Infinite-dimensional nonlinear stationary Fokker-Planck-Kolmogorov equations

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 $L_{b(p,\bullet )}^{\ast }(p\cdot \gamma )=0$ with $L_{b(p,\bullet )}\phi ={\Delta }_H\phi +(b(p,\bullet ),D_{{}_H}\phi {)}_{{}_H}$, 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.