Convolution Semigroups Of Probability Measures Research Articles

Real natural exponential families and generalized orthogonality

In this article, we use the notion of generalized orthogonality for a sequence of polynomials introduced by Bryc, Fakhfakh, and Mlotkowski (2019) to extend the characterizations of the Feinsilver, Meixner, and Shanbhag based on orthogonal polynomials. These new versions subsume the real natural exponential families (NEFs) having polynomial variance function in the mean of arbitrary degree. We also relate generalized orthogonality to Sheffer systems. We show that the generalized orthogonality of Sheffer systems occurs if and only if the associated classical additive convolution semigroup of probability measures generates NEFs with polynomial variance function. In addition, we use the raising and lowering operators for quasi-monomial polynomials associated with NEFs to give a characterization of NEFs with polynomial variance function of arbitrary degree.

Some results in Cauchy-Stieltjes kernel families

In this paper we present two different results in the theory of Cauchy-Stieltjes Kernel (CSK) families. We firstly provide the construction of free Sheffer systems with the theory of CSK families. We associate a free additive convolution semigroup of probability measures to any free Sheffer systems and we prove that this is the only one that leads to an orthogonal free Sheffer systems. We also show that the orthogonality of free Sheffer systems occurs if and only if the associated free additive convolution semigroup of probability measures generates CSK families with quadratic variance function. Secondly, we are interested in the study of boolean additive convolution. Based on the criteria of convergence for a sequence of variance functions we give an approximation of elements of the CSK family generated by the boolean Gaussian distribution and an approximation of elements of the CSK family generated by the boolean Poisson distribution.

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.

Asymptotic Behaviour and Estimates of Slowly Varying Convolution Semigroups

Abstract We study the asymptotic formulas and estimates for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R}^{d}$ under the assumption that its Lévy–Khintchine exponent varies slowly. We derive some new estimates of the transition densities and Green functions.

Probabilistic trace and Poisson summation formulae on locally compact abelian groups

We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the $d$--dimensional torus, and the adelic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on $L^{2}$-space. The Gaussian is a very important example. For rotationally invariant $\alpha$-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adeles, we first investigate these on the $p$--adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adeles whose densities fail to satisfy the probabilistic trace formula.

Convolution Semigroups of Probability Measures on Gelfand Pairs, Revisited

Our goal is to find classes of convolution semigroups on Lie groups G that give rise to interesting processes in symmetric spaces G/K. The K–bi–invariant convolution semigroups are a well–studied example. An appealing direction for the next step is to generalise to right K–invariant convolution semigroups, but recent work of Liao has shown that these are in one–to–one correspondence with K–bi–invariant convolution semigroups. We investigate a weaker notion of right K–invariance, but show that this is, in fact, the same as the usual notion. Another possible approach is to use generalised notions of negative definite functions, but this also leads to nothing new. We finally find an interesting class of convolution semigroups that are obtained by making use of the Cartan decomposition of a semisimple Lie group, and the solution of certain stochastic differential equations. Examples suggest that these are well–suited for generating random motion along geodesics in symmetric spaces.

Probabilistic trace and Poisson summation formulae on locally compact abelian groups

Abstract We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the d-dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on L 2 ${L^{2}}$ -space. The Gaussian is a very important example. For rotationally invariant α-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the p-adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula.

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.

Estimates of transition densities and their derivatives for jump Lévy processes

We give upper and lower estimates of densities of convolution semigroups of probability measures under explicit assumptions on the corresponding Lévy measure and the Lévy–Khinchin exponent. We obtain also estimates of derivatives of densities.

