# zbMATH — the first resource for mathematics

Strict concavity of the intersection exponent for Brownian motion in two and three dimensions. (English) Zbl 0909.60065
Summary: 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.

##### MSC:
 60J65 Brownian motion 60G17 Sample path properties 60H30 Applications of stochastic analysis (to PDEs, etc.)
Full Text: