Numerical methods for quasi-stationary distributions

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The stochastic decay towards absorbing (non-fluctuating) states is used to model plethora of phenomena, such as species extinction, the disappearance of infectious diseases and consensus in opinion dynamics, amongst others. In the study of stochastic processes with absorbing states, much information can be gained from the distribution of the process conditioned to never get absorbed, the so-called quasi-stationary distribution. This distribution characterizes the fluctuations in the dynamics towards absorption, allows to calculate average survival times, and can be used to sample rare events. Available numerical methods in the literature to obtain quasi-stationary probability distributions are restricted to specific families of processes. Motivated by the applications of quasi-stationary distributions, we have developed two numerical methods (an iterative algorithm and a simulation method) to obtain the quasi-stationary probability distribution for a completely general Markov stochastic process, either with discrete or continuous states, in continuous or discrete time, and not restricted to a specific number of absorbing states.

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Raúl Toral

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