## Nonlinear self-stabilizing processes. I: Existence, invariant probability, propagation of chaos.(English)Zbl 0932.60063

Nonlinear stochastic differential equations of the type $X_{t} = X_{0} + B_{t} - \frac 12\int ^{t}_{0} b(s, X_{s}) ds, \quad b(s,x) = \mathbf E\beta (x-X_{s}),\tag{1}$ are studied. Here $$B$$ is a standard one-dimensional Brownian motion and the function $$\beta :\mathbb{R}\to \mathbb{R}$$ is assumed to be odd, increasing, locally Lipschitz continuous of a polynomial growth (in the sense that $$|\beta (x)-\beta (y)|\leq (c+|x|^{r}+|y|^{r}) |x-y|$$ holds for some $$c>0$$, $$r\geq 0$$ and any $$x,y\in \mathbb{R}$$) and satisfies $$\beta (x)-\beta (y)\geq \beta_{1}(x-y) +\beta_{2}$$ for some $$\beta_{2}\in \mathbb{R}$$, $$\beta_{1}>0$$ and all $$x\geq y$$. Under these hypotheses it is proven that the equation (1) has a unique strong solution provided $$X_{0}\in L^{2(r+1)^{2}} (\mathbf P)$$. If $$\beta$$ is in addition convex on $$\mathbb{R}_{+}$$, then there exists an invariant density for (1). This invariant density is unique if $$\beta (x) = \beta_{0}(x) + \alpha x$$, $$\beta_{0}$$ being an odd, increasing, locally Lipschitz function with a polynomial growth satisfying $$\beta_{0}(x)/x\to 0$$ as $$x\to 0+$$, and $$\alpha >0$$ being a sufficiently large constant. Moreover, the uniqueness of an invariant density is established for $$\beta (x)=x^{3}$$ and $$\beta (x)=x^{5}$$, although the general theory is not applicable to these two cases. Finally, it is shown that (1) is related to the propagation of chaos property for a suitable system of particles. [For part II see below].

### MSC:

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

Zbl 0932.60064
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