Characterization of the marginal distributions of Markov processes used in dynamic reliability (Q2498196)

Consider a two-component Markov process \((I_t,X_t)\) with state space \(E \times \mathbb{R}^d\), \(E\) finite, and \(dX_t(\omega)/dt=v(i,X_ (\omega))\), given \(I_t (\omega)=i\). The marginal distribution of the process at time \(t\) is shown to be the unique measure solution of a set of integro-differential equations which may be viewed as a weak form of the Chapman-Kolmogorov equation.