general β jacobi corners process and the gaussian free field we prove that the two dimensional gaussian free field describes the asymptotics of global fluctuations of a multilevel extension of the general jacobi random matrix ensembles our approach is based on the connection of the jacobi ensembles to a degeneration of the macdonald processes that parallels the degeneration of the macdonald polynomials to the heckman opdam hypergeometric functions of type a we also discuss the limit 2015 wiley periodicals inc

a new integral formula for heckman opdam hypergeometric functions abstract we provide harish chandra type formulas for the multivariate bessel functions and heckman opdam hypergeometric functions as representation valued integrals over dressing orbits our expression is the quasi classical limit of the realization of macdonald polynomials as traces of intertwiners of quantum groups given by etingof and kirillov jr in 10 integration over the liouville tori of the gelfand tsetlin integrable system and adjunction for higher calogero moser hamiltonians recovers and gives a new proof of the integral realization over gelfand tsetlin polytopes which appeared in the recent work 4 of borodin and gorin on the jacobi corners ensemble

a finitization of the bead process the bead process is the particle system defined on parallel lines with underlying measure giving constant weight to all configurations in which particles on neighbouring lines interlace and zero weight otherwise motivated by the statistical mechanical model of the tiling of an abc hexagon by three species of rhombi a finitized version of the bead process is defined the corresponding joint distribution can be realized as an eigenvalue probability density function for a sequence of random matrices the finitized bead process is determinantal and we give the correlation kernel in terms of jacobi polynomials two scaling limits are considered a global limit in which the spacing between lines goes to zero and a certain bulk scaling limit in the global limit the shape of the support of the particles is determined while in the bulk scaling limit the bead process kernel of boutillier is reclaimed after appropriate identification of the anisotropy parameter therein

random matrix ensembles associated to compact symmetric spaces we introduce random matrix ensembles that correspond to the infinite families of irreducible riemannian symmetric spaces of type i in particular we recover the circular orthogonal and symplectic ensembles of dyson and find other families of unitary orthogonal and symplectic ensembles of jacobi type we discuss the universal and weakly universal features of the global and local correlations of the levels in the bulk and at the hard edge of the spectrum i e at the central points 1 on the unit circle previously known results are extended and we find new simple formulas for the bessel kernels that describe the local correlations at a hard edge

asymptotic couplings by reflection and applications for nonlinear monotone spdes abstract asymptotic couplings by reflection are constructed for a class of nonlinear monotone spdes stochastic partial differential equations as applications gradient holder estimates as well as exponential convergence are derived for the associated markov semigroup the main results are illustrated by stochastic generalized porous media equations stochastic p laplacian equations and stochastic generalized fast diffusion equations we emphasize that the gradient estimate is studied for the first time for these equations

continuum limits of random matrices and the brownian carousel we show that at any location away from the spectral edge the eigenvalues of the gaussian unitary ensemble and its general siblings converge to sine a translation invariant point process this process has a geometric description in term of the brownian carousel a deterministic function of brownian motion in the hyperbolic plane

moments of the position of the maximum for gue characteristic polynomials and for log correlated gaussian processes we study three instances of log correlated processes on the interval the logarithm of the gaussian unitary ensemble gue characteristic polynomial the gaussian log correlated potential in presence of edge charges and the fractional brownian motion with hurst index h rightarrow 0 fbm0 in previous collaborations we obtained the probability distribution function pdf of the value of the global minimum equivalently maximum for the first two processes using the freezing duality conjecture fdc here we study the pdf of the position of the maximum x m through its moments using replica this requires calculating moments of the density of eigenvalues in the beta jacobi ensemble using jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary beta 0 and positive integer n in terms of sums over partitions for positive moments this expression agrees with a very recent independent derivation by mezzadri and reynolds we check our results against a contour integral formula derived recently by borodin and gorin presented in the appendix 1 from these authors the duality necessary for the fdc to work is proved and on our expressions found to correspond to exchange of partitions with their dual performing the limit n rightarrow 0 and to negative dyson index beta rightarrow 2 we obtain the moments of x m and give explicit expressions for the lowest ones numerical checks for the gue polynomials performed independently by n simm indicate encouraging agreement some results are also obtained for moments in laguerre hermite gaussian as well as circular and related ensembles the correlations of the position and the value of the field at the minimum are also analyzed

