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.
Publié le : 1999-04-15
Classification:
Stochastic partial differential equation,
wave equation,
Gaussian noise,
Malliavin calculus,
existence and smoothness of the density,
60H15,
60H07
@article{1022677387,
author = {Millet, Annie and Sanz-Sol\'e, Marta},
title = {A Stochastic Wave Equation in Two Space Dimension: Smoothness of
the Law},
journal = {Ann. Probab.},
volume = {27},
number = {1},
year = {1999},
pages = { 803-844},
language = {en},
url = {http://dml.mathdoc.fr/item/1022677387}
}
Millet, Annie; Sanz-Solé, Marta. A Stochastic Wave Equation in Two Space Dimension: Smoothness of
the Law. Ann. Probab., Tome 27 (1999) no. 1, pp. 803-844. http://gdmltest.u-ga.fr/item/1022677387/