Polynomial bounds in the Ergodic Theorem for positive recurrent one-dimensional diffusions and integrability of hitting times

Abstract

Let $X$ be a one dimensional positive recurrent diffusion with initial distribution $\nu$ and invariant probability $\mu$. Suppose that for some $p> 1$, $\exists a\in\R$ such that $\forall x\in\R, \E_x T_a^p<\infty$ and $\E_\nu T_a^{p/2}<\infty$, where $T_a$ is the hitting time of $a$. For such a diffusion, we derive non asymptotic deviation bounds of the form $$\P_{\nu} (|\frac1t\int_0^tf(X_s)ds-\mu(f)|\geq\ge)\leq K(p)\frac1{t^{p/2}}\frac 1{\ge^p}A(f)^p.$$ Here $f$ bounded or bounded and compactly supported and $A(f)=\|f\|_{\infty}$ when $f$ is bounded and $A(f)=\mu(|f|)$ when $f$ is bounded and compactly supported. We also give, under some conditions on the coefficients of $X$, a polynomial control of $\E_xT_a^p$ from above and below. This control is based on a generalized Kac's formula (see theorem \ref{thm:mainKac}) for the moments $\E_x f(T_a)$ of a differentiable function $f$.