Convolution Semigroups

Updated 7 February 2026

Convolution semigroups are families of probability measures defined on groups and spaces that satisfy identity, associativity, and weak continuity, underpinning Lévy processes and Markov transitions.
They extend classical probability frameworks to noncommutative settings, enabling analysis on Lie groups, homogeneous spaces, and quantum groups through Lévy–Khintchine formulas and analytic transforms.
Their study reveals sharp support, density, and smoothing properties with applications in PDE analysis, symbolic calculus on groups like the Heisenberg group, and operator-valued probability.

A convolution semigroup is a family of probability measures or states on a given algebraic structure (typically a group, homogeneous space, or a noncommutative analogue) indexed by a non-negative real parameter and closed under convolution, satisfying an associativity and continuity property. They form the analytic backbone for the theory of Lévy processes, Markov semigroups, random walks, and noncommutative probability. Convolution semigroups unify a vast set of phenomena across classical, free, monotone, and Boolean probability, locally compact groups, Lie groups, quantum groups, and function spaces.

1. Foundational Definitions and Structures

Let $G$ be a locally compact group, and $\mathcal{P}(G)$ the space of Borel probability measures on $G$ . The convolution of $\mu, \nu \in \mathcal{P}(G)$ is

$(\mu * \nu)(A) = \int_G \int_G \mathbf{1}_A(xy)\, \mu(dx)\, \nu(dy),\qquad A\in \mathcal{B}(G),$

making $\mathcal{P}(G)$ a commutative semigroup with identity $\delta_e$ (Dirac mass at the identity). A convolution semigroup is a family $\{ \mu_t : t \geq 0 \}$ such that:

$\mu_0 = \delta_e$ ,
$\mu_s * \mu_t = \mu_{s+t}$ for all $s, t \geq 0$ ,
$t \mapsto \mu_t$ is weakly continuous: $\mu_t \Rightarrow \delta_e$ as $t \downarrow 0$ .

These properties also extend to homogeneous spaces $G/K$ , quantum groups, and beyond, with convolution defined in ways consistent with the algebraic structure (Liao, 2015, Applebaum, 2017, Skalski et al., 2023).

Convolution semigroups appear as the one-parameter laws of Lévy processes, and as transition kernels of translation-invariant Markov semigroups. Their analytic and algebraic structure is often encoded by Lévy–Khintchine-type formulas or by cumulant transforms (e.g., $R$ - or $S$ -transforms in free probability).

2. Structural and Classification Theorems

On commutative or Lie groups, convolution semigroups are classified by a Lévy–Khintchine representation (Liao, 2015, Applebaum, 2017): $\mathcal{F}[\mu_t](\xi) = e^{-t \Psi(\xi)},$ where $\Psi$ is the Lévy–Khintchine exponent determined by a drift, covariance, and Lévy measure (as in the Hunt generator for Lie groups).

On locally compact quantum groups, convolution semigroups of states have a precise correspondence with translation-invariant Markov semigroups and with noncommutative Dirichlet forms, generalizing the classical symmetry to KMS-invariance with respect to Haar weight (Skalski et al., 2017).

In free and other noncommutative probability theories, convolution semigroups are parameterized by analytic functions such as the Voiculescu, $R$ , $S$ , or $\Sigma$ transforms, with subordination techniques providing deep regularity and support structure for the convolution semigroup (see below) (Zhong, 2011, Huang et al., 2013, Deng et al., 2017).

