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},
journal={arXiv: Probability},
year={2012},
pages={69-89}
}
• Published 7 October 2010
• Mathematics
• arXiv: Probability
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.
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References

SHOWING 1-10 OF 27 REFERENCES
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
Parabolic Anderson model with voter catalysts: dichotomy in the behavior of Lyapunov exponents
• Mathematics
• 2012
We consider the parabolic Anderson model ∂u∕∂t=κΔu+γξu with u:$${\mathbb{Z}}^{d} \times{\mathbb{R}}^{+} \rightarrow{\mathbb{R}}^{+}$$, where $$\kappa\in{\mathbb{R}}^{+}$$ is the diffusion constant, Δ
Survival Probability of a Random Walk Among a Poisson System of Moving Traps
• Mathematics
• 2012
We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on $${\mathbb{Z}}^{d}$$, which can also be interpreted as the
Intermittency on catalysts: three-dimensional simple symmetric exclusion
• Mathematics
• 2008
We continue our study of intermittency for the parabolic Anderson model $\partial u/\partial t = \kappa\Delta u + \xi u$ in a space-time random medium $\xi$, where $\kappa$ is a positive diffusion
Parabolic problems for the Anderson model
• Mathematics
• 1990
Summary. This is a continuation of our previous work [6] on the investigation of intermittency for the parabolic equation (∂/∂t)u=Hu on ℝ+×ℤd associated with the Anderson Hamiltonian H=κΔ+ξ(·) for
Intermittency on catalysts : voter model
• Mathematics
• 2009
In this paper we study intermittency for the parabolic Anderson equation ?u/?t = ?? u + ??u with u: Zd × [0,8) ? R, where ? ? [0,8) is the diffusion constant, ? is the discrete Laplacian, ? ? (0,8)
Intermittency in a catalytic random medium
• Mathematics
• 2004
In this paper, we study intermittency for the parabolic Anderson equation ∂u/∂t=κΔu+ξu, where u:ℤd×[0, ∞)→ℝ, κ is the diffusion constant, Δ is the discrete Laplacian and ξ:ℤd×[0, ∞)→ℝ is a space-time
Parabolic problems for the Anderson model
• Mathematics
• 1990
The objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics, in particular in hydrodynamics. On a