2017/03/31 by F. Guarnieri, Filippo Guarnieri, W. Moon +3 · 17 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Bessel function #Bessel process #Brownian motion #Eigenfunction #Eigenvalues and eigenvectors #Fokker–Planck equation #Logarithm #Mathematical analysis #Mathematics #Orthogonal polynomials #Partial differential equation #Physics #Quantum mechanics #Schrödinger equation #Statistical Mechanics and Entropy #cond-mat.stat-mech #math-ph #math.MP #stochastic dynamics and bifurcation
paper · pdf · doi:10.1063/1.5000386
published in Journal of Mathematical Physics 58(9) (American Institute of Physics) · 21 pages, 8 figures
arxiv created 2017/03/31 · openalex publication_date 2017/09/01 · arxiv updated 2019/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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)=−[bln(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 Schrödinger 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.