Cylindrical Lévy processes in Banach spaces
Cylindrical Lévy processes in Banach spaces
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Banach 空间中的圆柱 Lévy 过程
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
M. Riedle
中科院分区：
文献类型：
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作者：
D. Applebaum;M. Riedle
圆柱概率测度是Banach空间上的σ可加测度，它具有到所有维欧氏空间的σ可加投影。它们自然地与弱（圆柱）随机变量的概念相关联，因此与弱（圆柱）随机过程相关联。本文主要研究圆柱Lévy过程。它们有（弱）Lévy-Itô分解和一个相关的Lévy-Khintchine公式。如果过程是弱平方可积的，则其协方差算子可以用于构造再生核希尔伯特空间，其中过程具有从不相关的真实一维Lévy过程序列构建的无穷级数的分解。这个系列是用来定义圆柱随机积分，从圆柱Ornstein-Uhlenbeck过程可以构造为相关的柯西问题的唯一解决方案。我们证明了这样的过程是圆柱马尔可夫过程，并研究其（圆柱）不变的措施。
Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma‐additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Lévy processes. These have (weak) Lévy–Itô decompositions and an associated Lévy–Khintchine formula. If the process is weakly square‐integrable, its covariance operator can be used to construct a reproducing kernel Hilbert space in which the process has a decomposition as an infinite series built from a sequence of uncorrelated bona fide one‐dimensional Lévy processes. This series is used to define cylindrical stochastic integrals from which cylindrical Ornstein–Uhlenbeck processes may be constructed as unique solutions of the associated Cauchy problem. We demonstrate that such processes are cylindrical Markov processes and study their (cylindrical) invariant measures.