Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
- König, Wolfgang
- Perkowski, Nicolas
- van Zuijlen, Willem
2010 Mathematics Subject Classification
- 60H17 60H25 60L40 82B4 35J10 35P15
- Parabolic Anderson model, Anderson Hamiltonian, white-noise potential, singular SPDE, paracontrolled distribution, regularization in two dimensions, intermittency, almost-sure large-time asymptotics, principal eigenvalue of random Schrödinger operator
We consider the parabolic Anderson model (PAM) in ℝ ² with a Gaussian (space) white-noise potential. We prove that the almost-sure large-time asymptotic behaviour of the total mass at time t is given asymptotically by Χ t log 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 principal Dirichlet of the eigenvalue the Anderson operator on the t by t box around zero asymptotically by Χ log t.