Quasi stationary distributions and Fleming-Viot processes in countable spaces
Pablo A. Ferrari, Universidade de Sao Paulo
Nevena Maric, Syracuse University
Abstract
We consider an irreducible pure jump Markov process with rates
Q on
S+0 where S is countable and
0 an absorbing state. A quasi-stationary distribution
(QSD) is a probability measure μ on
S that satisfies:
starting with μ, the conditional distribution at
time t,
given that at time t the process has not been absorbed, is still
μ.
A Fleming-Viot (FV) process is a system of N particles moving in
S. Each particle moves independently with rates Q until it hits the
absorbing state 0; but then instantaneously chooses one of the N-1 particles
remaining in S and jumps to its position. Between absorptions each
particle moves with rates Q independently.
Under the condition "α(ergodicity coefficient) > C (maximal absorbing rate)" we
prove existence of QSD for Q; uniqueness has been proven by Jacka and
Roberts. When α>0 the FV process is ergodic for each N. Under
α>C the mean normalized densities of the FV unique stationary measure
converge to the QSD of Q, as N goes to infinity; in this limit the variances
vanish.
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