An Absorbing Eigentime Identity

#### L. Miclo

2015, v.21, Issue 2, 249-262

ABSTRACT

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 proposes a new extension to the finite absorbing irreducible Markov framework, in continuous and discrete times.

Keywords: finite ergodic/absorbing Markov process, eigentime identity, invariant probability, quasi-invariant probability, first Dirichlet eigenvalue/eigenvector, algebraic spectrum