2025/11/28 by Bogachev, Vladimir I., Röckner, Michael, Shaposhnikov, Stanislav V.
Mathematics · Physics and Astronomy · #Gas Dynamics and Kinetic Theory #Statistical Mechanics and Entropy #Mathematical Biology Tumor Growth
paper · doi:10.48550/arxiv.2511.23058
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 L2(γ) consisting of probability densities. The equation has the form Lb(p,\bullet) ^*(p⋅ γ)=0 with Lb(p,\bullet)φ=ΔH φ+ (b(p,\bullet) , DHφ)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.