Regularity for the Solution of a Stochastic Partial Differential Equation with the Fractional Laplacian

Abstract

We study the regularity properties for the mild solution of a stochastic partial differential equation in ${\mathbf {R}}^d$ with the fractional Laplacian $-(-\varDelta )^{\frac{\gamma }{2}}$ under the condition where its solution exists uniquely as a function valued process. To show its regularity, we estimate the fundamental solution and use the Kolmogorov-Centsov theorem. Due to the unboundedness of the domain, we need to check the behavior of the fundamental solution for sufficiently large |x|, $x\in {\mathbf {R}}^d$.

Appendix

For the convenience of the readers, we briefly recall the Kolmogorov-Centsov theorem for random fields without its proof (see e.g. Theorem 1.4.4 in [10]).

Theorem 22.3.3

Let X(x, y), $x\in D_1$, $y\in D_2$ be a random field with values in a Banach space B with its norm $||\cdot ||$, where $D_1$, $D_2$ are domains in ${\mathbf {R}}^{d_1}$ and ${\mathbf {R}}^{d_2}$, respectively. Suppose that there exist constants C, $\gamma >0$ and $\alpha _1>d_1$, $\alpha _2>d_2$ such that

$$\begin{aligned}&{\mathbf {E}}\Bigl [||(X(x',y')-X(x,y'))-(X(x',y)-X(x,y))||^\gamma \Bigr ]\leqslant C |x-x'|^{\alpha _1}|y-y'|^{\alpha _2},\end{aligned}$$

(22.70)

$$\begin{aligned}&{\mathbf {E}}\Bigl [||X(x',y)-X(x,y)||^\gamma \Bigr ]\leqslant C |x-x'|^{\alpha _1},\end{aligned}$$

(22.71)

$$\begin{aligned}&{\mathbf {E}}\Bigl [||X(x,y')-X(x,y)||^\gamma \Bigr ]\leqslant C |y-y'|^{\alpha _2} \end{aligned}$$

(22.72)

hold for any $x,\,x'\in D_1$ and $y,\,y'\in D_2$. Then, X(x, y) has a continuous modification Y(x, y), which is $(\beta _1,\beta _2)$-Hölder continuous for any $\beta _1\in (0,\frac{\alpha _1-d_1}{\gamma })$ and $\beta _2\in (0,\frac{\alpha _2-d_2}{\gamma })$.