Infinite-dimensional nonlinear stationary Fokker-Planck-Kolmogorov equations
Vladimir I. Bogachev, Michael Röckner, Stanislav V. Shaposhnikov
TL;DR
The paper develops existence results for nonlinear stationary Fokker-Planck-Kolmogorov equations in infinite-dimensional settings driven by a Gaussian measure. It treats drift fields of the form $b(p,x)=-x+v(p,x)$ with $v$ bounded in the Cameron–Martin space and continuous in the density variable, constructing solutions as densities in $L^2(\gamma)$ via a Schauder fixed-point argument on the map $p\mapsto\varrho_p$, where $\varrho_p$ solves the corresponding linear equation. The authors extend the framework to Vlasov-type drifts, address parameter dependence with measurable selections, and discuss cases with unbounded components of $v$ under stronger continuity assumptions. The approach yields rigorous existence results and a robust method for handling nonlinear infinite-dimensional FPK equations relevant to statistical mechanics and stochastic analysis on function spaces.
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 $γ$ 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 $γ$ and $v$ is defined on the space $E\times X$, where is $E$ is the subset of $L^2(γ)$ consisting of probability densities. The equation has the form $L_{b(p,\bullet)} ^*(p\cdot γ)=0$ with $L_{b(p,\bullet)}\varphi =Δ_H \varphi + (b(p,\bullet) , D_{_H}\varphi)_{_H}$, 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.