenever |i − j| ≥ Λ, Before we prove Equation (40), we complete the proof of Equation (33) and thereby of Theorem 1.
It follows from Equation (39) that for all τ > 0, whenever i, j ∈ S n satisfy |j − i| ≥ Λ, Thus, by Equation (38), For the remaining covariance, use the fact that all Y i,n 's are either 0 or 1. In particular, Cov(Y i,n , Y j,n ) ≤ E[Y i,n ] = p n . Thus Var(S n ) ≤ τD 2 n p 2 n + D n p n Λ.

Conclusions
In this article, we have presented that the solutions to the fourth order time fractional SPIDEs and their gradient, driven by space-time white noise, are almost surely nowhere differentiable in all directions of space variable x. We have established the exact spatial moduli of non-differentiability, and been concerned with the small fluctuation behavior, with delicate analysis of regularities, for the above class of equations and their gradient. They complement Allouba's earlier works on the spatio-temporal Hölder regularity of time fractional SPIDEs and their gradient. Together with the Khinchin-type law of the iterated logarithm and the uniform modulus of continuity, they provide complete information on the regularity properties of time fractional SPIDEs and their gradient in space.