On the small balls problem for equivalent Gaussian measures

V. I. Bogachev

Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 683-705

Résumé

Let $\mu$ be a centred Gaussian measure in a linear space $X$ with Cameron-Martin space $H$, let $q$ be a $\mu$-measurable seminorm, and let $Q$ be a $\mu$-measurable second-order polynomial. We show that it is sufficient for the existence of the limit $\lim _{\varepsilon \to 0}\mathsf E(\exp Q|q\leqslant \varepsilon)$, where $E$ is the expectation with respect to $\mu$, that the second derivative $D_{\!H}^{\,2}Q$ of the function $Q$ be a nuclear operator on $H$. This condition is also necessary for the existence of the above-mentioned limit for all seminorms $q$. The problem under discussion can be reformulated as follows: study $\lim _{\varepsilon \to 0}\nu (q\leqslant \varepsilon )/\mu (q\leqslant \varepsilon )$ for Gaussian measures $\nu$ equivalent to $\mu$.