singular stochastic equations on hilbert spaces harnack inequalities for their transition semigroups we consider stochastic equations in hilbert spaces with singular drift in the framework of g da prato m rockner singular dissipative stochastic equations in hilbert spaces probab theory related fields 124 2 2002 261 303 we prove a harnack inequality in the sense of f y wang logarithmic sobolev inequalities on noncompact riemannian manifolds probab theory related fields 109 1997 417 424 for its transition semigroup and exploit its consequences in particular we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding kolmogorov operator has at most one infinitesimally invariant measure satisfying some mild integrability conditions finally we prove existence of such a measure for noncontinuous drifts

height fluctuations in the honeycomb dimer model we study a model of random surfaces arising in the dimer model on the honeycomb lattice for a fixed wire frame boundary condition as the lattice spacing 0 cohn kenyon and propp 3 showed the almost sure convergence of a random surface to a non random limit shape 0 in 12 okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions in particular constructing a natural conformal structure on them we show here that when 0 has no facets for a family of boundary conditions approximating the wire frame the large scale surface fluctuations height fluctuations about 0 converge as 0 to a gaussian free field for the above conformal structure we also show that the local statistics of the fluctuations near a given point x are as conjectured in 3 given by the unique ergodic gibbs measure on plane configurations whose slope is the slope of the tangent plane of 0 at x

liouville theorems for non local operators abstract the paper characterizes some classes of pseudo differential operators for which there are or there are not non constant bounded harmonic functions non local perturbations of ornstein uhlenbeck operators and operators with dissipative coefficients are considered the methods used are probabilistic and based on the concept of absorption function and on a new extension of the bismut elworthy li formula the probabilistic interpretation of the liouville theorem by means of absorption functions for general markov processes is given as well

plancherel representations of u infty and correlated gaussian free fields we study asymptotics of traces of noncommutative monomials formed by images of certain elements of the universal enveloping algebra of the infinitedimensional unitary group in its plancherel representations we prove that they converge to commutative moments of a gaussian process that can be viewed as a collection of simply yet nontrivially correlated two dimensional gaussian free fields the limiting process has previously arisen via the global scaling limit of spectra for submatrices of wigner hermitian random matrices the results of the present work were announced in 4

central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients we study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles we study those biorthogonal ensembles for which the underlying biorthogonal family s

integrability conditions for sdes and semilinear spdes by using the local dimension free harnack inequality established on incomplete riemannian manifolds integrability conditions on the coefficients are presented for sdes to imply the nonexplosion of solutions as well as the existence uniqueness and regularity estimates of invariant probability measures these conditions include a class of drifts unbounded on compact domains such that the usual lyapunov conditions cannot be verified the main results are extended to second order differential operators on hilbert spaces and semilinear spdes

thick points of the gaussian free field let u c be a bounded domain with smooth boundary and let f be an instance of the continuum gaussian free field on u with respect to the dirichlet inner product u f x g x dx the set t a u of a thick points of f consists of those z u such that the average of f on a disk of radius r centered at z has growth a log 1 r as r 0 we show that for each 0 a 2 the hausdorff dimension of t a u is almost surely 2 a that 2 a t a u when 0 2 furthermore we prove that t a u is invariant under conformal transformations in an appropriate sense the notion of a thick point is connected to the liouville quantum gravity measure with parameter y given formally by dz e 2 f z dz considered by duplantier and sheffield

product of random projections jacobi ensembles and universality problems arising from free probability we consider the product of two independent randomly rotated projectors the square of its radial part turns out to be distributed as a jacobi ensemble we study its global and local properties in the large dimension scaling relevant to free probability theory we establish asymptotics for one point and two point correlation functions as well as properties of largest and smallest eigenvalues

