Spatial averages for the Parabolic Anderson model driven by rough noise

Abstract

In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters $H_0$ in time and $H_1$ in space, satisfying $H_0\in (1/2,1)$, $H_1\in (0,1/2)$ and $H_0+H_1>3/4$. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.

References in corpus (5)

• Central limit theorems for sequences of multiple stochastic integrals
• Renormalized self-intersection local time for fractional Brownian motion
• Gaussian fluctuations for the stochastic heat equation with colored noise
• Averaging Gaussian functionals
• Spatial ergodicity of stochastic wave equations in dimensions 1,2 and 3

Cited by in corpus (4)

• Averaging Gaussian functionals
• The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications
• Quantitative central limit theorems for the parabolic Anderson model driven by colored noises
• Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise