Lévy Processes in Lie Groups

Liu, Ming

doi:10.1017/cbo9780511546624.003

Cited by 30 publications

(85 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…for some constants a ij with a ij = a ji and X ∈ p, where X l denote the left invariant vector field on G induced by X for any X ∈ g. The K-invariance of T implies that X = Ad(k)X for k ∈ K. Since K Ad(k)Xdk = 0 (see [8,Lemma 7.2]), where dk denotes the normalized Haar measure on K, we see that X = 0.…”

Section: Generator Of Lévy Processmentioning

confidence: 99%

“…In this case, G is a connected noncompact semisimple Lie group with a finite center, and K is a maximal compact subgroup (necessarily connected). The path limiting properties of the Lévy process x t in G/K may be derived directly from those of a Lévy process g t in G, established in Liao [8] following the basic ideas for G-valued random walks in Guivarc'h-Raugi [3]. In this paper, for the Lévy process x t in G/K, we will first obtain some of these properties, including the radial convergence, directly without resorting to the results in [8].…”

mentioning

confidence: 99%

“…The path limiting properties of the Lévy process x t in G/K may be derived directly from those of a Lévy process g t in G, established in Liao [8] following the basic ideas for G-valued random walks in Guivarc'h-Raugi [3]. In this paper, for the Lévy process x t in G/K, we will first obtain some of these properties, including the radial convergence, directly without resorting to the results in [8]. We will then derive the remaining path limiting properties from the results in [8].…”

mentioning

confidence: 99%

“…In this paper, for the Lévy process x t in G/K, we will first obtain some of these properties, including the radial convergence, directly without resorting to the results in [8]. We will then derive the remaining path limiting properties from the results in [8]. An important advantage of our approach is that the complicated conditions such as the total irreducibility and existence of a contracting sequence in [8,Theorem 6.4] need not be assumed, and the integrability condition in [8,Theorem 7.1] is also weakened.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Limiting Properties of Lévy Processes in Symmetric Spaces of Noncompact Type

Liu

Wang

2012

Stoch. Dyn.

View full text Add to dashboard Cite

show abstract

Section: Generator Of Lévy Processmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Limiting Properties of Lévy Processes in Symmetric Spaces of Noncompact Type

Liu

Wang

2012

Stoch. Dyn.

View full text Add to dashboard Cite

show abstract

“…We restrict our attention to the discrete-time case. Although some particular continuous-time stochastic models have been proposed for manifolds, e.g., Lévy processes in Lie groups [5], the specific AR discrete-time case has received little attention (also, there are some striking differences between continuous-time and discretetime stochastic processes on manifolds: see [4] for an example on the unit-circle).…”

Section: Introductionmentioning

confidence: 99%

On the Generalization of AR Processes To Riemannian Manifolds

Manton

2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings

View full text Add to dashboard Cite

The autoregressive (AR) process is fundamental to linear signal processing and is commonly used to model the behaviour of an object evolving on Euclidean space. In real life, there are myriad examples of objects evolving not on flat spaces but on curved spaces such as the surface of a sphere. For instance, wind-direction studies in meteorology and the estimation of relative rotations of tectonic plates based on observations on the Earth's surface deal with spherical data, while subspace tracking in signal processing is actually inference on the Grassmann manifold. This paper considers how to extend the AR process to one evolving on a curved space, or in a general, a manifold. Doing so is non-trivial, and in fact, several different extensions are proposed, along with their advantages and disadvantages. Algorithms for estimating the parameters of these generalized AR processes are derived.

show abstract

Markovian extensions of symmetric second order elliptic differential operators

Posilicano

2014

Math. Nachr.

View full text Add to dashboard Cite

Key words Self-adjoint extensions, Markovian operators, elliptic differential operators MSC (2010) Primary, 47B25, Secondary, 47B38, 35J25Let ⊂ R n be bounded with a smooth boundary and let S be the symmetric operator in L 2 ( ) given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self-adjoint extensions of S by providing an explicit one-to-one correspondence between such extensions and the class of Dirichlet forms in L 2 ( ) which are additively decomposable by the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Kreȋn's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell-type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.

show abstract

Lévy Processes in Lie Groups

Cited by 30 publications

References 0 publications

Limiting Properties of Lévy Processes in Symmetric Spaces of Noncompact Type

Limiting Properties of Lévy Processes in Symmetric Spaces of Noncompact Type

On the Generalization of AR Processes To Riemannian Manifolds

Markovian extensions of symmetric second order elliptic differential operators

Contact Info

Product

Resources

About