Remarks on the intersection local time of fractional Brownian motions

Let BHBH and B̃H be two independent dd-dimensional fractional Brownian motions with Hurst parameter H∈(0,1)H∈(0,1). Assume that d≥2d≥2. In this paper we consider the so-called intersection local time I(BH,B̃H)≡∫0T∫0Tδ(BtH−B̃sH)dsdt, where δδ denotes the Dirac delta function. We prove the existence of the random variable in L2L2. As a related problem, we also discuss the necessary and sufficient conditions for I(BH,B̃H) to be smooth in the sense of Meyer–Watanabe. The condition says that it is smooth if and only if H<2d+2.