Inversions of Lévy Measures and the Relation Between Long and Short Time Behavior of Lévy Processes

Abstract.For Hunt processes on Rd, strong and weak transience is defined by finiteness and infiniteness, respectively, of the expected last exit times from closed balls. Under some condition, which is satisfied by Lévy processes and Ornstein-Uhlenbeck type processes, this definition is expressed in terms of the transition probabilities. A criterion is given for strong and weak transience of Ornstein-Uhlenbeck type processes on Rd, using their Lévy measures and coefficient matrices of linear drift terms. An example is discussed.

Download Full-text

Typical long-time behavior of ground state-transformed jump processes

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500020 ◽

2019 ◽

Vol 22 (02) ◽

pp. 1950002 ◽

Cited By ~ 1

Author(s):

Kamil Kaleta ◽

József Lőrinczi

Keyword(s):

Ground State ◽

Lévy Processes ◽

Control Function ◽

Ground States ◽

Levy Processes ◽

Jump Processes ◽

Time Behavior ◽

Long Time Behavior ◽

Long Time

We consider a class of Lévy-type processes derived via a Doob transform from Lévy processes conditioned by a control function called potential. These ground state transformed processes (also called [Formula: see text]-processes) have position-dependent and generally unbounded components, with stationary distributions given by the ground states of the Lévy generators perturbed by the potential. We derive precise upper envelopes for the almost sure long-time behavior of these ground state-transformed Lévy processes, characterized through escape rates and integral tests. We also highlight the role of the parameters by specific examples.

Download Full-text

Infinite dimensional analysis of pure jump Lévy processes on the Poisson space

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-14994 ◽

2006 ◽

Vol 98 (2) ◽

pp. 237 ◽

Cited By ~ 6

Author(s):

Arne Løkka ◽

Frank Norbert Proske

Keyword(s):

White Noise ◽

Lévy Processes ◽

Levy Processes ◽

Wick Product ◽

Charlier Polynomials ◽

Starting Point ◽

Poisson Space ◽

Lévy Measures ◽

Distribution Spaces ◽

White Noise Calculus

We develop a white noise calculus for pure jump Lévy processes on the Poisson space. This theory covers the treatment of Lévy processes of unbounded variation. The starting point of the theory is the construction of a distribution space. This space has many of the same nice properties as the classical Schwartz space, but is modified in a certain way in order to be more suitable for pure jump Lévy processes. We apply Minlos's theorem to this space and obtain a white noise measure which satisfies the first condition of analyticity, and which is non-degenerate. Furthermore, we obtain generalized Charlier polynomials for all Lévy measures. We introduce Kondratiev test function and distribution spaces, the $\mathcal{S}$-transform and the Wick product. We proceed by using a transfer principle on Poisson spaces to establish a differential calculus.

Download Full-text

First Passage Problems over Increasing Boundaries for Lévy Processes with Exponentially Decayed Lévy Measures

Theory of Probability and Its Applications ◽

10.1137/s0040585x97t988071 ◽

2017 ◽

Vol 61 (1) ◽

pp. 140-151

Author(s):

Sh. Kaji

Keyword(s):

Lévy Processes ◽

Levy Processes ◽

First Passage ◽

Lévy Measures

Download Full-text

On association and other forms of positive dependence for Feller processes

Journal of Applied Probability ◽

10.1017/jpr.2019.36 ◽

2019 ◽

Vol 56 (2) ◽

pp. 624-646

Author(s):

Eddie Tu

Keyword(s):

State Space ◽

Lévy Processes ◽

Levy Processes ◽

Poisson Processes ◽

Jump Processes ◽

Uhlenbeck Process ◽

Positive Dependence ◽

Positive Orthant ◽

Ornstein Uhlenbeck Process ◽

Lévy Measures

AbstractWe characterize various forms of positive dependence, such as association, positive supermodular association and dependence, and positive orthant dependence, for jump-Feller processes. Such jump processes can be studied through their state-space dependent Lévy measures. It is through these Lévy measures that we will provide our characterization. Finally, we present applications of these results to stochastically monotone Feller processes, including Lévy processes, the Ornstein–Uhlenbeck process, pseudo-Poisson processes, and subordinated Feller processes.

Download Full-text