Intersection Probabilities

Abstract

We start the study of intersection probabilities for random walks. It will be useful to make some notational assumptions which will be used throughout this book for dealing with multiple random walks. Suppose we wish to consider k independent simple random walks S ¹,..., S ^k. Without loss of generality, we will assume that S ⁱ is defined on the probability space (Ω _i , P _i ) and that (Ω,P) = (Ω₁× ... × Ω_k, P ₁× ... ×P _k). We will use E _i for expectations with respect to P _i ; E for expectations with repect to P; ω_i for elements of Ω _i ; and ω = (ω₁,...,ω_k) for elements of Ω. We will write ${P^{{x_1},...,{x_k}}}$ and ${E^{{x_1},...,{x_k}}}$ to denote probabilities and expectations assuming S ¹ (0) = x ₁,..., S ^k (0) = x _k . As before, if the x ₁,..., x _k are missing then it is assumed that S¹ (0) = ... = S ^k (0) = 0. If σ ≤ τ are two times, perhaps random, we let

$$ {S^i}[\sigma ,\tau ] = \{ {S^i}\left( j \right):\sigma \le j \le \tau \} $$

,

$$ {S^i}[\sigma ,\tau ] = \{ {S^i}\left( j \right):\sigma < j < \tau \} $$

and similarly for S ⁱ (σ, τ] and S ⁱ [σ, τ).