Semiuniform semigroups and convolution

Asymptotic behaviour and estimates of slowly varying convolution semigroups

We prove the asymptotic formulas 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 also derive some new estimates of the transition densities and Green functions.

Density and tails of unimodal convolution semigroups

We give sharp bounds for the isotropic unimodal probability convolution semigroups when their Lévy-Khintchine exponent has Matuszewska indices strictly between 0 and 2.

Asymptotic behavior of densities of unimodal convolution semigroups

We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R}^d$ under the assumption that its Lévy--Khintchine exponent is regularly varying of index between $0$ and $2$.

(Co-)homology of Rees semigroup algebras

Let $S$ be a Rees semigroup, and let $\ell^1(S)$ be its convolution semigroup algebra. Using Morita equivalence we show that bounded Hochschild homology and cohomology of $\ell^1(S)$ is isomorphic to those of the underlying discrete group algebra.

One-parameter convolution semigroups of rapidly decreasing distributions

Let $\Mmm$ denote the set of $\mm$ matrices with complex entries, and let $\calG(\partial_1,...,\partial_n)$ be an $\mm$ matrix whose entries are partial differential operators on $\Rn$ with constant complex coefficients. It is proved that $\calG(\partial_1,...,\partial_n)\otimes δ$ is the generating distribution of a smooth one-parameter convolution semigroup of $\Mmm$-valued rapidly decreasing distributions on $\Rn$ if and only if $$\sup_{(ξ_1,...,ξ_n)\in\Rn}\hReσ(\calG(iξ_1,...,iξ_n))<\infty.$$ Applications to systems of partial differential operators with constant coefficients are consider…

Symbolic calculus and convolution semigroups of measures on the Heisenberg group

Let $P$ be a generalized laplacian on $R^{2n+1}$. It is known that $P$ is the generating functional of semigroups of measures $μ_{t}$ on the Heisenberg group $H^{n}$ and $ν_{t}$ on the Abelian group $R^{2n+1}$. Under some smoothness and growth conditions on the functional $P$ expressed in terms of its Abelian Fourier transform $\widehat{P}$ we show that the semigroup $μ_{t}$ is a kind of perturbation of the semigroup $ν_{t}$. More precisely, we give pointwise estimates for the difference of the densities of the measures $μ_{t}$ and $ν_{t}$. As a consequence we get a description of the asymptot…

Umbral Calculus and Cancellative Semigroup Algebras

We describe some connections between three different fields: combinatorics (umbral calculus), functional analysis (linear functionals and operators) and harmonic analysis (convolutions on group-like structures). Systematic usage of cancellative semigroup, their convolution algebras, and tokens between them provides a common language for description of objects from these three fields. Keywords: cancellative semigroups, umbral calculus, harmonic analysis, token, convolution algebra, integral transform

Preduals of semigroup algebras

For a locally compact group $G$, the measure convolution algebra $M(G)$ carries a natural coproduct. In previous work, we showed that the canonical predual $C_0(G)$ of $M(G)$ is the unique predual which makes both the product and the coproduct on $M(G)$ weak$^*$-continuous. Given a discrete semigroup $S$, the convolution algebra $\ell^1(S)$ also carries a coproduct. In this paper we examine preduals for $\ell^1(S)$ making both the product and the coproduct weak$^*$-continuous. Under certain conditions on $S$, we show that $\ell^1(S)$ has a unique such predual. Such $S$ include the free semigro…

Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms

The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a Lévy process on a locally compact group. In particular a definitive one-to-one correspondence between symmetric convolution semigroups of states and noncommutative Dirichlet forms satisfying the natural translation invariance property is established, extending earlier partial results and providing a powerful tool to analyse such semigroups. This is then applied to provide new characterisations of the Haagerup Property and Property …

On orbits of truncated convolution operators

We prove that a semigroup generated by a finitely many truncated convolution operators on $L^p[0,1]$ with $1\leq p<\infty$ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

Simplicial cohomology of band semigroup algebras

We establish simplicial triviality of the convolution algebra $\ell^1(S)$, where $S$ is a band semigroup. This generalizes results of the first author [Glasgow Math. J. 2005, Houston J. Math. 2010]. To do so, we show that the cyclic cohomology of this algebra vanishes in all odd degrees, and is isomorphic in even degrees to the space of continuous traces on $\ell^1(S)$. Crucial to our approach is the use of the structure semilattice of $S$, and the associated grading of $S$, together with an inductive normalization procedure in cyclic cohomology; the latter technique appears to be new, and its…

Free Brownian motion and free convolution semigroups: multiplicative case

We consider a pair of probability measures $μ,ν$ on the unit circle such that $Σ_λ(η_ν(z))=z/η_μ(z)$. We prove that the same type of equation holds for any $t\geq 0$ when we replace $ν$ by $ν\boxtimesλ_t$ and $μ$ by $\mathbb{M}_t(μ)$, where $λ_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $\mathbb{C}$ and $\mathbb{M}_t$ is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subord…

