Lévy processes with respect to the Whittaker convolution,Transactions of the American Mathematical Society

当前位置： X-MOL 学术 › Trans. Am. Math. Soc. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

Lévy processes with respect to the Whittaker convolution
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2021-01-21 , DOI: 10.1090/tran/8294
Rúben Sousa , Manuel Guerra , Semyon Yakubovich

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.

[1] D. Bakry and N. Huet, The Hypergroup Property and Representation of Markov Kernels, Séminaire de Probabilités XLI, Springer, Berlin, 2008, pp. 297-347, DOI 10.1007/978-3-540-77913-1_15.
[2]

中文翻译：

关于Whittaker卷积的Lévy过程

摘要：很自然地问，是否有可能构造像Lévy那样的过程，其中给定半群中随机元素的动作起增量作用。这样的半群在有限测度的空间中引发了卷积状的代数结构。在本文中，我们证明了与Shiryaev过程相关的Whittaker卷积算子产生了一种卷积测度代数，该代数具有概率测度的卷积是一种概率测度的性质。然后，我们针对Whittaker卷积介绍Lévy过程的类，并研究它们的基本性质。我们获得了Shiryaev过程的a表征，类似于Lévy对布朗运动的表征。