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abstract:
Using the fact that in many cases the characteristic exponent of a Levy process give rise to a metric we give a geometric description of a large class of transition functions of Levy processes. The diagonal behaviour is controlled by volume growth of balls measured by the intrinsic metric coming from the characteristic exponent whereas the off-diagonal decay is controlled by a further intrinsic metric and is then essentially as in the Gaussian case. The fact that the Gaussians are essentially fixed points of the Fourier transform leads to the untypical situation that both metrics coincide. The result naturally suggests how to handle more general processes, i.e. jump processes having a suitable symbol : Riemannian geometry should be replaced by a geometry induced by metrics (being x-dependent) being induced by continuous negative definite functions.
(This is joint work with Viktoria Knopova, Sandra Landwehr and Rene Schilling) |
