The Hölder and the Besov regularity of the density for the solution of a parabolic stochastic partial differential equation

Abstract

In this paper we prove that the density $p_{t,x}\left(y\right)$ of the solution of a white-noise-driven parabolic stochastic partial differential equation (SPDE) satisfying a strong ellipticity condition is $\frac{1}{2}$ Lipschitz continuous with respect to (w.r.t.) $t$ and $1-\epsilon$ Lipschitz continuous w.r.t. $x$ for all $\epsilon \in ]0,1[$. In addition, we show that it belongs to the Besov space $B_{1,\mathrm{\infty },\mathrm{\infty }}$ w.r.t. $x$. The proof is based on the Malliavin calculus of variations and on some refined estimates for the Green kernel associated with the SPDE.