Quasi-stationary distributions for stochastic processes with an absorbing state

Details Quasi-stationary distributions for stochastic processes with an absorbing state Quasi-stationary distributions for stochastic processes with an absorbing state

2001-10-26

arxiv.org/abs/cond-mat/0110557v1

J. Phys. A 35, 1147 (2002)

Physics

Condensed Matter

Statistical Mechanics

20 pages, 13 figures

Scientific paper

10.1088/0305-4470/35/5/303

We study the long-time behavior of stochastic models with an absorbing state, conditioned on survival. For a large class of processes, in which saturation prevents unlimited growth, statistical properties of the surviving sample attain time-independent limiting values. We may then define a quasi-stationary probability distribution as one in which the ratios p_n(t)/p_m(t) (for any pair of nonabsorbing states n and m), are time-independent. This is not a true stationary distribution, since the overall normalization decays as probability flows irreversibly to the absorbing state. We construct quasi-stationary solutions for the contact process on a complete graph, the Malthus-Verhulst process, Schlogl's second model, and the voter model on a complete graph. We also construct the master equation and quasi-stationary state in a two-site approximation for the contact process, and for a pair of competing Malthus-Verhulst processes.

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