Linearization of Nonlinear Fokker-Planck Equations and Applications

Abstract: We associate a coupled nonlinear Fokker-Planck equation on $\R^d$, i.e. with solution paths in $\scr P$, to a linear Fokker-Planck equation for probability measures on the product space $\R^d\times \scr P$, i.e. with solution paths in $\scr P(\R^d\times\scr P)$. We explicitly determine the corresponding linear Kolmogorov operator $\tilde{\mathbf{L}}_t$ using the natural tangent bundle over $\scr P$ with corresponding gradient operator $\nabla^\scr P$. Then it is proved that the diffusion process generated by $\tilde{\mathbf{L}}_t$ on $\R^d\times\scr P$ is intrinsically related to the solution of a McKean-Vlasov stochastic differential equation (SDE). We also characterize the ergodicity of the diffusion process generated by $\tilde{\mathbf{L}}_t$ in terms of asymptotic properties of the coupled nonlinear Fokker-Planck equation. Another main result of the paper is that the restricted well-posedness of the non-linear Fokker-Planck equation and its linearized version imply the (restricted) well-posedness of the McKean-Vlasov equation and that in this case the laws of the solutions have the Markov property. All this is done under merely measurebility conditions on the coefficients in their measure dependence, hence in particular applies if the latter is of "Nemytskii-type". As a consequence, we obtain the restricted weak well-posedness and the Markov property of the so-called nonlinear distorted Brownian motion, whose associated nonlinear Fokker-Planck equation is a porous media equation perturbed by a nonlinear transport term. This realizes a programme put forward by McKean in his seminal paper of 1966 for a large class of nonlinear PDEs. As a further application we obtain a probabilistic representation of solutions to Schr\"odinger type PDEs on $\R^d\times\scr P_2$, through the Feynman-Kac formula for the corresponding diffusion processes.