stochastic higher spin six vertex model and q taseps abstract we present two new connections between the inhomogeneous stochastic higher spin six vertex model in a quadrant and integrable stochastic systems from the macdonald processes hierarchy first we show how macdonald q difference operators with t 0 an algebraic tool crucial for studying the corresponding macdonald processes can be utilized to get q moments of the height function h in the higher spin six vertex model first computed in 21 using bethe ansatz this result in particular implies that for the vertex model with the step bernoulli boundary condition the value of h at an arbitrary point n 1 t z 2 z 1 has the same distribution as the last component n of a random partition under a specific t 0 macdonald measure on the other hand it is known that x n n n can be identified with the location of the nth particle in a certain discrete time q tasep started from the step initial configuration the second construction we present is a coupling of this q tasep and the higher spin six vertex model with the step bernoulli boundary condition along time like paths providing an independent probabilistic explanation of the equality of h n 1 t and x n n in distribution as an illustration of our main results we obtain gue tracy widom asymptotics of a certain discrete time q tasep with the step initial configuration and special jump parameters by means of schur measures which are t q macdonald measures this analysis combines our results with the identification of averages of observables between the stochastic higher spin six vertex model and schur measures obtained recently in 8

analysis on wiener spaces ii differential forms abstract the author gives the notions of p forms and p forms and shows some properties of them in particular a vanishing theorem of de rham cohomology of poincare type also he shows the hypoellipticity of certain elliptic operators on wiener space and a theorem of poinare inequality type

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

a parametrix for the neumann problem on pseudoconvex domains of finite type abstract we construct a parametrix for the neumann problem on any pseudoconvex domain of finite type which reduces completely the study of isotropic lp sobolev and holder estimates to those for b and b we also establish the maximal nonisotropic estimates in the case of d q domains and provide a precise comparison of the bergman and szego projections and of the canonical solutions to with respect to these two operators the argument is substantially simpler than the known ones for the cases of strongly pseudoconvex domains and pseudoconvex domains of finite type in c 2

ingham type theorems for vector valued functions and observability of coupled linear systems we propose a new approach to study the observability of coupled linear distributed systems it is based on a generalization of some classical theorems of nonharmonic analysis to vector valued functions applying this method we answer some questions of j l lions controlabilite exacte et stabilisation de systems distribues vol 2 masson 1988 on the boundary observability of coupled linear distributed systems in particular cases

logarithmic sobolev inequalities on noncompact riemannian manifolds this paper presents a dimension free harnack type inequality for heat semigroups on manifolds from which a dimension free lower bound is obtained for the logarithmic sobolev constant on compact manifolds and a new criterion is proved for the logarithmic sobolev inequalities abbrev lsi on noncompact manifolds as a result it is shown that lsi may hold even though the curvature of the operator is negative everywhere

gues and queues consider the process d k k 1 2 given by b i being independent standard brownian motions this process describes the limiting behavior near the edge in queues in series totally asymmetric exclusion processes or oriented percolation the problem of finding the distribution of d was posed in gw the main result of this paper is that the process d has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from gaussian unitary ensemble

boundary stabilization of solutions of a nonlinear system of timoshenko type in this paper we are concerned with a multidimensional timoshenko system subjected to boundary conditions of memory type we establish general rate decay results the usual exponential and polynomial decay rates are only special cases

asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star shaped domain we study the asymptotic stability and give the energy decay rates for solutions of the wave equation with boundary dissipation of the memory type

interpretations of some parameter dependent generalizations of classical matrix ensembles two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions pdfs as the parameter is varied one interpolates between the eigenvalue pdf for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue pdf for a single classical ensemble with unitary symmetry while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry we give interpretations of these pdfs in terms of probabilities associated to the continuous robinson schensted knuth correspondence between matrices with entries chosen from certain exponential distributions and non intersecting lattice paths and in the course of this probability measures on partitions and pairs of partitions are identified the latter are generalized by using macdonald polynomial theory and a particular continuum limit the jacobi limit of the resulting measures is shown to give pdfs related to those appearing in the work of anderson on the selberg integral and also in some classical work of dixon by interpreting anderson s and dixon s work as giving the pdf for the zeros of a certain rational function it is then possible to identify random matrices whose eigenvalue pdfs realize the original parameter dependent pdfs this line of theory allows sampling of the original parameter dependent pdfs their dixon anderson type generalizations and associated marginal distributions from the zeros of certain polynomials defined in terms of random three term recurrences

