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February, 1984 A Local Time Analysis of Intersections of Brownian Paths in the Plane
Donald Geman, Joseph Horowitz, Jay Rosen
Ann. Probab. 12(1): 86-107 (February, 1984). DOI: 10.1214/aop/1176993375

Abstract

We envision a network of $N$ paths in the plane determined by $N$ independent, two-dimensional Brownian motions $W_i(t), t \geq 0, i = 1, 2, \cdots, N$. Our problem is to study the set of "confluences" $z$ in $\mathbb{R}^2$ where all $N$ paths meet and also the set $M_0$ of $N$-tuples of times $\mathbf{t} = (t_1, \cdots, t_N)$ at which confluences occur: $M_0 = \{\mathbf{t}: W_1(t_1) = \cdots = W_N(t_N)\}$. The random set $M_0$ is analyzed by constructing a convenient stochastic process $X$, which we call "confluent Brownian motion", for which $M_0 = X^{-1}(0)$ and using the theory of occupation densities. The problem of confluences is closely related to that of multiple points for a single process. Some of our work is motivated by Symanzik's use of Brownian local time in quantum field theory.

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Donald Geman. Joseph Horowitz. Jay Rosen. "A Local Time Analysis of Intersections of Brownian Paths in the Plane." Ann. Probab. 12 (1) 86 - 107, February, 1984. https://doi.org/10.1214/aop/1176993375

Information

Published: February, 1984
First available in Project Euclid: 19 April 2007

zbMATH: 0536.60046
MathSciNet: MR723731
Digital Object Identifier: 10.1214/aop/1176993375

Subjects:
Primary: 60G15
Secondary: 60G17 , 60G60 , 60J65

Keywords: Confluent Brownian motion , Hausdorff dimension , Holder conditions , Local time , multiple intersections

Rights: Copyright © 1984 Institute of Mathematical Statistics

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Vol.12 • No. 1 • February, 1984
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