Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Lower tails of quadratic functionals of symmetric stable processes

Auteur(s):

Code(s) de Classification MSC:

• 60J30 Processes with independent increments

Résumé: Let $\xi_{\alpha,\mu}= \int_0^1 X_\alpha^2 \d \mu$, where $X_\alpha =\{ X_\alpha(t); \, t\in [0,1]\}$ is a symmetric stable process of index $\alpha \in (0,2]$, and $\mu$ is a finite measure on $[0,1]$. Under general assumptions upon $\mu$, we prove that $\lim_{x\to 0} \, x^{\alpha/2}\, \log \p( \xi_{\alpha, \mu} < x)= -c_{\alpha, \mu}$. The constant $c_{\alpha, \mu}$ is formulated as the solution to a variational problem, and its value is explicitly given when $X_\alpha$ is a symmetric Cauchy process ($\alpha=1$) or Brownian motion ($\alpha=2$).

Mots Clés: Small ball probability; stable process; quadratic functional

Date: 1999-02-02

Prépublication numéro: PMA-483