Self-intersection local time of fractional Brownian motions-via chaos expansion

Abstract

Let $B_{1,t}^{H},\ldots ,B_{d,t}^{H}$ be $d$ independent fractional Brownian motions with Hurst parameter $H\in (0, 1)$. Denote $X_{t}=(B_{1,t}^{H},\ldots ,B_{d,t}^{H})$ and let $\delta$ be the Dirac delta function. It is shown that when $H< \mathrm{min}(3/(2d), 2/(d+2))$, the (renormalized) self-intersection local time of fractional Brownian motion, $\int _{0}^{T}\int _{0}^{t}\delta (X_{t}-X_{s})dsdt-\mathbb{E}\int _{0}^{T}\int _{0}^{t}\delta (X_{t}-X_{s})dsdt$, is in $D_{1,2}$, where $D_{1,2}$ is the Meyer-Watanabe test functional space, i.e. the $L^{2}$ space of “differentiable” functionals, whose precise meaning is given in Section 2.

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 49 citations exist upstream.