Ergodic Theorems for an Infinite Particle System with Births and Deaths

Abstract

Let $p(x, y)$ be an irreducible symmetric transition function for a Markov chain on a countable set $S$. Let $\eta_t$ be the infinite particle system on $S$ with simple exclusion interaction modified to allow the spontaneous creation and destruction of particles in the system. A complete characterization of the invariant probability measures for this system is obtained in the case where the exponential rates of creation and destruction are independent of the configuration of the system. Furthermore, if $\mathscr{M}$ is the set of probability measures on the state space of $\eta_t$ and $S(t)$ is the semigroup on $\mathscr{M}$ determined by $$S(t)\mu(A) = \int P^\eta\lbrack\eta_t \in A\rbrack d\mu(\eta)$$ theorems concerning the weak convergence of $S(t)\mu$ to the invariant measures of $\eta_t$ are proved.