Search

We investigate the Courrège theorem in the context of linear operators that satisfy the positive maximum principle on a space of continuous functions over a symmetric space. Applications are given to Feller–Markov processes. We also introduce Gangolli operators, which satisfy the positive maximum principle, and generalise the form associated with the generator of a Lévy process on a symmetric space. When the space is compact, we show that Gangolli operators are pseudo-differential operators having scalar symbols.

We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an $${\mathbb {R}}^d$$ R d -valued Lévy process through rolling without slipping. We prove that the Feller semigroups associated with these processes extend to strongly continuous contraction semigroups on $$L^p$$ L p , for $$1\le p 1 ≤ p < ∞ . When $$p=2$$ p = 2 , we show that these semigroups are self-adjoint. If, in addition, the motion has a non-trivial Brownian part, we prove that the generator has a discrete spectrum of eigenvalues and that the semigroup is trace-class.

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.

Markov and Feller semigroups are introduced, together with the corresponding stochastic processes. As all generators of Feller semigroups satisfy the positive maximum principle, we focus on that property and discuss the associated Hille–Yosida–Ray theorem. The main result of the chapter is proof of the Courrege theorem, which gives a Levy–Khinchine representation (but with variable coefficients) for all linear operators satisfying the positive maximum principle. We conclude with a brief discussion of the martingale problem and sub-Feller semigroups.

The theory of semigroups of operators is one of the most important themes in modern analysis. Not only does it have great intellectual beauty, but also wide-ranging applications. In this book the author first presents the essential elements of the theory, introducing the notions of semigroup, generator and resolvent, and establishes the key theorems of Hille–Yosida and Lumer–Phillips that give conditions for a linear operator to generate a semigroup. He then presents a mixture of applications and further developments of the theory. This includes a description of how semigroups are used to solve parabolic partial differential equations, applications to Levy and Feller–Markov processes, Koopmanism in relation to dynamical systems, quantum dynamical semigroups, and applications to generalisations of the Riemann–Liouville fractional integral. Along the way the reader encounters several important ideas in modern analysis including Sobolev spaces, pseudo-differential operators and the Nash inequality.

We investigate the spectrum of the generator of a self-adjoint transition semigroup of a (symmetric) Lévy process taking values in d–dimensional space.

We find necessary and sufficient conditions for a finite $K$-bi-invariant measure on a compact Gelfand pair $(G, K)$ to have a square-integrable density. For convolution semigroups, this is equivalent to having a continuous density in positive time. When $(G,K)$ is a compact Riemannian symmetric pair, we study the induced transition density for $G$-invariant Feller processes on the symmetric space $X = G/K$. These are obtained as projections of $K$-bi-invariant L\'{e}vy processes on $G$, whose laws form a convolution semigroup. We obtain a Fourier series expansion for the density, in terms of spherical functions, where the spectrum is described by Gangolli's L\'evy-Khintchine formula. The density of returns to any given point on $X$ is given by the trace of the transition semigroup, and for subordinated Brownian motion, we can calculate the short time asymptotics of this quantity using recent work of Ba\~nuelos and Baudoin. In the case of the sphere, there is an interesting connection with the Funk-Hecke theorem.

We study recurrence and transience for Lévy processes induced by topological transformation groups acting on complete Riemannian manifolds. In particular the transience–recurrence dichotomy in terms of potential measures is established and transience is shown to be equivalent to the potential measure having finite mass on compact sets when the group acts transitively. It is known that all bi-invariant Lévy processes acting in irreducible Riemannian symmetric pairs of noncompact type are transient. We show that we also have ‘harmonic transience’, that is, local integrability of the inverse of the real part of the characteristic exponent which is associated to the process by means of Gangolli’s Lévy–Khinchine formula.

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.

International audience

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.

We investigate the perturbation of the nonlinear differential equation [Formula: see text] by random noise terms consisting of Brownian motion and an independent Poisson random measure. We find conditions under which the perturbed system is almost surely exponentially stable and estimate the corresponding Lyapunov exponents.

Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito decompositions and an associated Levy-Khintchine formula. If the process is weakly square integrable, its covariance operator can be used to construct a reproducing kernel Hilbert space in which the process has a decomposition as an infinite series built from a sequence of uncorrelated bona fide one-dimensional Levy processes. This series is used to define cylindrical stochastic integrals from which cylindrical Ornstein-Uhlenbeck processes may be constructed as unique solutions of the associated Cauchy problem. We demonstrate that such processes are cylindrical Markov processes and study their (cylindrical) invariant measures., Comment: 31 pages

Using key tools such as Itô's formula for general semimartingales, Kunita's moment estimates for Lévy-type stochastic integrals, and the exponential martingale inequality, we find conditions under which the solutions to the stochastic differential equations (SDEs) driven by Lévy noise are stable in probability, almost surely and moment exponentially stable.

