cores for feller semigroups with an invariant measure abstract let a i j 1 n a i j x d i j i 1 n b i x d i be an elliptic differential operator with unbounded coefficients on r n and assume that the associated feller semigroup t t t 0 has an invariant measure then t t t 0 extends to a strongly continuous semigroup t p t t 0 on l p l p r n for every 1 p we prove that under mild conditions on the coefficients of a the space of test functions c c r n is a core for the generator a p d p of t p t t 0 in l p for 1 p

semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

weak stability of ricci expanders with positive curvature operator we prove the l infty stability of expanding gradient ricci solitons with positive curvature operator and quadratic curvature decay at infinity

the equivalence of certain heat kernel and green function bounds abstract we show that certain upper and lower bounds on the green function and heat kernel of a second order elliptic operator in a bounded region are equivalent and imply a number of apparently stronger bounds we also show that these bounds are equivalent to the harnack inequality except for peculiar regions and are therefore almost always valid

some classes of non analytic markov semigroups we deal with markov semigroups tt corresponding to second order elliptic operators au u du f where f is an unbounded locally lipschitz vector field on rn we obtain new conditions on f under which tt is not analytic in cb rn in particular we prove that the one dimensional operator au u x3u with domain u c2 r u u x3u cb r is not sectorial in cb r under suitable hypotheses on the growth of f we introduce a class of non analytic markov semigroups in lp rn where is an invariant measure for tt

estimates of the derivatives for parabolic operators with unbounded coefficients we consider a class of second order uniformly elliptic operators a with unbounded coefficients in r n using a bernstein approach we provide several uniform estimates for the semigroup t t generated by the realization of the operator a in the space of all bounded and continuous or holder continuous functions in r n as a consequence we obtain optimal schauder estimates for the solution to both the elliptic equation u au f 0 and the nonhomogeneous dirichlet cauchy problem d t u au g then we prove two different kinds of pointwise estimates of t t that can be used to prove a liouville type theorem finally we provide sharp estimates of the semigroup t t in weighted l p spaces related to the invariant measure associated with the semigroup

cores for parabolic operators with unbounded coefficients abstract let a i j 1 n q i j s x d i j i 1 n b i s x d i be a family of elliptic differential operators with unbounded coefficients defined in r n 1 in m kunze l lorenzi a lunardi nonautonomous kolmogorov parabolic equations with unbounded coefficients trans amer math soc in press under suitable assumptions it has been proved that the operator g a d s generates a semigroup of positive contractions t p t in l p r n 1 for every 1 p where is an infinitesimally invariant measure of t p t here under some additional conditions on the growth of the coefficients of a which cover also some growths with an exponential rate at we provide two different cores for the infinitesimal generator g p of t p t in l p r n 1 for p 1 and we also give a partial characterization of d g p finally we extend the results so far obtained to the case when the coefficients of the operator a are t periodic with respect to the variable s for some t 0

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

an algorithm for computing weight multiplicities in irreducible modules for complex semisimple lie algebras abstract let g be a finite dimensional semisimple lie algebra over c having rank l and let v be an irreducible finite dimensional g module having highest weight computations of weight multiplicities in v usually based on freudenthal s formula are in general difficult to carry out in large ranks or for with large coefficients in terms of the fundamental weights in this paper we first show that in some situations these coefficients can be lowered in order to simplify the calculations we then investigate how this can be used to improve the aforementioned formula of freudenthal leading to a more efficient version of the latter in terms of complexity as well as to a way of dealing with certain computations in unbounded ranks we conclude by illustrating the last assertion with a concrete example

spectrum of ornstein uhlenbeck operators in lp spaces with respect to invariant measures abstract let a i j 1 n q ij d ij i j 1 n b ij x j d i be a possibly degenerate ornstein uhlenbeck operator in r n and assume that the associated markov semigroup has an invariant measure we compute the spectrum of a in l p for 1 p

nonsymmetric macdonald polynomials and demazure characters we establish a connection between a specialization of the nonsymmetric macdonald polynomials and the demazure characters of the corresponding affine kac moody algebra this allows us to obtain a representation theoretical interpretation of the coefficients of the expansion of the specialized symmetric macdonald polynomials in the basis formed by the irreducible characters of the associated finite lie algebra

