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

## TL;DR

This paper solves a Fokker-Planck equation with a logarithmic potential related to Bessel processes, revealing a mixed eigenvalue spectrum and introducing a new normalization technique for continuous eigenfunctions, with applications in physics, biology, and finance.

## Contribution

It presents a novel method to evaluate the normalization of continuous eigenfunctions in Fokker-Planck equations with mixed spectra, applied to Bessel-like processes with negative drift.

## Key findings

- Eigenfunction normalization technique based on asymptotic behavior
- Solution of Fokker-Planck with mixed eigenvalue spectrum
- Comparison with other analytical and numerical methods

## Abstract

Motivated by a problem in climate dynamics, we investigate the solution of a Bessel-like process with 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 Schroedinger 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 negative constant drift. We conclude with a comparison with other analytical methods and with numerical solutions.