Boundary behavior for random walks in cones

Abstract

We study the asymptotic behavior of zero-drift random walks confined to multidimensional convex cones, when the endpoint is close to the boundary. We derive a local limit theorem in the fluctuation regime.

Martin boundary of random walks in convex cones

“…Ignatiouk-Robert [36] shows the uniqueness of the harmonic function in a convex cone, under the assumption that the first exit time has infinite expectation. Finally, in the paper [45], the second and third authors derive a local limit theorem for zero-drift random walks confined to multidimensional convex cones, when the endpoint is close to the boundary.…”

“…Caravenna and Doney [12] have used this approach to obtain necessary and sufficient conditions for validity of the local renewal theorem for one-dimensional non-restricted random walks. To control local large deviation probabilities in our model, we use recent results obtained in Raschel and Tarrago [45] by using heat kernel estimates. These results, which are improvements of the local limit theorems of [16], lead to Theorem 1 (b) and to the first claim in Theorem 3.…”

“…In the proof of the following lemma, we use several estimates from [45] on the transition probabilities of a Brownian motion in a cone. Those estimates come from general Gaussian estimates for the heat kernel in a Lipschitz domain, see [33,Sec.…”

“…By the local Hölder continuity of the survival probability P(τ bm x > 1) in x (see [45,Prop. 18]), there exist α, χ, C α > 0 such that…”

“…Those bounds are used in the proof of our main results. The strategy of the proof is to compare the tangent cone at σ with some smaller cones included in K. Let us give a first recall a useful result from[45, Lem. 21].…”