Wiener’s test for space-time random walks and its applications

Abstract

Abstract. This paper establishes a criterion for whether a d-dimensional random walk on the integer lattice Z d visits a space-time subset infinitely often or not. It is a precise analogue of Wiener's test for regularity of a boundary point with respect to the classical Dirichlet problem. The test obtained is applied to strengthen the harder half of Kolmogorov's test for the random walk.

On large intersection and self-intersection local times in dimension five or more

“…2d γ∈S(Λ\{0}) |Λ\{0}| i=1 ϕ γ(i) − γ(i − 1)We first show that there is a constant κ d > 0, such that for any finite Λ ⊂ Z d , we havecap(Λ) ≥ κ d |Λ| 1− 2 d . (5.3)We use the variational characterisation of capacity (see the Appendix of[6]), which says that if G d is the Green kernel, µ is a non-negative measure on Λ, and c a positive constant ∀z ∈ Λ,z ′ ∈Λ G d (z, z ′ )µ(z ′ ) ≤ c =⇒ cap(Λ) ≥ z∈Λ µ(z) c .We have shown in the proof of Lemma 1.2 of[3], that there is κ d > 0 such that for any z ∈ Λz ′ ∈Λ G d (z, z ′ )µ(z ′ ) ≤ 1 κ d , with µ(z) = 1I Λ (z) |Λ| 2/d . (5.5)The desired bound (5.3) follows readily from (5.5).…”