局部紧量子群的 Rieffel 形变上的卷积半群

Convolution semigroups on Rieffel deformations of locally compact quantum groups

本文研究带闭经典阿贝尔子群和2-上循环的局部紧量子群的Rieffel形变，建立了原群与形变群上Γ-不变状态卷积半群之间的典范对应。

通过卷积半群的长时间行为研究局部紧量子群的可和性

Amenability of locally compact quantum groups via the long-time behavior of convolution semigroups

该研究刻画了第二可数局部紧量子群的可和性，结合其对偶量子群的逼近性质，构造了受时间两端行为控制的卷积半群，相关结果在经典局部紧群情形下也属首次。

Small time sharp bounds for kernels of convolution semigroups

We study small time bounds for transition densities of convolution semigroups corresponding to pure jump Lévy processes in $\mathbb{R}^{d}$, $d \geq 1$, including those with jumping kernels exponentially and subexponentially localized at infinity. For a large class of Lévy measures, non-necessarily symmetric nor absolutely continuous with respect to the Lebesgue measure, we find the optimal, both in time and space, upper bound for the corresponding transition kernels at infinity. In case of Lévy measures that are symmetric and absolutely continuous, with densities $g$ such that $g(x) \asymp f(…

由全序半格产生的加权卷积代数的Arens正则性

Arens regularity of weighted convolution algebras that arise from totally ordered semilattices

本文将Dales与Strauss关于全序半格卷积代数Arens正则性的无权结果推广到加权情形，并进一步研究其Banach代数前对偶的存在性。

Infinitely divisible central probability measures on compact Lie groups---regularity, semigroups and transition kernels

We introduce a class of central symmetric infinitely divisible probability measures on compact Lie groups by lifting the characteristic exponent from the real line via the Casimir operator. The class includes Gauss, Laplace and stable-type measures. We find conditions for such a measure to have a smooth density and give examples. The Hunt semigroup and generator of convolution semigroups of measures are represented as pseudo-differential operators. For sufficiently regular convolution semigroups, the transition kernel has a tractable Fourier expansion and the density at the neutral element may…

A note on stochastic Fubini's theorem and stochastic convolution

We provide a version of the stochastic Fubini's theorem which does not depend on the particular stochastic integrator chosen as far as the stochastic integration is built as a continuous linear operator from an $L^p$ space of Banach space-valued processes (the stochastically integrable processes) to an $L^p$ space of Banach space-valued paths (the integrated processes). Then, for integrators on a Hilbert space $H$, we consider stochastic convolutions with respect to a strongly continuous map $R:(0,T]\rightarrow L(H)$, not necessarily a semigroup. We prove existence of predictable versions of s…

随机卷积连续性的免费午餐不存在

No free lunch for continuity of stochastic convolutions

本文构造了指数稳定的 $C_0$-半群，使得随机卷积在正概率下无界，并利用必要条件与 Schauder 乘子给出反例，同时在更强条件下建立了连续性与极大估计。

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. Rosi{ń}ski.

Integral Equations in the Theory of Levy Processes

In this article we consider the Levy processes and the corresponding semigroup. We represent the generator of this semigroup in a convolution form. Using the obtained convolution form and the theory of integral equations we investigate the properties of a wide class of Levy processes (potential, quasi-potential, the probability of the Levy process remaining within the given domain, long time behavior, stable processes). We analyze in detail a number of concrete examples of the Levy processes (stable processes, the variance damped Levy processes, the variance gamma processes, the normal Gaussia…

Spatial asymptotics at infinity for heat kernels of integro-differential operators

We study a spatial asymptotic behaviour at infinity of kernels $p_t(x)$ for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits $$ \lim_{r \to \infty} \frac{p_t(rθ-y)}{t \, ν(rθ)}, \quad t \in T, \ \ θ\in E, \ \ y \in \mathbb R^d, $$ exist and can be effectively computed. Here $ν$ is the corresponding Lévy density, $T \subset (0,\infty)$ is a bounded time-set and $E$ is a subset of the unit sphere in $\mathbb R^d$, $d \geq 1$. Our results are local on the unit sphere. They apply to a wide class of convolution …

在非局部设置中对地面态变换半群的$ L^1 $平滑性研究

Smoothing on $L^1$ for ground state transformed semigroups in non-local settings

本文研究了非局部设置中地面态变换半群的$ L^1 $平滑性，揭示了其在奥里奇空间中的作用，并通过实例展示了正则化效应随时间增强的特性。

Generating functionals for locally compact quantum groups

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 c…

A maximal inequality for stochastic convolutions in 2-smooth Banach spaces

Let (e^{tA})_{t \geq 0} be a C_0-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t \geq 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{t \geq 0} be a progressively measurable process with values in the space γ(H,E) of all γ-radonifying operators from H to E. We prove that for all 0<p<\infty there exists a constant C, depending only on p and E, such that for all T \geq 0 we have \E \sup_{0\le t\le T} || \int_0^t e^{(t-s)A} g_s dW_s \ ||^p \leq C \mathbb{E} (\int_0^T || g_t ||_{γ(H,E)}^2 dt)^\frac{p}{2}. For p \geq 2 the proof is based on the observation …