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 R + × 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 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 ≥ 2. In addition, we show that this solution has a Hölder 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.MSC 2010: Primary 60H15; Secondary 60H07, 37H15.

Hölder Continuity for the Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise

“…By Lemma 3.8 of [1], we know that under Dalang's condition, H(t; γ) < ∞ and H(t; γ) < ∞ for any t > 0 and γ > 0. The following result is a particular case of Lemma 3.4 of [1].…”

“…In this section, we show the existence and uniqueness of the solution to equation (1) and its continuity in L p (Ω) on [0, ∞) × R d , using the method of [1] for the heat equation with initial condition given by a measure. We note that the existence of solution to equation (1) was established also in Theorem 3.2 of [5], using a different method.…”

“…Definition 1.1. A square-integrable process u = {u(t, x); t ≥ 0, x ∈ R d } with u(0, x) = 1 for all x ∈ R d is a (mild Skorohod) solution of equation (1) if u has a jointly measurable modification (denoted also by u) such that sup (t,x)∈[0,T ]×R d E|u(t, x)| 2 < ∞ for all T > 0, and for any t > 0 and x ∈ R d , the following equality holds in L 2 (Ω):…”

“…We refer the reader to [5,2,1] for more details regarding the concept of Skorohod solution, and to [7] for the background on Malliavin calculus.…”

“…Our proofs use simplified versions of the arguments contained in [1] for the heat equation with initial condition given by a measure. But the results that we present here do not follow directly from Theorems 1.1.…”