Intersections of Random Walks pp 87-113 | Cite as

# Intersection Probabilities

Chapter

## 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 ,
and similarly for

*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*) = (Ω_{1}× ... × 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 = (ω_{1},...,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 S^{1}(0) = ... =*S*^{ k }(0) = 0. If σ ≤ τ are two times, perhaps random, we let$$ {S^i}[\sigma ,\tau ] = \{ {S^i}\left( j \right):\sigma \le j \le \tau \} $$

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

*S*^{ i }(σ, τ] and*S*^{ i }[σ, τ).## Keywords

Random Walk Killing Rate Range Intersection Invariant Probability Measure Simple Random Walk
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1991