Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
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| Format: | Preprint |
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2020
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| _version_ | 1866916007299252224 |
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| author | König, Wolfgang Perkowski, Nicolas van Zuijlen, Willem |
| author_facet | König, Wolfgang Perkowski, Nicolas van Zuijlen, Willem |
| contents | We consider the parabolic Anderson model (PAM) $\partial_t u = \frac12 Δu + ξu$ in $\mathbb R^2$ 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 $\log U(t)\sim χt \log t$ for $t \to \infty$, 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 $\boldsymbol λ_1(Q_t)\simχ\log t$ of the principal eigenvalue $\boldsymbolλ_1(Q_t)$ of the Anderson operator with Dirichlet boundary conditions on the box $Q_t= [-\frac{t}{2},\frac{t}{2}]^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_11611 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential König, Wolfgang Perkowski, Nicolas van Zuijlen, Willem Probability We consider the parabolic Anderson model (PAM) $\partial_t u = \frac12 Δu + ξu$ in $\mathbb R^2$ 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 $\log U(t)\sim χt \log t$ for $t \to \infty$, 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 $\boldsymbol λ_1(Q_t)\simχ\log t$ of the principal eigenvalue $\boldsymbolλ_1(Q_t)$ of the Anderson operator with Dirichlet boundary conditions on the box $Q_t= [-\frac{t}{2},\frac{t}{2}]^2$. |
| title | Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential |
| topic | Probability |
| url | https://arxiv.org/abs/2009.11611 |