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

TL;DR

This paper develops a method to construct weak solutions for a class of distribution-dependent stochastic differential equations with possibly degenerate diffusion, by solving associated nonlinear Fokker-Planck equations and applying the superposition principle.

Contribution

It introduces a novel approach combining nonlinear Fokker-Planck equations and superposition to solve distribution-dependent SDEs with degenerate diffusion matrices.

Findings

Successfully constructs weak solutions for a broad class of distribution-dependent SDEs.

Extends existing methods to handle degenerate diffusion matrices.

Provides a framework linking nonlinear Fokker-Planck equations to SDE solutions.

Abstract

We construct weak solutions to a class of distribution dependent SDE, of type $dX(t)=b\left( X(t), \displaystyle\frac{d\mathcal{L}_{X(t)}}{dx}(X(t))\right) dt+\sigma\left( X(t),\displaystyle\frac{d\mathcal{L}_{X(t)}}{dt}(X(t))\right) dW(t)$ 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.