An absorbing eigentime identity

International audienceConsider a finite irreducible Markov process $X$. Sampling two points $x$ and $y$ independently according to the invariant measure, the eigentime identity states that the expected time for $X$ to go from $x$ to $y$ is equal to the sum of the inverses of the non-zero eigenvalues of the (opposite of the) underlying generator. This short paper gives a simple proof of this equality and propose a new extension to the finite absorbing irreducible Markov framework, in continuous and discrete times