Transition asymptotics for reaction-diffusion in random media

Details Transition asymptotics for reaction-diffusion in random media Transition asymptotics for reaction-diffusion in random media

2005-10-24

arxiv.org/abs/math/0510519v2

Mathematics

Probability

To appear in: Probability and Mathematical Physics: A Volume in Honor of Stanislav Molchanov, Editors - AMS | CRM, (2007)

Scientific paper

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 $2d\kappa$, simple, symmetric, with $\kappa \ge 0$. 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 $F_\theta(t):=\frac{H_1((1+\theta)t)-(1+\theta)H_1(t)}{\theta}$, and assume that $\frac{F_{2\theta}(t)-F_\theta(t)}{\theta\log(\kappa t+e)}\to\infty$ for $|\theta|>0$ small enough, where $H_1(t):=\log < m(0,t)>$ and $$ 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_\epsilon(t)}$ for some $\epsilon >0$ and large enough $t$, a law of large numbers is satisfied; if $ L(t)\ge e^{\frac{1}{d} F_\epsilon (2t)}$ for some $\epsilon>0$ and large enough $t$, a CLT is satisfied. These statements are violated if the reversed inequalities are satisfied for some negative $\epsilon$. Applications to potentials with Weibull, Frechet and double exponential tails are given.

Affiliated with

Also associated with

No associations

Rating

Transition asymptotics for reaction-diffusion in random media does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Transition asymptotics for reaction-diffusion in random media, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Transition asymptotics for reaction-diffusion in random media will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-430546