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

TL;DR

This paper investigates the modulus of continuity of Brownian local times, establishing limit theorems involving Riesz potentials and extending results from 1D to 2D using advanced stochastic calculus techniques.

Contribution

It introduces new limit theorems for the L2 modulus of continuity of Brownian local times, including Gaussian mixture limits and extensions to higher dimensions.

Findings

Limit law as Gaussian mixture for Brownian modulus of continuity with Riesz potentials

Central limit theorems for L2 modulus of continuity projections in 1D

Extension of CLT results to 2D Brownian motion

Abstract

This article is concerned with modulus of continuity of Brownian local times. Specifically, we focus on 3 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 L2 modulus of continuity for a 1-dimensional Brownian motion. (c) Extension of the second result to a 2-dimensional Brownian motion. Our proofs rely on a combination of stochastic calculus and Malliavin calculus tools, plus a thorough analysis of singular integrals.