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 L2 modulus of continuity for a one-dimensional Brownian motion. (c) Extens...

Let τD(Z) is the first exit time of iterated Brownian motion from a domain D ⊂ Rn started at z ∈ D and let Pz [τD(Z) > t] be its distribution. In this paper we establish the exact asymptotics of Pz[τD(Z) > t] over bounded domains as an extension of the result in DeBlassie [14], for z ∈ D Pz[τD(Z) > t] ≈ t exp(− 3 2 πλ 2/3 D t ), as t → ∞. We also study asymptotics of the life time of Brownian-t...

Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton– Watson forests with i.i.d. exponential branch lengths. We give ...

We give potential theoretic estimates for the probability that a set A contains a double point of planar Brownian motion run for unit time. Unlike the probability for A to intersect the range of a Markov process, this cannot be estimated by a capacity of the set A. Instead, we introduce the notion of a capacity with respect to two gauge functions simultaneously. We also give a polar decompositi...

Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forest-valued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton-Watson forests with i.i.d. exponential branch lengths. We give e...

We work out the relation between Chern-Simons, 2d Yang-Mills on the cylinder, and Brownian motion. We show that for the unitary, orthogonal and symplectic groups, various observables in Chern-Simons theory on S3 and lens spaces are exactly given by counting the number of paths of a Brownian particle wandering in the fundamental Weyl chamber of the corresponding Lie algebra. We construct a fermi...

G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion. Its quadratic variation process is also a continuous process with independent and stationary increments. We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limi...

We investigate the sample path properties of Martin-Löf random Brownian motion. We show (1) that many classical results which are known to hold almost surely hold for every Martin-Löf random Brownian path, (2) that the effective dimension of zeroes of a Martin-Löf random Brownian path must be at least 1/2, and conversely that every real with effective dimension greater than 1/2 must be a zero o...

A theorem characterizing fractional Brownian motion by Index Terms -Wavelet transform, fractional Brownian motion.the covariance structure of its wavelet transform is established.

Abstract By the work of P. Lévy, sample paths Brownian motion are known to satisfy a certain Hölder regularity condition almost surely. This was later improved by Ciesielski, who studied these in Besov and Besov-Orlicz spaces. We review results propose new function spaces type, strictly smaller than those Ciesielski which surely lie. In same spirit, we extend Kamont, investigated question for m...