## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 1 (1984), 86-107.

### A Local Time Analysis of Intersections of Brownian Paths in the Plane

Donald Geman, Joseph Horowitz, and Jay Rosen

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 12, Number 1 (1984), 86-107.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993375

**Digital Object Identifier**

doi:10.1214/aop/1176993375

**Mathematical Reviews number (MathSciNet)**

MR723731

**Zentralblatt MATH identifier**

0536.60046

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G15: Gaussian processes

Secondary: 60G17: Sample path properties 60G60: Random fields 60J65: Brownian motion [See also 58J65]

**Keywords**

Confluent Brownian motion multiple intersections Hausdorff dimension local time Holder conditions

#### Citation

Geman, Donald; Horowitz, Joseph; Rosen, Jay. A Local Time Analysis of Intersections of Brownian Paths in the Plane. Ann. Probab. 12 (1984), no. 1, 86--107. doi:10.1214/aop/1176993375. https://projecteuclid.org/euclid.aop/1176993375