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.

We apply Peter–Weyl theory to obtain necessary and sufficient conditions for a probability measure on a compact group to have a square-integrable density. Applications are given to measures on the d-dimensional torus.

In 1964 R. Gangolli published a Lévy–Khintchine type formula which characterised $K$-bi-invariant infinitely divisible probability measures on a symmetric space $G/K$. His main tool was Harish-Chandra’s spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this paper we use generalised spherical functions (or Eisenstein integrals) and extensions of these which we construct using representation theory to obtain such a characterisation for arbitrary infinitely divisible probability measures on a non-compact symmetric space. We consider the example of hyperbolic space in some detail.

We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by Levy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.

We study Hilbert space valued Ornstein–Uhlenbeck processes (Y(t), t ≥ 0) which arise as weak solutions of stochastic differential equations of the type dY = JY + CdX(t) where J generates a C0 semigroup in the Hilbert space H, C is a bounded operator and (X(t), t ≥ 0) is an H-valued Levy process. The associated Markov semigroup is of generalised Mehler type. We discuss an analogue of the Feller property for this semigroup and explicitly compute the action of its generator on a suitable space of twice-differentiable functions. We also compare the properties of the semigroup and its generator with respect to the mixed topology and the topology of uniform convergence on compacta.

N. Jacob et ses collaborateurs ont recemment etudie des processus de Markov a valeurs dans des espaces prehilbertiens dont le generateur infinitesimal est un operateur pseudo-differentiel au sens de Kohn Nirenberg. Nous souhaitons generaliser cette etude au groupe de Heisenberg quand on utilise le calcul de Weyl pour construire les operateurs pseudo-differentiels. Nous commencons par le cas des processus de Levy, et obtenons une forme generale pour le symbole de leur generateur infinitesimal. Ensuite nous considerons des processus dont les coordonnees dans l'espace de phase sont des processus de Levy, et dont la partie reelle est l'aire de Levy associee. Ceci permet de clarifier une preuve probabiliste de la formule de Mehler, due a Gaveau. Dans une deuxieme partie, nous decrivons quelques proprietes du generateur de la representation de Schrodinger. Dans le cas ou ces operateurs sont symetriques, positifs, nous montrons qu'ils ne correspondent pas toujours a une forme de Dirichlet. Cependant dans le cas ou ils correspondent a une forme de Dirichlet, on donne une description du processus de Hunt associe sous certaines conditions qui sont precisees.

We consider a class of Feller semigroups on Lie groups which fail to commute with left translation due to the existence of a cocycle h which is identically one for Levy processes. Under certain conditions, we are able to show that the infinitesimal generator of such a semigroup has the Levy–Khintchine–Hunt form but with variable characteristics, thus we obtain an extension of classical work in Euclidean space by Courrege.

We give a short summary of some of Varopoulos’ Hardy-Littlewood-Sobolev inequalities for self-adjoint \(C_{0}\) semigroups and give a new probabilistic representation of the classical fractional integral operators on \(\mathbb {R}^n\) as projections of martingale transforms. Using this formula we derive a new proof of the classical Hardy-Littlewood-Sobolev inequality based on Burkholder-Gundy and Doob’s inequalities for martingales.

Representation theory is a deep and beautiful subject. Our goal in this chapter is to develop those concepts and results that we need for applications to Fourier analysis on compact groups and hence to probability theory. In the first part, we give a self-contained account of key aspects of the representation theory of compact groups, including proofs of Schur’s lemma, the Schur orthogonality relations and the Peter-Weyl theorem. We also introduce the Fourier transform for suitable functions on the group and establish some of its elementary properties. In the second part of the chapter, we introduce weights and roots and sketch proofs of the Weyl character and Weyl dimension formulae. This part is far less rigorous and many proofs are omitted. The key point that readers need to absorb is that irreducible representations are in one-to-one correspondence with highest weights and, as we will see in later chapters, this enables us to carry out a finer analysis of functions and measures in “Fourier space”. Finally we illustrate the abstract theory by finding all the irreducible representations of \(SU(2)\). (We use a lot of elementary Hilbert space ideas in this chapter. Readers requiring a reminder of key concepts should consult a standard text such as Debnath and Mikusinski [55] or Reed and Simon [166]).

