Portal de Periódicos da CAPES

Random invariant measures for Markov chains, and independent particle systems

1978; Springer Nature; Volume: 45; Issue: 4 Linguagem: Inglês

10.1007/bf00537539

0044-3719

Thomas M. Liggett,

Random Matrices and Applications

Let P be the transition operator for a discrete time Markov chain on a space S. The object of the paper is to study the class of random measures on S which have the property that MP=M in distribution. These will be called random invariant measures for P. In particular, it is shown that MP=M in distribution implies MP=M a.s. for various classes of chains, including aperiodic Harris recurrent chains and aperiodic irreducible random walks. Some of this is done by exploiting the relationship between random invariant measures and entrance laws. These results are then applied to study the invariant probability measures for particle systems in which particles move independently in discrete time according to P. Finally, it is conjectured that every Markov chain which has a random invariant measure also has a deterministic invariant measure.