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

Aurélien Deya (1) , David Nualart (2) , Samy Tindel (3, 1)

1 IECL - Institut Élie Cartan de Lorraine
2 Department of Mathematics [Lawrence, Kansas]
3 BIGS - Biology, genetics and statistics

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.

Mots clés: Malliavin calculus; Brownian motion; local time

Domaines: Probabilités [math.PR]; Mathématiques générales [math.GM]

Aurélien Deya, David Nualart, Samy Tindel. On L2 modulus of continuity of Brownian local times and Riesz potentials. Annals of Probability, 2015, 43 (3), pp.1493-1534. ⟨10.1214/13-AOP904⟩. ⟨hal-00814171v2⟩