intrinsic ultracontractivity of a schrodinger semigroup in rn abstract we give a possibly sharp sufficient condition on the electric potential q r n 0 in the schrodinger operator a q x on l 2 r n that guarantees that the schrodinger heat semigroup e a t t 0 on l 2 r n generated by a is intrinsically ultracontractive moreover if q x q x is radially symmetric we show that our condition on q is also necessary i e truly sharp it reads r 0 q r 1 2 d r for some r 0 0 our proofs make essential use of techniques based on a logarithmic sobolev inequality rosen s inequality proved via a new fenchel young inequality and a very precise asymptotic formula due to hartman and wintner

schauder theorems for ornstein uhlenbeck equations in infinite dimension abstract we prove schauder type estimates for stationary and evolution equations driven by the classical ornstein uhlenbeck operator in a separable banach space endowed with a centered gaussian measure

l p uniqueness for elliptic operators with unbounded coefficients in r n let a be an elliptic operator with unbounded and sufficiently smooth coefficients and let be a sub invariant measure of the operator a in this paper we give sufficient conditions guaranteeing that the closure of the operator a cc rn generates a sub markovian strongly continuous semigroup of contractions in lp rn applications are given in the case when a is a generalized schrodinger operator

spectral properties of metastable markov semigroups abstract it is shown that if a symmetric markov semigroup e ht on the hilbert space l 2 x is hypercontractive then the approximate degeneracy of the ground state has several consequences concerning other parts of the spectrum of h and concerning the unitary group e iht in particular in the presence of a space inversion symmetry all the eigenvalues occur in pairs with gaps comparable to the gap between the bottom two eigenvalues

intrinsic ultracontractivity and the dirichlet laplacian we show that if d x y 0 x 1 f x y 1 where f is negative upper semicontinuous and bounded below then the semigroup associated with the dirichlet laplacian on d is intrinsically ultracontractive analogs in higher dimensions are proved we construct domains of infinite area with this property

nonuniqueness of the positive dirichlet problem for parabolic equations in cylinders abstract we investigate uniqueness of nonnegative solutions to parabolic equations in cylinders of r n 1 with zero dirichlet boundary condition where bases of cylinders are unbounded domains we give necessary and or sufficient conditions for the uniqueness in terms of geometric properties of the bases

elliptic problems on ℝ n with unbounded coefficients in classical sobolev spaces we discuss the existence and uniqueness in h1 n and the h2 n regularity of the solutions of au f when f l2 n and a is a second order linear elliptic operator whose first and zeroth order coefficients may be unbounded at infinity we also investigate whether a generates a c0 or analytic semigroup on l2 the approach in this nonweighted setting is based on a new and general method the idea consists in embedding a into a suitable one parameter family of operators as s with a0 a the properties of as when s 0 make it possible to prove that the boundary integrals arising from simple integration by parts over balls with increasing radius tend to 0 at infinity this provides the needed estimates for uniqueness and regularity

gradient estimates for dirichlet parabolic problems in unbounded domains we consider autonomous parabolic dirichlet problems in a regular unbounded open set rn involving second order operator a with possibly unbounded coefficients we determine new conditions on the coefficients of a yielding global gradient estimates for the bounded classical solution

on the ornstein uhlenbeck operator in spaces of continuous functions we study the realization of the ornstein uhlenbeck operator a in the space of the uniformly continuous and bounded functions in rn we prove that it generates a semigroup which is neither strongly continuous nor analytic but enjoys nice smoothing properties such properties are used to prove a schauder type theorem for the elliptic equation u au f

on schrodinger semigroups and related topics abstract this paper deals with two related subjects in the first part we give generation theorems relying on weak compactness arguments for perturbed positive semigroups in general ordered banach spaces with additive norm on the positive cone the second part provides new functional analytic developments on semigroup theory for schrodinger operators in l p spaces with l 1 bounded potentials without restriction on the l 1 bound in particular our formalism enlarges a priori the classical kato class and its subsequent refinements the connection with form perturbation theory is also dealt with

