Intersection Local Time for Two Independent Fractional Brownian Motions

Abstract

Let B H and e B H be two independent, d-dimensional fractional Brownian motions with Hurst parameter H ∈ (0, 1) . Assume d ≥ 2. We prove that the intersection local time of B H and e B H I(B H , e B H ) = Z T 0 Z T 0 δ(B H t − e B H s )dsdt exists in L 2 if and only if Hd < 2.

Smoothness for the collision local time of two multidimensional bifractional Brownian motions

“…we show that it exists in L 2 if and only if HK < 2/d (this result is in accordance with the paper Nualart et al [11]), and it is smooth in the sense of Meyer-Watanabe if and only if HK < 2/(d + 2).…”

Collision local times of two independent fractional Brownian motions

“…On the other hand, renormalized functions of local time need be subtracted because the functions in high dimensions fail to exist without subtractions (see [9,14]). …”

Limit theorems for functionals of two independent Gaussian processes

“…Let X be an independent copy of X. When X and X are fBms, we know that the intersection local time of X and X does not exist if Hd = 2 ( [10,15]), and this is called the critical case. If X and X are fBms with H ≤ 1/2, the following convergence in law was obtained in [2].…”