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Qualitative behaviour of solutions of stochastic reaction-diffusion equations. (English) Zbl 0761.60055

Via the setting of G. Da Prato and J. Zabczyk [Differ. Integral Equ. 1, No. 2, 143-155 (1988; Zbl 0721.60068)] the author gives a qualitative study of stochastic equations \[ d\xi_ t=A\xi_ t dt+f(t,\xi_ t)dt+dw_ t(*) \] in a Banach space \(E\). Under certain conditions the authors prove the so-called strong Feller property in the narrow sense (for the transition probability functions \(P(t,s,x,\cdot)\)), and as consequences, the mutual absolute continuity of \(P(t,s,x,\cdot)\) and the strong law of large numbers (which essentially generalizes results of J. Zabczyk (1989) in the case \(f(t,u)=f(u)\) and of B. Maslowski (1989)). Under modified conditions they prove a Girsanov type theorem, and as a consequence, get analogous results for this case. An application to a general reaction-diffusion equation (which is a model of phase transitions in synergetics and quantum field theory) is given. In the proofs there is largely used the approximation of J. Zabczyk [Stochastic partial differential equations and applications II, Proc. 2nd Conf., Trento/Italy 1988, Lect. Notes Math. 1390, 237-256 (1989; Zbl 0701.60060)] for solutions of \((*)\) in \(E\) by solutions of finite- dimensional systems.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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