Geodesic
On the small balls problem for equivalent Gaussian measures
Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 683-705 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\mu$. 
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     title = {On the small balls problem for equivalent {Gaussian} measures},
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     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_5_a2/}
}
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V. I. Bogachev. On the small balls problem for equivalent Gaussian measures. Sbornik. Mathematics, Tome 189 (1998) no. 5, pp. 683-705. http://geodesic.mathdoc.fr/item/SM_1998_189_5_a2/

[1] Sytaya G. N., “Ob asimptotike mery malykh sfer dlya gaussovskikh protsessov, ekvivalentnykh brounovskomu”, Teoriya veroyatn. i matem. statist., 1977, no. 19, 128–133 | MR

[2] Nagaev S. V., “Ob asimptotike vinerovskoi mery uzkoi polosy”, Teoriya veroyatn. i ee primen., 26:3 (1981), 630 | MR

[3] Novikov A. A., “Malye ukloneniya gaussovskikh protsessov”, Matem. zametki, 29:2 (1981), 291–301 | MR | Zbl

[4] Li W., “Comparison results for the lower tail of Gaussian seminorm”, J. Theoret. Probab., 5 (1992), 1–32 | DOI | MR

[5] Lifshits M. A., Gaussovskie sluchainye funktsii, TViMS, Kiev, 1995 | Zbl

[6] Linde W., “Comparison results for the small ball behavior of Gaussian random variables”, Probability in Banach spaces, 9, eds. J. Hoffmann-Jorgensen, J. Kuelbs, M. B. Marcus, Birkhäuser, Boston, 1994, 273–297 | MR

[7] Bogachev V. I., “Funktsii Onzagera–Maklupa dlya gaussovskikh mer”, Dokl. AN, 344:4 (1995), 439–441 | MR | Zbl

[8] Bogachev V. I., Gaussovskie mery, Fizmatlit, M., 1997 | MR | Zbl

[9] Kuelbs J., Li W., “Metric entropy and the small ball problem for Gaussian measures”, J. Funct. Anal., 116:1 (1993), 133–157 | DOI | MR | Zbl

[10] Shao Q.-M., Wang D., “Small ball probabilities of Gaussian fields”, Probab. Theory Related Fields, 102 (1995), 511–517 | DOI | MR | Zbl

[11] Shi Z., “Small ball probabilities for a Wiener process under weighted sup-norms, with an application to the supremum of Bessel local times”, J. Theoret. Probab., 9:4 (1996), 915–929 | DOI | MR | Zbl

[12] Stolz W., “Some small ball probabilities for Gaussian processes under nonuniform norms”, J. Theoret. Probab., 9:3 (1996), 613–630 | DOI | MR | Zbl

[13] Cirelson B. S., Ibragimov I. A., Sudakov V. N., “Norms of Gaussian sample functions”, Lect. Notes in Math., 550, 1976, 20–41 | MR | Zbl

[14] Shiryaev A. N., Veroyatnost, Nauka, M., 1989 | MR

[15] Vakhaniya N. N., Tarieladze V. I., Chobanyan S. A., Veroyatnostnye raspredeleniya v banakhovykh prostranstvakh, Nauka, M., 1984 | MR | Zbl

[16] Titchmarsh E., Teoriya funktsii, Nauka, M., 1980 | MR | Zbl

[17] Shepp L. A., Zeitouni O., “A note on conditional exponential moments and the Onsager Machlup functional”, Ann. Probab., 20:1 (1992), 652–654 | DOI | MR | Zbl

[18] Shepp L. A., “Radon–Nikodym derivatives of Gaussian measures”, Ann. Math. Statist., 37:2 (1966), 321–354 | DOI | MR | Zbl

[19] Shepp L. A., Zeitouni O., “Exponential estimates for convex norms and some applications”, Barcelona Seminar on Stochastic Analysis, Progr. Probab., 32, Birkhäuser, Boston, 1993, 203–215 | MR | Zbl

[20] Šidák Z., “On multivariate normal probabilities of rectangles: their dependence on correlations”, Ann. Math. Statist., 39:5 (1968), 1425–1434 | MR

[21] Preiss D., “Gaussian measures and covering theorems”, Comment. Math. Univ. Carolin., 20:1 (1979), 95–99 | MR | Zbl

[22] Preiss D., “Gaussian measures and the density theorems”, Comment. Math. Univ. Carolin., 22:1 (1981), 181–193 | MR | Zbl

[23] Preiss D., “Differentiation of measures in infinitely dimensional spaces”, Topol. and Meas. III, Proc. Conf., Part 2 (Vitte/Hiddensee, Oct. 19–25, 1980), Wissen. Beitrage d. Greifswald Univ., Greifswald, 1982, 201–207 | MR | Zbl