An absorbing eigentime identity

Laurent Miclo (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Consider 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.

Mots clés: algebraic spectrum; eigentime identity; invariant probability; quasi-invariant probability; first Dirichlet eigenvalue/eigenvector; algebraic spectrum.; finite ergodic/absorbing Markov process

Domaines: Probabilités [math.PR]

Laurent Miclo. An absorbing eigentime identity. Markov Process. Related Fields, 2015. ⟨hal-01067907v2⟩