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

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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.

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  1. Physics Department, University of Houston, Houston, Texas, 77204-5005, USA

    J. L. McCauley

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McCauley, J.L. Non-Markov stochastic processes satisfying equations usually associated with a Markov process. Eur. Phys. J. Spec. Top. 204, 133–143 (2012). https://doi.org/10.1140/epjst/e2012-01557-7

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