×

An application of the Gaussian correlation inequality to the small deviations for a Kolmogorov diffusion. (English) Zbl 1490.60069

A Kolmogorov diffusion \((X_t)_{t\in[0,T]}\) is the stochastic process \(X_t=(X_1(t),\ldots,X_d(t))\) on \(\mathbb R^d\) whose marginals are iterated integrated Brownian motions given recursively by \(X_1(t)=B_t\) and \(X_{k+1}(t)=\int_0^t X_k(s)\,ds\), where \((B_t)_{t\geq0}\) is a standard one-dimensional Brownian motion. The author proves by an application of the Gaussian correlation inequality that the Kolmogorov diffusion fulfills the same small deviation principle as \(d\)-dimensional Brownian motion. As a consequence, Chung’s law of the iterated logarithm as \(T\to\infty\) and \(T\downarrow 0\) are provided for Kolmogorov diffusion. It turns out that as \(T\downarrow0\) the rate and constant in Chung’s law are the same as for \(d\)-dimensional Brownian motion, whereas rate and constant coincide with those of the iterated integrated Brownian motion \(X_d(t)\) as \(T\to\infty\).

MSC:

60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems
60G15 Gaussian processes
60J65 Brownian motion
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Gérard Ben Arous and Jing Wang, Very rare events for diffusion processes in short time, 2019.
[2] Marco Carfagnini, Juraj Földes, and David P. Herzog, A functional law og the iterated logarithm for weakly hypoelliptic diffusions at time zero, 2021. · Zbl 1490.60201
[3] Marco Carfagnini and Maria Gordina, Small deviations and chung’s law of iterated logarithm for a hypoelliptic brownian motion on the heisenberg group, 2020. · Zbl 1493.60064
[4] Xia Chen and Wenbo V. Li, Quadratic functionals and small ball probabilities for the m-fold integrated Brownian motion, Ann. Probab. 31 (2003), no. 2, 1052-1077. · Zbl 1030.60026
[5] Alejandro de Acosta, Small deviations in the functional central limit theorem with applications to functional laws of the iterated logarithm, Ann. Probab. 11 (1983), no. 1, 78-101. · Zbl 0504.60033
[6] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. · Zbl 0156.10701
[7] Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, second ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam, 1989. · Zbl 0684.60040
[8] Davar Khoshnevisan and Zhan Shi, Chung’s law for integrated Brownian motion, Trans. Amer. Math. Soc. 350 (1998), no. 10, 4253-4264. · Zbl 0902.60031
[9] A. Kolmogoroff, Über das Gesetz des iterierten Logarithmus, Ann. of Math. 35 (1934), no. 1, 116-117. · Zbl 0008.39906
[10] James Kuelbs and Wenbo V. Li, Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal. 116 (1993), no. 1, 133-157. · Zbl 0799.46053
[11] James Kuelbs and Wenbo V. Li, Small ball estimates for Brownian motion and the Brownian sheet, J. Theoret. Probab. 6 (1993), no. 3, 547-577. · Zbl 0780.60079
[12] RafałLatał a and Dariusz Matlak, Royen’s proof of the Gaussian correlation inequality, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2169, Springer, Cham, 2017, pp. 265-275. · Zbl 1366.60058
[13] W. V. Li and Q.-M. Shao, Gaussian processes: inequalities, small ball probabilities and applications, Stochastic processes: theory and methods, Handbook of Statist., vol. 19, North-Holland, Amsterdam, 2001, pp. 533-597. · Zbl 0987.60053
[14] Wenbo V. Li, A Gaussian correlation inequality and its applications to small ball probabilities, Electron. Comm. Probab. 4 (1999), 111-118. · Zbl 0937.60026
[15] Wenbo V. Li and Werner Linde, Approximation, metric entropy and small ball estimates for Gaussian measures, Ann. Probab. 27 (1999), no. 3, 1556-1578. · Zbl 0983.60026
[16] Ditlev Monrad and Holger Rootzén, Small values of Gaussian processes and functional laws of the iterated logarithm, Probab. Theory Related Fields 101 (1995), no. 2, 173-192. · Zbl 0821.60043
[17] Bruno Rémillard, On Chung’s law of the iterated logarithm for some stochastic integrals, Ann. Probab. 22 (1994), no. 4, 1794-1802. · Zbl 0840.60030
[18] Thomas Royen, A simple proof of the Gaussian correlation conjecture extended to some multivariate gamma distributions, Far East J. Theor. Stat. 48 (2014), no. 2, 139-145. · Zbl 1314.60070
[19] Qi Man Shao, A note on small ball probability of a Gaussian process with stationary increments, J. Theoret. Probab. 6 (1993), no. 3, 595-602. · Zbl 0776.60050
[20] Michel Talagrand, The small ball problem for the Brownian sheet, Ann. Probab. 22 (1994), no. 3, 1331-1354. · Zbl 0835.60031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.