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Self-intersection local time of fractional Brownian motions – via chaos expansion. (English) Zbl 1008.60091
The article is devoted to the fractional Brownian motion X in R d , which is defined as d one-dimensional independent fractional Brownian motions with the Hurst parameter H(0,1)· The main object is the renormalized self-intersection local time 0 T δ(X t -X s )dsdt-E 0 T δ(X t -X s )dsdt. The author proves, that under the condition H<min(3/(2d),2/(d+2)) the renormalized local time lies in the space D 1,2 of smooth Gaussian functionals. This result is obtained via using the estimation of the norm in D 1,2 for the random variable F by the derivative d/du(Γ(u)F). Here Γ is the operator of the second quantization. The author gives conditions for the convergence in D 1,2 of the random variables 0 T P ε (X t -X s )dsdt-E 0 T P ε (X t -X s )dsdt. Here P ε is the Gaussian density, which approximates the delta-function. The proof of convergence uses the local nondeterminism property of the fractional Brownian motion.
MSC:
60J65Brownian motion
60J55Local time, additive functionals
60H07Stochastic calculus of variations and the Malliavin calculus