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Symmetric solutions of semilinear stochastic equations. (English) Zbl 0701.60060
Stochastic partial differential equations and applications II, Proc. 2nd Conf, Trento/Italy 1988, Lect. Notes Math. 1390, 237-256 (1989).
[For the entire collection see Zbl 0669.00018.]
Consider the following linear equation of the form $(1)\quad dX=(AX+U'(X))dt+dW,\quad X(0)=x,$ on a Banach space E contained in a Hilbert space H and assume that the cylindrical Wiener process W evolves on H, A is an unbounded linear operator, and $$U: E\to {\mathbb{R}}$$ is H- differentiable. The author proves the following main result. Assume that following conditions are satisfied:
(i) E is a Banach space embedded continuously and as a Borel subset into a separable Hilbert space H; the operator A generates on E a $$C_ 0$$- semigroup S(t), $$t>0$$, extendable on H to a $$C_ 0$$-semigroup $$S_ 0(t)$$, $$t>0$$, with infinitesimal generator $$A_ 0;$$
(ii) the operator $$A_ 0$$ is self-adjoint negative definite, the operator $$A_ 0^{-1}$$ is nuclear and the measure $$m=N(0,\Gamma)$$, where $$\Gamma =-2^{-1}A_ 0^{-1}$$, is concentrated on E;
(iii) equation (1) has a mild solution if there exists an Ornstein- Uhlenbeck process Z, E-continuous and Markov with respect to an increasing family of $$\sigma$$-fields, $${\mathcal F}_ t$$, $$t\geq 0$$, and with transition semigroup $$Q_ t$$, $$t\geq 0.$$
(iv) the H-derivative of a bounded from above function U exists and is locally Lipschitz.
(v) the mild solution of the equation (1) exists for arbitrary $$x\in E.$$
Then the family $$P_ t$$, $$t\geq 0$$, is a Markovian and M-symmetric semigroup, where: $M(dx)=ke^{2U(x)}m(dx),\quad k=(\int_{E}e^{2U(x)}m(dx))^{-1}.$ This theorem generalizes the results of P. L. Chow [Stochastic partial differential equations and applications, Proc. Conf. Trento/Italy 1985, Lect. Notes Math. 1236, 40-56 (1987; Zbl 0618.60076)], T. Funaki [Nagoya Math. J. 89, 129- 193 (1983; Zbl 0531.60095)] and R. Marcus [Trans. Amer. Math. Soc. 198, 177-190 (1974; Zbl 0293.60056)].
Reviewer: S.Wedrychowicz

MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60J60 Diffusion processes