Nonlinear self-stabilizing processes. II: Convergence to invariant probability. (English) Zbl 0932.60064

[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.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory


Zbl 0932.60063
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