On Multi-Particle Brownian Survivals and the Spherical Laplacian

by Bannur S. Balakrishna of Unaffiliated

February 18, 2013

Abstract: The probability density function for survivals, that is for transitions without hitting a barrier, for a collection of particles driven by correlated Brownian motions is analyzed. The analysis is known to lead to a study of the spectrum of the Laplacian on domains on the sphere in higher dimensions. The first eigenvalue of the Laplacian governs the large time behavior of the probability density function and the asymptotics of the hitting time distribution. It is found that the solution leads naturally to a spectral function, a 'generating function' for the eigenvalues and multiplicities of the Laplacian. Analytical properties of the spectral function suggest a simple scaling procedure for determining the eigenvalues, readily applicable for a homogeneous collection of correlated particles. Comparison of the first eigenvalue with the available theoretical and numerical results for some specific domains shows remarkable agreement.

JEL Classification: G13.

AMS Classification: 60J60, 91B25.

Keywords: Survival Probability, Hitting Time, Correlation, Laplacian, Spherical Domain.

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