Lévy processes with respect to the Whittaker convolution

Sousa, Rúben; Guerra, Manuel; Yakubovich, Semyon

doi:10.1090/tran/8294

Lévy processes with respect to the Whittaker convolution
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by Rúben Sousa, Manuel Guerra and Semyon Yakubovich PDF

Trans. Amer. Math. Soc. 374 (2021), 2383-2419 Request permission

Abstract:

It is natural to ask whether it is possible to construct Lévy-like processes where actions by random elements of a given semigroup play the role of increments. Such semigroups induce a convolution-like algebra structure in the space of finite measures.

In this paper, we show that the Whittaker convolution operator, related with the Shiryaev process, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. We then introduce the class of Lévy processes with respect to the Whittaker convolution and study their basic properties. We obtain a martingale characterization of the Shiryaev process analogous to Lévy’s characterization of Brownian motion.

Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed for differential operators whose associated convolution does not satisfy the usual compactness assumption on the support of the convolution.

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