## 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.

### MSC:

 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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### References:

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