Non-Markov stochastic processes satisfying equations usually associated with a Markov process

Abstract

There are non-Markov Ito processes that satisfy the Fokker-Planck, backward time Kolmogorov, and Chapman-Kolmogorov equations. These processes are non-Markov in that they may remember an initial condition formed at the start of the ensemble. Some may even admit 1-point densities that satisfy a nonlinear 1-point diffusion equation. However, these processes are linear, the Fokker-Planck equation for the conditional density (the 2-point density) is linear. The memory may be in the drift coefficient (representing a flow), in the diffusion coefficient, or in both. We illustrate the phenomena via exactly solvable examples. In the last section we show how such memory may appear in cooperative phenomena.