49 research outputs found
On Mean Field Convergence and Stationary Regime
Assume that a family of stochastic processes on some Polish space
converges to a deterministic process; the convergence is in distribution (hence
in probability) at every fixed point in time. This assumption holds for a large
family of processes, among which many mean field interaction models and is
weaker than previously assumed. We show that any limit point of an invariant
probability of the stochastic process is an invariant probability of the
deterministic process. The results are valid in discrete and in continuous
time
Stochastic Persistence
Let be a continuous time Markov process on some metric
space leaving invariant a closed subset called the {\em
extinction set}. We give general conditions ensuring either "Stochastic
persistence" (Part I) : Limit points of the occupation measure are invariant
probabilities over or "Extinction" (Part II) : a.s. In the persistence case we also discuss conditions
ensuring the a.s convergence (respectively exponential convergence in total
variation) of the occupation measure (respectively the distribution) of
toward a unique probability on These results extend and generalize
previous results obtained for various stochastic models in population dynamics,
given by stochastic differential equations, random differential equations, or
pure jump processes
Deterministic Approximation of Stochastic Evolution in Games
This paper provides deterministic approximation results for stochastic processes that arise when finite populations recurrently play finite games. The deterministic approximation is defined in continuous time as a system of ordinary differential equations of the type studied in evolutionary game theory. We establish precise connections between the long-run behavior of the stochastic process, for large populations, and its deterministic approximation. In particular, we show that if the deterministic solution through the initial state of the stochastic process at some point in time enters a basin of attraction, then the stochastic process will enter any given neighborhood of that attractor within a finite and deterministic time with a probability that exponentially approaches one as the population size goes to infinity. The process will remain in this neighborhood for a random time that almost surely exceeds an exponential function of the population size. During this time interval, the process spends almost all time at a certain subset of the attractor, its so-called Birkhoff center. We sharpen this result in the special case of ergodic processes. Game Theory; Evolution; Approximation