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

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Let X be a one dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p> 1, ∃ a∈ such that ∀ x∈, ₓ Tₐᵖ<∞ and νTₐᵖ/²<∞, where Tₐ is the hitting time of a. For such a diffusion, we derive non asymptotic deviation bounds of the form ¶ν (|1t∫₀ᵗf(Xₛ)ds-μ(f)|≥≥)≤ K(p)1tᵖ/² 1≥ᵖA(f)ᵖ. Here f bounded or bounded and compactly supported and A(f)=f∞ when f is bounded and A(f)=μ(|f|) when f is bounded and compactly supported. We also give, under some conditions on the coefficients of X, a polynomial control of ₓTₐᵖ from above and below. This control is based on a generalized Kac's formula (see theorem thm:mainKac) for the moments ₓ f(Tₐ) of a differentiable function f.