## Bernoulli

• Bernoulli
• Volume 5, Number 2 (1999), 275-298.

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

Pierre-Luc Morien

#### Abstract

In this paper we prove that the density $p t ,x (y)$ of the solution of a white-noise-driven parabolic stochastic partial differential equation (SPDE) satisfying a strong ellipticity condition is $1 2$ Lipschitz continuous with respect to (w.r.t.) $t$ and $1 -ε$ Lipschitz continuous w.r.t. $x$ for all $ε ∈]0,1[$. In addition, we show that it belongs to the Besov space $B 1 ,∞,∞$ 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.

#### Article information

Source
Bernoulli, Volume 5, Number 2 (1999), 275-298.

Dates
First available in Project Euclid: 5 March 2007

https://projecteuclid.org/euclid.bj/1173147907

Mathematical Reviews number (MathSciNet)
MR1681699

Zentralblatt MATH identifier
0932.60072

#### Citation

Morien, Pierre-Luc. The Hölder and the Besov regularity of the density for the solution of a parabolic stochastic partial differential equation. Bernoulli 5 (1999), no. 2, 275--298. https://projecteuclid.org/euclid.bj/1173147907

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