Maximal regularity for stochastic integral equations

Abstract

We examine the stochastic parabolic integral equation of convolution type where takes values in with a σ-finite measure space, and . The linear operator A maps into , is nonnegative and admits a bounded H ∞ -calculus on . The kernels are powers of t , with , , and , . We show that, in the maximal regularity case, where , one has the estimate where c is independent of G . Here and denotes fractional integration if , and fractional differentiation if , both with respect to the t -variable. The proof relies on recent work on stochastic differential equations by van Neerven, Veraar and Weis, and extends their maximal regularity result to the integral equation case.