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Some results on stochastic partial differential equations by the stochastic characteristics method. (English) Zbl 0647.60070
Consider the following simple stochastic partial differential equation, \[ (*)\quad du_ t=A(t)u_ tdt+\sum^{N}_{i=1}B_ i(t)u_ tdW_ i(t),\quad u_ 0=u\quad 0, \] where u:\(\Omega\times [0,T]\times {\mathbb{R}}\) \(d\to {\mathbb{R}}^ d.\) \(W=(W_ 1,...,W_ N)\) is a standard N-dimensional Wiener process on a probability space (\(\Omega\), \({\mathcal F}\), \({\mathbb{P}})\), and \(A(t)\) and \(B_ i(t)\) are nonrandom linear differential operators of second and first-order, respectively, on \({\mathbb{R}}^ d\) with coefficients depending on t. This problem can be transformed by using the so-called stochastic characteristics method developed by H. Kunita [Stochastic analysis. Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982 - North Holland Math. Libr. 32, 249-269 (1984; Zbl 0545.60061)] into a family of deterministic parabolic problems \[ (**)\quad \partial v/\partial t=L(t)v,\quad v(0)=u_ 0, \] where L(t) denotes a second- order differential operator with random coefficients. The author solved (**) using some recent results in the theory of abstract evolution equations.
Note that (*) has been studied by other authors, see E. Pardoux [Thèse d’Etat, Univers. Paris Sud, Orsay (1975)], G. Da Prato [Stochastic Anal. Appl. 1, 57-88 (1983; Zbl 0511.60055)] and N. V. Krylov and B. L. Rozovskij [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 1329-1347 (1977; Zbl 0371.60076)]. The equivalence between (*) and (**) has been stated by B. L. Rozovskij [Evolutionary stochastic systems. (1983; Zbl 0525.60063)].
Reviewer: R.Manthey

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H25 Random operators and equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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