scholarly journals ASPECTS OF RECURRENCE AND TRANSIENCE FOR LÉVY PROCESSES IN TRANSFORMATION GROUPS AND NONCOMPACT RIEMANNIAN SYMMETRIC PAIRS

2013 ◽  
Vol 94 (3) ◽  
pp. 304-320 ◽  
Author(s):  
DAVID APPLEBAUM
Keyword(s):  
Riemannian Manifolds ◽  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Transformation Groups ◽  
Compact Sets ◽  
Topological Transformation ◽  
Local Integrability ◽  
Potential Measure ◽  
Complete Riemannian Manifolds

AbstractWe study recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. In particular the transience–recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of noncompact type are transient. We show that we also have ‘harmonic transience’, that is, local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli’s Lévy–Khinchine formula.

scholarly journals The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of Lévy Processes

2019 ◽  
Vol 169 (1) ◽  
pp. 59-77
Author(s):  
Loïc Chaumont ◽  
Jacek Małecki
Keyword(s):  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Time Variable ◽  
Stable Processes ◽  
The Laplace Transform ◽  
Entrance Law ◽  
Generalized Eigenfunctions ◽  
Positive Half ◽  
Integral Formulae

Abstract We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.

scholarly journals Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions

2010 ◽  
Vol 47 (04) ◽  
pp. 1023-1033 ◽  
Author(s):  
A. Kuznetsov
Keyword(s):  
Lévy Processes ◽  
Series Representation ◽  
Theta Functions ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Transcendental Equation ◽  
Simple Expression ◽  
Lévy Measure ◽  
Infinite Products ◽  
Wide Range

In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.

scholarly journals Isotropic Lévy processes on Riemannian manifolds

2000 ◽  
Vol 28 (1) ◽  
pp. 166-184 ◽  
Author(s):  
D. Applebaum ◽  
A. Estrade
Keyword(s):  
Riemannian Manifolds ◽  
Lévy Processes ◽  
Levy Processes

scholarly journals Wiener-Hopf Factorization for a Family of Lévy Processes Related to Theta Functions

2010 ◽  
Vol 47 (4) ◽  
pp. 1023-1033 ◽  
Author(s):  
A. Kuznetsov
Keyword(s):  
Lévy Processes ◽  
Series Representation ◽  
Theta Functions ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Transcendental Equation ◽  
Simple Expression ◽  
Lévy Measure ◽  
Infinite Products ◽  
Wide Range

In this paper we study the Wiener-Hopf factorization for a class of Lévy processes with double-sided jumps, characterized by the fact that the density of the Lévy measure is given by an infinite series of exponential functions with positive coefficients. We express the Wiener-Hopf factors as infinite products over roots of a certain transcendental equation, and provide a series representation for the distribution of the supremum/infimum process evaluated at an independent exponential time. We also introduce five eight-parameter families of Lévy processes, defined by the fact that the density of the Lévy measure is a (fractional) derivative of the theta function, and we show that these processes can have a wide range of behavior of small jumps. These families of processes are of particular interest for applications, since the characteristic exponent has a simple expression, which allows efficient numerical computation of the Wiener-Hopf factors and distributions of various functionals of the process.

scholarly journals Exit problems for general draw-down times of spectrally negative Lévy processes

2019 ◽  
Vol 56 (2) ◽  
pp. 441-457 ◽  
Author(s):  
Bo Li ◽  
Nhat Linh Vu ◽  
Xiaowen Zhou
Keyword(s):  
Lévy Processes ◽  
Exit Time ◽  
Levy Processes ◽  
Laplace Transforms ◽  
Scale Functions ◽  
Dynamic Level ◽  
Running Maximum ◽  
Potential Measure ◽  
Exit Problems ◽  
Spectrally Negative Lévy Processes

AbstractFor spectrally negative Lévy processes, we prove several fluctuation results involving a general draw-down time, which is a downward exit time from a dynamic level that depends on the running maximum of the process. In particular, we find expressions of the Laplace transforms for the two-sided exit problems involving the draw-down time. We also find the Laplace transforms for the hitting time and creeping time over the running-maximum related draw-down level, respectively, and obtain an expression for a draw-down associated potential measure. The results are expressed in terms of scale functions for the spectrally negative Lévy processes.

scholarly journals More on hypergeometric Lévy processes

2016 ◽  
Vol 48 (A) ◽  
pp. 153-158
Author(s):  
Emma L. Horton ◽  
Andreas E. Kyprianou
Keyword(s):  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Stable Processes ◽  
Short Article ◽  
The Family ◽  
Self Similar ◽  
Definition Of ◽  
Complex Valued ◽  
Independent Parameters

AbstractKuznetsov and co-authors in 2011‒14 introduced the family of hypergeometric Lévy processes. They appear naturally in the study of fluctuations of stable processes when one analyses stable processes through the theory of positive self-similar Markov processes. Hypergeometric Lévy processes are defined through their characteristic exponent, which, as a complex-valued function, has four independent parameters. In 2014 it was shown that the definition of a hypergeometric Lévy process could be taken to include a greater range of the aforesaid parameters than originally specified. In this short article, we push the parameter range even further.

scholarly journals Estimates of heat kernels of non-symmetric Lévy processes

2021 ◽  
Vol 33 (5) ◽  
pp. 1207-1236
Author(s):  
Tomasz Grzywny ◽  
Karol Szczypkowski
Keyword(s):  
Lévy Processes ◽  
Characteristic Exponent ◽  
Levy Processes ◽  
Heat Kernels ◽  
Probability Measures ◽  
Convolution Semigroups

Abstract We investigate densities of vaguely continuous convolution semigroups of probability measures on ℝ d {{\mathbb{R}^{d}}} . First, we provide results that give upper estimates in a situation when the corresponding jump measure is allowed to be highly non-symmetric. Further, we prove upper estimates of the density and its derivatives if the jump measure compares with an isotropic unimodal measure and the characteristic exponent satisfies a certain scaling condition. Lower estimates are discussed in view of a recent development in that direction, and in such a way to complement upper estimates. We apply all those results to establish precise estimates of densities of non-symmetric Lévy processes.

On correlating Lévy processes

2010 ◽  
Vol 13 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Ernst Eberlein ◽  
Dilip Madan
Keyword(s):  
Lévy Processes ◽  
Levy Processes
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