Brownian motion and Random Walk above Quenched Random Wall

TL;DR

This paper investigates the persistence probability of one random walk staying above another, revealing that the decay rate depends on their variance ratio and is generally faster than the classical 1/2 exponent, with extensions to continuous processes.

Contribution

It establishes the persistence exponent for two dependent or independent random walks above a quenched random wall, showing dependence on variance ratio and extending results to continuous processes.

Findings

Persistence exponent exceeds 1/2 for non-trivial random walls.

Exponent depends only on the variance ratio of the walks.

In continuous settings, the probability decays exponentially.

Abstract

We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\{B_n\}$ and $\{W_n\}$ be two centered, weakly dependent random walks. We establish that $\mathbb{P}(\forall_{n\leq N} B_n \geq W_n|W) = N^{-\gamma + o(1)}$ for a non-random $\gamma\geq 1/2$. In the classical setting, $W_n \equiv 0$, it is well-known that $\gamma = 1/2$. We prove that for any non-trivial $W$ one has $\gamma>1/2$ and the exponent $\gamma$ depends only on $\text{Var}(B_1)/\text{Var}(W_1)$. Our result holds also in the continuous setting, when $B$ and $W$ are independent and possibly perturbed Brownian motions or Ornstein-Uhlenbeck processes. In the latter case the probability decays at exponential rate.