gaussian free field in an interlacing particle system with two jump rates we study the fluctuations of a random surface in a stochastic growth model on a system of interlacing particles placed on a two dimensional lattice there are two different types of particles one with a low jump rate and the other with a high jump rate in the large time limit the random surface has a deterministic shape due to the different jump rates the limit shape and the domain on which it is defined are not smooth the main result is that the fluctuations of the random surface are governed by the gaussian free field 2012 wiley periodicals inc

levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator we establish by simple semigroup arguments a levy gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the gaussian measure this produces in particular a new proof of the gaussian isoperimetric inequality this isoperimetric inequality strengthens the classical logarithmic sobolev inequality in this context a local version for the heat kernel measures is also proved which may then be extended into an isoperimetric inequality for the wiener measure on the paths of a riemannian manifold with bounded ricci curvature

on existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions abstract we study the global existence of solutions of the nonlinear degenerate wave equation 0 x y y 0 in 0 y 0 on 1 0 y y f y g y 0 on 0 0 y x 0 y 0 y x 0 y 1 x in where y denotes the derivative of y with respect to parameter t f s c0 s s and g is a nondecreasing c1 function such that k1 s 2 g s s k2 s 2 for some k1 k2 0 with 0 0 if n 1 2 the existence of solutions is proved by means of the faedo galerkin method furthermore when 0 the uniform decay is obtained by making use of the perturbed energy method

on the kirchhoff plates equations with thermal effects and memory boundary conditions in this paper we study the existence of weak and strong global solutions and uniform decay of the energy to the kirchhoff plates equations with thermal effect and memory conditions working at the boundary we show that the dissipation produced by the memory effect not depend on the present values of temperature gradient that is we show that the dissipation produced by memory effect is strong enough to produce exponential decay of the solution provided the relaxation functions also decays exponentially when the relaxation functions decays polynomially we show that the solution decays polynomially with the same rate

existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary in this article we study the degenerate system r 1 r 2 0 subject to memory conditions on the boundary given email protected 1 x u t email protected email protected u v email protected 0 r 2 x v t email protected a u v email protected 0 email protected 0 email protected 0 tg 1 t s email protected email protected s email protected 1x 0 email protected 0 email protected 0 tg 2 t s email protected email protected s email protected 1x 0 u 0 v 0 u 0 v 0 r 1u t 0 r 2v t 0 r 1u 1 r 2v 1 email protected where w is a bounded region in r n whose boundary is partitioned into disjoint sets c 0 c 1 we prove that the dissipations given by the memory terms are strong enough to guarantee exponential or polynomial decay provided the relaxation functions also decay exponentially or polynomially and with the same rate of decay

harnack and functional inequalities for generalized mehler semigroups abstract harnack inequalities are established for a class of generalized mehler semigroups which in particular imply upper bound estimates for the transition density moreover poincare and log sobolev inequalities are proved in terms of estimates for the square field operators furthermore under a condition well known in the gaussian case we prove that generalized mehler semigroups are strong feller the results are illustrated by concrete examples in particular we show that a generalized mehler semigroup with an stable part is not hyperbounded but exponentially ergodic and that the log sobolev constant obtained by our method in the special gaussian case can be sharper than the one following from the usual curvature condition moreover a harnack inequality is established for the generalized mehler semigroup associated with the dirichlet heat semigroup on 0 1 we also prove that this semigroup is not hyperbounded

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

quantum knizhnik zamolodchikov equations and affine root systems quantum difference knizhnik zamolodchikov equations s1 fr are generalized for ther matrices from ch1 with the arguments in arbitrary root systems and their formal counterparts in particular qkz equations with certain boundary conditions are introducted the self consistency of the equations from fr and the cross derivative integrability conditions for ther matrix kz equations from ch2 are obtained as corollaries a difference counterpart of the quantum many body problem connected with macdonald s operators is defined as an application

