Local H\"{o}lder continuity property of the Densities of Solutions of SDEs with Singular Coefficients

TL;DR

This paper proves that solutions to certain elliptic SDEs with smooth diffusion and H"older continuous drift have H"older continuous densities, highlighting the influence of drift regularity on density smoothness.

Contribution

It establishes the H"older continuity of the density for SDE solutions with locally smooth diffusion and H"older continuous drift, extending previous results.

Findings

Density of solutions is H"older continuous under specified conditions

Highlights the role of drift regularity in density smoothness

Complements existing literature on density existence for SDEs

Abstract

We prove that the weak solution of a uniformly elliptic stochastic differential equation with locally smooth diffusion coefficient and H\"{o}lder continuous drift has a H\"{o}lder continuous density function. This result complements recent results of Fournier-Printems \cite{F1}, where the density is shown to exist if both coefficients are H\"{o}lder continuous and exemplifies the role of the drift coefficient in the regularity of the density of a diffusion.