Characterization of the marginal distributions of Markov processes used in dynamic reliability

In dynamic reliability, the evolution of a system is described by a piecewise deterministic Markov process (It,Xt) t≥0 with state-space E×ℝ d, where E is finite. The main result of the present paper is the characterization of the marginal distribution of the Markov process (It,Xt) t≥0 at time t, as the unique solution of a set of explicit integro-differential equations, which can be seen as a weak form of the Chapman–Kolmogorov equation. Uniqueness is the difficult part of the result.