Brownian motion and Random Walk above Quenched Random Wall

Abstract : We study the probability of a random walk staying above a trajectory of another random walk. More precisely, let {Bn} n∈N and {Wn} n∈N be two centered random walks (subject to moment conditions). We establish that P (∀ n≤N Bn ≥ Wn|W) ~ N −γ , where γ is a non-random exponent and ~ is understood on the log scale. In the classical setting (i.e. Wn ≡ 0) it is well-known that γ = 1/2. We prove that for any non-trivial wall W one has γ > 1/2 and the exponent γ depends only on Var(B1)/Var(W1). Further, we prove that these results still hold if B depends weakly on W , this problem naturally emerges in studies of branching random walks in a time-inhomogenous random environment. They are valid also in the continuous time setting, when B and W are (possibly perturbed) Brownian motions. Finally, we present an analogue for Ornstein-Uhlenbeck processes. This time the decay is exponential exp(−γN).
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https://hal.archives-ouvertes.fr/hal-01322463
Contributor : Bastien Mallein <>
Submitted on : Friday, May 27, 2016 - 11:12:34 AM
Last modification on : Saturday, May 25, 2019 - 1:18:43 AM
Long-term archiving on : Sunday, August 28, 2016 - 10:29:45 AM

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Bastien Mallein, Piotr Miłoś. Brownian motion and Random Walk above Quenched Random Wall. 2015. ⟨hal-01322463v1⟩

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