# Precise Asymptotics for the Parabolic Anderson Model with a Moving Catalyst or Trap

@article{Schnitzler2012PreciseAF, title={Precise Asymptotics for the Parabolic Anderson Model with a Moving Catalyst or Trap}, author={Adrian Schnitzler and Tilman Wolff}, journal={arXiv: Probability}, year={2012}, pages={69-89} }

We consider the solution \(u: [0,\infty ) \times{\mathbb{Z}}^{d} \rightarrow[0,\infty )\) to the parabolic Anderson model, where the potential is given by \((t,x)\mapsto \gamma {\delta }_{{Y }_{t}}\left (x\right )\) with Y a simple symmetric random walk on \({\mathbb{Z}}^{d}\). Depending on the parameter γ∈[−∞,∞), the potential is interpreted as a randomly moving catalyst or trap.

## 16 Citations

Parabolic Anderson model with a finite number of moving catalysts

- Mathematics
- 2010

We consider the parabolic Anderson model (PAM) which is given by the equation ∂u∕∂t=κΔu+ξu with \(u: \,{\mathbb{Z}}^{d} \times[0,\infty ) \rightarrow\mathbb{R}\), where κ∈[0,∞) is the diffusion…

Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment

- Mathematics
- 2010

We continue our study of the parabolic Anderson equation ∂u ∕ ∂t = κΔu + γξu for the space–time field \(u: \,{\mathbb{Z}}^{d} \times [0,\infty ) \rightarrow \mathbb{R}\), where κ ∈ [0, ∞) is the…

The Parabolic Anderson Model

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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 ℤd. We first introduce the model…

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- 2016

In this chapter, we explain what the asymptotics of the logarithm of the moments of the total mass U(t) of the solution u(t, ⋅ ) of the PAM in ( 1.1)– ( 1.2) are determined by, and how they can be…

The parabolic Anderson model and long-range percolation

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- 2014

This thesis has two parts. The first part deals with the parabolic Anderson model, which is a stochastic differential equation. It models the evolution of a field of particles performing independent…

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- Mathematics
- 2011

The parabolic Anderson model is the heat equation on the lattice with a random potential. A characteristic feature of the large time asymptotics of the solution is the occurrence of small islands…

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- MathematicsSpringer Proceedings in Mathematics & Statistics
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There have been extensive studies of a random walk among a field of immobile traps (or obstacles), where one is interested in the probability of survival as well as the law of the random walk…

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- 2016

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The Parabolic Anderson Model with Long Range Basic Hamiltonian and Weibull Type Random Potential

- Mathematics
- 2012

We study the quenched and annealed asymptotics for the solutions of the lattice parabolic Anderson problem in the situation in which the underlying random walk has long jumps and belongs to the…

Time-Dependent Potentials

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Of fundamental importance is the parabolic Anderson model in (1.1) also if the random potential is allowed to be time-dependent. Here we consider the Cauchy problem

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We consider the parabolic Anderson model (PAM) which is given by the equation ∂u∕∂t=κΔu+ξu with \(u: \,{\mathbb{Z}}^{d} \times[0,\infty ) \rightarrow\mathbb{R}\), where κ∈[0,∞) is the diffusion…

Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment

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We continue our study of the parabolic Anderson equation ∂u ∕ ∂t = κΔu + γξu for the space–time field \(u: \,{\mathbb{Z}}^{d} \times [0,\infty ) \rightarrow \mathbb{R}\), where κ ∈ [0, ∞) is the…

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