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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 $H\in \left(0,1\right)·$ The main object is the renormalized self-intersection local time ${\int }_{0}^{T}\delta \left({X}_{t}-{X}_{s}\right)dsdt-E{\int }_{0}^{T}\delta \left({X}_{t}-{X}_{s}\right)dsdt$. The author proves, that under the condition $H 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 $d/du\left(\parallel {\Gamma }\left(\sqrt{u}\right)F\parallel \right)$. Here ${\Gamma }$ is the operator of the second quantization. The author gives conditions for the convergence in ${D}_{1,2}$ of the random variables ${\int }_{0}^{T}{P}_{\epsilon }\left({X}_{t}-{X}_{s}\right)dsdt-E{\int }_{0}^{T}{P}_{\epsilon }\left({X}_{t}-{X}_{s}\right)dsdt$. Here ${P}_{\epsilon }$ is the Gaussian density, which approximates the delta-function. The proof of convergence uses the local nondeterminism property of the fractional Brownian motion.
##### MSC:
 60J65 Brownian motion 60J55 Local time, additive functionals 60H07 Stochastic calculus of variations and the Malliavin calculus