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

Abstract

We consider the parabolic Anderson model (PAM) ${\partial }_{t}u=\frac{1}{2}\Delta u+\xi u$ in ${\mathbb{R}}^2$ with a Gaussian (space) white-noise potential $\xi$. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time $t$, written $U(t)$, is given by $\log U(t)\sim \chi t\log t$ for $t\to \infty$, with the deterministic constant $\chi$ identified in terms of a variational formula. In earlier work of one of the authors this constant was used to describe the asymptotic behaviour ${\mathbf{\lambda }}_1(Q_t)\sim \chi \log t$ of the principal eigenvalue ${\mathbf{\lambda }}_1(Q_t)$ of the Anderson operator with Dirichlet boundary conditions on the box $Q_t=[-\frac{t}{2},\frac{t}{2}{]}^2$.

References in corpus (10)

• A theory of regularity structures
• KPZ reloaded
• A simple construction of the continuum parabolic Anderson model on ${\mathbf{R}}^2$
• The continuous Anderson hamiltonian in $d\le 3$
• Quenched asymptotics for Brownian motion in generalized Gaussian potential
• Localization of the continuous Anderson Hamiltonian in $1$-d
• From Extreme Values of I.I.D. Random Fields to Extreme Eigenvalues of Finite-volume Anderson Hamiltonian
• Quantitative heat kernel estimates for diffusions with distributional drift
• Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails
• Phase Transitions in Asymptotically Singular Anderson Hamiltonian and Parabolic Model

Cited by in corpus (2)

• Integrated density of states of the Anderson Hamiltonian with two-dimensional white noise
• Phase Transitions in Asymptotically Singular Anderson Hamiltonian and Parabolic Model