Brownian motion and Random Walk above Quenched Random Wall

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 $\mathbb{P}({\forall }_{n\le N}{B}_n\ge W_n|W)=N^{-\gamma +o(1)}$ for a non-random $\gamma \ge 1/2$. 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.

To appear in Ann. Inst. Henri Poincaré Probab. Stat

References in corpus (1)

• Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment

Cited by in corpus (2)

• Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment
• Tightness for branching random walk in time-inhomogeneous random environment