Abstract
We consider the parabolic Anderson problem $\partial_{t}u = \kappa\Delta u + \xi(x)u$ on $\mathbb{R}_+ \times \mathbb{R}^d$ with initial condition $u(0, x) = 1$. Here $\xi(\cdot)$ is a random shift-invariant potential having high $\delta$-like peaks on small islands. We express the second-order asymptotics of the $p$th moment $(p \in [1, \infty))$ of $u(t,0)$ as $t \to \infty$ in terms of a variational formula involving an asymptotic description of the rescaled shapes of these peaks via their cumulant generating function. This includes Gaussian potentials and high Poisson clouds.
Citation
Jürgen Gärtner. Wolfgang König. "Moment asymptotics for the continuous parabolic Anderson model." Ann. Appl. Probab. 10 (1) 192 - 217, February 2000. https://doi.org/10.1214/aoap/1019737669
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