Convergence of Semigroups of Complex Measures on a Lie Group (original) (raw)
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Convergence of semigroups of measures on a Lie group
A theorem of Siebert asserts that if µn(t) are semigroups of probability measures on a Lie group G, and Pn are the corresponding generating functionals, then µn(t), f − → n µ 0 (t), f , f ∈ C b (G), t > 0, implies π Pn u, v − → n π P 0 u, v , u ∈ C ∞ (π), v ∈ X, for every unitary representation π of G on a Hilbert space X, where C ∞ (π, X) denotes the space of smooth vectors for π. The aim of this note is to give a simple proof of the theorem and propose some improvements. In particular, we completely avoid employing unitary representations by showing simply that under the same hypothesis Pn, f − → n P 0 , f , f ∈ C 2 b (G). As a corollary, the above thesis of Siebert is extended to strongly continuous representations of G on Banach spaces.
0 Convergence of Semigroups of Measures on a Lie Group
2016
A theorem of Siebert asserts that if µn(t) are semigroups of probability measures on a Lie group G, and Pn are the corresponding generating functionals, then µn(t), f − → n µ 0 (t), f , f ∈ C b (G), t > 0, implies π Pn u, v − → n π P 0 u, v , u ∈ C ∞ (π), v ∈ X, for every unitary representation π of G on a Hilbert space X, where C ∞ (π, X) denotes the space of smooth vectors for π. The aim of this note is to give a simple proof of the theorem and propose some improvements. In particular, we completely avoid employing unitary representations by showing simply that under the same hypothesis Pn, f − → n P 0 , f , f ∈ C 2 b (G). As a corollary, the above thesis of Siebert is extended to strongly continuous representations of G on Banach spaces.
Vector Measures of Bounded Semivariation and Associated Convolution Operators
Glasgow Mathematical Journal, 2010
Let G be a compact metrizable abelian group, and let X be a Banach space. We characterize convolution operators associated with a regular Borel X-valued measure of bounded semivariation that are compact (resp; weakly compact) from L1(G), the space of integrable functions on G into L1(G) X, the injective tensor product of L1(G) and X. Along the way we prove a Fourier Convergence theorem for vector measures of relatively compact range that are absolutely continuous with respect to the Haar measure.
Convolution of Probability Measures on Lie Groups and Homogenous Spaces
Potential Analysis, 2015
We study (weakly) continuous convolution semigroups of probability measures on a Lie group G or a homogeneous space G/K, where K is a compact subgroup. We show that such a convolution semigroup is the convolution product of its initial measure and a continuous convolution semigroup with initial measure at the identity of G or the origin of G/K. We will also obtain an extension of Dani-McCrudden's result on embedding an infinitely divisible probability measure in a continuous convolution semigroup on a Lie group to a homogeneous space.
1997
Two types of conditions have been significant when considering the convergence of convolution products of non-identical probability measures on group and semigroups. The essential points of a sequence of measures have been useful in characterizing the supports of the limit measures. Also, enough mass eventually on an idempotent has proven sufficient for convergence in a number of structures. In this paper, both of these types of conditions are analyzed in the context of discrete non-abelian semigroups. In addition, an application to the convergence of non-homogeneous Markov chains is given.
A new approach to the existence of invariant measures for Markovian semigroups
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2019
We give a new, two-step approach to prove existence of finite invariant measures for a given Markovian semigroup. First, we fix a convenient auxiliary measure and then we prove conditions equivalent to the existence of an invariant finite measure which is absolutely continuous with respect to it. As applications, we obtain a unifying generalization of different versions for Harris' ergodic theorem which provides an answer to an open question of Tweedie. Also, we show that for a nonlinear SPDE on a Gelfand triple, the strict coercivity condition is sufficient to guarantee the existence of a unique invariant probability measure for the associated semigroup, once it satisfies a Harnack type inequality with power. A corollary of the main result shows that any uniformly bounded semigroup on L p possesses an invariant measure and we give some applications to sectorial perturbations of Dirichlet forms. Résumé. On établit une approche en deux étapes pour démontrer l'existence des mesure invariantes finies pour un semigroupe de Markov donné. En fixant d'abord une mesure auxiliaire convenable, on démontre ensuite des conditions équivalentes à l'existence d'une mesure invariante finie qui est absolument continue par rapport à elle. Comme applications, on obtient une généralisation unificatrice des diverses versions du théorème ergodique de Harris et on fournit une réponse à une question ouverte de Tweedie. On montre aussi que pour une EDP stochastique sur un triplet de Gelfand, la condition de coercivité stricte est suffisante pour garantir l'existence d'une seule mesure de probabilité pour le semigroupe associé, si une inégalité de type Harnack avec puissance est satisfaite. Un corollaire du résultat central montre que tout semigroupe uniformément borné sur L p possède une mesure invariante ; on donne des applications aux perturbations sectorielles des formes de Dirichlet.
Approximation-theoretic aspects of probabilistic representations for operator semigroups
Journal of Approximation Theory, 1985
In the present paper explicit sharp estimations are provided for the rate of convergence of basically all known representation formulae for operator semigroups which in an earlier paper have been shown to arise from a single general probabilistic representation theorem based on a special version of the weak law of large numbers. As a main tool, sharp estimations for moments and momentgenerating functions of suitable random variables are used. Some of the results are applied to exponential operators as well as to a class of Poisson approximation theorems in probability theory.