Parabolic Anderson model with rough dependence in space

Abstract

This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$ in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the $n$th moment of the solution.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1505.04924

References in corpus (2)

• Feynman-Kac formula for heat equation driven by fractional white noise
• The Continuum Directed Random Polymer