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

schauder estimates for drifted fractional operators in the supercritical case abstract we consider a non local operator l which is the sum of a fractional laplacian 2 0 1 plus a first order term which is measurable in the time variable and locally holder continuous in the space variables importantly the fractional laplacian 2 does not dominate the first order term we show that global parabolic schauder estimates hold even in this case under the natural condition 1 thus the constant appearing in the schauder estimates is in fact independent of the l norm of the first order term in our approach we do not use the so called extension property and we can replace 2 with other operators of stable type which are somehow close including the relativistic stable operator moreover when 1 2 1 we can prove schauder estimates for more general stable type operators like the singular cylindrical one i e when 2 is replaced by a sum of one dimensional fractional laplacians k 1 d x k x k 2 2

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

a priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains we investigate quantitative properties of the nonnegative solutions u t x geq 0 to the nonlinear fractional diffusion equation partial t u mathcal l u m 0 posed in a bounded domain x in omega subset mathbb r n with m 1 for t 0 as mathcal l we use one of the most common definitions of the fractional laplacian delta s 0 s 1 in a bounded domain with zero dirichlet boundary conditions we consider a general class of very weak solutions of the equation and obtain a priori estimates in the form of smoothing effects absolute upper bounds lower bounds and harnack inequalities we also investigate the boundary behaviour and we obtain sharp estimates from above and below in addition we obtain similar estimates for fractional semilinear elliptic equations either the standard laplacian case s 1 or the linear case m 1 are recovered as limits the method is quite general suitable to be applied to a number of similar problems

uniform bounds for solutions to quasilinear parabolic equations abstract we consider a class of quasilinear parabolic equations whose model is the heat equation corresponding to the p laplacian operator u pu di 1 i u p 2 iu with p 2 d on a domain d r d of finite measure we prove that u t x c d t u0 r for all t 0 x d and for all initial data u0 lr d provided r is not smaller than a suitable r0 where are positive constants explicitly computed in terms of d p r the nonlinear cases associated with the case p 2 display exactly the same contractivity properties which hold for the linear heat equation we also show that the nonlinear evolution considered is contractive on any lq space for any q 2

bounds on short cylinders and uniqueness in cauchy problem for degenerate kolmogorov equations abstract we consider the cauchy problem for degenerate kolmogorov equations in the form t u i j 1 m a i j x t x i x j u j 1 m a j x t x j u i j 1 n b i j x i x j u x t r n 0 t 1 m n as well as in its divergence form we prove that if u x t m exp a t x 2 for some positive constants a m and 0 1 and u 0 0 then u 0 the proof of the main result is based on some previous uniqueness result and on the application of some estimates in short cylinders previously used by ferretti in the study of uniformly parabolic operators

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

the boundary behavior of heat kernels of dirichlet laplacians we prove the following complete and qualitatively sharp description of heat kernels g of dirichlet laplacians on bounded c1 1 domains d there exist positive constants c1 c2 and t 0 depending on d such that for x dist x d x y t 1c1tn 2e c2 x y 2 t g x t y 0 x y t 11c1tn 2e x y 2 c2t for all x y d and 0 t are known bounds when d is unbounded are also given

gaussian estimates for elliptic operators with unbounded drift we consider a strictly elliptic operator au ij di aij dj u b u div c u vu

a latent manifold markovian dynamics gaussian process in this paper we propose a gaussian process gp model for analysis of nonlinear time series formulation of our model is based on the consideration that the observed data are functions of latent variables with the associated mapping between observations and latent representations modeled through gp priors in addition to capture the temporal dynamics in the modeled data we assume that subsequent latent representations depend on each other on the basis of a hidden markov prior imposed over them derivation of our model is performed by marginalizing out the model parameters in closed form using gp priors for observation mappings and appropriate stick breaking priors for the latent variable markovian dynamics this way we eventually obtain a nonparametric bayesian model for dynamical systems that accounts for uncertainty in the modeled data we provide efficient inference algorithms for our model on the basis of a truncated variational bayesian approximation we demonstrate the efficacy of our approach considering a number of applications dealing with real world data and compare it with the related state of the art approaches

