From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE

Abstract

We construct weak solutions to a class of distribution dependent SDE, of type $dX(t)=b\left(X(t),\frac{d{\mathcal{L}}_{X(t)}}{dx}(X(t))\right)dt+\sigma \left(X(t),\frac{d{\mathcal{L}}_{X(t)}}{dt}(X(t))\right)dW(t)$ for possibly degenerate diffusion matrices $\sigma$ with $X(0)$ having a given law, which has a density with respect to Lebesgue measure, $dx$. Here ${\mathcal{L}}_{X(t)}$ denotes the law of $X(t)$. Our approach is to first solve the corresponding nonlinear Fokker-Planck equations and then use the well known superposition principle to obtain weak solutions of the above SDE.

Cited by in corpus (1)

• Stochastic nonlinear Fokker-Planck equations