Renormalized self-intersection local time of bifractional Brownian motion

Abstract

Abstract Let BH,K={BH,K(t),t≥0} $B^{H,K}=\{B^{H,K}(t), t \geq 0\}$ be a d-dimensional bifractional Brownian motion with Hurst parameters H∈(0,1) $H\in (0,1)$ and K∈(0,1] $K\in (0,1]$. Assuming d≥2 $d\geq 2$, we prove that the renormalized self-intersection local time ∫0T∫0tδ(BH,K(t)−BH,K(s))dsdt−E(∫0T∫0tδ(BH,K(t)−BH,K(s))dsdt) $$\begin{aligned} \int^{T}_{0} \int^{t}_{0}\delta \bigl(B^{H,K}(t)-B^{H,K}(s) \bigr)\,ds\,dt-\mathbb{E} \biggl( \int^{T}_{0} \int^{t}_{0}\delta \bigl(B^{H,K}(t)-B^{H,K}(s) \bigr)\,ds\,dt \biggr) \end{aligned}$$ exists in L2 $L^{2}$ if and only if HKd<3/2 $HKd< 3/2$, where δ denotes the Dirac delta function. Our work generalizes the result of the renormalized self-intersection local time for fractional Brownian motion.

Keywords

• Bifractional Brownian motion
• Self-intersection local time
• Renormalization
• Existence