Solvable models of the Fokker-Planck equation: An approach based on the Gel'fand-Levitan method

Abstract

From a given solvable Fokker-Planck equation one can construct a number of other solvable models for diffusion in a stable or bistable potential fields using the Gel'fand-Levitan method of the inverse scattering theory. The simplest way of achieving this is to change the lowest eigenvalue and/or the normalization of the lowest eigenfunction of the ordinary differential equation obtained by separating the time-dependent part. For these cases it is shown that the new probability distribution is expressible in terms of integrals involving the original probability distribution and the Gel'fand-Levitan kernel. The possibility of changing the lowest eigenvalue enables one to find bistable potential fields which would correspond to a well-defined long time relaxation rate for the probability current.