|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'501'711
Articles rated: 2609
19 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment | | Jürgen Gärtner
; Frank den Hollander
; Grégory Maillard
; | | Date: |
2 Nov 2010 | | Abstract: | We continue our study of the parabolic Anderson equation $partial u/partial
t = kappaDelta u + gammaxi u$ for the space-time field $ucolon,^d imes
[0,infty) oR$, where $kappa in [0,infty)$ is the diffusion constant,
$Delta$ is the discrete Laplacian, $gammain (0,infty)$ is the coupling
constant, and $xicolon,^d imes [0,infty) oR$ 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 $^d$. 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 $^d$ 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. | | Source: | arXiv, 1011.0541 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER