Vol. 27, No. 3, 1968

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Feller boundary induced by a transition operator

S. P. Lloyd

Vol. 27 (1968), No. 3, 547–566

A transition operator T is a nonnegative contraction on an AL space L such that = μfor μ 0. The set = {f L : Tf = f} of invariant functions of the adojoint T turns out to be lattice isomorphic to C(B) for a certain hyperstonian compact Hausdorff B. For the transition operator of a countable state Markov chain, B is the Feller boundary of the process, and in the general case we call B the Feller boundary induced by T. For the general case we exhibit several Markov processes associated with T such that B appears as a subset of the state space. These processes involve the potential theory of T. When L is separable there is a quotient space B0 of B and a measure μ0 with B0 as closed support such that is isomorphic to L(B00). There is also a Markov process whose paths converge to B0 with probability 1. However, we do not obtain the kernel representation of superharmonic functions as with the Martin-Doob boundary.

Mathematical Subject Classification
Primary: 60.60
Received: 17 November 1967
Published: 1 December 1968
S. P. Lloyd