uniform positivity improving property sobolev inequalities and spectral gaps abstract we introduce a certain kind of strong ergodicity condition to study the existence of spectral gap for markov generator we can estimate the spectral gap using the ergodicity condition and a sobolev type inequality we apply our results to the dirichlet form on wiener spaces riemannian manifolds and loop spaces

essential spectral radius for markov semigroups i discrete time case using two new measures of non compactness p and w p for a positive kernel p on a polish space e we obtain a new formula of nussbaum gelfand type for the essential spectral radius ress p on b using that formula we show that different known sufficient conditions for geometric ergodicity such as doeblin s condition drift condition by means of lyapunov function geometric recurrence etc lead to variational formulas of the essential spectral radius all those can be easily transported on the weighted space bu some related results on l2 are also obtained especially in the symmetric case moreover we prove that for a strongly feller and topologically transitive markov kernel the large deviation principle of donsker varadhan for occupation measures of the associated markov process holds if and only if the essential spectral radius is zero this result allows us to show that the sufficient condition of donsker varadhan for the large deviation principle is in fact necessary the knowledge of ress p allows us to estimate eigenvalues of p in l2 in the symmetric case and to estimate the geometric convergence rate by means of that in the metric of wasserstein applications to different concrete models are provided for illustrating those general results

from concentration to logarithmic sobolev and poincare inequalities abstract we give a new proof of the fact that gaussian concentration implies the logarithmic sobolev inequality when the curvature is bounded from below and also that exponential concentration implies poincare inequality under null curvature condition our proof holds on non smooth structures such as length spaces and provides a universal control of the constants we also give a new proof of the equivalence between dimension free gaussian concentration and talagrand s transport inequality

coercive inequalities for gibbs measures abstract we prove the generalized nash and logarithmic nash inequalities for gibbs measures with dirichlet form associated to the kawasaki dynamics

estimation of the first eigenvalue of second order elliptic operators this note studies the first non trivial eigenvalue of second order self adjoint elliptic operators in rd some lower bounds of the eigenvalue are obtained by using a probabilistic approach and some geometric consideration in one dimensional case an analytic proof is also presented the resulting bounds can be sharp

robust dimension free isoperimetry in gaussian space we prove the first robust dimension free isoperimetric result for the standard gaussian measure n and the corresponding boundary measure n in rn the main result in the theory of gaussian isoperimetry proven in the 1970s by sudakov and tsirelson and independently by borell states that if n a 1 2 then the surface area of a is bounded by the surface area of a half space with the same measure n a 2 1 2 our results imply in particular that if a rn satisfies n a 1 2 and n a 2 1 2 then there exists a half space b rn such that n a b clog 1 2 1 for an absolute constant c since the gaussian isoperimetric result was established only recently a robust version of the gaussian isoperimetric result was obtained by cianchi et al who showed that n a b c n for some function c n with no effective bounds compared to the results of cianchi et al our results have optimal i e no dependence on the dimension but worse dependence on

a poincare inequality on loop spaces we show that the laplacian on the loop space over a class of riemannian manifolds has a spectral gap the laplacian is defined using the levi civita connection the brownian bridge measure and the standard bismut tangent spaces

approach to equilibrium of particles diffusing on curved surfaces abstract we present a simple numerical analysis of the diffusion on a curved surface given by the equation r 0 in a finite domain d r 3 the first non vanishing eigenvalue of the beltrami laplace operator with the reflecting boundary conditions is determined in our simulations for the p d g s s1 and i wp nodal periodic surfaces where d is their respective cubic unit cell we observe that the first eigenvalue for the surfaces of simple topology p d g i wp is smaller than for the surfaces of complex topology s s1

sharp dimension free quantitative estimates for the gaussian isoperimetric inequality we provide a full quantitative version of the gaussian isoperimetric inequality the difference between the gaussian perimeter of a given set and a half space with the same mass controls the gap between the norms of the corresponding barycenters in particular it controls the gaussian measure of the symmetric difference between the set and the half space oriented so to have the barycenter in the same direction of the set our estimate is independent of the dimension sharp on the decay rate with respect to the gap and with optimal dependence on the mass

on order preservation and positive correlations for multidimensional diffusion processes as a continuation of the study by herbst and pitt 1991 this note presents two criteria the first one is on the order preservation for two may be different multidimensional diffusion processes the second one is on the preservation of positive correlations for a diffusion process

an isoperimetric inequality for uniformly log concave measures and uniformly convex bodies abstract we prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log concave measures that we introduce these inequalities imply up to universal constants the log sobolev inequalities proved by bobkov ledoux s g bobkov m ledoux from brunn minkowski to brascamp lieb and to logarithmic sobolev inequalities geom funct anal 10 5 2000 1028 1052 and the isoperimetric inequalities due to bakry ledoux d bakry m ledoux levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator invent math 123 2 1996 259 281 and bobkov zegarlinski s g bobkov b zegarlinski entropy bounds and isoperimetry mem amer math soc 176 829 2005 x 69 we also recover a concentration inequality for uniformly convex bodies similar to that proved by gromov milman m gromov v d milman generalization of the spherical isoperimetric inequality to uniformly convex banach spaces compos math 62 3 1987 263 282

