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Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum
Stockholm University, Faculty of Science, Department of Mathematics. Stockholm University, Nordic Institute for Theoretical Physics (Nordita). British Antarctic Survey, United Kingdom.
Number of Authors: 32017 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 58, no 9, article id 093301Article in journal (Refereed) Published
Abstract [en]

Motivated by a problem in climate dynamics, we investigate the solution of a Bessel-like process with a negative constant drift, described by a Fokker-Planck equation with a potential V(x) = -[b ln(x) + a x], for b > 0 and a < 0. The problem belongs to a family of Fokker-Planck equations with logarithmic potentials closely related to the Bessel process that has been extensively studied for its applications in physics, biology, and finance. The Bessel-like process we consider can be solved by seeking solutions through an expansion into a complete set of eigenfunctions. The associated imaginary-time Schrodinger equation exhibits a mix of discrete and continuous eigenvalue spectra, corresponding to the quantum Coulomb potential describing the bound states of the hydrogen atom. We present a technique to evaluate the normalization factor of the continuous spectrum of eigenfunctions that relies solely upon their asymptotic behavior. We demonstrate the technique by solving the Brownian motion problem and the Bessel process both with a constant negative drift. We conclude with a comparison to other analytical methods and with numerical solutions.

Place, publisher, year, edition, pages
2017. Vol. 58, no 9, article id 093301
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-148905DOI: 10.1063/1.5000386ISI: 000412102600025OAI: oai:DiVA.org:su-148905DiVA, id: diva2:1158445
Available from: 2017-11-20 Created: 2017-11-20 Last updated: 2017-11-20Bibliographically approved

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