Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum

Guarnieri, F.; Moon, Woosok; Wettlaufer, J. S.. 2017 Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum. Journal of Mathematical Physics, 58 (9), 093301. https://doi.org/10.1063/1.5000386

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Abstract/Summary

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.

Item Type:	Publication - Article
Digital Object Identifier (DOI):	https://doi.org/10.1063/1.5000386
ISSN:	0022-2488
Additional Keywords:	Arctic sea-ice, regulated Brownian-motion, thickness distribution, transient behaviour, Bessel processes, model mechanics
NORA Subject Terms:	Mathematics
Date made live:	07 Nov 2017 10:47 +0 (UTC)
URI:	https://nora.nerc.ac.uk/id/eprint/518316

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