Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise and Rough Initial Condition

Abstract

In this article, we study the Parabolic Anderson Model driven by a space-time homogeneous Gaussian noise on $\mathbb{R}_{+} \times \mathbb{R}^d$, whose covariance kernels in space and time are locally integrable non-negative functions, which are non-negative definite (in the sense of distributions). We assume that the initial condition is given by a signed Borel measure on $\mathbb{R}^d$, and the spectral measure of the noise satisfies Dalang's (1999) condition. Under these conditions, we prove that this equation has a unique solution, and we investigate the magnitude of the $p$-th moments of the solution, for any $p \geq 2$. In addition, we show that this solution has a H\"older continuous modification with the same regularity and under the same condition as in the case of the white noise in time, regardless of the temporal covariance function of the noise.