Hölder Continuity for the Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise
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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 Lp (Ω). Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, 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 [6, 10] under more restrictive conditions, and with sub-optimal exponents for Hölder continuity.
Key wordsGaussian noise stochastic partial differential equations Malliavin calculus
2010 MR Subject Classification60H15 60H07
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