A Sobolev space theory for the Stochastic Partial Differential Equations with space-time non-local operators

Abstract

We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes $${\partial }_t^{\alpha }u=\left(\varphi (\Delta )u+f(u)\right)+{\partial }_t^{\beta }\sum _{k=1}^{\infty }{\int }_0^t{g}^k(u)d{w}_s^k,t>0,x\in {\mathbb{R}}^d;u(0,\cdot )=u_0$$ as well as the SPDE driven by space-time white noise $${\partial }_t^{\alpha }u=\varphi (\Delta )u+f(u)+{\partial }_t^{\beta -1}h(u)\dot{W},t>0,x\in {\mathbb{R}}^d;u(0,\cdot )=u_0.$$ Here, $\alpha \in (0,1),\beta \in (-\infty ,\alpha +1/2)$, $\{w_t^k:k=1,2,\cdots \}$ is a family of independent one-dimensional Wiener processes, and $\dot{W}$ is a space-time white noise defined on $[0,\infty )\times {\mathbb{R}}^d$. The time non-local operator ${\partial }_t^{\gamma }$ denotes the Caputo fractional derivative if $\gamma >0$ and the Riemann-Liouville fractional integral if $\gamma \le 0$. The the spatial non-local operator $\varphi (\Delta )$ is a type of integro-differential operator whose symbol is $-\varphi (|\xi {|}^2)$, where $\varphi$ is a Bernstein function satisfying \begin{equation*} κ_0\left(\frac{R}{r}\right)^{δ_{0}} \leq \frac{ϕ(R)}{ϕ(r)}, \qquad \forall\,\, 0<r<R<\infty \end{equation*} with some constants ${\kappa }_0>0$ and ${\delta }_0\in (0,1]$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity results of solutions.

arXiv comment50 pages