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

isoperimetric constants for product probability measures a dimension free lower bound is found for isoperimetric constants of product probability measures from this some analytic inequalities are derived

analysis and geometry on configuration spaces the gibbsian case using a natural riemannian geometry like structure on the configuration space over rd we prove that for a large class of potentials the corresponding canonical gibbs measures on 1 can be completely characterized by an integration by parts formula that is if is the gradient of the riemannian structure on one can define a corresponding divergence div such that the canonical gibbs measures are exactly those measures for which and div are dual operators onl2 one consequence is that for such the corresponding dirichlet forms e are defined in addition each of them is shown to be associated with a conservative diffusion process on with invariant measure the corresponding generators are extensions of the operator div the diffusions can be characterized in terms of a martingale problem and they can be considered as a brownian motion on perturbed by a singular drift another main result of this paper is the following if is a canonical gibbs measure then it is extreme or a pure phase if and only if the corresponding weak sobolev spacew1 2 on is irreducible as a consequence we prove that for extreme canonical gibbs measures the above mentioned diffusions are time ergodic in particular this holds for tempered grand canonical gibbs measures ruelle measures provided that the activity constant is small enough we also include a complete discussion of the free case i e 0 where the underlying space rdis even replaced by a riemannian manifoldx

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

slow decay of gibbs measures with heavy tails we consider glauber dynamics reversible with respect to gibbs measures with heavy tails in the case when spins are unbounded the interactions are bounded and of finite range the self potential enters into two classes of measures concave probability measures and sub exponential laws for which it is known that no exponential decay can occur using coercive inequalities we prove that for concave probability measures the associated infinite volume semi group decays to equilibrium polynomially and stretched exponentially for sub exponential laws this improves and extends previous results by bobkov and zegarlinski

symmetry of minimizers of a gaussian isoperimetric problem we study an isoperimetric problem described by a functional that consists of the standard gaussian perimeter and the norm of the barycenter the second term is in competition with the perimeter balancing the mass with respect to the origin and because of that the solution is not always the half space we characterize all the minimizers of this functional when the volume is close to one by proving that the minimizer is either the half space or the symmetric strip depending on the strength of the barycenter term as a corollary we obtain that the symmetric strip is the solution of the gaussian isoperimetric problem among symmetric sets when the volume is close to one as another corollary we obtain the optimal constant in the quantitative gaussian isoperimetric inequality

exponential convergence of coupled diffusion processes the coupled diffusion process which describes brownian motors is an important model in the physics related to biophenomena we address the exponential convergence of the markov semigroup of a coupled diffusion process and show the spectral gap inequality and log sobolev inequality by comparing them with those of related diffusion processes at the end of the paper we provide sufficient conditions for a coupled diffusion process to converge exponentially

extremal properties of central half spaces for product measures we deal with the isoperimetric and the shift problem for subsets of measure 1 2 in product probability spaces we prove that the canonical central half spaces are extremal in particular cases products of log concave measures on the real line satisfying precise conditions and products of uniform measures on spheres or balls as a corollary we improve the known log sobolev constants for euclidean balls we also give some new results about the related question of estimating the volume of sections of unit balls of lp sums of minkowski spaces

estimation of spectral gap for elliptic operators a variational formula for the lower bound of the spectral gap of an elliptic operator is presented in the paper for the first time the main known results are either recovered or improved a large number of new examples with sharp estimate are illustrated moreover as an application of the march coupling the poincare inequality with respect to the absolute distribution of the process is also studied

uniqueness of generalized schrodinger operators and applications abstract we prove necessary and sufficient conditions on r d r 0 a e so that the operator s 2 1 w1 n has exactly one self adjoint extension on w1 p which generates a sub markovian semigroup on w1 n these conditions are shown to be always fulfilled if d 1 and are verified for a large class of functions if d 1 we also prove an infinite dimensional analogue of this result where r d is replaced by some topological vector space e and the lebesgue measure dx by a gaussian measure the role of is taken by some ornstein uhlenbeck operator with general linear drift and dom s bounded smooth cylinder functions both results are derived as special cases from a general perturbation result we also give the corresponding probabilistic version i e we prove uniqueness for the associated martingale problem we present in some detail applications to the case where is a ground state of a hamiltonian in quantum field theory settling a problem which has been open for some time furthermore we apply our results to prove uniqueness in distribution of the diffusion first constructed by g jona lasinio and p k mitter for the stochastic quantization of f 2 field theory