nonsymmetric koornwinder polynomials and duality in the fundamental work of lusztig l on affine hecke algebras a special role is played by the root system of type cn the affine hecke algebra is a deformation of the group algebra of an affine weyl group which usually depends on as many parameters as there are distinct root lengths i e one or two for an irreducible root system however in the cn case the hecke algebra h has three parameters corresponding to the fact that there is a simple coroot which is divisible by 2 recently cherednik c1 c3 has introduced the notion of a double affine hecke algebra and has used it to prove several conjectures on macdonald polynomials these polynomials and cherednik s double affine hecke algebra involve two or three parameters i e one more than the number of root lengths

gradient estimates for diffusion semigroups with singular coefficients abstract uniform gradient estimates are derived for diffusion semigroups possibly with potential generated by second order elliptic operators having irregular and unbounded coefficients we first consider the r d case by using the coupling method due to the singularity of the coefficients the coupling process we construct is not strongly markovian so that additional difficulties arise in the study then more generally we treat the case of a possibly unbounded smooth domain of r d with dirichlet boundary conditions we stress that the resulting estimates are new even in the r d case and that the coefficients can be holder continuous our results also imply a new liouville theorem for space time bounded harmonic functions with respect to the underlying diffusion semigroup

rational approximation schemes for bi continuous semigroups abstract this paper extends the hille phillips functional calculus and rational approximations results due to r hersh t kato p brenner and v thomee to generators of bi continuous semigroups the method yields error estimates for rational time discretization schemes for such semigroups in particular for dual semigroups feller semigroups such as the ornstein uhlenbeck semigroup the heat semigroup semigroups induced by nonlinear flows implemented semigroups and evolution semigroups furthermore the results provide error estimates for a new class of inversion formulas for the laplace transform

nonsymmetric ornstein uhlenbeck semigroups in banach spaces letebe a separable real banach space and letq l e e be positive and symmetric lets s t t 0be ac0 semigroup onewe study the relations between the reproducing kernel hilbert spaces associated with the operatorsqt t0 s s qs s ds under the assumption that these operators are the covariances of centered gaussian measures tone we also study equivalence t sfor different values of s and t and we calculate their radon nikodym derivatives

hierarchical dirichlet processes we consider problems involving groups of data where each observation within a group is a draw from a mixture model and where it is desirable to share mixture components between groups we assume that the number of mixture components is unknown a priori and is to be inferred from the data in this setting it is natural to consider sets of dirichlet processes one for each group where the well known clustering property of the dirichlet process provides a nonparametric prior for the number of mixture components within each group given our desire to tie the mixture models in the various groups we consider a hierarchical model specifically one in which the base measure for the child dirichlet processes is itself distributed according to a dirichlet process such a base measure being discrete the child dirichlet processes necessarily share atoms thus as desired the mixture models in the different groups necessarily share mixture components we discuss representations of hierarchical dirichlet processes

mild solutions of semilinear elliptic equations in hilbert spaces abstract this paper extends the theory of regular solutions c 1 in a suitable sense for a class of semilinear elliptic equations in hilbert spaces the notion of regularity is based on the concept of g derivative which is introduced and discussed a result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property that is it maps continuous functions into g differentiable ones the validity of this smoothing assumption is fully discussed for the case of the ornstein uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature it is shown that the results apply to hamilton jacobi bellman hjb equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that in particular they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

ultraboundedness for parabolic equations in convex domains without boundary conditions abstract we consider the operator a u 1 2 u d u d u where u is a convex real function defined in a convex open set o r n and lim x u x lim x o u x we prove that the associated markov semigroup is ultrabounded with respect to the gibbs measure e 2 u x d x

a markov random field regulated pitman yor process prior for spatially constrained data clustering in this work we propose a markov random field regulated pitman yor process mrf pyp prior for nonparametric clustering of data with spatial interdependencies the mrf pyp is constructed by imposing a pitman yor process over the distribution of the latent variables that allocate data points to clusters model states the discount hyperparameter of which is regulated by an additionally postulated simplified pointwise markov random field gibbsian distribution with a countably infinite number of states further based on the stick breaking construction of the pitman yor process we derive an efficient truncated variational bayesian algorithm for model inference we examine the efficacy of our approach by considering an unsupervised image segmentation application using a real world dataset we show that our approach completely outperforms related methods from the field of bayesian nonparametrics including the recently proposed infinite hidden markov random field model and the dirichlet process prior

