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Regularity properties for stochastic partial differential equations of parabolic type. (English) Zbl 0770.60062
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 D. A. Dawson [J. Multivariate Anal. 5, 1-52 (1975; Zbl 0299.60050)] via the semigroup method, of E. Pardoux [Sémin. Équat. Dériv. part., Part III, Coll. Fr. 1974-75, Exposé No. II (1975; Zbl 0363.60041)], N. V. Krylov and 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 existence and uniqueness of solutions by the usual method of successive approximation.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35K55 Nonlinear parabolic equations