Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic
  random environment   We continue our study of the parabolic Anderson equation $\partial u/\partial
t = \kappa\Delta u + \gamma\xi u$ for the space-time field $u\colon\,\Z^d\times
[0,\infty)\to\R$, where $\kappa \in [0,\infty)$ is the diffusion constant,
$\Delta$ is the discrete Laplacian, $\gamma\in (0,\infty)$ is the coupling
constant, and $\xi\colon\,\Z^d\times [0,\infty)\to\R$ 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 $\Z^d$. 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
\emph{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 \emph{quenched} Lyapunov exponent, i.e., the exponential
growth rate of $u$ conditional on $\xi$. We first prove existence and derive
some qualitative properties of the quenched Lyapunov exponent for a general
$\xi$ that is stationary and ergodic w.r.t.\ translations in $\Z^d$ and
satisfies certain noisiness conditions. After that we focus on the three
particular choices for $\xi$ mentioned above and derive some more detailed
properties. We close by formulating a number of open problems.

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The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph