催化随机介质中的间歇性
Intermittency in a catalytic random medium
摘要：
本文研究了抛物型Anderson方程$\partial u/\partial t=κΔu+ξu$的非线性性，其中$u:\mathbb{Z}^d\times [0,\infty)\to\mathbb{R}$，$κ$为扩散常数，$Δ$为离散拉普拉斯算子，$ξ:\mathbb{Z}^d\times[0,\infty)\to\mathbb {R}$为时空随机介质.我们专注于$ξ$的情况下，$γ$倍的随机介质，通过运行独立的简单随机游走与扩散常数$ρ$从泊松随机场强度$ν$。在本文中，我们假设$κ,γ,ρ,ν\in (0,\infty)$。方程的解描述了在"催化剂“$ξ$的影响下”反应物“$u$的演变。我们认为退火的李雅普诺夫指数，也就是说，指数增长率的连续时刻的$u$，并表明，他们显示出一个有趣的依赖于尺寸$d$和参数$κ,γ,ρ,ν$，与定性不同的行为在$d=1,2$，在$d=3$和$d\geq4$。特别注意这些李雅普诺夫指数的渐近性$κ\downarrow0$和$κ\to\infty$。
英文摘要：
In this paper, we study intermittency for the parabolic Anderson equation $\partial u/\partial t=κΔu+ξu$, where $u:\mathbb{Z}^d\times [0,\infty)\to\mathbb{R}$, $κ$ is the diffusion constant, $Δ$ is the discrete Laplacian and $ξ:\mathbb{Z}^d\times[0,\infty)\to\mathbb {R}$ is a space-time random medium. We focus on the case where $ξ$ is $γ$ times the random medium that is obtained by running independent simple random walks with diffusion constant $ρ$ starting from a Poisson random field with intensity $ν$. Throughout the paper, we assume that $κ,γ,ρ,ν\in (0,\infty)$. The solution of the equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $ξ$. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$, and show that they display an interesting dependence on the dimension $d$ and on the parameters $κ,γ,ρ,ν$, with qualitatively different intermittency behavior in $d=1,2$, in $d=3$ and in $d\geq4$. Special attention is given to the asymptotics of these Lyapunov exponents for $κ\downarrow0$ and $κ\to\infty$.
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