## The Annals of Probability

- Ann. Probab.
- Volume 27, Number 2 (1999), 803-844.

### A Stochastic Wave Equation in Two Space Dimension: Smoothness of the Law

Annie Millet and Marta Sanz-Solé

#### Abstract

We prove the existence and uniqueness, for any time, of a real-valued process solving a nonlinear stochastic wave equation driven by a Gaussian noise white in time and correlated in the two-dimensional space variable. We prove that the solution is regular in the sense of the Malliavin calculus. We also give a decay condition on the covariance function of the noise under which the solution has Hölder continuous trajectories and show that, under an additional ellipticity assumption, the law of the solution at any strictly positive time has a smooth density.

#### Article information

**Source**

Ann. Probab. Volume 27, Number 2 (1999), 803-844.

**Dates**

First available in Project Euclid: 29 May 2002

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1022677387

**Digital Object Identifier**

doi:10.1214/aop/1022677387

**Mathematical Reviews number (MathSciNet)**

MR1698971

**Zentralblatt MATH identifier**

0944.60067

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60]

Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

**Keywords**

Stochastic partial differential equation wave equation Gaussian noise Malliavin calculus existence and smoothness of the density

#### Citation

Millet, Annie; Sanz-Solé, Marta. A Stochastic Wave Equation in Two Space Dimension: Smoothness of the Law. Ann. Probab. 27 (1999), no. 2, 803--844. doi:10.1214/aop/1022677387. http://projecteuclid.org/euclid.aop/1022677387.