Lévy–Khinchin Formula and Existence of Densities for Convolution Semigroups on Symmetric Spaces

Liu, Ming; Wang, Longmin

doi:10.1007/s11118-007-9048-2

Cited by 16 publications

(29 citation statements)

References 9 publications

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“…An interesting case is obtained by assuming that the standard irreducibility conditions hold. It then follows by Theorems 3.3 (3) and 3.3 (see also arguments in [1], [16] and section 5.5 in [15]) that the generator…”

Section: Applications To Feller Processesmentioning

confidence: 97%

The positive maximum principle on symmetric spaces

Applebaum

Ngan

2020

Positivity

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We investigate the Courrège theorem in the context of linear operators that satisfy the positive maximum principle on a space of continuous functions over a symmetric space. Applications are given to Feller-Markov processes. We also introduce Gangolli operators, which satisfy the positive maximum principle, and generalise the form associated with the generator of a Lévy process on a symmetric space. When the space is compact, we show that Gangolli operators are pseudodifferential operators having scalar symbols.

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Section: Applications To Feller Processesmentioning

confidence: 97%

The positive maximum principle on symmetric spaces

Applebaum

Ngan

2020

Positivity

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“…for each σ ∈ G, k ∈ K where Ad is the adjoint representation of G. It follows from work of Gangolli [10] (see also [17] and [1]) that b = 0, the matrix a = cI where c ≥ 0 and ν is a spherical measure i.e. that ν(…”

Section: Case 2: Compact Symmetric Pairsmentioning

confidence: 99%

“…There is a one to one correspondence between convolution semigroups (µ t , t ≥ 0) of spherical measures on G and convolution semigroups (κ t , t ≥ 0) of K-invariant measures on G/K (in the sense of [17]) given by…”

mentioning

confidence: 99%

Some L 2 properties of semigroups of measures on Lie groups

Applebaum

2008

Semigroup Forum

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We investigate the induced action of convolution semigroups of probability measures on Lie groups on the L 2 -space of Haar measure. Necessary and sufficient conditions are given for the infinitesimal generator to be self-adjoint and the associated symmetric Dirichlet form is constructed. We show that the generated Markov semigroup is traceclass if and only if the measures have a square-integrable density. Two examples are studied in some depth where the spectrum can be explicitly computed, these being the n-torus and Riemannian symmetric pairs of compact type.

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“…His approach makes extensive use of Fourier analysis -especially the Peter Weyl theorem -and requires the hypoellipticity assumption to hold for the diffusion part. More recently, working together with L.Wang [15], he has obtained densities of convolution semigroups of probability measures on symmetric spaces. The main purpose of this note is to adapt Liao's approach to arbitrary probability measures on a compact group.…”

Section: Introductionmentioning

confidence: 99%