Complete localisation in the parabolic Anderson model with Pareto-distributed potential

Abstract

The parabolic Anderson problem is the Cauchy problem for the heat equation ${\partial }_{t}u(t,z)=\Delta u(t,z)+\xi (z)u(t,z)$ on $(0,\infty )\times {\mathbb{Z}}^d$ with random potential $(\xi (z):z\in {\mathbb{Z}}^d)$. We consider independent and identically distributed potential variables, such that Prob$(\xi (z)>x)$ 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 $(Z_t:t\ge 0)$ with values in ${\mathbb{Z}}^d$ such that ${\lim }_{t\uparrow \infty }u(t,Z_t)/{\sum }_{z\in {\mathbb{Z}}^d}u(t,z)=1,$ in probability. We also identify the asymptotic behaviour of $Z_t$ in terms of a weak limit theorem.

20 pages

References in corpus (1)

• Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials