Measurable centres in convolution semigroups

In certain convolution semigroups over locally compact groups, the only measurable translations are those defined by Radon measures. In other words, the measurable centre of every such convolution semigroup consists of Radon measures.     

Perturbation theory for convolution semigroups

We deal with convolution semigroups (not necessarily symmetric) in Lp(RN) and provide a general perturbation theory of their generators by indefinite singular potentials. Such semigroups arise in the theory of Lévy processes and cover many examples such as Gaussian semigroups, α-stable semigroups, relativistic Schrödinger semigroups, etc. We give new generation theorems and Feynman-Kac formulas. In particular, by using weak compactness methods in L¹, we enlarge the extended Kato class potentials used in the theory of Markov processes. In L² setting, Dirichlet form-perturbation theory is finely related to L¹-theory and the extended Kato class measures is also enlarged. Finally, various perturbation problems for subordinate semigroups are considered.     

On the continuity of convolution semigroups

Communicated by Karl H. Hofmann     

Continuity of convolution semigroups on hypergroups

LetK be a commutative hypergroup with the property that either the identity character is contained in the support of the Plancherel measure onK ^{^}, or the identity character is not isolated inK ^{^} and all characters sufficiently close (but not equal) to the identity character vanish at infinity. We present a shift compactness theorem forK and use this to prove that every symmetric convolution semigroup of probability measures onK is continuous.     

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.     

Zeta function of a convolution

The zeta function of the convolution ,convergent in the domain s > 1, is extended meromorphically into the left half-plane. The explicit form of this extension is found.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 22–48, 1990.     

Continuous state branching semigroups

Summary This paper is concerned with Markov processes with continuous creation where the phase space is a general separable compact metric space. The transition probabilities for such a process determine a semigroup of operators acting on a function space over the collection of bounded Borel measures on the phase space. Such a semigroup is characterized by a particular convolution condition and is called a continuous state branching semigroup. A connection is established between continuous state branching semigroups and certain semigroups of nonlinear operators and then this connection is exploited to establish an existence theorem for the former.Research associated with a project in probability at Princeton University supported by the Office of Army Research.     

Asymptotic ordering of distribution functions and convolution semigroups

It is well-known that the setH of distribution functions on [0,∞] supplied with the convolution product * is a semigroup. We introduce a preorder on (H,*) with attractive algebraic properties. The corresponding equivalence relation ≈^W extends the concept of tail-equivalence of distribution functions. We show that the idempotents of the factorsemigroupH _W form a subsemigroup ofH _W.     

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 R ^d , d ≥ 1, including the processes with jump measures which are exponentially and subexponentially localized at ∞. For a large class of Lévy measures, not necessarily symmetric or absolutely continuous with respect to Lebesgue measure, we find the optimal upper bound in both time and space for the corresponding heat kernels at ∞. In case of Lévy measures that are symmetric and absolutely continuous with densities g such that g(x) ? f(|x|) for non-increasing profile functions f, we also prove the full characterization of the sharp two-sided transition densities bounds of the form

$${p_t}\left( x \right) \asymp h{\left( t \right)^{ - d}} \cdot {1_{\left\{ {\left| x \right| \leqslant \theta h\left( t \right)} \right\}}} + tg\left( x \right) \cdot {1_{\left\{ {\left| x \right| \geqslant \theta h\left( t \right)} \right\}}},t \in \left( {0,{t_0}} \right),{t_0} > 0,x \in {\mathbb{R}^d}.$$

This is done for small and large x separately. Mainly, our argument is based on new precise upper bounds for convolutions of Lévy measures. Our investigations lead to a surprising dichotomy correspondence of the decay properties at ∞ for transition densities of pure jump Lévy processes. All results are obtained solely by analytic methods, without use of probabilistic arguments.

     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

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)\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).