logarithmic sobolev inequalities and exponential integrability we prove the exponential integrability of lipschitz functions in abstract wiener spaces by using logarithmic sobolev inequalities we introduce two notions of lipschitz continuity and apply the method of davis and simon j funct anal 59 1984 335 395 to prove our main theorem moreover we give several conditions under which the function is exponentially integrable

on gradient bounds for the heat kernel on the heisenberg group it is known that the couple formed by the two dimensional brownian motion and its levy area leads to the heat kernel on the heisenberg group which is one of the simplest sub riemannian space the associated diffusion operator is hypoelliptic but not elliptic which makes difficult the derivation of functional inequalities for the heat kernel however driver and melcher and more recently h q li have obtained useful gradient bounds for the heat kernel on the heisenberg group we provide in this paper simple proofs of these bounds and explore their consequences in terms of functional inequalities including cheeger and bobkov type isoperimetric inequalities for the heat kernel

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

a class of degenerate elliptic equations and a dido s problem with respect to a measure in this paper we consider the following class of linear elliptic problems div a x u xnkexp x 22 f x in u 0on xn 0 where k 0 is a domain possibly unbounded of r n x x1 xn rn xn 0 f belongs to a suitable weighted lebesgue space and a x aij x ij is a symmetric matrix with measurable coefficients satisfying xnkexp x 22 2 aij x i j cxnkexp x 22 2 we compare the solution to such a problem with the solution to a symmetric one dimensional problem belonging to the same class our approach use classical symmetrization methods adapted to a relative isoperimetric inequality with respect to a measure related to the structure of the equation

l_2 rates of convergence for attractive reversible nearest particle systems the critical case reversible nearest particle systems are certain one dimensional interacting particle systems whose transition rates are determined by a probability density beta n with finite mean on the positive integers the reversible measure for such a system is the distribution nu of the stationary renewal process corresponding to this density in an earlier paper we proved under certain regularity conditions that the system converges exponentially rapidly in l 2 nu if and only if the system is supercritical this in turn is equivalent to beta n having exponential tails in this paper we consider the critical case and give moment conditions on beta n which are separately necessary and sufficient for the convergence of the process in l 2 nu at a specified algebraic rate in order to do so we develop conditions on the generator of a general markov process which correspond to algebraic l 2 convergence of the process the use of these conditions is also illustrated in the context of birth and death chains on the positive integers

gradient estimates of dirichlet heat semigroups and application to isoperimetric inequalities by using probabilistic approaches some uniform gradient estimates are obtained for dirichlet heat semigroups on a riemannian manifold with boundary as an application lower bound estimates of isoperimetric constants are presented in terms of functional inequalities

functional inequalities and subordination stability of nash and poincare inequalities we show that certain functional inequalities e g nash type and poincare type inequalities for infinitesimal generators of c0 semigroups are preserved under subordination in the sense of bochner our result improves earlier results by bendikov and maheux trans am math soc 359 3085 3097 2007 theorem 1 3 for fractional powers and it also holds for non symmetric settings as an application we will derive hypercontractivity supercontractivity and ultracontractivity of subordinate semigroups

application of coupling methods to the neumann eigenvalue problem by using coupling methods some lower bounds are obtained for the first neumann eigenvalue on riemannian manifolds this method is new and the results improve some known estimates an example shows that our estimates can be sharp

bv functions in abstract wiener spaces abstract functions of bounded variation in an abstract wiener space i e an infinite dimensional banach space endowed with a gaussian measure and a related differential structure have been introduced by m fukushima and m hino using dirichlet forms and their properties have been studied with tools from analysis and stochastics in this paper we reformulate in an integral geometric vein and with purely analytical tools the definition and the main properties of bv functions and investigate further properties

hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n sphere a logarithmic sobolev inequality with respect to the probability measure a 1 x2 12 dx on 1 1 is proved for all 12 from this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n sphere

studying discrete dynamical systems through differential equations abstract in this paper we consider dynamical systems generated by a diffeomorphism f defined on u an open subset of r n and give conditions over f which imply that their dynamics can be understood by studying the flow of an associated differential equation x x x also defined on u in particular the case where f has n 1 functionally independent first integrals is considered in this case x is constructed by imposing that it shares with f the same set of first integrals and that the functional equation f x det d f x x x u has some non zero solution several examples for n 2 3 are presented most of them coming from several well known difference equations

