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2 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.

Second order abstract differential equations of elliptic type set in R+

Eltaief A., Maingot S.

Demonstratio Mathematica

2013

Vol. 46, nr 4

709--727

In this paper we give some new results on complete abstract second order differential equations of elliptic type set in R+. In the framework of UMD spaces, we use the celebrated Dore-Venni Theorem to prove existence and uniqueness for the strict solution. We will use also the Da Prato-Grisvard Sum Theory to furnish results when the space is not supposed to be a UMD space.