H ̈older continuity for the Parabolic Anderson Modelwith space-time homogeneous Gaussian noise

In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang's condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in $L^p(\Omega)$. Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, with optimal exponents, and under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time). This improves similar results which were obtained in Hu, Huang, Nualart and Tindel (2015), and Song (2017) under more restrictive conditions, and with sub-optimal exponents for Hölder continuity.

Hölder continuity for the Parabolic Anderson Model with space-time homogeneous Gaussian noise arXiv:1807.05420v1 [math.PR] 14 Jul 2018 Raluca M. Balan∗ Lluı́s Quer-Sardanyons† Jian Song‡ July 15, 2018 Abstract In this article, we consider…

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