Unifying stochastic Markov process and its transition probability density function.

Abstract

A stochastic Markov process is defined which generalizes a class of processes previously considered by the author [Phys. Rev. A 42, 4485 (1990)]. The enlarged class unifies such apparently unrelated processes as the Ornstein-Uhlenbeck process, the generalized Verhulst-Landau processes, the generalized Rayleigh process, the hyperbolic tangent processes, and many other cases of physico-mathematical interest. The Fokker-Planck equation associated with this unifying stochastic process is solved analytically for the transition probability density function, using a similar constructive solution method as in the author's previous work. The result is obtained as an eigenfunction expansion over a generally mixed spectrum. The discrete eigenfunctions are related (but not identical) to Jacobi polynomials. It is shown that the results for the above-mentioned processes are obtainable as limiting cases, and that some additional solvable equivalent Schr\"odinger problems can be defined.