Deviation bounds for additive functionals of Markov process

Abstract

In this paper we derive non asymptotic deviation bounds for $${\P }_{\nu }(|\frac{1}{t}{\int }_0^{t}V(X_s)ds-\int Vd\mu |\ge R)$$ where $X$ is a $\mu$ stationary and ergodic Markov process and $V$ is some $\mu$ integrable function. These bounds are obtained under various moments assumptions for $V$, and various regularity assumptions for $\mu$. Regularity means here that $\mu$ may satisfy various functional inequalities (F-Sobolev, generalized Poincaré etc...).