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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 $H\in (0,1).$ The main object is the renormalized self-intersection local time $\int_0^T\delta (X_t-X_s) ds dt- E\int_0^T\delta (X_t-X_s) ds dt$. 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 $d/du(\|\Gamma (\sqrt{u})F\|)$. Here $\Gamma$ is the operator of the second quantization. The author gives conditions for the convergence in $D_{1,2}$ of the random variables $\int_0^T P_\varepsilon (X_t-X_s) ds dt- E\int_0^T P_\varepsilon (X_t-X_s) ds dt$. Here $P_\varepsilon $ is the Gaussian density, which approximates the delta-function. The proof of convergence uses the local nondeterminism property of the fractional Brownian motion.

60J65Brownian motion
60J55Local time, additive functionals
60H07Stochastic calculus of variations and the Malliavin calculus