Solvability of Parabolic Anderson Equation with Fractional Gaussian Noise

Abstract

This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model $\frac{\partial u}{\partial t}=\frac{1}{2}\Delta +u{\dot{W}}$ on $[0, \infty )\times {{\mathbb {R}}}^d $ with $d\ge 1$ has a unique random field solution, where W(t, x) is a fractional Brownian sheet on $[0, \infty )\times {{\mathbb {R}}}^d$ and formally $\dot{W} =\frac{\partial ^{d+1}}{\partial t \partial x_1 \cdots \partial x_d} W(t, x)$. When the noise W(t, x) is white in time, our condition is both necessary and sufficient when the initial data u(0, x) is bounded between two positive constants. When the noise is fractional in time with Hurst parameter $H_0>1/2$, our sufficient condition, which improves the known results in the literature, is different from the necessary one.

Keywords

Stochastic heat equation / Fractional Brownian fields / Wiener chaos expansion / Random field solution / Necessary condition / sufficient condition / Moment bounds

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Funding

Simons Foundation

Canadian Network for Research and Innovation in Machining Technology

Natural Sciences and Engineering Research Council of Canada