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Pré-Publication, Document De Travail Année : 2015

Brownian motion and Random Walk above Quenched Random Wall

Résumé

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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Dates et versions

hal-01322463 , version 1 (27-05-2016)
hal-01322463 , version 2 (18-05-2019)

Identifiants

  • HAL Id : hal-01322463 , version 1

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