Periodic solutions of Fokker-Planck equations
This paper deals with problem of existence of periodic solution in distribution to the stochastic differential equation \(dx(t) = f (t, x(t))dt + \sigma(t, x(t))dW(t)\), where \(W(t)\) is an \(l\)-dimensional standard Brownian motion, \(f: \mathbb R^{+}\times \mathbb R^{d}\to \mathbb R^{d}\), \(\sigma: \mathbb R^{+}\times \mathbb R^{d}\to\mathbb R^{d\times l}\) are Borel-measurable, \(\theta\)-periodic on \(t\) functions. The Fokker-Planck equation for the probability density \(p(t, x)\) of Ito diffusion \(x(t)\) satisfying considered stochastic differential equations is \(\partial p(t, x)/\partial t=-\sum_{i}\partial(f_{i} (t, x)p(t, x))/\partial x_{i} + {1\over 2}\sum_{ij}\partial^2 ((\sigma\sigma^{T})_{ij}(t, x)p(t, x))/\partial x_{i}\partial x_{j}\). The authors prove that if \(f(t,x)\) and \(\sigma(t,x)\) are continuous, \(\theta\)-periodic in the \(t\) functions satisfying the Lipschitz condition and Lyapunov conditions hold, then the Fokker-Plank equation admits a \(\theta\)-periodic solution. It is proved existence of \(L^2\)-bounded \(\theta\)-periodic in distribution solution to the considered Ito equation. The theorem on existence of unique uniformly asymptotically stable, \(\theta\)-periodic in distribution solution to the considered Ito equation is proved.
scientific article; zbMATH DE number 6712344
scientific article