On the fractional derivative of Brownian local times

Local times of fractional Brownian sheets

Let BH 0 = fBH 0 (t); t 2 RN+g be a real-valued fractional Brownian sheet. Consider the (N; d) Gaussian random eld BH de ned by BH(t) = (BH 1 (t); : : : ; BH d (t)) (t 2 RN+ ); where BH 1 ; : : : ; BH d are independent copies of BH 0 . In this paper, the existence and joint continuity of the local times of BH are established. Running Title: Local Times of Fractional Brownian Sheets

Joint continuity of the local times of fractional Brownian sheets

Let B = {B(t), t∈ RN+} be an (N,d)-fractional Brownian sheet with index H = (H1, . . . ,HN) ∈ (0,1) N defined by B(t) = (B 1 (t), . . . ,B H d (t)) (t∈ R N + ), where B H 1 , . . . ,B H d are independent copies of a real-valued fractional Brownian sheet B 0 . We prove that if d < ∑ N l=1 H l , then the local times of B are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab...

Regularity of Intersection Local Times of Fractional Brownian Motions

Let Bi be an (Ni, d)-fractional Brownian motion with Hurst index αi (i = 1,2), and let B1 and B2 be independent. We prove that, if N1 α1 + N2 α2 > d , then the intersection local times of B1 and B2 exist, and have a continuous version. We also establish Hölder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and int...

Large deviations for the local and intersection local times of fractional Brownian motions

Large deviation principle for the non-linear functionals of non-Markovian models is a challenging subject. A class of such models are Gaussian processes. Among them, the fractional Brownian motions are perhaps the most important processes. In this talk, I will talk about some recent progress achieved in the large deviations for local times and intersection local times of fractional Brownian mot...

The Supremum of Brownian Local Times