On a stochastic wave equation in two space dimensions: regularity of the solution and its density

We pursue the investigation started in a recent paper by Millet and Sanz-Solé (1999, Ann. Probab. 27, 803-844) concerning a non-linear wave equation driven by a Gaussian white noise in time and correlated in the two-dimensional space variable. Under more restrictive conditions on the covariance function of the noise, we prove Hölder-regularity properties for both the solution and its density. For the latter, we adapt the method used in a paper by Morien (1999, Bernoulli: Official J. Bernoulli Soc. 5(2), 275-298) based on the Malliavin calculus.