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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.

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