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)].

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 |