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

harnack inequality and applications for stochastic generalized porous media equations by using coupling and girsanov transformations the dimension free harnack inequality and the strong feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations as applications explicit upper bounds of the l p norm of the density as well as hypercontractivity ultracontractivity and compactness of the corresponding semigroup are derived

harnack inequality and strong feller property for stochastic fast diffusion equations abstract as a continuation to f y wang harnack inequality and applications for stochastic generalized porous media equations ann probab 35 2007 1333 1350 where the harnack inequality and the strong feller property are studied for a class of stochastic generalized porous media equations this paper presents analogous results for stochastic fast diffusion equations since the fast diffusion equation possesses weaker dissipativity than the porous medium one does some technical difficulties appear in the study as a compensation to the weaker dissipativity condition a sobolev nash inequality is assumed for the underlying self adjoint operator in applications some concrete examples are constructed to illustrate the main results

log harnack inequality for stochastic burgers equations and applications abstract by proving an l 2 gradient estimate for the corresponding galerkin approximations the log harnack inequality is established for the semigroup associated to a class of stochastic burgers equations as applications we derive the strong feller property of the semigroup the irreducibility of the solution the entropy cost inequality for the adjoint semigroup and entropy upper bounds of the transition density

upper semicontinuity of pullback attractors for a non autonomous viscoelastic equation this paper is concerned with upper semicontinuity of pullback attractors for a non autonomous viscoelastic equation with a past history with a nonautonomous perturbation in particular under some proper assumptions we prove that the pullback attractor a e t t r of eq 1 1 with e 0 and the global attractor a of eq 1 1 with e 0 satisfy lim e 0 dist x a e t a 0 for any email protected r

beta ensembles stochastic airy spectrum and a diffusion we prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low lying eigenvalues of the random schrodinger operator d2 dx2 x 2 bx restricted to the positive half line where b x is white noise in doing so we extend the definition of the tracy widom distributions to all 0 and also analyze their tails last in a parallel development we provide a second characterization of these laws in terms of a one dimensional diffusion the proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit in particular we show how tracy widom laws arise from a functional central limit theorem

structural properties of semilinear spdes driven by cylindrical stable processes we consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise we investigate structural properties of the solutions like markov irreducibility stochastic continuity feller and strong feller properties and study integrability of trajectories the obtained results are applied to semilinear stochastic heat equations with dirichlet boundary conditions and bounded and lipschitz nonlinearities

generalized mehler semigroups and applications we construct and study generalized mehler semigroups p t t 0 and their associated markov processesm the construction methods for p t t 0 are based on some new purely functional analytic results implying in particular that any strongly continuous semigroup on a hilbert spaceh can be extended to some larger hilbert spacee with the embeddingh e being hilbert schmidt the same analytic extension results are applied to construct strong solutions to stochastic differential equations of typedx t c dw t ax t dt with possibly unbounded linear operatorsa andc onh on a suitably chosen larger spacee for gaussian generalized mehler semigroups p t t 0 with corresponding markov processm the associated non symmetric dirichlet forms ed e are explicitly calculated and a necessary and sufficient condition for path regularity ofm in terms of e d e is proved then using dirichlet form methods it is shown thatm weakly solves the above stochastic differential equation if the state spacee is chosen appropriately finally we discuss the differences between these two methods yielding strong resp weak solutions

airy kernel with two sets of parameters in directed percolation and random matrix theory we introduce a generalization of the extended airy kernel with two sets of real parameters we show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

convergence in variation of solutions of nonlinear fokker planck kolmogorov equations to stationary measures abstract we study convergence in variation of probability solutions of nonlinear fokker planck kolmogorov equations to stationary solutions we obtain sufficient conditions for the exponential convergence of solutions to the stationary solution in case of coefficients that can have an arbitrary growth at infinity and depend on the solutions through convolutions with unbounded discontinuous kernels in addition we study a more difficult case where the nonlinear equation has several stationary solutions and convergence to a stationary solution depends on initial data finally we obtain sufficient conditions for solvability of nonlinear fokker planck kolmogorov equations

