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1 2013

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Maximal regularity for stochastic integral equations

Desch G, Londen S.-O

Journal of Applied Analysis

2013

Vol. 19, nr 1

125--140

We examine the stochastic parabolic integral equation of convolution type U(t)+A∫t0k1(t-s)U(s)ds=∫t0k2(t-s)G(s)dWH(s), t≥0, where U(t) takes values in Lq(O; R) with O a σ-finite measure space, and q∈[2, ∞). The linear operator A maps D(A)⊂Lq(O; R) into Lq(O; R), is nonnegative and admits a bounded H∞-calculus on Lq(O; R). The kernels are powers of t, with k1(t)=1/Γ(α) tα-1, k2(t)=1/Γ(β) tβ-1, and α∈(0, 2), β∈(1/2, 2). We show that, in the maximal regularity case, where β-αθ-η=1/2, one has the estimate ║AθDηtU║Lp(R+xΩ;Lq(O;R))≤C║G║Lp(R+xΩ;Lq(O;H)), where c is independent of G. Here θ ∈(0, 1) and Dηt denotes fractional integration if η∈(-1, 0), and fractional differentiation if η∈(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.