Smoothness of the density for McKean-Vlasov SDEs with measurable kernel

TL;DR

This paper proves that the probability density of solutions to certain McKean-Vlasov SDEs becomes smoother over time, with regularity depending on the kernel's properties and noise type, surpassing classical SDE regularity.

Contribution

It establishes new regularity results for densities of McKean-Vlasov SDEs with measurable and distributional kernels, including cases with stable noise, showing enhanced smoothness compared to standard SDEs.

Findings

Density is continuously differentiable with Hölder continuous gradient for bounded measurable kernels.

Density becomes infinitely differentiable when the kernel is continuously differentiable.

Similar smoothing phenomena occur for singular kernels and stable noise processes.

Abstract

Consider the McKean-Vlasov SDE $$ dX_t=\langle b(X_t-\cdot),\mu_t\rangle dt+dW_t,\quad \mu_t=\operatorname{Law}(X_t), $$ where $W$ is the $n$-dimensional Brownian motion and $b:\mathbb{R}^d\to\mathbb{R}^d$ is a measurable function. First assuming $b\in L^\infty$, we prove that the law $\mu_t$ of $X_t$ has a density $p_t$ with respect to the Lebesgue measure, which is continuously differentiable with gradient being $\gamma$-H\"older continuous for each $\gamma\in(0,1)$. Assume further that $b\in \mathcal{C}_b^1$, we prove that the density $p_t$ is infinitely differentiable. In the regularization by noise perspective, this shows McKean-Vlasov SDEs tend to have a smoother density function than SDEs without density dependence, under the same regularity assumption of the coefficients. We observe similar phenomenon for singular interaction kernels satisfying Krylov's integrability condition, for distributional kernels $b\in B_{\infty,\infty}^\alpha$, $\alpha\in(-1,0)$, and for processes driven by an $\alpha$-stable noise for $\alpha\in(1,2)$.