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

Let $X$ be a one dimensional positive recurrent diffusion with initial distribution $\nu$ and invariant probability $\mu$. Suppose that for some $p> 1$,$\exists a\in\R$suchthat$\forall x\in\R, \E_x T_a^p<\infty$and$\E_\nu T_a^{p/2}<\infty$,where$T_a$isthehittingtimeof$a$. For such a diffusion, we derive non asymptotic deviation bounds of the form $${\mathrm{\P }}_{\nu }(\mathrm{\mid }\frac{1}{t}{\int }_0^{t}f(X_s)ds-\mu (f)\mathrm{\mid }\ge \ge )\le K(p)\frac{1}{t^{p\mathrm{/}2}}\frac{1}{{\ge }^p}A(f{)}^p\mathrm{.}$$ Here $f$ bounded or bounded and compactly supported and $A(f)=\mathrm{\parallel }f{\mathrm{\parallel }}_{\mathrm{\infty }}$ when $f$ is bounded and $A(f)=\mu (\mathrm{\mid }f\mathrm{\mid })$ when $f$ is bounded and compactly supported. We also give, under some conditions on the coefficients of $X$, a polynomial control of ${\text{E}}_x{T}_a^p$ from above and below. This control is based on a generalized Kac’s formula (see theorem \ref{thm:mainKac}) for the moments $\E_x f(T_a)$ofadifferentiablefunction$f$.