Abstract
Let SH and S ̃ H be 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 d s d t , T > 0 , where δ denotes the Dirac delta function at zero. By elementary inequalities, we show that ℓ T exists in L2 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.2010 Mathematics Subject Classification: 60G15; 60F25; 60G18; 60J55.
Highlights
The intersection properties of Brownian motion paths have been investigated since the forties, and since a large number of results on intersection local times of Brownian motion have been accumulated
The intersection local time of independent fractional Brownian motions has been studied by Chen and Yan [7], Nualart et al [8], Rosen [9], Wu and Xiao [10] and the references therein
As for applications in physics, the Edwards’ model of long polymer molecules by Brownian motion paths uses the intersection local time to model the ‘excluded volume’ effect: different parts of the molecule should not be located at the same point in space, while Symanzik [11], Wolpert [12] introduced the intersection local time as a tool in constructive quantum field theory
Summary
The intersection properties of Brownian motion paths have been investigated since the forties (see [1]), and since a large number of results on intersection local times of Brownian motion have been accumulated (see Wolpert [2], Geman et al [3], Imkeller et al [4], de Faria et al [5], Albeverio et al [6] and the references therein). Δ SHt − S Hs dsdt, T > 0, where δ denotes the Dirac delta function at zero. We give the regularity of the intersection local time process. The intersection local time of independent fractional Brownian motions has been studied by Chen and Yan [7], Nualart et al [8], Rosen [9], Wu and Xiao [10] and the references therein.
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