On multi-particle Brownian survivals and the spherical Laplacian

B S, Balakrishna (2013): On multi-particle Brownian survivals and the spherical Laplacian.

Abstract

The probability density function for survivals, that is for transitions without hitting the barrier, for a collection of particles driven by correlated Brownian motions is analyzed. The analysis is known to lead one 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 one naturally to a generating function for the eigenvalues and multiplicities of the Laplacian. Analytical properties of the generating function suggest a simple scaling procedure for determining the eigenvalues readily applicable for a homogeneous collection correlated particles. Comparison of the first eigenvalue with the theoretical and numerical results of Ratzkin and Treibergs for some special domains shows remarkable agreement.

• Item Type: MPRA Paper
• Language: English
• Keywords: Brownian; Survival Probability; Hitting Time; Correlation; Laplacian; Spherical Domain; Eigenvalue
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing

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