# Basic question on continuous time Markov chain

For some reason I cannot find a worked example for reference, so any help in that direction (even just suggestions for a resource) is appreciated. Here is the question:

Let $$X_t$$ be a continuous time Markov chain and suppose $$\{1,\dots,n\}$$ is the finite state space of the process. Suppose that $$X_0=1$$ and let $$Q$$ be the rate matrix and $$P$$ the transition probability matrix. Is there an obvious way to compute $$\mathcal{L}(X_t)$$, the distribution law of $$X_t$$, in terms of $$P$$ and/or $$Q$$ as with discrete time Markov chains? Note that I am not interested in the limiting case of the stationary distribution here.

In the discrete time case, we have something like $$P^k\mu_0 = \mu_k$$ where $$\mu_0$$ is the distribution of $$X_0$$ and similarly $$\mu_k$$ for $$X_k$$.

Edit: I originally wrote $$\mathbb{E}(X_t)$$ instead of $$\mathcal{L}(X_t)$$, but answers in either direction are appreciated.

The continuous case is quite analogous; it is simply $$\pi_0 e^{tQ}$$ (I think it is customary to put the law on the left and the functions on the right by adjointness). That is because the diffusion has to solve Kolmogorov's forward-backward equations. ($$P(t) = e^{tQ}$$ is your transition matrix for time $$t$$, and it satisfies $$\frac{\partial P(t)}{\partial t} = QP(t) = P(t)Q$$). Durrett's Essentials of Stochastic Processes has a discussion of this in chapter 4 (and is available for free online).