LARGE DEVIATIONS FOR LOCAL TIMES AND INTERSECTION LOCAL TIMES OF FRACTIONAL BROWNIAN MOTIONS AND RIEMANN―LIOUVILLE PROCESSES

In this paper, we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann― Liouville processes. We also show that a fractional Brownian motion and the related Riemann―Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of exponentially randomized local time of a fractional Brownian motion; (2) logarithmic subadditivity of a normalized sequence of moments of exponentially randomized intersection local time of Riemann―Liouville processes; (3) comparison of local and intersection local times based on embedding of a part of a fractional Brownian motion into the reproducing kernel Hilbert space of the Riemann―Liouville process.

[1]  Wenbo V. Li,et al.  Large and moderate deviations for intersection local times , 2004 .

[2]  W. König,et al.  The Universality Classes in the Parabolic Anderson Model , 2005, math/0504102.

[3]  Xia Chen,et al.  Random Walk Intersections: Large Deviations and Related Topics , 2010 .

[4]  T. W. Anderson,et al.  An Introduction to Multivariate Statistical Analysis , 1959 .

[5]  A. Berlinet,et al.  Reproducing kernel Hilbert spaces in probability and statistics , 2004 .

[6]  D. Nualart,et al.  Integral representation of renormalized self-intersection local times , 2008, 0806.3706.

[7]  D. Geman,et al.  A Local Time Analysis of Intersections of Brownian Paths in the Plane , 1984 .

[8]  S. Berman.  Local nondeterminism and local times of Gaussian processes , 1973 .

[9]  I. M. Pyshik,et al.  Table of integrals, series, and products , 1965 .

[10]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[11]  Moderate deviations for a random walk in random scenery , 2006, math/0604293.

[12]  Regularity of Intersection Local Times of Fractional Brownian Motions , 2009, 0904.0949.

[13]  F. Castell,et al.  Random walk in random scenery and self-intersection local times in dimensions d ≥  5 , 2007 .

[14]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[15]  N. Madras,et al.  THE SELF-AVOIDING WALK , 2006 .

[16]  Hölder properties of local times for fractional Brownian motions , 2009 .

[17]  Alan D. Sokal,et al.  Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory , 1992 .

[18]  Peter Mörters.  Random walk intersections: large deviations and related topics (Mathematical Surveys and Monographs 157) , 2011 .

[19]  H. Kesten,et al.  A large-deviation result for the range of random walk and for the Wiener sausage , 2001 .

[20]  J. -. Gall,et al.  Propriétés d'intersection des marches aléatoires , 1986 .

[21]  A. W. Vaart,et al.  Reproducing kernel Hilbert spaces of Gaussian priors , 2008, 0805.3252.

[22]  Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks , 2004, math/0503593.

[23]  Xia Chen,et al.  Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion , 2003 .

[24]  S. Varadhan,et al.  The polaron problem and large deviations , 1981 .

[25]  Yimin Xiao.  Properties of Local Nondeterminism of Gaussian and Stable Random Fields and Their Applications , 2006 .

[26]  Renormalized self-intersection local time for fractional Brownian motion , 2005, math/0506592.

[27]  Wenbo V. Li,et al.  Large Deviations for Local Times of Stable Processes and Stable Random Walks in 1 Dimension , 2005 .

[28]  Intersection Local Time for Two Independent Fractional Brownian Motions , 2007, 0704.1259.

[29]  Q. Shao,et al.  Gaussian processes: Inequalities, small ball probabilities and applications , 2001 .

[30]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[31]  Yimin Xiao.  Strong Local Nondeterminism and Sample Path Properties of Gaussian Random Fields , 2006 .

[32]  Self-intersection local time: Critical exponent, large deviations, and laws of the iterated logarithm , 2004, math/0503592.

[33]  S. Berman.  Local times and sample function properties of stationary Gaussian processes , 1969 .

[34]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[35]  Werner Linde,et al.  Approximation, metric entropy and small ball estimates for Gaussian measures , 1999 .

[36]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[37]  LI WENBOV.,et al.  QUADRATIC FUNCTIONALS AND SMALL BALL PROBABILITIES FOR THE m-FOLD INTEGRATED BROWNIAN MOTION , 2000 .

[38]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[39]  M. Talagrand,et al.  Probability in Banach Spaces: Isoperimetry and Processes , 1991 .

[40]  Parkpoom Phetpradap,et al.  Intersections of random walks , 2011 .

[41]  Deconvolution of fractional brownian motion , 2002 .

[42]  I︠U︡lii︠a︡ S. Mishura.  Stochastic Calculus for Fractional Brownian Motion and Related Processes , 2008 .

[43]  T. Mountford,et al.  A law of the iterated logarithm for fractional Brownian motions , 2008 .

[44]  W. König,et al.  Brownian intersection local times: Upper tail asymptotics and thick points , 2002 .

[45]  J. Rosen.  The intersection local time of fractional Brownian motion in the plane , 1987 .

[46]  J. Kuelbs.  Probability on Banach spaces , 1978 .

[47]  L. Werner.  Existence of small ball constants for fractional Brownian motions , 1998 .

[48]  Zhan Shi,et al.  Annealed deviations of random walk in random scenery , 2004, math/0408327.