## Appearance of Forests and Trees in the Solution of the Poisson Equation for 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.

Area: CS52 - Trees and combinatorial probability (Michele Aleandri)

Keywords: Matrix tree theorem, Matrix forest theorem. Poisson equation

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