Existence of periodic solutions of an abstract stochastic equation. Asymptotic periodicity of solutions of the Cauchy problem. (Russian) Zbl 0669.60056

Dorogovtsev, A. Ya.

Teor. Veroyatn. Mat. Stat., Kiev 39, 47-52 (1988).

The following equation is considered: \[ \dot x(t)+Ax(t)=f(t,x(t))+\sigma (t,x(t))\dot w(t),\quad t\in {\mathbb{R}}. \] Here x: \({\mathbb{R}}\to H\), f: \({\mathbb{R}}\times H\to H\), \(\sigma\) : \({\mathbb{R}}\times H\to {\mathcal L}(H,H)\), H is a separable Hilbert space, w a Wiener process in H, A a linear possibly unbounded operator in H. Conditions are given providing existence of strong and generalized solutions of the above equation and of their periodicity in the sense of finite dimensional distributions periodicity. These conditions are expressed in terms of the spectrum of the operator A, constants in the Lipschitz conditions for the functions f and \(\sigma\), and the correlation operator of w(1). Under these conditions and with proper initial value the solution of the Cauchy problem for the above equation is asymptotically periodic with probability one.

Reviewer: A.Ya.Dorogovtsev

MSC:

60H10

Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

strong and generalized solutions; finite dimensional distributions periodicity; Cauchy problem; asymptotically periodic with probability one