The Annals of Probability

Ergodic Theorems for an Infinite Particle System with Births and Deaths

Diane Schwartz

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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.

Article information

Source
Ann. Probab., Volume 4, Number 5 (1976), 783-801.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995984

Digital Object Identifier
doi:10.1214/aop/1176995984

Mathematical Reviews number (MathSciNet)
MR418293

Zentralblatt MATH identifier
0357.60050

JSTOR
links.jstor.org

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Infinite particle systems invariant measures ergodic theorems

Citation

Schwartz, Diane. Ergodic Theorems for an Infinite Particle System with Births and Deaths. Ann. Probab. 4 (1976), no. 5, 783--801. doi:10.1214/aop/1176995984. https://projecteuclid.org/euclid.aop/1176995984


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