Regularity Estimates for Singular Density Dependent SDEs

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.

💡 Research Summary

This paper investigates a class of stochastic differential equations (SDEs) on ℝⁿ whose drift depends pointwise on the probability density of the solution itself – a Nemytskii‑type, density‑dependent SDE. Unlike the classical McKean‑Vlasov framework, where the coefficients depend on global distributional functionals (e.g., expectations), here the drift bₜ(x, r, ρ) may be singular both in the space‑time variables (t, x) and in the density argument ρ. The authors allow bₜ to be merely locally integrable in (t, x) and to exhibit strong singularities with respect to ρ, measured by a family of localized L̃ᵏ‑norms. The diffusion matrix σₜ(x) is allowed to be non‑uniform but satisfies uniform ellipticity and Hölder continuity of its covariance aₜ = σₜσₜᵗ.

The main contributions are threefold:

1. Well‑posedness and super‑continuity – By freezing the density path γ(t) and solving the resulting SDE with a fixed density, the authors construct a map Φ_μ that sends a candidate density trajectory to the law of the corresponding solution. Using a Banach fixed‑point argument together with refined heat‑kernel estimates for the reference diffusion (with drift b^{(1)}ₜ) and a Duhamel‑type representation, they prove existence and uniqueness of strong (or weak) solutions for any initial law μ belonging to a suitable L̃ᵖ‑class. Moreover, they establish a super‑continuity estimate (2.4)–(2.10) showing that for any p ≤ k, \