Maximal regularity for stochastic integral equations

• Autorzy: Desch, G. , Londen, S.-O.
• Języki publikacji: EN

Abstrakty

EN

We examine the stochastic parabolic integral equation of convolution type U(t)+A∫^t₀k₁(t-s)U(s)ds=∫^t₀k₂(t-s)G(s)dW_H(s), t≥0, where U(t) takes values in L_q(O; R) with O a σ-finite measure space, and q∈[2, ∞). The linear operator A maps D(A)⊂L_q(O; R) into L_q(O; R), is nonnegative and admits a bounded H∞-calculus on L_q(O; R). The kernels are powers of t, with k₁(t)=1/Γ(α) t^α-1, k₂(t)=1/Γ(β) t^β-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 ║A^θD^η_tU║L_p(R⁺xΩ;L_q(O;R))≤C║G║L_p(R⁺xΩ;L_q(O;H)), where c is independent of G. Here θ ∈(0, 1) and D^η_t 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.

Słowa kluczowe

PL

stochastyczne równanie całkowe; regularność maksymalna; H-nieskończoność

EN

stochastic integral equation; maximal regularity; H-infinity calculus

• Czasopismo: Journal of Applied Analysis
• Rocznik: 2013
• Tom: Vol. 19, nr 1
• Strony: 125--140
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Desch, G.

• Department of Mathematics and Scientific Computing, Karl-Franzens-University Graz, Heinrichstrasse 36, 8010 Graz, Austria,

autor

Londen, S.-O.

• Department of Mathematics, Aalto University, PO Box 11100, FI-00076 Aalto, Finland,

Bibliografia

• [1] C. Martinez Carracedo and M. Sanz Alix, The Theory of Fractional Powers of Operators, North-Holland Math. Stud. 187, North-Holland, Amsterdam, 2001.
• [2] P. Clément, S.-O. Londen and G. Simonett, Quasilinear evolutionary equations and continuous interpolation spaces, J. Differential Eqs. 196 (2004), 418-447.
• [3] W. Desch and S.-O. Londen, A generalization of an inequality by N. V. Krylov, J. Evolution Eqs. 9 (2009), 525-560.
• [4] W. Desch and S.-O. Londen, An Lp-theory for stochastic integral equations, J. Evolution Eqs. 11 (2011), 287-317.
• [5] N. V. Krylov, A parabolic Littlewood-Paley inequality with applications to parabolic equations, Topol. Methods Nonlinear Anal. 4 (1994), 355-364.
• [6] N. V. Krylov, An analytic approach to SPDEs, in: Stochastic Partial Differential Equations: Six Perspectives, Math. Surveys Monogr. 64, American Mathematical Society, Providence (1999), 185-242.
• [7] J. van Neerven, M. Veraar and L. Weis, Stochastic maximal Lp-regularity, Ann. Probab. 40 (2012), 788-812.
• [8] L. W. Weis, The H^∞ holomorphic functional calculus for sectorial operators - A survey, in: Partial Differential Equations and Functional Analysis, Oper. Theory Adv. Appl. 168, Birkhäuser, Basel (2006), 263-294.
• [9] R. Zacher, Quasilinear parabolic problems with nonlinear boundary conditions, dissertation, Martin-Luther-Universität Halle-Wittenberg, 2003.