Nonlinear self-stabilizing processes. I: Existence, invariant probability, propagation of chaos
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].
scientific article; zbMATH DE number 1364478
scientific article