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

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.