## 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)

### Citations:

Zbl 0721.60068; Zbl 0701.60060
Full Text:

### References:

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