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Local time for Gaussian processes as an element of Sobolev space. (English) Zbl 1331.60154
Summary: We consider local time for a Gaussian process with values in \(\mathbb R^d\). We define it as a limit of the standard approximations in Sobolev space. We also study renormalization of local time, by which we mean the modification of the standard approximations by subtracting a finite number of the terms of its Itō-Wiener expansion. We prove that renormalized local time exists and is continuous in Sobolev space under a certain condition on the covariation of the process (the condition is general and includes the non-renormalized local time case). This condition is also necessary for the existence of local time if we consider renormalized local time at zero for a zero-mean Gaussian process. We use our general result to obtain a necessary and sufficient condition for the existence of renormalized local time and self-intersection local time for fractional Brownian motion in \(\mathbb R^d\).

60J55 Local time and additive functionals
60G15 Gaussian processes
60G22 Fractional processes, including fractional Brownian motion
46N30 Applications of functional analysis in probability theory and statistics
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60H07 Stochastic calculus of variations and the Malliavin calculus