Regularity properties for stochastic partial differential equations of parabolic type
type: Item
instance of: scholarly article
title: Regularity properties for stochastic partial differential equations of parabolic type
zbMATH Open document ID: 0770.60062
author: Tadahisa Funaki
published in: Osaka Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: This article is a first work devoting a study of regularity properties (such as differentiability, Hölder continuity) of solutions and their derivatives on more ``nice'' spaces for semilinear stochastic PDE's \[ dS_ t(x)=-{\mathcal A}S_ t(x)dt+\sum^ J_{j=1}{\mathcal B}_ j\{B_ j(x,S_ t)\}dt+\sum^ J_{j=1}{\mathcal C}_ jdw^ j_ t(x),\quad x\in\mathbb{R}^ d,\;t>0, \tag{1} \] where \({\mathcal A}\), \({\mathcal B}_ j\), \({\mathcal C}_ j\) are differential operators of arbitrary orders; \(\partial/\partial t+{\mathcal A}\) is uniformly parabolic in the sense of Petrovskiȷ; \(\{w^ j_ t(x)\}\) is a system of \(J\) independent \(\{{\mathcal F}_ t\}\)-cylindrical Brownian motions. The main crucial nontrivial tools consist in establishing numerous estimates in corresponding norms and metrics for deterministic and stochastic parts of the integral equation equivalent to the equation (1). The author also introduces and proves for solutions of (1) the weak differentiability playing a rôle to determine a sufficient number of boundary data. In the last section the author introduces the martingale problem associated with (1) and establishes its well-posedness. This article is motivated by a study of the time-dependent Ginzburg-Landau equations of non- conservative and conservative types. Remark that solutions established in the work of \textit{D. A. Dawson} [J. Multivariate Anal. 5, 1-52 (1975; Zbl 0299.60050)] via the semigroup method, of \textit{E. Pardoux} [Sémin. Équat. Dériv. part., Part III, Coll. Fr. 1974-75, Exposé No. II (1975; Zbl 0363.60041)], \textit{N. V. Krylov} and \textit{B. L. Rozovskij} [J. Sov. Math. 16, 1233-1277 (1981); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 14, 71-146 (1979; Zbl 0436.60047)] via the variational (monotone operators) method, belong to rather large ``not so nice'' spaces, although their results are available for a rather large class of nonlinear stochastic partial differential equations. The author establishes the results on e…
reviewed by: Hong Thai Nguyen
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 59704
zbMATH Keywords: differentiability
zbMATH Keywords: Hölder continuity of solutions
zbMATH Keywords: regularity properties
zbMATH Keywords: martingale problem
zbMATH Keywords: Ginzburg-Landau equations
zbMATH Keywords: method of successive approximation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 59704