Convolution Semigroups Research Articles

Density of the Free Additive Convolution of Multi-cut Measures

Abstract We consider the free additive convolution semigroup $\lbrace \mu ^{\boxplus t}:\,t\ge 1\rbrace $ and determine the local behavior of the density of $\mu ^{\boxplus t}$ at the endpoints and at any singular point of its support. We then study the free additive convolution of two multi-cut probability measures and show that its density decays either as a square root or as a cubic root at any endpoint of its support. The probability measures considered in this paper satisfy a power law behavior with exponents strictly between $-1$ and $1$ at the endpoints of their supports.

Convolution semigroups on Rieffel deformations of locally compact quantum groups

Convolution semigroups on Rieffel deformations of locally compact quantum groups

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.

Categorial Independence and Lévy Processes

We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence.

Regularity results for free Lévy processes

Given a free additive convolution semigroup (μt)t≥0 and a probability measure ν on R, we find the necessary and sufficient conditions for the process μt⊞ν to be Lebesgue absolutely continuous with a positive and analytic density throughout R at all time t>0. For semigroups without this property, we find the necessary and sufficient conditions for the density of μt⊞ν to be analytic at its zeros. These results are quantified by the Lévy measure of the semigroup, making it fairly easy to construct many concrete examples. Finally, we show that μt⊞ν has a finite number of connected components in its support if both the Lévy measure of (μt)t≥0 and the initial law ν do.

Definable convolution and idempotent Keisler measures

We initiate a systematic study of the convolution operation on Keisler measures, generalizing the work of Newelski in the case of types. Adapting results of Glicksberg, we show that the supports of definable and finitely satisfiable (or just definable, assuming NIP) measures are nice semigroups, and classify idempotent measures in stable groups as invariant measures on type-definable subgroups. We establish left-continuity of the convolution map in NIP theories, and use it to show that the convolution semigroup on finitely satisfiable measures is isomorphic to a particular Ellis semigroup in this context.

On extremal domains and codomains for convolution of distributions and fractional calculus

It is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution of distributions a well-defined bilinear mapping. The distribution spaces are systematically endowed with topologies and bornologies that make convolution hypocontinuous whenever defined. Largest modules and smallest algebras for convolution semigroups are constructed along the same lines. The fact that extremal domains and codomains for convolution exist within this class of spaces is fundamentally related to quantale theory. The quantale theoretic residual formed from two c-closed spaces is characterized as the largest c-closed subspace of the corresponding space of convolutors. The theory is applied to obtain maximal distributional domains for fractional integrals and derivatives, for fractional Laplacians, Riesz potentials and for the Hilbert transform. Further, maximal joint domains for families of these operators are obtained such that their composition laws are preserved.

Convolution inequalities for Besov and Triebel–Lizorkin spaces, and applications to convolution semigroups

We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel–Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{

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.

Invariant Markov semigroups on quantum homogeneous spaces

Invariance properties of linear functionals and linear maps on algebras of functions on quantum homogeneous spaces are studied, in particular for the special case of expected co-ideal *-subalgebras. Several one-to-one correspondences between such invariant functionals are established. Adding a positivity condition, this yields one-to-one correspondences of invariant quantum Markov semigroups acting on expected co-ideal *-subalgebras and certain convolution semigroups of states on the underlying compact quantum group. This gives an approach to classifying invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces.

Probability density functions and the dynamics of complex systems associated to some classes of non-archimedean pseudo-differential operators

In this article, we study certain p-adic master equations which describe the dynamics of a large class of complex systems such as glasses, macromolecules and proteins. These equations are naturally associated to certain non-archimedean pseudo-differential operators whose symbols are connected via Fourier transform with radial probability density functions defined on the p-adic numbers. We show that the fundamental solutions of these equations are probability measures and determine a convolution semigroup on the p-adic numbers. Also, we show that the classical solution of this equations preserves the mass and satisfies the comparison principle. Moreover, we study some strong Markov processes corresponding to radial probability density functions of linear and logarithmic type.