local existence and regularity for a class of degenerate parabolic equations on etudie des equations paraboliques scalaires a une variable d espace de la forme u t f u u x uu xx ou f f u p s est une fonction lisse de u p s r 3 avec f s strictement positive

a harnack type inequality for non symmetric markov semigroups abstract for a strong feller and irreducible markov semigroup on a locally compact polish space the harnack type inequality 1 1 holds if and only if the semigroup has a unique invariant probability measure and is ultracontractive moreover new sufficient conditions for this inequality to hold as well as upper bound estimates of the underlying constant are presented for diffusion semigroups on riemannian manifolds

l1 properties of intrinsic schrodinger semigroups abstract we investigate the l 1 properties of the intrinsic markov semigroup associated with a schrodinger operator on r n which possesses a positive ground state we discover cases for which this semigroup is norm analytic for positive times and others for which the semigroup is norm discontinuous in the strongest possible sense in the case of the harmonic oscillator we show that the generator of the intrinsic semigroup has totally different spectrum depending on whether one works in l 1 r e x 2 dx or l 2 r e x 2 dx in more general cases we show that the equality of the l 1 and l 2 spectrum is closely related to whether the schrodinger semigroup is intrinsically ultracontractive

the infinite hidden markov random field model hidden markov random field hmrf models are widely used for image segmentation as they appear naturally in problems where a spatially constrained clustering scheme is asked for a major limitation of hmrf models concerns the automatic selection of the proper number of their states i e the number of region clusters derived by the image segmentation procedure existing methods including likelihood or entropy based criteria and reversible markov chain monte carlo methods usually tend to yield noisy model size estimates while imposing heavy computational requirements recently dirichlet process dp infinite mixture models have emerged in the cornerstone of nonparametric bayesian statistics as promising candidates for clustering applications where the number of clusters is unknown a priori infinite mixture models based on the original dp or spatially constrained variants of it have been applied in unsupervised image segmentation applications showing promising results under this motivation to resolve the aforementioned issues of hmrf models in this paper we introduce a nonparametric bayesian formulation for the hmrf model the infinite hmrf model formulated on the basis of a joint dirichlet process mixture dpm and markov random field mrf construction we derive an efficient variational bayesian inference algorithm for the proposed model and we experimentally demonstrate its advantages over competing methodologies

multiplicity formulas for fundamental strings of representations of classical lie algebras we call the p fundamental string of a complex simple lie algebra to the sequence of irreducible representations having highest weights of the form k 1 p for k 0 where j denotes the jth fundamental weight of the associated root system for a classical complex lie algebra we establish a closed explicit formula for the weight multiplicities of any representation in any p fundamental string

sharp bounds for ground state eigenfunctions on domains with horns and cusps abstract we prove a lower bound on the ground state eigenfunction of the dirichlet laplacian for horn shaped and cusped domains the novelty of the method is that for horn shaped domains it implies a new result for dimension two and more importantly is valid for higher dimensions moreover the same technique applies just as easily to cusps finally we construct an example to show that the estimates are sharp

variational bayesian filtering the use of the variational bayes vb approximation in bayesian filtering is studied both as a means to accelerate marginalized particle filtering and as a deterministic local one step approximation the vb method of approximation is reviewed together with restrictions that allow various computational savings to be achieved these variants provide a range of algorithms that can be used in a principled tradeoff between quality of approximation and computational cost in combination with marginalized particle filtering they generalize previously published work on variational filtering and extend currently available methods for speeding up stochastic approximations in bayesian filtering in particular the free form nature of the vb approximation allows optimal selection of moments which summarize the particles other bayesian filtering schemes are developed by replacing the marginalization operator in bayesian filtering with vb marginals this leads to further computational savings at the cost of quality of approximation the performance of the various vb filtering schemes is illustrated in the context of a gaussian model with a nonlinear substate and a hidden markov model

spectral properties of random non self adjoint matrices and operators i describe some numerical experiments which determine the degree of spectral instability of medium sized randomly generated matrices which are far from self adjoint the conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size i also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace but the eigenvalues do not determine the spectrum my results imply that the spectrum of the non self adjoint anderson model changes suddenly on passing to the infinite volume limit

