Strong Markov Consistency of Multivariate Markov Families and Processes

In this chapter the concept of strong Markov consistency for multivariate Markov families and for multivariate Markov processes is introduced and studied. Strong Markov consistency of a multivariate Markov family/process, if satisfied, provides for invariance of the Markov property under coordinate projections, a property that is important in various practical applications. We only consider conservative Markov processes and Markov families. In Section 2.1, we study the so-called strong Markov consistency for multivariate Markov families and multivariate Markov processes taking values in an arbitrary metric space. This study is geared towards formulating a general framework within which the strong Markov consistency can be conveniently analyzed. In Section 2.2, we specify our study of the strong Markov consistency to the case of multivariate Feller-Markov families taking values in R n . The analysis is first carried in the time-inhomogeneous case, and then in the time homogeneous case where a more comprehensive study can be done.