On the Poisson equation for nonreversible Markov jump processes

We study the solution V of the Poisson equation LV + f = 0 where L is the backward generator of an irreducible (finite) Markov jump process and f is a given centered state function. Bounds on V are obtained using a graphical representation derived from the Matrix Forest Theorem and using a relation with mean first-passage times. Applications include estimating time-accumulated differences during relaxation toward a steady nonequilibrium regime. That includes obtaining bounds for the quasipotential which controls the thermal response.