Related papers: A scaling limit theorem for the parabolic Anderson…

The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically…

The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_tu(t,z)=\Delta u(t,z)+\xi(z)u(t,z)$ on $(0,\infty)\times {\mathbb{Z}}^d$ with random potential $(\xi(z):z\in{\mathbb{Z}}^d)$. We consider independent and…

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 $(\xi(z) \colon z\in {\mathbb Z}^d)$. We consider…

The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential $\xi$. We consider the case when $\{\xi(z):z\in\mathbb{Z}^d\}$ is a collection of independent identically distributed…

This is a survey on the intermittent behavior of the parabolic {Anderson} model, which is the Cauchy problem for the heat equation with random potential on the lattice $\Z^d$. We first introduce the model and give heuristic explanations of…

We study the parabolic Anderson problem, that is, the heat equation $\partial_tu=\Delta u+\xi u$ on $(0,\infty)\times{\mathbb{Z}}^d$ with independent identically distributed random potential $\{\xi(z):z\in{\mathbb{Z}}^d\}$ and localized…

We consider the solution to the parabolic Anderson model with homogeneous initial condition in large time-dependent boxes. We derive stable limit theorems, ranging over all possible scaling parameters, for the rescaled sum over the solution…

We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in $Z^d$. We use i.i.d. potentials $\xi: Z^d \to \R$ in the third universality class, namely the class of almost bounded potentials, in…

We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on $\Z^d$. We consider general i.i.d. potentials and show that exactly \emph{four} qualitatively different…

In [1] a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton-Watson random tree with an i.i.d. random potential whose marginal distribution is…

The parabolic Anderson problem with a random potential obtained by attaching a long tailed potential around a randomly perturbed lattice is studied. The moment asymptotics of the total mass of the solution is derived.The results show that…

The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that…

We establish the exact quenched asymptotic growth of the solution to the parabolic Anderson model (PAM) in the hyperbolic space with a regular, stationary, time-independent Gaussian potential. More precisely, we show that with probability…

We study positive solutions of the pseudoparabolic equation with a sublinear source in $\mathbb{R}^n$. In this work, the source coefficient could be unbounded and time-dependent. Global existence of solutions to the Cauchy problem is…

We study the solutions $u=u(x,t)$ to the Cauchy problem on $\mathbb Z^d\times(0,\infty)$ for the parabolic equation $\partial_t u=\Delta u+\xi u$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $\Delta$ is the discrete Laplacian on $\mathbb…

We consider the parabolic Anderson model (PAM) which is given by the equation $\partial u/\partial t = \kappa\Delta u + \xi u$ with $u\colon\, \Z^d\times [0,\infty)\to \R$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$…

We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, roughly speaking of the type of the exponential of a power of the distance from the origin. We…

We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a…

This paper is concerned with the hypercoercivity property of solutions to the Cauchy problem on the linear Boltzmann equation with a confining potential force. We obtain the exponential time rate of solutions converging to the steady state…

We continue our study of the parabolic Anderson equation $\partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t)$, $x\in\Z^d$, $t\geq 0$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete…