Large and moderate deviations for intersection local times

Abstract

We study the large and moderate deviations for intersection local times generated by, respectively, independent Brownian local times and independent local times of symmetric random walks. Our result in the Brownian case generalizes the large deviation principle achieved in Mansmann (1991) for the L 2 -norm of Brownian local times, and coincides with the large deviation obtained by Csörgö, Shi and Yor (1999) for self intersection local times of Brownian bridges. Our approach relies on a Feynman-Kac type large deviation for Brownian occupation time, certain localization techniques from Donsker-Varadhan (1975) and Mansmann (1991), and some general methods developed along the line of probability in Banach space. Our treatment in the case of random walks also involves rescaling, spectral representation and invariance principle. The law of the iterated logarithm for intersection local times is given as an application of our deviation results.

Exponential asymptotics for intersection local times of stable processes and random walks

“…In this case the large deviations and law of the iterated logarithm have been established for a α p (·) in recent work [4] for Brownian motion and [5] for the symmetric stable processes.…”

Random walk in random scenery and self-intersection local times in dimensions d ≥ 5

“…For large deviations of SILT in d = 1, we refer the reader to Mansmann [19], and Chen and Li [11], while in d = 2, this problem is treated in Bass and Chen [5], and in Bass et al [4]. We first present large deviations estimates for the SILT.…”

Exponential moments of self-intersection local times of stable random walks in subcritical dimensions

“…This part is inspired by the proof of Theorem 1.3 in [14]. Let us assume for a while the following theorem:…”

“…This confinement happens with probability of order exp(−t 1−(αq)/d r (αq)/(pd) t ). Precise logarithmic asymptotics were first proved in d = 1 in [14] for α = 2, and later extended in [15] for α > 1. Very recently, the case d ≥ 2, α = 2 was treated in [7] (with the restriction d < 2/(p − 1) < 2q), and [27].…”

“…Typically, this can be done in "small" dimension using the regularity of local times. Roughly speaking, this is the way followed by Bass, Chen & Rosen in a series of works starting with the paper of Chen & Li [14] about the large deviations of the self-intersections (p = 2) of Brownian motion or random walk in d = 1. Later, they studied the cases d = 1, α > d in [15],…”