Steady state distribution for Markov semigroup with self-adjoint infinitesimal matrix

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Let $P(t)$ be a differentiable semigroup of Markov transition matrices, $Q$ — its infinitesimal generator matrix. Assuming that $Q$ is a self-adjoint matrix, how can I find the steady state distribution for $P(t)$? And is it true, that there is only one steady distribution for this semigroup?