Maximal Regularity For Stochastic Integral Equations

We examine the stochastic parabolic integral equation of convolution typeU(t) + A integral(t)(0) k1(t-s)U(s)ds = integral(t)(0) k(2)(t-s)G(s)dW(H)(s), t >= 0,where U(t ) takes values in L-q(O; R) with O a sigma-finite measure space, and q is an element of [2, infinity). The linear operator A maps D(A) subset of L-q(O; k) into L-q(O; R), is nonnegative and admits a bounded H-infinity-calculus on L-q(O; R). The kernels are powers of t, with k(1) (t) = 1/Gamma(alpha)t(alpha-1), k(2)(t) = 1/Gamma(beta)t(beta-1), and alpha is an element of(0, 2), beta is an element of (1/2, 2). We show that, in the maximal regularity case, where beta - alpha theta - eta = 1/2, one has the estimateparallel to A(theta)D(t)(eta)U parallel to(Lp(R+x Omega;Lq(O;R)))<= C parallel to G parallel to(Lp(R+x Omega;Lq(O;H))),where c is independent of G. Here theta is an element of (0, 1) and D-t(eta) denotes fractional integration if eta is an element of (-1, 0), and fractional differentiation if eta is an element of (0, 1), 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.