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Two and Three Dimensions

  • Gregory F. Lawler
Part of the Probability and Its Applications book series (PA)

Abstract

In this chapter we study
$$ f\left( n \right) = P\{ {S^1}\left( {0,n} \right) \cap {S^2}(0,n] = \phi \} $$
where S 1, S 2 are independent simple random walks in Z 2 or Z 3. By (3.29),
$$ {c_1}{n^{\left( {d - 4} \right)/2}} \le f\left( n \right) \le {c_2}{n^{\left( {d - 4} \right)/4}} $$
(5.1)
so we would expect that
$$ f\left( n \right) \approx {n^{ - \zeta }} $$
for some ζ; = ζ d . We show that this is the case and that the exponent is the same as an exponent for intersections of Brownian motions. Let B 1, B 2 be independent Brownian motions in R d starting at distinct points x, y. It was first proved in [19] that if d < 4,
$${P^{x,y}}\left\{ {{B^1}\left[ {0,\infty } \right) \cap {B^2}\left[ {0,\infty } \right) \ne \phi } \right\} = 1$$

Keywords

Brownian Motion Random Walk Variational Formulation Conformal Invariance Harmonic Measure 
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

Authors and Affiliations

  • Gregory F. Lawler
    • 1
  1. 1.Department of MathematicsDuck UniversityDurhamUSA

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