= Transition semigroup of a continuous-time Markov chain
{c}
{title2=$P(t)$}
= Transition probability matrix
{synonym}
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 https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/245cbb8888672b1b902c308fd6e2468f_MIT18_445S15_lecture21.pdf[Hao Wu's MIT lecture 21, transition semigroups] and https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/f6b1c125c2b471928e960a1d2c8ea6b5_MIT18_445S15_lecture22.pdf[lecture 22, infinitesimal rates].
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