Brownian motion and Random Walk above Quenched Random Wall

Bastien Mallein (1, 2) , Piotr Miłoś (3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 UW - University of Warsaw

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).

Domaines: Probabilités [math.PR]

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