Nonlinear self-stabilizing processes. II: Convergence to invariant probability
[For part I see above.] Let us consider nonlinear stochastic differential equations \[ X_{t} = X_{0} + B_{t} - \frac 12\int ^{t}_{0} b(s, X_{s}) \roman ds, \quad b(s,x) = \mathbf E\beta (x-X_{s}),\tag{1} \] where \(B\) is a standard Brownian motion and \(\beta :\mathbb{R}\to \mathbb{R}\) is an odd, increasing, locally Lipschitz continuous function of a polynomial growth, convex on \(\mathbb{R}_{+}\). In the first part of the paper the authors proved that there exists a unique invariant density \(u\) for the equation (1) if \(\beta (x)=\beta_{0}(x)+\alpha x\), where \(\alpha >0\) is large enough, and \(\beta_{0}\) is an odd, increasing, locally Lipschitz function with a polynomial growth, satisfying \(|\beta_{0} (x)|\geq k|x|^\varrho \) for some \(\varrho >1\) and any \(|x|\geq 1\). In the second part the weak convergence of the solution \(X_{t}\) to the invariant probability \(u(x) dx\) as \(t\to \infty \) is established under the above hypotheses.
scientific article; zbMATH DE number 1364479