dynamic nonparametric bayesian models for analysis of music the dynamic hierarchical dirichlet process dhdp is developed to model complex sequential data with a focus on audio signals from music the music is represented in terms of a sequence of discrete observations and the sequence is modeled using a hidden markov model hmm with time evolving parameters the dhdp imposes the belief that observations that are temporally proximate are more likely to be drawn from hmms with similar parameters while also allowing for innovation associated with abrupt changes in the music texture the sharing mechanisms of the time evolving model are derived and for inference a relatively simple markov chain monte carlo sampler is developed segmentation of a given musical piece is constituted via the model inference detailed examples are presented on several pieces with comparisons to other models the dhdp results are also compared with a conventional music theoretic analysis all the supplemental materials used by this paper are available online

structure of certain weyl modules for the spin groups abstract let k be an algebraically closed field of characteristic p 0 and let w be a finite dimensional k vector space of dimension greater than or equal to 5 in this paper we give the structure of certain weyl modules for g spin w in the case where p 2 as well as the dimension of the corresponding irreducible finite dimensional rational kg modules in addition we determine the composition factors of the restriction of certain irreducible finite dimensional rational k sl w modules to so w

a new proof of doobʼs theorem abstract we prove that every bounded positive irreducible stochastically continuous semigroup on the space of bounded measurable functions which is strong feller consists of kernel operators and possesses an invariant measure converges pointwise this differs from doob s theorem in that we do not require the semigroup to be markovian and request a fairly weak kind of irreducibility in addition we elaborate on the various notions of kernel operators in this context show the stronger result that the adjoint semigroup converges strongly and discuss as an example diffusion equations on rough domains the proofs are based on the theory of positive semigroups and do not use probability theory

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

kolmogorov equation associated to a stochastic navier stokes equation abstract a direct solution of the kolmogorov equation associated to a stochastic navier stokes equation is given with restriction to two space dimensions and periodic boundary conditions the existence of a variational solution is proved using a special property of the nonlinear operator

global l p estimates for degenerate ornstein uhlenbeck operators we consider a class of degenerate ornstein uhlenbeck operators in mathbb r n of the kind mathcal a equiv sum i j 1 p 0 a ij partial x i x j 2 sum i j 1 n b ij x i partial x j where aij bij are constant matrices aij is symmetric positive definite on mathbb r p 0 p0 n and bij is such that mathcal a is hypoelliptic for this class of operators we prove global lp estimates 1 p of the kind left vert partial x i x j 2 u right vert l p left mathbb r n right leq c left left vert mathcal a u right vert l p left mathbb r n right left vert u right vert l p left mathbb r n right right rm for i j 1 2 ldots p 0 and corresponding weak type 1 1 estimates this result seems to be the first case of global estimates in lebesgue lp spaces for complete hormander s operators sum x i 2 x 0 proved in absence of a structure of homogeneous group we obtain the previous estimates as a byproduct of the following one which is of interest in its own left vert partial x i x j 2 u right vert l p left s right leq c left vert lu right vert l p left s right for any u in c 0 infty left s right where s is the strip mathbb r n times left 1 1 right and l is the kolmogorov fokker planck operator mathcal a partial t to get this estimate we use in a crucial way the left invariance of l with respect to a lie group structure in mathbb r n 1 and some results on singular integrals on nonhomogeneous spaces recently proved in bramanti revista matematica iberoamericana 2009 in press

an invariance principle for branching diffusions in bounded domains we study branching diffusions in a bounded domain d of mathbb r d in which particles are killed upon hitting the boundary partial d it is known that any such process undergoes a phase transition when the branching rate beta exceeds a critical value a multiple of the first eigenvalue of the generator of the diffusion we investigate the system at criticality and show that the associated genealogical tree when the process is conditioned to survive for a long time converges to aldous continuum random tree under appropriate rescaling the result holds under only a mild assumption on the domain and is valid for all branching mechanisms with finite variance and a general class of diffusions

nonparametric mixtures of gaussian processes with power law behavior gaussian processes gps constitute one of the most important bayesian machine learning approaches based on a particularly effective method for placing a prior distribution over the space of regression functions several researchers have considered postulating mixtures of gps as a means of dealing with nonstationary covariance functions discontinuities multimodality and overlapping output signals in existing works mixtures of gps are based on the introduction of a gating function defined over the space of model input variables this way each postulated mixture component gp is effectively restricted in a limited subset of the input space in this paper we follow a different approach we consider a fully generative nonparametric bayesian model with power law behavior generating gps over the whole input space of the learned task we provide an efficient algorithm for model inference based on the variational bayesian framework and prove its efficacy using benchmark and real world datasets

