the matrix-exponential transition function
The matrix-exponential transition function is the formula P_t = e^(tQ) giving the t-step transition probabilities of a continuous-time Markov chain directly from its generator Q. It is the continuous-time counterpart of P^n = (one-step matrix)^n for discrete chains, with the integer power replaced by a matrix exponential. It is both the solution of the Kolmogorov equations and the cleanest statement of how a CTMC's distribution propagates over any time interval.
For a finite state space the matrix exponential is the convergent power series e^(tQ) = sum_{n>=0} (tQ)^n / n! = I + tQ + t^2 Q^2/2 + ..., and this P_t solves both Kolmogorov equations dP_t/dt = Q P_t = P_t Q with P_0 = I. The family {P_t : t >= 0} is a semigroup: P_{s+t} = P_s P_t, which is exactly the Chapman-Kolmogorov / Markov property, and Q = P'_0 is its generator. Computationally, when Q is diagonalizable as Q = V D V^{-1} with eigenvalues on the diagonal of D, one gets e^(tQ) = V e^(tD) V^{-1}, where e^(tD) just exponentiates the eigenvalues; the eigenvalue 0 (with eigenvector the stationary distribution as a left eigenvector) gives the equilibrium part, and the other eigenvalues, all with nonpositive real part, give the decaying transients whose rates are the relaxation times of the chain.
This formula is why CTMCs are so tractable: distributions, expectations and correlation functions are all read off from e^(tQ), and the spectral gap (the smallest nonzero |Re(eigenvalue)|) controls the speed of convergence to equilibrium. Two honest cautions. On an infinite state space Q is an unbounded operator and the term-by-term power series may diverge; e^(tQ) must then be defined by semigroup theory (Hille-Yosida), and it equals the minimal transition function only when the chain is non-explosive — otherwise the rows of P_t can sum to less than 1, the missing mass being the probability of having already exploded. Also, do not confuse e^(tQ) with the entrywise exponential of tQ; it is the matrix exponential, an entirely different object.
For the two-state chain with Q-rows ( -lambda, lambda ) and ( mu, -mu ), diagonalizing gives eigenvalues 0 and -(lambda+mu). Then e^(tQ) has the stationary row (mu, lambda)/(lambda+mu) as the t -> infinity limit, plus a transient decaying like e^(-(lambda+mu)t); the relaxation time is 1/(lambda+mu).
Eigenvalue 0 carries the equilibrium; the nonzero eigenvalue's magnitude is the rate of relaxation to it.
On infinite state spaces e^(tQ) is not the term-by-term power series of an unbounded operator; it is defined via semigroup theory and coincides with the minimal transition function only when the chain is non-explosive. And it is the matrix exponential, never the entrywise exponential of tQ.