3. Support, Regularity, and Fine Properties

Support Monotonicity: In multiplicative free convolution semigroups, for $\mu_t$ constructed from an initial measure $\mu$ by iterated convolution (free or Boolean), the number of connected components of the support of $\mu_t$ decreases as $t$ increases. This support evolves in a highly regular fashion, and for $t>1$ , the number of components is finite and nonincreasing (Huang et al., 2013).
No Gap Phenomenon: If $\mu_t$ has two atoms at $a < b$ , then $\mu_t((a,b)) > 0$ ; in other words, intervals with atoms at both ends cannot have zero mass in between—ensuring supports cannot be arbitrarily “pinned” at two points (Zhong, 2011).
Continuity: The family $\{\operatorname{supp}(\mu_t)\}_{t > 1}$ varies continuously in the Hausdorff metric, and atoms move continuously as $t$ changes (Deng et al., 2017).
Analyticity and Density: The density of $\mu_t$ is real analytic inside the support arcs, and atomic parts can be determined by explicit analytic conditions (Huang et al., 2013, Zhong, 2011).

4. Convolution Semigroups Beyond the Classical Setting

Quantum and Noncommutative Frameworks

Locally Compact Quantum Groups: On a von Neumann algebra $(M,\Delta)$ with coassociative coproduct, convolution semigroups of states are families $\{ \mu_t \}$ with $\mu_{s+t} = \mu_s * \mu_t$ via $(\mu_s \otimes \mu_t) \circ \Delta$ and $*-$ weak continuity. There is a one-to-one correspondence between $w^*$ -continuous symmetric convolution semigroups and invariant noncommutative Dirichlet forms (Skalski et al., 2017).
Rieffel Deformations: Under Rieffel deformation by a $2$-cocycle, convolution semigroups on quantum groups deform canonically, with a bijection between invariant convolution semigroups pre/post deformation (Skalski et al., 2023).
Operator-valued Probability: Monotone convolution semigroups on operator algebras are implemented by composition semigroups of analytic maps, with Berkson–Porta-type generators corresponding to Lévy–Khintchine data. Monotonically infinitely divisible operator-valued distributions sit in such semigroups (Anshelevich et al., 2014).

Special Geometric and Functional-analytic Settings

Lie Groups and Homogeneous Spaces: On $G/K$ , convolution semigroups factor canonically into an idempotent initial measure (Haar on a compact subgroup) and a continuous semigroup at the identity, and every infinitely divisible law is embedded in a continuous convolution semigroup (Liao, 2015). In the case of symmetric spaces/Gelfand pairs, convolution semigroups are closely related to spherical analysis and generalized Lévy–Khintchine decompositions (Applebaum, 2017).
Heisenberg Group: Via symbolic calculus for convolution operators, semigroups generated by generalized Laplacians on $H^n$ are analyzed as perturbations of Abelian convolution semigroups. Density estimates near the origin and at infinity are controlled sharply via symbol classes (Bekała, 2015).

5. Moment Functionals and Rigidity Results

Monotone homomorphisms—maps $\phi$ from the semigroup of probability measures on $\mathbb{R}$ to $\mathbb{R}$ satisfying both

$\phi(\mu * \nu) = \phi(\mu) + \phi(\nu)$ and monotonicity with respect to stochastic order—are extremely rigid. For $L^p$ -moment semigroups ( $1 \leq p < \infty$ ), the only such homomorphisms are scalar multiples of the mean: $\phi(\mu) = c\,\mathbb{E}[\mu], \quad c \geq 0.$

For the full semigroup $\mathcal{P}(\mathbb{R})$ , or for heavier-tailed classes $L^p$ with $p<1$ , no nontrivial monotone additive functionals exist (Fritz et al., 2019). Intermediate semigroups may admit functionals reflecting tail masses, but such cases are severely restricted.

This rigidity underlines that, for convolution semigroups of interest in analysis and stochastics, the mean is essentially the only order-preserving, additive measure of central tendency.

6. Analytic, PDE, and Functional Inequalities Aspects

Convolution semigroups act as smoothing/propagator operators on various function spaces. Convolution inequalities in Besov and Triebel–Lizorkin scales provide mapping and smoothing properties for semigroups, encompassing Gaussian, poly-harmonic, and stable laws. Acted as linear smoothing flows, these semigroups propagate and possibly improve spatial regularity, quantified by sharp norm estimates (Kühn et al., 2021). This framework applies not only in probability but also in the analysis of PDEs of parabolic and fractional type.

