339 research outputs found
Feller semigroups, Lp-sub-Markovian semigroups, and applications to pseudo-differential operators with negative definite symbols
The question of extending L-p-sub-Markovian semigroups to the spaces L-q, q > P, and the interpolation of LP-sub-Markovian semigroups with Feller semigroups is investigated. The structure of generators of L-p-sub-Markovian semigroups is studied. Subordination in the sense of Bochner is used to discuss the construction of refinements of L-p-sub-Markovian semigroups. The role played by some function spaces which are domains of definition for L-p-generators is pointed out. The problem of regularising powers of generators as well as some perturbation results are discussed
Spectral expansions of non-self-adjoint generalized Laguerre semigroups
We provide the spectral expansion in a weighted Hilbert space of a
substantial class of invariant non-self-adjoint and non-local Markov operators
which appear in limit theorems for positive-valued Markov processes. We show
that this class is in bijection with a subset of negative definite functions
and we name it the class of generalized Laguerre semigroups. Our approach,
which goes beyond the framework of perturbation theory, is based on an in-depth
and original analysis of an intertwining relation that we establish between
this class and a self-adjoint Markov semigroup, whose spectral expansion is
expressed in terms of the classical Laguerre polynomials. As a by-product, we
derive smoothness properties for the solution to the associated Cauchy problem
as well as for the heat kernel. Our methodology also reveals a variety of
possible decays, including the hypocoercivity type phenomena, for the speed of
convergence to equilibrium for this class and enables us to provide an
interpretation of these in terms of the rate of growth of the weighted Hilbert
space norms of the spectral projections. Depending on the analytic properties
of the aforementioned negative definite functions, we are led to implement
several strategies, which require new developments in a variety of contexts, to
derive precise upper bounds for these norms.Comment: 162page
Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of L\'{e}vy processes
We introduce a class of L\'{e}vy processes subject to specific regularity
conditions, and consider their Feynman-Kac semigroups given under a Kato-class
potential. Using new techniques, first we analyze the rate of decay of
eigenfunctions at infinity. We prove bounds on -subaveraging
functions, from which we derive two-sided sharp pointwise estimates on the
ground state, and obtain upper bounds on all other eigenfunctions. Next, by
using these results, we analyze intrinsic ultracontractivity and related
properties of the semigroup refining them by the concept of ground state
domination and asymptotic versions. We establish the relationships of these
properties, derive sharp necessary and sufficient conditions for their validity
in terms of the behavior of the L\'{e}vy density and the potential at infinity,
define the concept of borderline potential for the asymptotic properties and
give probabilistic and variational characterizations. These results are amply
illustrated by key examples.Comment: Published at http://dx.doi.org/10.1214/13-AOP897 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Boas-type formulas in Banach spaces with applications to analysis on manifolds
The paper contains Boas-type formulas for trajectories of one-parameter
groups of operators in Banach spaces. The results are illustrated using
one-parameter groups of operators which appear in representations of Lie
groups.Comment: submitte
Contractivity and ground state domination properties for non-local Schr\"odinger operators
We study supercontractivity and hypercontractivity of Markov semigroups
obtained via ground state transformation of non-local Schr\"odinger operators
based on generators of symmetric jump-paring L\'evy processes with Kato-class
confining potentials. This class of processes has the property that the
intensity of single large jumps dominates the intensity of all multiple large
jumps, and the related operators include pseudo-differential operators of
interest in mathematical physics. We refine these contractivity properties by
the concept of -ground state domination and its asymptotic version, and
derive sharp necessary and sufficient conditions for their validity in terms of
the behaviour of the L\'evy density and the potential at infinity. As a
consequence, we obtain for a large subclass of confining potentials that, on
the one hand, supercontractivity and ultracontractivity, on the other hand,
hypercontractivity and asymptotic ultracontractivity of the transformed
semigroup are equivalent properties. This is in stark contrast to classical
Schr\"odinger operators, for which all these properties are known to be
different.Comment: 15 page
-modules, Bernstein-Sato polynomials and -invariants of direct summands
We study the structure of -modules over a ring which is a direct
summand of a polynomial or a power series ring with coefficients over a
field. We relate properties of -modules over to -modules over . We
show that the localization and the local cohomology module
have finite length as -modules over . Furthermore, we show the existence
of the Bernstein-Sato polynomial for elements in . In positive
characteristic, we use this relation between -modules over and to
show that the set of -jumping numbers of an ideal is
contained in the set of -jumping numbers of its extension in . As a
consequence, the -jumping numbers of in form a discrete set of
rational numbers. We also relate the Bernstein-Sato polynomial in with the
-thresholds and the -jumping numbers in .Comment: 24 pages. Comments welcome
Heat kernel generated frames in the setting of Dirichlet spaces
Wavelet bases and frames consisting of band limited functions of nearly
exponential localization on Rd are a powerful tool in harmonic analysis by
making various spaces of functions and distributions more accessible for study
and utilization, and providing sparse representation of natural function spaces
(e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in
more general homogeneous spaces, on the interval and ball. The purpose of this
article is to develop band limited well-localized frames in the general setting
of Dirichlet spaces with doubling measure and a local scale-invariant
Poincar\'e inequality which lead to heat kernels with small time Gaussian
bounds and H\"older continuity. As an application of this construction, band
limited frames are developed in the context of Lie groups or homogeneous spaces
with polynomial volume growth, complete Riemannian manifolds with Ricci
curvature bounded from below and satisfying the volume doubling property, and
other settings. The new frames are used for decomposition of Besov spaces in
this general setting
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