Regularity Estimates for Singular Density Dependent SDEs

TL;DR

This paper develops regularity estimates for singular density-dependent SDEs, linking entropy measures with Wasserstein distances, and introduces a refined Khasminskii estimate for such processes.

Contribution

It provides new entropy estimates for singular SDEs and a refined Khasminskii estimate, extending classical inequalities to more singular and density-dependent cases.

Findings

Relative entropy estimates relate to Wasserstein distances of initial distributions.

In one dimension, entropy estimates match classical entropy-cost inequalities.

Introduces a refined Khasminskii estimate for singular SDEs.

Abstract

Consider the density dependent (i.e. Nemytskii-type) SDEs on $\mathbb R^d$, where the drift $b_t(x,\rho(x),\rho)$ is locally integrable in $(t,x)\in [0,\infty)\times \mathbb R^d$ and may be singular in the distribution density function $\rho$. The relative/Renyi entropies between two time-marginal distributions are estimated by using the Wasserstein distance of initial distributions. When $d=1$ and $b_t$ decays at $t=0$ with rate $t^{\frac 1 2+}$, our the relative entropy estimate coincides with the classical entropy-cost inequality for elliptic diffusion processes. To estimate the Renyi entropy, a refined Khasminskii estimate is presented for singular SDEs which may be interesting by itself.