Lluís Quer-Sardanyons

Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation

Abstract We present new results regarding the existence of density of the real-valued solution to a 3-dimensional stochastic wave equation. The noise is white in time and with a spatially homogeneous correlation whose spectral measure μ satisfies that ∫ R 3 μ(dξ)(1+|ξ| 2 ) −η , for some η∈(0, 1 2 ) . Our approach is based on the mild formulation of the equation given by means of Dalangs extended version of Walshs stochastic integration; we use the tools of Malliavin calculus. Let S 3 be the fundamental solution to the 3-dimensional wave equation. The assumption on the noise yields upper and lower bounds for the integral ∫ 0 t ds ∫ R 3 μ(dξ)| F S 3 (s)(ξ)| 2 and upper bounds for ∫ 0 t ds ∫ R 3 μ(dξ)|ξ|| F S 3 (s)(ξ)| 2 in terms of powers of t . These estimates are crucial in the analysis of the Malliavin variance, which can be done by a comparison procedure with respect to smooth approximations of the distribution-valued function S 3 ( t ) obtained by convolution with an approximation of the identity.

OPTIMAL GAUSSIAN DENSITY ESTIMATES FOR A CLASS OF STOCHASTIC EQUATIONS WITH ADDITIVE NOISE

In this note, we establish optimal lower and upper Gaussian bounds for the density of the solution to a class of stochastic integral equations driven by an additive spatially homogeneous Gaussian random field. The proof is based on the techniques of the Malliavin calculus and a density formula obtained by Nourdin and Viens. Then, the main result is applied to the mild solution of a general class of SPDEs driven by a Gaussian noise which is white in time and has a spatially homogeneous correlation. In particular, this covers the case of the stochastic heat and wave equations in

Existence of Weak Solutions for a Class of Semilinear Stochastic Wave Equations

\mathbb{R}^d

Gaussian Upper Density Estimates for Spatially Homogeneous SPDEs

with

Stochastic integrals for spde"s: A comparison

d\geq 1

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

and

The 1-d stochastic wave equation driven by a fractional Brownian sheet

d\leq 3

A stochastic wave equation in dimension 3: smoothness of the law

, respectively. The upper and lower Gaussian bounds have the same form and are given in terms of the variance of the stochastic integral term in the mild form of the equation.

Space Semi-Discretisations for a Stochastic Wave Equation

We prove existence of weak solutions (in the probabilistic sense) for a general class of stochastic semilinear wave equations on bounded domains of

Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations

R^d