Properties of convolution semigroups of probability measures and associated Hunt semigroups of operators on general Lie groups are described. We outline a proof of Hunt’s theorem which gives a Levy-Khintchine type representation for the infinitesimal generator of such semigroups. Then we study \(L^{2}\)-properties of these semigroups, specifically finding conditions which guarantee self-adjointness, injectivity, analyticity and the trace-class property. We then use the Fourier transform on a compact Lie group to find a direct analogue of the Levy-Khintchine formula. We prove the central limit theorem for Gaussian semigroups of measures. Then we describe the technique of subordination, and discuss regularity of densities for some semigroups of measures that are obtained by subordinating the standard Gaussian. Finally we investigate the small-time asymptotic behaviour of some subordinated densities.

We study the Laplacian from an analytic viewpoint as a self-adjoint operator with discrete eigenvalues given by the Casimir spectrum. This leads naturally to a study of Sobolev spaces, which are also characterised from a Fourier analytic viewpoint. We introduce Sugiura’s zeta function as a tool to study regularity of Fourier series on groups. In particular, we find conditions for absolute and uniform convergence, and for smoothness. Smoothness is characterised by means of the Sugiura space of rapidly decreasing functions defined on the space of highest weights, and we will utilise this in the next chapter to study probability measures on groups that have smooth densities.

We review the scheme introduced by Kim and Richards for deconvolution density estimation on compact Lie groups. The Fourier transform is used to express the problem in terms of products of matrices of irreducible representations. We introduce a sequence of consistent estimators, by taking cut-offs in a certain Fourier expansion. Some smoothness classes of noise are discussed and used to find optimal rates of convergence of the estimators.

We employ the interlacing construction to show that the solutions of stochastic differential equations on manifolds which are written in Marcus canonical form and driven by infinite-dimensional semimartingales with jumps give rise to stochastic flows of diffeomorphisms.

A sample path description of compound Poisson processes on groups is given and applied to represent Levy processes on connected Lie groups as almost sure limits of sequences of Brownian motions with drift interlaced with random jumps. We obtain spherically symmetric Levy processes in Riemannian symmetric spaces of the form M=G\K, where G is a semisimple Lie group and K is a compact subgroup by projection of symmetric horizontal Levy processes in G and give a straightforward proof of Gangolli's Levy–Khintchine formula for their spherical transform. Finally, we show that such processes can be realized in Fock space in terms of creation, conservation, and annihilation processes. They appear as generators of a new class of factorizable representations of the current group C(R+, G). In the case where M is of noncompact type, these Fock space processes are indexed by the roots of G and the natural action of the Weyl group of G induces a (send quantized) unitary equivalence between them.

The hand has remained a considerable treatment challenge, as new soft tissue fillers have arrived in the esthetic marketplace. The challenge has been the result of both the multiple visits required for treatment in, for example, autologous fat grafting and the simple management of pain in the innervated areas of the hand between the bones. This paper introduces a novel, noticeably less painful approach to treatment of the hand with calcium hydroxylapatite (CaHA; Radiesse®, BioForm Medical, San Mateo, CA). Anesthetic is added to the compound prior to injection, resulting in a homogenous admixture of CaHA and anesthetic. A bolus of the mixture is injected into the skin, using tenting, and then spread throughout the hand. The result of this approach – mixing anesthetic with CaHA – is treatment that is easier to massage and disseminate, less painful to the patient than conventional hand injection, and characterized by less swelling and bruising, with minimal post-treatment downtime.

A concept of quantum martingale measure is introduced and examples are constructed as quantum stochastic spectral integrals in Fock space. These are then utilized as space–time noise to drive a parabolic stochastic partial differential equation (spde). We establish the existence and uniqueness of the solutions as families of densely defined closable operators in Fock space that are jointly continuous in time and space variables and satisfy a Markov property.

A quantum stochastic calculus for fermions is developed where the basic integrators are based on Dirac fields and the charge operator. The associated Ito formula has seven non-trivial correction terms. Conditions are found for the solutions of stochastic differential equations to be unitary and it is shown that the corresponding quantum stochastic flow manifests a broken symmetry whereby the particle and antiparticle noises no longer balance each other. An abstract theory of such flows is then developed. By employing the unification between boson and fermion stochastic calculi, we are able to develop the entire theory using boson Fock spaces.

This paper constructs a class of martingale transforms based on L\'evy processes on Lie groups. From these, a natural class of bounded linear operators on the $L^p$-spaces of the group (with respect to Haar measure) for $1

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 be expressed as the trace of the Hunt semigroup. We compute the short time asymptotics of the density at the neutral element for the Cauchy distribution on the $d$-torus, on SU(2) and on SO(3), where we find markedly different behaviour than is the case for the usual heat kernel., Comment: Published in at http://dx.doi.org/10.1214/10-AOP604 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The index of a Fredholm operator associated to aθ-summable Fredholm module is expressed in terms of the vacuum expectation value of a unitary operator-valued stochastic process which satisfies a stochastic differential equation with unbounded coefficients driven by fermion noise.