State spaces for Markov chains

If p(t, i,j) is the transition probability (from i to j in time t) of a continuous parameter Markov chain, with p(O +, i, i) = 1, entrance and exit spaces for p are defined. If L [L*] is an entrance [exit] space, the functionp(-, *, j) [p(., i, *)/h(-)] has a continuous extension to (0, oo) x L [(O, oo) x L*, for a certain norming function h on L*]. It is shown that there is always a space which is both an entrance and exit space. On this space one can define right continuous strong Markov processes, for the parameter interval [0, b], with the given transition function as conditioned by specification of the sample function limits at 0 and b. Summary. Let p be a stochastic transition function on the set I of integers: p(t, i, j) is the probability of a transition from i to j in time t. An entrance space for p is a topological space LI with the property that p(., , j) has a continuous extension to (0, oo) x L, and that the transition function together with any absolute probability function f determines a Markov process with state space L, parameter interval (0, oo), right continuous with left limits, with limit at 0 iff=p(., 6, -) for a point 6 in L. An exit space L* is a space which, together with an adjoined isolated point, is an entrance space for a certain (not uniquely determined) dual transition function, and has among other properties the property that, with a certain normalizing function h on I, the function (t, j) p(t, i, j)/h(j) has a continuous finite extension to (0, oo) x L*. Previous treatments (see for example [2], [4], [5]) have concentrated attention on a certain entrance space, but in the course of an analysis of entrance and exit spaces it is shown in the present paper that there is a space which is both an entrance and an exit space for p. Space time (x excessive functions and measures on entrance, exit, and combined spaces are studied. On the appropriate state spaces the natural supermartingales defined in terms of p and its extensions have smooth sample functions. If L is both an entrance and an exit space it is possible to define Markov processes {x(t), 0 < t < b} with state space L, which are Markov processes having the given transition function, conditioned by specification of x(0 +) and x(b-).