On L2 modulus of continuity of Brownian local times and Riesz potentials

Abstract

This article is concerned with modulus of continuity of Brownian local times. Specifically, we focus on three closely related problems: (a) Limit theorem for a Brownian modulus of continuity involving Riesz potentials, where the limit law is an intricate Gaussian mixture. (b) Central limit theorems for the projections of L 2 modulus of continuity for a one-dimensional Brownian motion. (c) Extension of the second result to a two-dimensional Brownian motion. Our proofs rely on a combination of stochastic calculus and Malliavin calculus tools, plus a thorough analysis of singular integrals.1. Introduction. Let {B t , 0 ≤ t ≤ 1} be a standard linear Brownian motion defined on some complete probability space (Ω, F, P). In the sequel, we denote by L t (x) the local time of B at a given point x ∈ R, defined for t ∈ [0, 1]. A nice combination of stochastic calculus, stochastic analysis and evaluation of singularities associated with heat kernels have recently led to a number of interesting limit theorems for quantities related to the family {L t (x); t ∈ [0, 1], x ∈ R}. Let us quote, for instance, the use of Malliavin and stochastic calculus tools in order to get suitably normalized limits for L 2 modulus of continuity (see [6,13]) or third moment in space (cf. [7]) of Brownian local time. Malliavin calculus tools have also been essential in order to generalize the notion of self-intersection local time [5,8] and to obtain central limit theorems for additive functionals [9] of fractional Brownian motion.The current article proposes to take another step into the relationships between Brownian local time and stochastic analysis. Specifically, we shall handle the following problems:

Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion

“…It is surprising to remark that the limit behavior of the chaotic components of α ε is different from that of the whole sequence. This phenomenon was observed, for instance, in the central limit theorem for the second spatial moment of Brownian local time increments (see [4]). However, in this case the limit of the whole sequence is a mixture of Gaussian distributions, whereas in the present paper the normalization of α ε converges to a Gaussian law.…”