Feller boundary induced by a transition operator
S. P. Lloyd · Pacific Journal of Mathematics · 1968
A transition operator T is a nonnegative contraction on an AL space L such that || Tμ\\ = \\μ\\ for μ ^ 0. The set ^ = {fe L*: T*f = /} of invariant functions of the ado joint 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 B Q of B and a measure μ 0 with B o as closed support such that ^ is isomorphic to Loo(Bo 9 μ 0 ).There is also a Markov process whose paths converge to B o with probability 1.However, we do not obtain the kernel representation of superharmonic functions as with the Martin-Doob boundary.