随机介质中反应扩散的过渡渐近性
Transition asymptotics for reaction-diffusion in random media
摘要：
我们描述了一个普遍的过渡机制，其特征在于通过退火的行为和制度，这种行为的波动是高斯的，为长期渐近的经验平均数的期望值的随机游走分支和湮灭${\mathbb Z}^d$，与固定的随机率。随机游动是独立的，连续的时间速率$2dκ$，简单的，对称的，与$κ\ge 0$。一个以$x\in{\mathbb Z}^d$的随机游走，以$v_+(x)$的速率二进制分支，并以$v_-(x)$的速率零化。随机环境$w$具有坐标$w(x)=(v_-(x),v_+(x))$，其为i.i.d.我们确定了一种自然的方式来描述退火高斯跃迁机制在温和的条件下的速率。实际上，我们引入指数$F_θ(t):=\frac{H_1((1+θ)t)-(1+θ)H_1(t)}θ$，并假设$|θ|>0$的$\frac{F_{2θ}(t)-F_θ(t)}{θ\log(κt+e)}\to\infty$足够小，其中$H_1(t):=\log < m(0,t)>$和$<m(0,t)>$表示在时间$t$和速率$w$的环境下粒子数$m(0,t,w)$的期望值的平均值，假定最初只有一个粒子为0。然后$m(x,t,w)$在一个侧$L(t)$框上的经验平均值具有不同的行为：如果$ L(t)\ge e^{\frac{1}{d} F_ε(t)}$对于一些$ε>0$和足够大的$t$，则满足大数定律;如果$ L(t)\ge e^{\frac{1}{d} F_ε(2t)}$对于一些$ε>0$和足够大的$t$，则满足CLT。如果对某些负的$ε$满足反向不等式，则违反这些陈述。应用威布尔，Frechet和双指数尾的潜力。
英文摘要：
We describe a universal transition mechanism characterizing the passage to an annealed behavior and to a regime where the fluctuations about this behavior are Gaussian, for the long time asymptotics of the empirical average of the expected value of the number of random walks which branch and annihilate on ${\mathbb Z}^d$, with stationary random rates. The random walks are independent, continuous time rate $2dκ$, simple, symmetric, with $κ\ge 0$. A random walk at $x\in{\mathbb Z}^d$, binary branches at rate $v_+(x)$, and annihilates at rate $v_-(x)$. The random environment $w$ has coordinates $w(x)=(v_-(x),v_+(x))$ which are i.i.d. We identify a natural way to describe the annealed-Gaussian transition mechanism under mild conditions on the rates. Indeed, we introduce the exponents $F_θ(t):=\frac{H_1((1+θ)t)-(1+θ)H_1(t)}θ$, and assume that $\frac{F_{2θ}(t)-F_θ(t)}{θ\log(κt+e)}\to\infty$ for $|θ|>0$ small enough, where $H_1(t):=\log < m(0,t)>$ and $<m(0,t)>$ denotes the average of the expected value of the number of particles $m(0,t,w)$ at time $t$ and an environment of rates $w$, given that initially there was only one particle at 0. Then the empirical average of $m(x,t,w)$ over a box of side $L(t)$ has different behaviors: if $ L(t)\ge e^{\frac{1}{d} F_ε(t)}$ for some $ε>0$ and large enough $t$, a law of large numbers is satisfied; if $ L(t)\ge e^{\frac{1}{d} F_ε(2t)}$ for some $ε>0$ and large enough $t$, a CLT is satisfied. These statements are violated if the reversed inequalities are satisfied for some negative $ε$. Applications to potentials with Weibull, Frechet and double exponential tails are given.
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