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Intersection local time for two independent fractional Brownian motions. (English) Zbl 1154.60028

For two independent \(d\)-dimensional fractional Brownian motions \(B^1, B^2\) with Hurst parameter \(H\in(0,1)\) it is shown that, for \(d\geq 2\) the mutual intersection local time exists as \(L^2\)-limit \[ \lim_{\epsilon\downarrow 0} \int_0^T\int_0^T p_\epsilon(B_t^1-B_s^2)\, ds dt, \] where \(p_\epsilon\) is the Gaussian density with variance \(\epsilon\), if and only if \(dH<2\).

MSC:

60G15 Gaussian processes
60F25 \(L^p\)-limit theorems
60G18 Self-similar stochastic processes
60J55 Local time and additive functionals
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