On the density of singular SDEs with fractional noise and applications to McKean-Vlasov equations

TL;DR

This paper studies the properties of solutions to stochastic differential equations driven by fractional Brownian noise with singular drifts, focusing on the regularity and density of their laws, and applies these findings to McKean-Vlasov equations.

Contribution

It provides new insights into the regularity and density of the law of solutions to fractional SDEs with singular drifts, and establishes existence and uniqueness results for McKean-Vlasov equations.

Findings

Law of solutions has Gaussian tails.

Quantified spatial regularity of the law.

Proved existence and uniqueness for McKean-Vlasov equations.

Abstract

We investigate properties of the (conditional) law of the solution to SDEs driven by fractional Brownian noise with a singular, possibly distributional, drift. Our results on the law are twofold: i) we quantify the spatial regularity of the law, while keeping track of integrability in time, and ii) we prove that it has a density with Gaussian tails. Then the former result is used to establish novel results on existence and uniqueness of solutions to McKean-Vlasov equations of convolutional type.