scholarly journals Lyapunov-type conditions for non-strong ergodicity of Markov processes

2021 ◽  
Vol 58 (1) ◽  
pp. 238-253
Author(s):  
Yong-Hua Mao ◽  
Tao Wang
Keyword(s):  
Markov Processes ◽  
Riemannian Manifolds ◽  
Diffusion Processes ◽  
Stable Processes ◽  
Strong Ergodicity ◽  
Stable Noise

AbstractWe present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed, including diffusion processes on Riemannian manifolds and Ornstein–Uhlenbeck processes driven by symmetric $\alpha$-stable processes. In particular, we show that any process of d-dimensional Ornstein–Uhlenbeck type driven by $\alpha$-stable noise is not strongly ergodic for every $\alpha\in (0,2]$.

SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 11 (01) ◽  
pp. 21-31 ◽  
Author(s):  
ILYA V. TELYATNIKOV
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Riemannian Manifolds ◽  
Marginal Distribution ◽  
Diffusion Processes ◽  
Ambient Space ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

Proportional intensities and strong ergodicity for Markov processes

1983 ◽  
Vol 20 (01) ◽  
pp. 185-190 ◽  
Author(s):  
Mark Scott ◽  
Dean L. Isaacson
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Homogeneous Markov Process ◽  
Strong Ergodicity ◽  
Transition Functions ◽  
The Matrix ◽  
Intensity Functions ◽  
Time Point

By assuming the proportionality of the intensity functions at each time point for a continuous-time non-homogeneous Markov process, strong ergodicity for the process is determined through strong ergodicity of a related discrete-time Markov process. For processes having proportional intensities, strong ergodicity implies having the limiting matrix L satisfy L · P(s, t) = L, where P(s, t) is the matrix of transition functions.

The age distribution of Markov processes

1977 ◽  
Vol 14 (3) ◽  
pp. 492-506 ◽  
Author(s):  
Benny Levikson
Keyword(s):  
Differential Equations ◽  
Markov Processes ◽  
Difference Equations ◽  
Diffusion Processes ◽  
Genetic Model ◽  
Branching Processes ◽  
Age Distribution ◽  
Renewal Theory ◽  
Gene Frequencies ◽  
Birth And Death Processes

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

scholarly journals A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 279
Author(s):  
Enrica Pirozzi
Keyword(s):  
Probability Density ◽  
Markov Processes ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
First Passage ◽  
Transition Probability Density ◽  
Symmetry Properties ◽  
Time Density

Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the first passage time density through specific boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the first passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.

scholarly journals Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach

2019 ◽  
Vol 53 (4) ◽  
pp. 1347-1375
Author(s):  
Andreas E. Kyprianou ◽  
Victor Rivero ◽  
Weerapat Satitkanitkul
Keyword(s):  
Stationary Distribution ◽  
Markov Processes ◽  
Stable Process ◽  
Stable Processes ◽  
Excursion Theory ◽  
Self Similar ◽  
Isotropic Stable Process

AbstractWe compute explicitly the distribution of the point of closest reach to the origin in the path of any d-dimensional isotropic stable process, with d ≥ 2. Moreover, we develop a new radial excursion theory, from which we push the classical Blumenthal–Getoor–Ray identities for first entry/exit into a ball (cf. Blumenthal et al. Trans. Amer. Math. Soc., 99, 540–554 1961) into the more complex setting of n-tuple laws for overshoots and undershoots. We identify explicitly the stationary distribution of any d-dimensional isotropic stable process when reflected in its running radial supremum. Finally, for such processes, and as consequence of some of the analysis of the aforesaid, we provide a representation of the Wiener–Hopf factorisation of the MAP that underlies the stable process through the Lamperti–Kiu transform. Our analysis continues in the spirit of Kyprianou (Ann. Appl. Probab., 20(2), 522–564 2010) and Kyprianou et al. (2015) in that our methodology is largely based around treating stable processes as self-similar Markov processes and, accordingly, taking advantage of their Lamperti-Kiu decomposition.

The age distribution of Markov processes

1977 ◽  
Vol 14 (03) ◽  
pp. 492-506 ◽  
Author(s):  
Benny Levikson
Keyword(s):  
Differential Equations ◽  
Markov Processes ◽  
Difference Equations ◽  
Diffusion Processes ◽  
Genetic Model ◽  
Branching Processes ◽  
Age Distribution ◽  
Renewal Theory ◽  
Gene Frequencies ◽  
Birth And Death Processes

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

Strong ergodicity for Markov processes by coupling methods

2002 ◽  
Vol 39 (4) ◽  
pp. 839-852 ◽  
Author(s):  
Yong-Hua Mao
Keyword(s):  
Markov Processes ◽  
Sufficient Conditions ◽  
Essential Spectra ◽  
Strong Ergodicity ◽  
One Dimensional ◽  
Coupling Methods ◽  
Explicit Criteria ◽  
Birth Death

In this paper, we apply coupling methods to study strong ergodicity for Markov processes, and sufficient conditions are presented in terms of the expectations of coupling times. In particular, explicit criteria are obtained for one-dimensional diffusions and birth-death processes to be strongly ergodic. As a by-product, strong ergodicity implies that the essential spectra of the generators for these processes are empty.

Sign in / Sign up

Export Citation Format

Share Document