State spaces for Markov chains

Abstract:

If $p(t,i,j)$ is the transition probability (from i to j in time t) of a continuous parameter Markov chain, with $p(0 + ,i,i) = 1$, entrance and exit spaces for p are defined. If $L[{L^ \ast }]$ is an entrance [exit] space, the function $p( \cdot , \cdot ,j)[p( \cdot ,i, \cdot )/h( \cdot )]$ has a continuous extension to $(0,\infty ) \times L[(0,\infty ) \times {L^ \ast }$, for a certain norming function h on ${L^ \ast }$]. 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.