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The Recurrence Structure of General Markov Processes
Author(s) -
Tuominen Pekka,
Tweedie Richard L.
Publication year - 1979
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-39.3.554
Subject(s) - mathematics , markov chain , semigroup , combinatorics , markov kernel , convolution (computer science) , lattice (music) , pure mathematics , lebesgue measure , discrete mathematics , lebesgue integration , markov model , variable order markov model , statistics , physics , machine learning , artificial neural network , computer science , acoustics
We study the recurrence properties of a Markov process ( X t ) with transition semigroup { P t (x, A) } in terms of the properties of the associated Markov chains with one‐step transition probabilitiesK F ( x , A ) = ∫ P t ( x , A ) F ( d t )where F is a distribution on [0, ∞) with finite mean. We show that if ( X t ) is a Hunt process, then ( X t ) and K F have the same recurrence structures if F has some convolution power non‐singular with respect to Lebesgue measure; and if P t is continuous in t this extends to lattice F , and to arbitrary F if the continuity is uniform in the neighbourhoods of infinity. This extends and unifies known results for resolvent chains ( F exponential) and skeleton chains ( F concentrated at h > 0). Using these results, the second part of the paper finds topological conditions on the kernels K F which ensure that ( X t ) admits a decomposition into a countable number of recurrent sets and a transient part of the space. These conditions are analogues of those previously shown to hold for discrete time Markov chains.