Electronic Journal of Probability

Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise

Raluca M. Balan, Lluís Quer-Sardanyons, and Jian Song

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In this article, we consider the stochastic wave equation on $\mathbb{R} _{+} \times \mathbb{R} $, driven by a linear multiplicative space-time homogeneous Gaussian noise whose temporal and spatial covariance structures are given by locally integrable functions $\gamma $ (in time) and $f$ (in space), which are the Fourier transforms of tempered measures $\nu $ on $\mathbb{R} $, respectively $\mu $ on $\mathbb{R} $. Our main result shows that the law of the solution $u(t,x)$ of this equation is absolutely continuous with respect to the Lebesgue measure.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 106, 43 pp.

Received: 17 May 2018
Accepted: 18 September 2019
First available in Project Euclid: 1 October 2019

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Primary: Primary 60H15 60H07: Stochastic calculus of variations and the Malliavin calculus

Gaussian noise stochastic partial differential equations Malliavin calculus

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Balan, Raluca M.; Quer-Sardanyons, Lluís; Song, Jian. Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise. Electron. J. Probab. 24 (2019), paper no. 106, 43 pp. doi:10.1214/19-EJP363. https://projecteuclid.org/euclid.ejp/1569895475

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