Continuous Convolution Semigroups Research Articles

Lévy processes: Concentration function and heat kernel bounds

We investigate densities of vaguely continuous convolution semigroups of probability measures on the Euclidean space. We expose that many typical conditions on the characteristic exponent repeatedly used in the literature of the subject are equivalent to the behaviour of the maximum of the density as a function of time variable. We also prove qualitative lower estimates under mild assumptions on the corresponding jump measure and the characteristic exponent.

Convolution of Probability Measures on Lie Groups and Homogenous Spaces

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.

Uniqueness of the embedding continuous convolution semigroup of a Gaussian probability measure on the affine group and an application in mathematical finance

Let \(\{\mu _{t}^{(i)}\}_{t\ge 0}\) (\(i=1,2\)) be continuous convolution semigroups (c.c.s.) of probability measures on \(\mathbf{Aff(1)}\) (the affine group on the real line). Suppose that \(\mu _{1}^{(1)}=\mu _{1}^{(2)}\). Assume furthermore that \(\{\mu _{t}^{(1)}\}_{t\ge 0}\) is a Gaussian c.c.s. (in the sense that its generating distribution is a sum of a primitive distribution and a second-order differential operator). Then \(\mu _{t}^{(1)}=\mu _{t}^{(2)}\) for all \(t\ge 0\). We end up with a possible application in mathematical finance.

Mehler semigroups, Ornstein-Uhlenbeck processes and background driving Lévy processes on locally compact groups and on hypergroups

For finite dimensional vector spaces it is well-known that there exists a 1-1-correspondence between distributions of Ornstein-Uhlenbeck type processes (w.r.t. a fixed group of automorphisms) and (background driving) Lévy processes, hence between M- or skew convolution semigroups on the one hand and continuous convolution semigroups on the other. An analogous result could be proved for simply connected nilpotent Lie groups. Here we extend this correspondence to a class of commutative hypergroups.

Convolution semigroups of states

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C 0-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.

Equivalent weights and standard homomorphisms for convolution algebras on ℝ+

AbstractWe take a second look at two basic topics in the study of weighted convolution algebras L1(ω) on ℝ+. An early result showed that one could replace the weight $\omega$ with a very well-behaved weight without changing the space L1(ω) as long as L1(ω) was an algebraffi We prove the analogous result for measure algebras when M(ω) is an algebraffi This allows us to preserve not only the norm topology but also the relative weak* topology on L1(ω). A homomorphism between weighted convolution algebras is said to be standard if it preserves generators of dense principal ideals. The original proofs of standardness and its variants are all based on finding the generator of a particular strongly continuous convolution semigroup. In this paper we give much simpler direct proofs of these results. We also improve the statement and proof of the theorem, giving useful properties equivalent to the standardness of a homomorphism.

Uniqueness of Embedding into a Gaussian Semigroup and a Poisson Semigroup with Determinate Jump Law on a Simply Connected Nilpotent Lie Group

Let {μ (i) t } t≥0 (i=1,2) be continuous convolution semigroups (c.c.s.) on a simply connected nilpotent Lie group G. Suppose that μ (1)1 =μ (2)1 . Assume furthermore that one of the following two conditions holds: (i) The c.c.s. {μ (1) t } t≥0 is a Gaussian semigroup (in the sense that its generating distribution just consists of a primitive distribution and a second-order differential operator) (ii) The c.c.s. {μ (i) t } t≥0 (i=1,2) are both Poisson semigroups, and the jump measure of {μ (1) t } t≥0 is determinate (i.e., it possesses all absolute moments, and there is no other nonnegative bounded measure with the same moments). Then μ (1) t =μ (2) t for all t≥0. As a complement, we show how our approach can be directly used to give an independent proof of Pap’s result on the uniqueness of the embedding Gaussian semigroup on simply connected nilpotent Lie groups. In this sense, our proof for the uniqueness of the embedding semigroup among all c.c.s. of a Gaussian measure can be formulated self-contained.

Uniqueness of embedding of Gaussian probability measures into a continuous convolution semigroup on simply connected nilpotent Lie groups

Let { μ t ( i ) } t ⩾ 0 ( i = 1 , 2 ) be continuous convolution semigroups on a simply connected nilpotent Lie group G. Suppose that μ 1 ( 1 ) = μ 1 ( 2 ) and that { μ t ( 1 ) } t ⩾ 0 is a Gaussian semigroup (in the sense that its generating distribution just consists of a primitive distribution and a second order differential operator). Then μ t ( 1 ) = μ t ( 2 ) for all t ⩾ 0 . To cite this article: D. Neuenschwander, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The central limit problem on locally compact groups

It is shown that the limit μ of a commutative infinitesimal triangular system Δ on a totally disconnected locally compact groupG is embeddable in a continuous one-parameter convolution semigroup if either (1)G is a compact extension of a closed solvable normal subgroup or (2)G is discrete and Δ is normal or (3)G is a discrete linear group over a field of characteristic zero. For a special triangular system of convolution powers\(\left( {\mu _\nu ^{\kappa _\nu } \to \mu ,\mu _n \to \delta _3 } \right)\), the above is shown to hold without any of the conditions (1)–(3). For a general locally compact groupG necessary conditions are obtained for the embeddability of a shift of limit μ of Δ; in particular, the conditions are trivially satisfied whenG is abelian. Also, the embedding of a limit of a symmetric system onG is shown to hold under condition (1) as above.