on the lp independence of the spectrum of the hodge laplacian on non compact manifolds abstract the central aim of this paper is the study of the spectrum of the hodge laplacian on differential forms of any order k in l p the underlying space is a c smooth open manifold m n with ricci curvature bounded below and uniformly subexponential volume growth it will be demonstrated that on such manifolds the l p spectrum of the hodge laplacian on differential k forms is independent of p for 1 p whenever the weitzenbock tensor on k forms is also bounded below it follows as a corollary that the isolated eigenvalues of finite multiplicity are l p independent the proof relies on the existence of a gaussian upper bound for the heat kernel of the hodge laplacian by considering the l p spectra on the hyperbolic space h n 1 we conclude that the subexponential volume growth condition is necessary in the case of one forms as an application we will show that the spectrum of the laplacian on one forms has no gaps on certain manifolds with a pole or that are in a warped product form this will be done under less strict curvature restrictions than what has been known so far and it was achieved by finding the l 1 spectrum of the laplacian

music analysis using hidden markov mixture models we develop a hidden markov mixture model based on a dirichlet process dp prior for representation of the statistics of sequential data for which a single hidden markov model hmm may not be sufficient the dp prior has an intrinsic clustering property that encourages parameter sharing and this naturally reveals the proper number of mixture components the evaluation of posterior distributions for all model parameters is achieved in two ways 1 via a rigorous markov chain monte carlo method and 2 approximately and efficiently via a variational bayes formulation using dp hmm mixture models in a bayesian setting we propose a novel scheme for music analysis highlighting the effectiveness of the dp hmm mixture model music is treated as a time series data sequence and each music piece is represented as a mixture of hmms we approximate the similarity of two music pieces by computing the distance between the associated hmm mixtures experimental results are presented for synthesized sequential data and from classical music clips music similarities computed using dp hmm mixture modeling are compared to those computed from gaussian mixture modeling for which the mixture modeling is also performed using dp the results show that the performance of dp hmm mixture modeling exceeds that of the dp gaussian mixture modeling

bayesian density estimation and inference using mixtures abstract we describe and illustrate bayesian inference in models for density estimation using mixtures of dirichlet processes these models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions efficient simulation methods are used to approximate various prior posterior and predictive distributions this allows for direct inference on a variety of practical issues including problems of local versus global smoothing uncertainty about density estimates assessment of modality and the inference on the numbers of components also convergence results are established for a general class of normal mixture models

maximum and anti maximum principlesand eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations in this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations we continue the study of some different notions of small perturbations and discuss their relations to comparisons of green s functions refined maximum and anti maximum principles ground state and the decay of eigenfunctions in particular we show that if v is a positive function which is a semismall perturbation of a subcritical schrodinger operator h defined on a domain omega subset mathbb r d and phi k k geq 0 are the dirichlet eigenfunctions of the equation hu lambda vu then for any k geq 0 the function phi k phi 0 is bounded and has a continuous extension up to the martin boundary of the pair omega h where phi 0 is the ground state of h with a principal eigenvalue lambda 0

trotter kato theorems for bi continuous semigroups and applications to feller semigroups abstract we give results on the convergence of bi continuous semigroups introduced and studied by kuhnemund as a consequence we obtain a lie trotter product formula and apply it to feller semigroups generated by second order elliptic differential operators with unbounded coefficients in c b r n

long time behavior of heat kernels of operators with unbounded drift terms abstract given a second order elliptic operator on r d with bounded diffusion coefficients and unbounded drift which is the generator of a strongly continuous semigroup on l 2 r d represented by an integral we study the time behavior of the integral kernel and prove estimates on its decay at infinity if the diffusion coefficients are symmetric a local lower estimate is also proved

analytic semigroups and degenerate elliptic operators with unbounded coefficients a probabilistic approach abstract by using techniques derived from the theory of stochastic differential equations we prove that a class of second order degenerate elliptic operators having unbounded coefficients generates analytic semigroups in c b r d the space of uniformly continuous and bounded functions from r d into r

