Transition semigroup of a continuous-time Markov chain ($P(t)$)

For a time-homogeneous continuous-time Markov chain, $P_{ij}(t)={\mathbb{P}}_i(X_t=j)$ and the matrices satisfy $P(s+t)=P(s)P(t)$, the Chapman-Kolmogorov equation. On a finite state space with Q-matrix $Q$, the matrix exponential gives $P(t)=e^{tQ}$.

The distinction from a discrete-time transition matrix is the elapsed-time parameter: $P(t)$ describes where the process is after $t$ time units, allowing any number of intervening jumps. Each $P(t)$ is a stochastic matrix, with $P(0)=I$. A Q-matrix instead records instantaneous rates and has row sums zero. Sampling at a fixed spacing $\Delta$ yields a Discrete-time Markov chain with transition matrix $P(\Delta )$. See Hao Wu's MIT lecture 21, transition semigroups and lecture 22, infinitesimal rates.

Sample a finite-state continuous-time Markov chain with generator $Q$ at the independent arrival times of a rate-$\lambda$ Poisson process. The sampled states form a discrete-time Markov chain with transition matrixIts invariant distributions are exactly those of the continuous-time chain.

For a finite-state continuous-time Markov chain with Q-matrix $Q$, the backward and forward equations are $P^{\prime }(t)=QP(t)$ and $P^{\prime }(t)=P(t)Q$, respectively.