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Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment. (English) Zbl 1252.39027
Deuschel, Jean-Dominique (ed.) et al., Probability in complex physical systems. In honour of Erwin Bolthausen and Jürgen Gärtner. Selected papers based on the presentations at the two 2010 workshops. Berlin: Springer (ISBN 978-3-642-23810-9/hbk; 978-3-642-23811-6/ebook). Springer Proceedings in Mathematics 11, 159-193 (2012).
Summary: We continue our study of the parabolic Anderson equation $$\partial u / \partial t = \kappa \Delta u + \gamma \xi u$$ for the space-time field $$u: \mathbb Z^d \times [0,\infty) \to \mathbb R$$, where $$\kappa \in [0, \infty)$$ is the diffusion constant, $$\Delta$$ is the discrete Laplacian, $$\gamma \in (0, \infty )$$ is the coupling constant, and $$x: \mathbb Z^d \times [0,\infty) \to \mathbb R$$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a “reactant” $$u$$ under the influence of a “catalyst” $$\xi$$, both living on $$\mathbb Z^d$$.
In earlier work we considered three choices for $$\xi$$: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $$u$$ w.r.t. $$\xi$$, and showed that these exponents display an interesting dependence on the diffusion constant $$\kappa$$, with qualitatively different behavior in different dimensions $$d$$. In the present paper we focus on the quenched Lyapunov exponent, i.e., the exponential growth rate of $$u$$ conditional on $$\xi$$.
We first prove existence and derive qualitative properties of the quenched Lyapunov exponent for a general $$\xi$$ that is stationary and ergodic under translations in space and time and satisfies certain noisiness conditions. After that we focus on the three particular choices for $$\xi$$ mentioned above and derive some further properties. We close by formulating open problems.
For the entire collection see [Zbl 1235.60005].

##### MSC:
 39A50 Stochastic difference equations 35K10 Second-order parabolic equations 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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