arXiv 0711.48722007-11math.PR

时空积环境中随机游动的大偏差

Large deviations for random walk in a space--time product environment

Atilla Yilmaz

备注：: Published in at http://dx.doi.org/10.1214/08-AOP400 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
发表信息：: Annals of Probability 2009, Vol. 37, No. 1, 189-205
DOI：: 10.1214/08-AOP400

摘要：

本文考虑时空积环境$ω\inΩ$中$\mathbb{Z}^d$上的随机游动$(X_n)_{n\geq0}$.我们从粒子的角度出发，研究了环境马尔可夫链$(T_{n,X_n}ω)_{n\geq0}$，其中$T$表示$Ω$上的位移.在粒子的渐近平均速度等于任意给定$ξ$的条件下，证明了环境马氏链的经验过程在平均测度下收敛到平稳过程$μ_ξ^{\infty}$.当$d\geq3$和$ξ$足够接近于典型速度时，我们证明了平均大偏差和淬灭大偏差是等价的，并且当粒子具有渐近平均速度$ξ$时，环境马尔可夫链的经验过程在淬灭测度下也收敛到$μ_ξ^{\infty}$.在这种情况下，我们表明，$μ_ξ^{\infty}$是一个平稳的马尔可夫过程，其内核是从原始内核的Doob $h$变换。

英文摘要：

We consider random walk $(X_n)_{n\geq0}$ on $\mathbb{Z}^d$ in a space--time product environment $ω\inΩ$. We take the point of view of the particle and focus on the environment Markov chain $(T_{n,X_n}ω)_{n\geq0}$ where $T$ denotes the shift on $Ω$. Conditioned on the particle having asymptotic mean velocity equal to any given $ξ$, we show that the empirical process of the environment Markov chain converges to a stationary process $μ_ξ^{\infty}$ under the averaged measure. When $d\geq3$ and $ξ$ is sufficiently close to the typical velocity, we prove that averaged and quenched large deviations are equivalent and when conditioned on the particle having asymptotic mean velocity $ξ$, the empirical process of the environment Markov chain converges to $μ_ξ^{\infty}$ under the quenched measure as well. In this case, we show that $μ_ξ^{\infty}$ is a stationary Markov process whose kernel is obtained from the original kernel by a Doob $h$-transform.

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摘要：

英文摘要：

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