Complete localisation in the parabolic Anderson model with Pareto-distributed potential
| dc.creator | Konig, Wolfgang | |
| dc.creator | Morters, Peter | |
| dc.creator | Sidorova, Nadia | |
| dc.date | 2006-08-22 | |
| dc.date.accessioned | 2026-07-07T07:22:02Z | |
| dc.date.available | 2026-07-07T07:22:02Z | |
| dc.description | The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_t u(t,z)=Δu(t,z)+ξ(z) u(t,z)$ on $(0,\infty)\times {\mathbb Z}^d$ with random potential $(ξ(z) \colon z\in {\mathbb Z}^d)$. We consider independent and identically distributed potential variables, such that Prob$(ξ(z)>x)$ decays polynomially as $x\uparrow\infty$. If $u$ is initially localised in the origin, i.e. if $u(0,x)=\one_0(x)$, we show that, at any large time $t$, the solution is completely localised in a single point with high probability. More precisely, we find a random process $(Z_t \colon t\ge 0)$ with values in $\Z^d$ such that $\lim_{t \uparrow\infty} u(t,Z_t)/\sum_{z\in\Z^d} u(t,z) =1,$ in probability. We also identify the asymptotic behaviour of $Z_t$ in terms of a weak limit theorem. | |
| dc.description | 20 pages | |
| dc.identifier | https://arxiv.org/abs/math/0608544 | |
| dc.identifier | http://arxiv.org/abs/math/0608544 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/115514 | |
| dc.subject | Probability | |
| dc.subject | 60H25; 82C44; 60F10 | |
| dc.title | Complete localisation in the parabolic Anderson model with Pareto-distributed potential | |
| dc.type | text |