Necessary and sufficient condition for the smoothness of intersection local time of subfractional Brownian motions

摘要

Let S~H and s~H °e two independent d-dimensional sub-fractional Brownian motions with indices H ∈ (0, 1). Assume d ≥ 2, we investigate the intersection local time of subfractional Brownian motions ℓ_t-=∫_0 ~T ∫_0~T δ(S_t~H-S_s~H)dsdt, T > 0, where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that l_T exists in L~2 if and only if Hd <2 and it is smooth in the sense of the Meyer-Watanabe if and only if H < 2/d+2. As a related problem, we give also the regularity of the intersection local time process.