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.