Regularity Estimates for Singular Density Dependent SDEs

TL;DR

This work develops regularity and entropy estimates for singular density-dependent SDEs of Nemytskii type, where the drift can be highly singular in the distribution density. It proves well-posedness and a super-continuity property for the evolution of time-marginal laws in $\tt L^k$-type metrics, using a fixed-point strategy with frozen-density SDEs. The authors derive explicit relative-entropy and Renyi-entropy bounds between time marginals, articulating how initial Wasserstein distances and density-norms control entropy growth, including sharp small-time rates and 1D simplifications. A refined Khasminskii estimate underpins these results, providing exponential moment bounds for singular drift functionals and enabling the entropy comparisons that extend entropy-cost-type inequalities to singular density-dependent dynamics.

Abstract

Consider the density dependent (i.e. Nemytskii-type) SDEs on $\mathbb R^d$, where the drift $b_t(x,ρ(x),ρ)$ is locally integrable in $(t,x)\in [0,\infty)\times \mathbb R^d$ and may be singular in the distribution density function $ρ$. 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.

Theorems & Definitions (13)

• Definition 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['T1']}
• Theorem 3.1
• proof
• Theorem 4.1