Kolmogorov equations
The Kolmogorov equations describe how CTMC transition probabilities change through continuous time.
Transition matrix
Let \(P(t)\) have entries \(p_{ij}(t)=P(X(t)=j\mid X(0)=i)\), and let \(Q\) be the generator.
Forward equation
\[\frac{dP(t)}{dt}=P(t)Q.\]Backward equation
\[\frac{dP(t)}{dt}=QP(t).\]For a finite time-homogeneous CTMC, both are satisfied and \(P(t)=e^{tQ}\).
Probability-vector form
For a row probability vector \(\boldsymbol\pi(t)\),
\[\frac{d\boldsymbol\pi(t)}{dt}=\boldsymbol\pi(t)Q.\]Key idea. The generator specifies instantaneous rates; the Kolmogorov equations convert those rates into time-dependent state probabilities.