Exactly solvable Fokker-Planck equation with time-dependent nonlinear drift and diffusion coefficients – the Lie-algebraic approach

Abstract

In this paper we have analytically solved the Fokker-Planck equation (FPE) associated with a fairly large class of multiplicative stochastic processes with time-varying nonliner drift and diffusion coefficients, which has wide applicability in various areas of physics, e.g. nonlinear optics and chemical reaction dynamics. By exploiting the dynamical symmetry of the FPE, we apply the Lie-algebraic approach to derive the time-dependent analytical closed-form solutions. The derived solutions fall into two different categories, namely (i) one with a moving absorbing boundary, and (ii) one with a fixed absorbing boundary at the origin, depending upon the model parameters. The corresponding escape (or survival) probabilities are also evaluated analytically. We believe that not only our analytically exact results can serve as standard models upon which the discussion of more complicated problems can be based, but they can also be useful as a benchmark to test approximate numerical or analytical procedures.