7. Asymptotics, Estimates, and Classifications

Small Time/Space: Detailed asymptotics for transition densities $p_t(x)$ are established for isotropic unimodal convolution semigroups, especially in the slowly varying or scaling regime, e.g.,

$p(t,x) \asymp t\,|x|^{-d}\,\ell(|x|^{-1})\,e^{-t\psi(|x|^{-1})},$

with explicit constants and de Haan-class $\ell$ (Grzywny et al., 2016).

Sharp Two-sided Estimates: For pure-jump Lévy processes, small-time, small/large-space two-sided bounds are given: $p_t(x) \asymp h(t)^{-d}\,\mathbf{1}_{|x| \leq \theta h(t)} + t\,g(x)\,\mathbf{1}_{|x| \geq \theta h(t)},$ where $h(t)$ is determined by the Lévy symbol scaling (Kaleta et al., 2014).
Multiplicative Free Case: In the free multiplicative setting, connections between support, density, and regularity are quantified, with phase transitions in the support as $t$ varies (Huang et al., 2013, Deng et al., 2017).

8. Interactions between Classical, Free, and Other Probability Theories

There exist bijections between classes of classical and free convolution semigroups through complete Bernstein functions:

Bondesson class (classical subordinators with completely monotone Lévy densities) and free-regular convolution semigroups are both classified by complete Bernstein functions (with zero drift) (1810.07044).
Explicit Laplace-transform identities bridge free and classical domains, enabling transfer of results, e.g.,

$\int_0^\infty e^{-w x} p_t(f; dx) = \frac{1}{w} \int_0^\infty v_{tw}^*(f; [0,y]) e^{-y} dy,$

relating the Laplace transforms of free and classical semigroup measures.

9. Remaining Directions and Open Problems

Characterization of non-negative, non-monotone additive functionals (e.g., variance) on semigroups remains open (Fritz et al., 2019).
Extension of classification and rigidity theorems to multivariate, infinite-dimensional, or noncommutative settings.
Classification of intermediate semigroups and the functional-analytic structure in boundary regimes between $L^p$ and $\mathcal{P}(\mathbb{R})$ (Fritz et al., 2019).
Markovian and Dirichlet form structure for quantum groups with more general symmetry or non-invariant types (Skalski et al., 2017).
Non-classical convolution semigroups on non-commutative or deformed homogeneous spaces, including geometric/probabilistic flows on manifolds.

References:

"Monotone homomorphisms on convolution semigroups" (Fritz et al., 2019)
"Convolution of probability measures on Lie groups and homogeneous spaces" (Liao, 2015)
"Convolution Semigroups of Probability Measures on Gelfand Pairs, Revisited" (Applebaum, 2017)
"Small time sharp bounds for kernels of convolution semigroups" (Kaleta et al., 2014)
"Asymptotic behaviour and estimates of slowly varying convolution semigroups" (Grzywny et al., 2016)
"On regularity for measures in multiplicative free convolution semigroups" (Zhong, 2011)
"On the support of measures in multiplicative free convolution semigroups" (Huang et al., 2013)
"Continuity and growth of free multiplicative convolution semigroups" (Deng et al., 2017)
"On free regular and Bondesson convolution semigroups" (1810.07044)
"Operator-Valued Monotone Convolution Semigroups and an Extension of the Bercovici-Pata Bijection" (Anshelevich et al., 2014)
"Symbolic calculus and convolution semigroups of measures on the Heisenberg group" (Bekała, 2015)
"Semigroups of distributions with linear Jacobi parameters" (Anshelevich et al., 2010)
"Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms" (Skalski et al., 2017)
"Convolution semigroups on Rieffel deformations of locally compact quantum groups" (Skalski et al., 2023)
"Measurable centres in convolution semigroups" (Pachl, 2011)
"Convolution inequalities for Besov and Triebel--Lizorkin spaces, and applications to convolution semigroups" (Kühn et al., 2021)