poincare s inequalities and talagrand s concentration phenomenon for the exponential distribution we present a simple proof based on modified logarithmic sobolev inequalities of talagrand s concentration inequality for the exponential distribution we actually observe that every measure satisfying a poincare inequality shares the same concentration phenomenon we also discuss exponential integrability under poincare inequalities and its consequence to sharp diameter upper bounds on spectral gaps

bayesian nonparametric inference of switching dynamic linear models many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes we consider two such models the switching linear dynamical system slds and the switching vector autoregressive var process our bayesian nonparametric approach utilizes a hierarchical dirichlet process prior to learn an unknown number of persistent smooth dynamical modes we additionally employ automatic relevance determination to infer a sparse set of dynamic dependencies allowing us to learn slds with varying state dimension or switching var processes with varying autoregressive order we develop a sampling algorithm that combines a truncated approximation to the dirichlet process with efficient joint sampling of the mode and state sequences the utility and flexibility of our model are demonstrated on synthetic data sequences of dancing honey bees the ibovespa stock index and a maneuvering target tracking application

on diagonal estimates on schrodinger semigroup kernels and reduced heat kernels we prove various estimates for the kernels of semigroups generated by schrodinger operators with magnetic field and potential of polynomial growth we also investigate the reduced heat kernels

global properties of generalized ornstein uhlenbeck operators on lp rn rn with more than linearly growing coefficients we show that the realization ap of the elliptic operator au div q u f u vu in lp rn rn p 1 generates a strongly continuous semigroup and we determine its domain d ap u w2 p rn rn f u vu lp rn rn if 1 p the diffusion coefficients q qij are uniformly elliptic and bounded together with their first order derivatives the drift coefficients f can grow as x log x and v can grow logarithmically our approach relies on the monniaux pruss theorem on the sum of noncommuting operators we also prove lp lq estimates and under somewhat stronger assumptions we establish pointwise gradient estimates and smoothing of the semigroup in the spaces w p rn rn 0 1 where 1 p

a recursion and a combinatorial formula for jack polynomials the jack polynomials j x form a remarkable class of symmetric polynomials they are indexed by a partition and depend on a parameter one of their properties is that several classical families of symmetric functions can be obtained by specializing e g the monomial symmetric functions m the elementary functions e 0 the schur functions s 1 and nally the two classes of zonal polynomials 2 1 2 the jack polynomials can be de ned in various ways e g a as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b as simultaneous eigenfunctions of certain di erential operators the sekiguchi debiard operators recently opdam o constructed a similar family f x of nonsymmetric polynomials the index runs now through all compositions nn they are de ned in a completely similar fashion e g the sekiguchi debiard operators are being replaced by the cherednik di erential re ection operators see sect 3 it is becoming more and more clear that these polynomials are as important as their symmetric counterparts the purpose of this paper is to add to the existing characterizations of jack polynomials two further ones c a recursion formula among the f together with two formulas to obtain j from them d combinatorial formulas of both j and f in terms of certain generalized tableaux

on modified logarithmic sobolev inequalities for bernoulli and poisson measures we show that for any positive functionfon the discrete cube 0 1 n ent np f pqe np1f df 2where npis the product measure of the bernoulli measure with probability of successp as well as related inequalities which may be shown to imply in the limit the classical gaussian logarithmic sobolev inequality as well as a logarithmic sobolev inequality for poisson measure we further investigate modified logarithmic sobolev inequalities to analyze integrability properties of lipschitz functions on discrete spaces in particular we obtain under modified logarithmic sobolev inequalities some concentration results for product measures that extend the classical exponential inequalities for sums of independent random variables

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

an existence theorem for the navier stokes flow in the exterior of a rotating obstacle we consider the three dimensional navier stokes initial value problem in the exterior of a rotating obstacle it is proved that a unique solution exists locally in time if the initial velocity possesses the regularity l 1 2 this regularity assumption is the same as that in the famous paper of fujita kato an essential step for the proof is the deduction of a certain smoothing property together with estimates near t 0 of the semigroup which is not an analytic one generated by the operator in the space l 2 where stands for the angular velocity of the rotating obstacle and p denotes the projection associated with the helmholtz decomposition