anisotropic 2 1 d growth and gaussian limits of q whittaker processes we consider a discrete model for anisotropic 2 1 dimensional growth of an interface height function owing to a connection with q whittaker functions this system enjoys many explicit integral formulas by considering certain gaussian stochastic differential equation limits of the model we are able to prove a space time limit of covariances to those of the 2 1 dimensional additive stochastic heat equation or edwards wilkinson equation along characteristic directions in particular the bulk height function converges to the gaussian free field which evolves according to this stochastic pde

log gases on quadratic lattices via discrete loop equations and q boxed plane partitions abstract we study a general class of log gas ensembles on shifted quadratic lattices we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are gaussian with a universal covariance we apply our general results to analyze the asymptotic behavior of a q boxed plane partition model introduced by borodin gorin and rains in particular we show that the global fluctuations of the height function on a fixed slice are described by a one dimensional section of a pullback of the two dimensional gaussian free field our approach is based on a q analogue of the schwinger dyson or loop equations which originate in the work of nekrasov and his collaborators and extends the methods developed by borodin gorin and guionnet to quadratic lattices

difference equation for the heckman opdam hypergeometric function and its confluent whittaker limit abstract we present an explicit difference equation for the heckman opdam hypergeometric function associated with root systems via a confluent hypergeometric limit an analogous difference equation is obtained for the class one whittaker function diagonalizing the open quantum toda chain

uniform decay of the solution to a wave equation with memory conditions on the boundary abstract a wave equation with variable coefficients in principal part and memory conditions on the boundary is considered the riemannian geometry method is applied to prove the exponential decay of the energy provided the relaxation function also decays exponentially

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

harnack inequalities for stochastic equations driven by levy noise abstract by using coupling argument and regularization approximations of the underlying subordinator dimension free harnack inequalities are established for a class of stochastic equations driven by a levy noise containing a subordinate brownian motion the harnack inequalities are new even for linear equations driven by levy noise and the gradient estimate implied by our log harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for levy processes or linear equations driven by levy noise the main results are also extended to semilinear stochastic equations in hilbert spaces

interlacing adjacent levels of beta jacobi corners processes we study the asymptotics of the global fluctuations for the difference between two adjacent levels in the beta jacobi corners process multilevel and general beta extension of the classical jacobi ensemble of random matrices the limit is identified with the derivative of the 2d gaussian free field our main tools are integral forms for the macdonald type difference operators originating from the shuffle algebra

levy ornstein uhlenbeck transition semigroup as second quantized operator abstract let be an invariant measure for the transition semigroup p t of the markov family defined by the ornstein uhlenbeck type equation d x a x d t d l on a hilbert space e driven by a levy process l it is shown that for any t 0 p t considered on l 2 is a second quantized operator on a poisson fock space of e at from this representation it follows that the transition semigroup corresponding to the equation on e r driven by an stable noise l 0 2 is neither compact nor symmetric

fluctuations of eigenvalues of random normal matrices in this article we consider a fairly general potential in the plane and the corresponding boltzmann gibbs distribution of eigenvalues of random normal matrices as the order of the matrices tends to infinity the eigenvalues condensate on a certain compact subset of the plane the droplet we prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a gaussian field and we discuss various ramifications of this result

on the stochastic p laplace equation the p laplace equation with random perturbation is studied for the singular case 1 p 2 in this paper some properties of the invariant measure and transition semigroups are obtained the main tool is the dimension free harnack inequality which is established by using the coupling argument as consequences some ergodicity compactness and contractive properties are derived for the associated transition semigroups the main results are applied to stochastic reaction diffusion equations and the stochastic p laplace equation in hilbert space

the heckman opdam markov processes we introduce and study the natural counterpart of the dunkl markov processes in a negatively curved setting we give a semimartingale decomposition of the radial part and some properties of the jumps we prove also a law of large numbers a central limit theorem and the convergence of the normalized process to the dunkl process eventually we describe the asymptotic behavior of the infinite loop as it was done by anker bougerol and jeulin in the symmetric spaces setting in iberoamericana 18 41 97 2002