MPEJ Volume 1, No.4, 35pp
Received: August 25, 1995, Revised: November 16, 1995, Accepted: November 22, 1995
Gregory F. Lawler
Nonintersecting Planar Brownian Motions
ABSTRACT: In this paper we construct a measure on pairs of Brownian motions
starting at the same point conditioned so their paths do not intersect.
The construction of this measure is a start towards the rigorous
understanding of nonintersecting Brownian motions as a conformal field.
Let $B^1,B^2$ be independent Brownian motions in $\R^2$
starting at distinct points on the unit circle.
Let $T_r^j$ be the first time
that the $j$th Brownian motion reaches distance $r$
and let $D_r$ be the event
$$ D_r = \{B^1[0,T_{e^r}^1] \cap B^2[0,T_{e^r}^2] = \emptyset \} . $$
We construct the measure by considering the limit
of the measure induced by Brownian motions conditioned on the event $D_r$.