Large Deviations for Local Times of Stable Processes and Stable Random Walks in 1 Dimension

Abstract

In Chen and Li (2004), large deviations were obtained for the spatial L p norms of products of independent Brownian local times and local times of random walks with finite second moment. The methods of that paper depended heavily on the continuity of the Brownian path and the fact that the generator of Brownian motion, the Laplacian, is a local operator. In this paper we generalize these results to local times of symmetric stable processes and stable random walks.

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

“…Once again this is dealt with in [5]. However, in the case β d considered in this paper, where local times do not exist, if in (1.5) we use a single process rather than p independent processes, the limit blows up.…”

“…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.…”

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

“…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].…”

“…More precisely, I t /t p−(p−1)/α converges in distribution to the L p -norm of the stable limiting process local time (see lemme 6 in [23], or lemma 14 in [15]). …”

“…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],…”