Infinite-dimensional nonlinear stationary Fokker-Planck-Kolmogorov equations

TL;DR

This paper proves the existence of solutions to a complex nonlinear stationary Fokker-Planck-Kolmogorov equation in infinite-dimensional spaces, with applications to Vlasov-type drifts and bounded drift components.

Contribution

It establishes the existence of probability solutions for a class of nonlinear infinite-dimensional Fokker-Planck equations with specific drift structures and continuity conditions.

Findings

Existence of solutions for nonlinear infinite-dimensional Fokker-Planck equations.

Application to Vlasov-type drifts via convolution.

Results for cases with bounded drift components under stronger conditions.

Abstract

We prove existence of a probability solution to the nonlinear stationary Fokker-Planck-Kolmogorov equation on an infinite dimensional space with a centered Gaussian measure $\gamma$ with a unit diffusion operator and a drift of the form $-x+v(p,x)$, where $v$ is a bounded mapping with values in the Cameron-Martin space $H$ of $\gamma$ and $v$ is defined on the space $E\times X$, where is $E$ is the subset of $L^2(\gamma)$ consisting of probability densities. The equation has the form $L_{b(p,\bullet)} ^*(p\cdot \gamma)=0$ with $L_{b(p,\bullet)}\varphi =\Delta_H \varphi + (b(p,\bullet) , D_{_H}\varphi)_{_H}$, so that the drift coefficient depends on the unknown solution, which makes the equation nonlinear. This dependence is assumed to satisfy a suitable continuity condition. This result is applied to drifts of Vlasov type defined by means of the convolution of a vector field with the solution. In addition, we consider a more general situation where only the components of $v$ are uniformly bounded and prove the existence of a probability solution under some stronger continuity condition on the drift.