MPEJ Volume 4, No.5, 67pp
Received: Apr 1, 1998, Revised: Oct 3, 1998, Accepted: Oct 5, 1998
Gregory F. Lawler
Strict Concavity of the Intersection Exponent for Brownian Motion
in Two and Three Dimensions
ABSTRACT: The intersection exponent for Brownian motion is a measure of how
likely Brownian motion paths in two and three dimensions do not intersect.
We consider the intersection exponent $\xi(\lambda) = \xi_d(k,\lambda)$
as a function of $\lambda$ and show that $\xi$ has a continuous, negative
second derivative. As a consequence, we improve some estimates for the
intersection exponent; in particular, we give the first proof that the
intersection exponent $\xi_3(1,1)$ is strictly greater than the mean field
prediction. The results here are used in a later paper to analyze the
multifractal spectrum of the harmonic measure of Brownian motion paths.