Analyticity and Injectivity of Convolution Semigroups on Lie Groups

We prove that all continuous convolution semigroups of probability distributions on an arbitrary Lie group are injective. Let {μt, t>0} be a continuous convolution semigroup of probability distributions on a Lie group G. For each t>0, we set Ttf(x)=∫Gf(xy)μt(dy) for a bounded continuous function f. We show that Ttf=0 holds if and only if f=0. This fact will be applied in proving the unique divisibleness of the convolution product for a certain distribution. We show that ν*ξ=ν*ξ′ implies ξ=ξ′, provided that ν is an infinitely divisible distribution on a simply connected nilpotent Lie group.

On the uniqueness problem for continuous convolution semigroups of probability measures on simply connected nilpotent Lie groups

On the uniqueness problem for continuous convolution semigroups of probability measures on simply connected nilpotent Lie groups

Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups

The goal of this paper is to characterise certain probability laws on a class of quantum groups or braided groups that we will call nilpotent. First we introduce a braided analogue of the Heisenberg–Weyl group, which shall serve as standard example. We introduce Gaussian functionals on quantum groups or braided groups as functionals that satisfy an analogue of the Bernstein property, i.e. that the sum and difference of independent random variables are also independent. The corresponding functionals on the braided line, braided plane and a braided q-Heisenberg–Weyl group are determined. Section 5 deals with continuous convolution semigroups on nilpotent quantum groups and braided groups. We extend recent results proving the uniqueness of the embedding of an infinitely divisible probability law into a continuous convolution semigroup for simply connected nilpotent Lie groups to nilpotent quantum groups and braided groups. Finally, in Section 6 we give some indications how the semigroup approach of Heyer and Hazod to the Bernstein theorem on groups can be extended to quantum groups and braided groups.

Gauss laws in the sense of Bernstein and uniqueness of embedding into convolution semigroups on quantum groups and braided groups

The aim of this note is to characterize certain probability laws on a class of quantum groups or braided groups that will be called nilpotent. First, we introduce a braided analogue of the Heisenberg-Weyl group, which shall serve as a standard example. We determine functional on the braided line and on this group satisfying an analogue of the Bernstein property ( see [3]). i.e. that the sum and difference of independent Gaussian random variables are also independent. We also study continuous convolution semigroups on nilpotent quantum groups and braided groups. We extend to nilpotent quantum groups and braided groups recent results proving the uniqueness of the embedding of an infinitely divisible probability law in a continuous convolution semigroup for simply connected nilpotent Lie groups.

$$D$$ -Domains of attraction of stable measures on stratified Lie groupsof attraction of stable measures on stratified Lie groups

LetG be a stratified Lie group and (μt)t ⩾ 0 be a continuous convolution semigroup of probability measures onG. A probability measurev is said to belong to the\(D\)-domain of attraction of μ1, if there exists a sequence (a n ) of positive real numbers such that\(\delta _{a_n } v^n \to \mu _1 \) weakly, where δ1 denotes the natural dilation onG. We prove convergence criteria for discrete convolution semigroups. These are used to obtain a simple necessary and sufficient condition for the existence of sucha n if (μt)t ⩾ 0 has no Gaussian component. For the proof we introduce the notion of regularly varying measures onG and develop the necessary theory of regular variation.

Hemigroups of probability measures on a locally compact group

In contrast to the widely elaborated theory of continuous convolution semigroups of probability measures on a locally compact group G there are only a few results concerning two-parameter semigroups of probability measures («hemigroups»), i.e. families (μ(s,t)) 0≤s≤t of probability measures on G such that (s,t)→μ(s,t) is weakly continous, μ(t,t) is the Dirac measure in the identity of G and μ(r,s)*μ(s,t)=μ(r,t) holds for all 0≤r≤s≤t. We discuss the characterization and generation of continuous hemigroups of probability measures on not necessarily abelian or compact locally compact groups by families of generators of continous convolution semigroups. Our results are based on the corresponding results for evolution families

On local integrability and transience of convolution semigroups of measures

The analytic proof due to M. Ito of the Kesten-Ornstein transience criterion for continuous convolution semigroups of nonnegative contraction measures on a compactly generated Abelian locally compact group has been reworked and given a self-contained form. The new proof still relies on the existence of the equilibrium measure but dispenses with the complete maximum principle.

Measures of finite energy on a commutative hypergroup

Some aspects of potential theory of invariant Dirichlet forms on a commutative hypergroupK are enlarged and extended. In particular measures of finite energy are characterized, and they are related to the problem of finding sufficient conditions for a continuous convolution semigroup onK to be transient.

An explicit L�vy-Hin?in formula for convolution semigroups on locally compact groups

LetG be a locally compact group and (μt)t ⩾ 0 a continuous convolution semigroup of probability measures onG. We show that an operatorN is the infinitesimal generator of (μt)t ⩾ 0 iffN is defined at least on the spaceC2(G) of twice right differentiable functions and if $$\begin{gathered} Nf(x) = \sum\limits_{i \in I} {r_i X_i f(x) + } \sum\limits_{i, j \in I} {a_{\ddot y} X_i X_j f(x)} \hfill \\ + \int_{G\backslash \{ e\} } {\left[ {f(xy) - f()y - \sum\limits_{i \in I} {k_i (y)X_i f(x)} } \right]} \eta (dy) \hfill \\ \end{gathered} $$ holds for allf∈C2(G) andx∈G. Here (Xi)ieI is a projective basis of the Lie algebra ofG and (ki)i∈I is a weak coordinate system ofG with respect to (Xi)i∈I. In analogy to the case of Lie groups,N is determined by a vector r ∈ℝ1, a positive semidefinite symmetric real-valuedI×I-matrix α and a Levy measure η.

Continuous convolution semigroups with unbounded L�vy measures on locally compact groups

Continuous convolution semigroups with unbounded L�vy measures on locally compact groups