Non-archimedean generalized Bessel potentials and their applications

This article describes a class of pseudo-differential operators(Aαφ)(x)=Fξ→x−1([max⁡{|ψ1(||ξ||p)|,|ψ2(||ξ||p)|}]−αφˆ(ξ)),φ∈D(Qpn) and α∈C; here [max⁡{|ψ1(||ξ||p)|,|ψ2(||ξ||p)|}]−α is the symbol of the operator Aα. These operators can be seen as a generalization of the Bessel potentials in the p-adic context. We show that the family (Kα)α>0 of convolution kernels attached to generalized Bessel potentials Aα, α>0, determine a convolution semigroup on Qpn. Imposing certain conditions we have that Kα, α>0, is a probability measure on Qpn. Moreover, we will study certain properties corresponding to the Green function of the operator Aα and we show that heat equations, naturally associated to these operators, describes the cooling (or loss of heat) in a given region over time.

Max-convolution semigroups and extreme values in limit theorems for the free multiplicative convolution

We investigate relations between additive convolution semigroups and max-convolution semigroups through the law of large numbers for the free multiplicative convolution. Based on these relations, we give a formula related with the Belinschi–Nica semigroup and the max-Belinschi–Nica semigroup. Finally, we give several limit theorems for classical, free and Boolean extreme values.

Generating Functionals for Locally Compact Quantum Groups

Abstract Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital *-subalgebra with core-like properties in its domain. On the other hand we prove that every normalised, symmetric, hermitian conditionally positive functional on a dense *-subalgebra of the unitisation of the universal C$^*$-algebra of a locally compact quantum group, satisfying certain technical conditions, extends in a canonical way to a generating functional. Some consequences of these results are outlined, notably those related to constructing cocycles out of convolution semigroups.

The log-Lévy moment problem via Berg–Urbanik semigroups

We consider the Stieltjes moment problem for the Berg–Urbanik semigroups which form a class of multiplicative convolution semigroups on $\mathbb R _+$ that is in bijection with the set of Bernstein functions. Berg and Durán (2004) proved that the law of s

A two-parameter extension of Urbanik’s product convolution semigroup

We prove that sna, b = Γan + b/Γb, n = 0, 1, . . ., is an infinitely divisible Stieltjes moment sequence for arbitrary a, b > 0. Its powers sna, bc, c > 0, are Stieltjes determinate if and only if ac ≤ 2. The latter was conjectured in a paper by Lin 2019 in the case b = 1. We describe a product convolution semigroup τca, b, c > 0, of probability measures on the positive half-line with densities eca, b and having the moments sna, bc. We determine the asymptotic behavior of eca, bt for t → 0 and for t → ∞, and the latter implies the Stieltjes indeterminacy when ac > 2. The results extend the previous work of the author and Lopez 2015 and lead to a convolution semigroup of probability densities gca, bxc>0 on the real line. The special case gca, 1xc>0 are the convolution roots of the Gumbel distribution with scale parameter a > 0. All the densities gca, bx lead to determinate Hamburger moment problems.

Logarithmic-Sobolev and multilinear Hölder’s inequalities via heat flow monotonicity formulas

Heat flow monotonicity formulas have evolved in recent years as a powerful tool in deriving functional and geometric inequalities which are in turn useful in mathematical analysis and applications. This paper aims mainly at proving Logarithmic Sobolev and multilinear Hölder’s inequalities through the heat flow method. Precisely, two entropy monotonicity formulas are constructed via the heat flow. It is shown that the first entropy monotonicity formula is intimately related to the concavity of the power of Shannon entropy and Fisher Information, from which the associated logarithmic Sobolev inequality for probability measure in Euclidean setting is recovered. The second monotonicity formula combines very well with convolution and diffusion semigroup properties of the heat kernel to establish the proof of the multilinear Hölder inequalities.

The positive maximum principle on Lie groups

We extend a classical theorem of Courr\`{e}ge to Lie groups in a global setting, thus characterising all linear operators on the space of smooth functions of compact support that satisfy the positive maximum principle. We show that these are L\'{e}vy type operators (with variable characteristics), and pseudo--differential operators when the group is compact. If the characteristics are constant, then the operator is the generator of the contraction semigroup associated to a convolution semigroup of sub--probability measures.

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.

A semigroup approach to nonlinear Lévy processes

We study the relation between Lévy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear Lévy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators (Aλ)λ∈Λ of linear Lévy processes which guarantees the existence of a nonlinear Lévy process such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE ∂tu=supλ∈ΛAλu. The results are illustrated with several examples.