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 derivative(alpha)(t)u = (phi(Delta)u + f (u)) + delta(beta)(t) Sigma(infinity)(k=1) integral(t)(0) g(k) (u)dw(s)(k), t > 0, x is an element of R-d as well as the SPDE driven by space-time white noise partial derivative(alpha)(t)u = (phi(Delta)u + f (u)) + partial derivative(beta-1)(t), t > 0, x. R-d. Here, alpha is an element of (0, 1), ss < alpha + 1/2, {w(t)(k) : k = 1, 2,...} is a family of independent one-dimensional Wiener processes and. (W) over dot is a space-timewhite noise defined on [0,infinity)xR(d). The time non-local operator partial derivative(alpha)(t) denotes the Caputo fractional derivative of order alpha, the function phi is a Bernstein function, and the spatial non-local operator phi(Delta) is the integro-differential operator whose symbol is -phi(vertical bar xi vertical bar(2)). In other words, phi(Delta) is the infinitesimal generator of the d-dimensional subordinate Brownian motion. We prove the uniqueness and existence results in Sobolev spaces and obtain the maximal regularity results of solutions.