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Intersection local times and Tanaka formulas. (English) Zbl 0798.60072

Double points of three-dimensional Brownian motion and \(k\)-multiple points of two-dimensional Brownian motion for every positive integer \(k\) are considered. Then a local time for these multiple points, that is, a functional that increases only at the times when Brownian motion has a multiple point, is constructed. A new approach to intersection local times (ILT) of Brownian motion is given by using additive functionals of a single Markov process and stochastic calculus. New results include the Tanaka formula for the \(k\)-multiple points of self-intersection local time. It is also proved that one can renormalize ILT for \(k\)-multiple points in terms of lower order ILTs in such a way that the renormalized ILT is jointly Hölder continuous in every variable almost surely for every \(k>1\).

MSC:

60J55 Local time and additive functionals
60J65 Brownian motion
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