Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential

2020/09/24 by König, Wolfgang, Perkowski, Nicolas, van Zuijlen, Willem · 4 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2009.11611

Abstract

We consider the parabolic Anderson model (PAM) ∂_t u = \frac12 Δu + ξu in \mathbb R² with a Gaussian (space) white-noise potential ξ. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t, written U(t), is given by log U(t)∼ χt log t for t → ∞, with the deterministic constant χ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour \boldsymbol λ₁(Q_t)∼χlog t of the principal eigenvalue \boldsymbolλ₁(Q_t) of the Anderson operator with Dirichlet boundary conditions on the box Q_t= [-(t)/(2),(t)/(2)]².

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