A Dirichlet principle for non reversible Markov chains and some recurrence theorems

Alexandre Gaudilliere (1, 2) , Claudio Landim (2, 3)

1 I2M - Institut de Mathématiques de Marseille
2 IMPA - Instituto Nacional de Matemática Pura e Aplicada
3 LMRS - Laboratoire de Mathématiques Raphaël Salem

We extend the Dirichlet principle to non-reversible Markov processes on countable state spaces. We present two variational formulas for the solution of the Poisson equation or, equivalently, for the capacity between two disjoint sets. As an application we prove some recurrence theorems. In particular, we show the recurrence of two-dimensional cycle random walks under a second moment condition on the winding numbers.

Mots clés: Recurrence; Markov process; Potential theory; Non-reversible; Dirichlet principle; Recurrence.

Domaines: Probabilités [math.PR]

Alexandre Gaudilliere, Claudio Landim. A Dirichlet principle for non reversible Markov chains and some recurrence theorems. Probability Theory and Related Fields, 2014, 158, pp.55-89. ⟨hal-00648042⟩