Stationary solutions of a nonlinear two-dimensional stochastic differential equation in Banach space. (English. Ukrainian original) Zbl 0942.60060

Gorodnij, M. F.

Theory Probab. Math. Stat. 53, 43-49 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 39-45 (1995).

The author obtains sufficient conditions for the existence of a bounded solution and a stationary solution of the equation in Banach space \[ \begin{aligned} Ax(\mu,\nu)&= x(\nu+1,\mu+1)+x(\nu+1,\mu-1)+x(\nu-1,\mu+1)+x(\nu-1,\mu-1)-4x(\nu, \mu)+\\ &\;+\sum_{k=2}^{\infty}G_{k}(x(\nu,\mu),\ldots,x(\nu,\mu))+y(\nu,\mu),\quad (\nu,\mu)\in Z^2, \end{aligned} \] where \(A\) is a closed operator, \(G_{k}\) is a \(k\)-linear form, \(y(\nu,\mu)\) is a bounded sequence or a stationary random field.

Reviewer: A.D.Borisenko (Kyïv)

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
34G20	Nonlinear differential equations in abstract spaces
60H20	Stochastic integral equations
34K30	Functional-differential equations in abstract spaces

Keywords:

stochastic difference equation; Banach space; existence theorem; bounded and stationary solution