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 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.

• 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

References

Carslaw, H. S. and Jaeger, J. C., (1959), ``Conduction of heat in solids'', Oxford University Press, Oxford (1959).

Cheeger, J., (1983), ``Spectral geometry of singular Riemannian spaces'', Journal of Differential Geometry, 18-4 (1983), 575-657.

DeBlassie, R. D., (1987), ``Exit times from cones in R^n of Brownian motion'', Probability Theory and Related Fields, 74-1 (1987), 1-29.

Kac, M., (1966), ``Can one hear the shape of a drum?'', American Mathematical Monthly, 73-4 (1966) 1-23.

Lame, G., (1833), ``Memoire sur la propagation de la chaleur dans les polyedres'', Journal de l'Ecole Polytechnique, 22 (1833), 194-251.

Ratzkin, J. and Treibergs, A., (2009), ``A capture problem in Brownian motion and eigenvalues of spherical domains'', Transactions of the American Mathematical Society, 361 (2009), 391-405.

Sommerfeld, A., (1894), ``Zur analytischen Theorie der Warmeleitung'', Mathematische Annalen, 45-2 (1894), 263-277.

Vassilevich, D. V., (2003), ``Heat kernel expansion: user's manual'', Physics Reports, 388 (2003), 279-360.

Zhou, C., (2001), ``An analysis of default correlations and multiple defaults'', Review of Financial Studies, 14-2 (2001), 555-576.