On quantum symmetries of compact metric spaces

An action of a compact quantum group on a compact metric space $(X,d)$ is (D)-isometric if the distance function is preserved by a diagonal action on $X\times X$. We show that an isometric action in this sense has the following additional property: The corresponding action on the algebra of continuous functions on $X$ by the convolution semigroup of probability measures on the quantum group contracts Lipschitz constants. It is, in other words, isometric in another sense due to H. Li, J. Quaegebeur and M. Sabbe; this partially answers a question of D. Goswami. We also introduce other possible notions of isometric quantum action, in terms of the Wasserstein $p$-distances between probability measures on $X$ for $p\ge 1$, used extensively in optimal transportation. It turns out all of these definitions of quantum isometry fit in a tower of implications, with the two above at the extreme ends of the tower. We conjecture that they are all equivalent.

Quantum dynamical semigroups, group representations and convolution semigroups

Semigroups of operators are known to play an important role in theoretical physics. In particular, quantum dynamical semigroups are fundamental in the theory of open quantum systems. We will describe a class of semigroups of operators which has interesting applications, for instance, in quantum information science. Each of these semigroups of operators is generated, in a suitable way, by a representation (or an antirepresentation) of a group in a Banach space and by a convolution semigroup of probability measures on that group. Some significant examples—including a remarkable type of quantum dynamical semigroups introduced by Kossakowski in the pioneering times of the theory of open quantum systems—and their mutual relations will be discussed.

On the embeddability of certain infinitely divisible probability measures on Lie groups

We describe certain sufficient conditions for an infinitely divisible probability measure on a class of connected Lie groups to be embeddable in a continuous one-parameter convolution semigroup of probability measures. (Theorem 1.3). This enables us in particular to conclude the embeddability of all infinitely divisible probability measures on certain Lie groups, including the so called Walnut group (Corollary 1.5). The embeddability is concluded also under certain other conditions (Corollary 1.4 and Theorem 1.6).

Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

We study algebraic and topological properties of the convolution semigroup of probability measures on a topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P(G) over some topological group G if and only if S embeds into the semigroup \(\exp(G)\) of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup S embeds into the functor-semigroup F(G) over a suitable compact topological group G for each weakly normal monadic functor F in the category of compacta such that F(G) contains a G-invariant element (which is an analogue of the Haar measure on G).

On a Certain Class of Semigroups of Operators

We define an interesting class of semigroups of operators in Banach spaces, namely, the randomly generated semigroups. This class contains as a remarkable subclass a special type of quantum dynamical semigroups introduced in the early 1970s by Kossakowski. Each randomly generated semigroup is associated, in a natural way, with a pair formed by a representation or an antirepresentation of a locally compact group in a Banach space and by a convolution semigroup of probability measures on this group. Examples of randomly generated semigroups having important applications in physics are briefly illustrated.

Transition density estimates for jump Lévy processes

Upper estimates of densities of convolution semigroups of probability measures are given under explicit assumptions on the corresponding Lévy measure and the Lévy–Khinchin exponent.

Brownian motion on Lie groups and open quantum systems

We study the twirling semigroups of (super) operators, namely certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.

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.

Estimates of Tempered Stable Densities

Estimates of densities of convolution semigroups of probability measures are given under specific assumptions on the corresponding Lévy measure and the Lévy–Khinchin exponent. The assumptions are satisfied, e.g., by tempered stable semigroups of J. Rosiński.

Some L 2 properties of semigroups of measures on Lie groups

We investigate the induced action of convolution semigroups of probability measures on Lie groups on the L2-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 trace-class 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.

On d-orthogonality of the Sheffer systems associated to a convolution semigroup

In this note we investigate which Sheffer polynomials can be associated to a convolution semigroup of probability measures, usually induced by a stochastic process with stationary and independent increments. From a recent kind of d-orthogonality ( d ∈ { 2 , 3 , … } ), we characterize the associated d-orthogonal polynomials by the class of generating probability measures, which belongs to the natural exponential family with polynomial variance functions of exact degree 2 d - 1 . This extends some results of (classical) orthogonality; in particular, some new sets of martingales are then pointed out. For each integer d ⩾ 2 we completely illustrate polynomials with ( 2 d - 1 )-term recurrence relation for the families of positive stable processes.