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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.

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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).

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    $\begingroup$ +1 thank you. This is exactly what I was looking for $\endgroup$ Commented May 25, 2021 at 16:33

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