Intersection Probabilities

Part of the Probability and Its Applications book series (PA)


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× ... × Ωk, P 1× ... ×P k). We will use E i for expectations with respect to P i ; E for expectations with repect to P; ωi for elements of Ω i ; and ω = (ω1,...,ωk) 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).


Random Walk Killing Rate Range Intersection Invariant Probability Measure Simple Random Walk 
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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  1. 1.Department of MathematicsDuck UniversityDurhamUSA

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