  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.
