Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times

Löcherbach, Eva; Loukianova, Dasha; Loukianov, Oleg

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 2, pp. 425-449.

Considérons une diffusion récurrente positive avec loi initiale ν et probabilité invariante μ. Pour tout a∈ℝ, soit Ta le temps d'atteinte du point a. Supposons qu'il existe p>1 et un point a∈ℝ tels que pour tout x∈ℝ, et . Alors nous obtenons l'inégalité de déviation non-asymptotique suivante: ℙν(|(1/t)∫0tf(Xs) ds-μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p, où f est une fonction bornée ou une fonction bornée à support compact. Ici, A(f)=‖f‖∞ dans le cas d'une fonction bornée et A(f)=μ(|f|) dans le cas d'une fonction bornée à support compact. De plus, sous certaines conditions sur les coefficients de la diffusion, nous obtenons une minoration et majoration, polynomiale en x, de . Ce résultat est basé sur une formule de Kac généralisée (voir théorème 4.1) pour les moments où f est une fonction dérivable.

Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p>1, ∃a∈ℝ such that ∀x∈ℝ, and , where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙν(|(1/t)∫0tf(Xs) ds-μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(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 from above and below. This control is based on a generalized Kac's formula (see Theorem 4.1) for the moments of a differentiable function f.

Classification : 60F99, 60J55, 60J60
Mots clés : diffusion process, recurrence, additive functionals, ergodic theorem, polynomial convergence, hitting times, Kac formula, deviations inequalities

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