Sinai行走的一些可加泛函的持久性
Persistence of some additive functionals of Sinai's walk
摘要：
我们有兴趣在西奈半岛的步行$(S\_n)\_{n\in\mathbb{N}}$。我们证明了退火概率$\sum\_{k=0}^n f(S\_k)$是严格正的所有$n\in[1,N]$等于$1/(\log N)^{\frac{3-\sqrt{5}}{2}+o(1)}$，为一大类的功能$f$，特别是$f(x)=x$。持久性指数$\frac{3-\sqrt{5}}{2}$首次出现在Le Doussal、Monthus和Fischer的一篇非严格论文中，其动机来自物理学。证明依赖于技术的本地化西奈半岛的步行和使用的结果Cheliotis的符号变化的谷底的两侧布朗运动。
英文摘要：
We are interested in Sinai's walk $(S\_n)\_{n\in\mathbb{N}}$. We prove that the annealed probability that $\sum\_{k=0}^n f(S\_k)$ is strictly positive for all $n\in[1,N]$ is equal to $1/(\log N)^{\frac{3-\sqrt{5}}{2}+o(1)}$, for a large class of functions $f$, and in particular for $f(x)=x$. The persistence exponent $\frac{3-\sqrt{5}}{2}$ first appears in a non-rigorous paper of Le Doussal, Monthus and Fischer, with motivations coming from physics. The proof relies on techniques of localization for Sinai's walk and uses results of Cheliotis about the sign changes of the bottom of valleys of a two-sided Brownian motion.
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