Infinite-dimensional nonlinear stationary Fokker–Planck–Kolmogorov equations

Abstract

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 × X , where is E is the subset of L 2 ( γ ) consisting of probability densities. The equation has the form L b ( p , ⋅ ) * ( p ⋅ γ ) = 0 ${L}_{b\left(p,\cdot \right)}^{{\ast}}\left(p\cdot \gamma \right)=0$ with L b ( p , ⋅ ) φ = Δ H φ + ( b ( p , ⋅ ) , D H φ ) H ${L}_{b\left(p,\cdot \right)}\varphi ={{\Delta}}_{H}\varphi +{\left(b\left(p,\cdot \right),{D}_{H}\varphi \right)}_{H}$ , 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.