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

an estimate of the gap of spectrum of schrodinger operators which generate hyperbounded semigroups abstract logarithmic sobolev inequalities with potential functions on loop spaces were proved by l gross 1991 j funct anal 102 265 313 s aida 1996 in proceedings of the fifth gregynog symposium stochastic analysis and applications pp 1 19 world scientific singapore and f z gong and z m ma 1998 j funct anal 599 623 the generators of the dirichlet forms are schrodinger operators on loop spaces and they generate hyperbounded semigroups recently f z gong m rockner and l m wu 2000 poincare inequality for weighted first order sobolev spaces on loop spaces submitted proved the existence of the gap of spectrum at the lowest eigenvalue of the schrodinger operators by using m hino s exponential decay estimates of the semigroup 2000 osaka j math 37 603 624 and the author s strong ergodicity property of the semi group s aida 1998 j funct anal 158 152 185 in this paper we will give a lower bound on the gap by using a weak poincare inequality which was introduced by m rockner and f y wang 2000 weak poincare inequalities and l 2 convergence rates of markov semigroups preprint also we will give estimates on the distribution function of ground states using the weak poincare inequality

functional inequalities on abstract hilbert spaces and applications we study the essential spectrum and the semigroup property for self adjoint operators on abstract hilbert spaces by using functional inequalities some known results obtained on the l 2 space w r t a measure space are generalized the functional inequality is also used to study non symmetric semigroups

k integrals and k lie symmetries in discrete dynamical systems abstract we generalize the concept of symplectic maps to that of k symplectic maps maps whose kth iterates are symplectic similarly k symmetries and k integrals are symmetries resp integrals of the kth iterate of the map it is shown that k symmetries and k integrals are related by the k symplectic structure as in the k 1 continuous case noether s theorem examples are given of k integrals and their related k symmetries for k 1 4

a functional form of the isoperimetric inequality for the gaussian measure abstract letgbe a smooth function on r nwith values in 0 1 using the isoperimetric property of the gaussian measure itis proved that 1 eg e 1 g e g conversely this inequality implies the isoperimetric property of the gaussian measure

on the isoperimetric deficit in gauss space we prove a sharp quantitative version of the isoperimetric inequality in the space bbb r n endowed with the gaussian measure

upper bounds on the first eigenvalue for a diffusion operator via bakry emery ricci curvature let l delta nabla varphi cdot nabla be a symmetric diffusion operator with an invariant measure d mu e varphi dx on a complete riemannian manifold in this paper we prove li yau gradient estimates for weighted elliptic equations on the complete manifold with nabla varphi leq theta and dimensional bakry emery ricci curvature bounded below by some negative constant based on this we give an upper bound on the first eigenvalue of the diffusion operator l on this kind manifold and thereby generalize a cheng s result on the laplacian case math z 143 289 297 1975

on some algebraic difference equations un 2un ψ un 1 in r related to families of conics or cubics generalization of the lyness sequences abstract in this paper and in a forthcoming one we study difference equations in r of the types 2 u n 2 u n a b u n 1 u n 1 2 4 u n 2 u n a b u n 1 c u n 1 2 c u n 1 6 u n 2 u n a b u n 1 c u n 1 2 c d u n 1 u n 1 2 which are linked to families of conics cubics and quartics respectively these equations generalize lyness one u n 2 u n a u n 1 studied in several papers whose behavior was completely elucidated in g bastien m rogalski in press through methods which are transposed in the present paper for the study of 1 and 2 and in the forthcoming one for 3 in particular we prove in the present paper a form of chaotic behavior for solutions of difference equations 1 and 2 and find all the possible periods for these solutions

theoretical guarantees for approximate sampling from smooth and log concave densities sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators test procedures or confidence intervals in many situations the exact sampling from a given distribution is impossible or computationally expensive and therefore one needs to resort to approximate sampling strategies however there is no well developed theory providing meaningful nonasymptotic guarantees for the approximate sampling procedures especially in the high dimensional problems this paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log concave density defined on rp for some integer p 0 we establish nonasymptotic bounds for the error of approximating the true distribution by the one obtained by the langevin monte carlo method and its variants we illustrate the effectiveness of the established guarantees with various experiments underlying our analysis are insights from the theory of continuous time diffusion processes which may be of interest beyond the framework of distributions with log concave densities considered in the present work

from u bounds to isoperimetry with applications to h type groups abstract in this paper we study u bounds in relation to l 1 type coercive inequalities and isoperimetric problems for a class of probability measures on a general metric space r n d we prove the equivalence of an isoperimetric inequality with several other coercive inequalities in this general framework the usefulness of our approach is illustrated by an application to the setting of h type groups and an extension to infinite dimensions

pringsheim s theorem revisited abstract in this paper we prove a generalization to higher order linear recurrence relations of pringsheim s theorem on the convergence of ordinary continued fractions

new convergence results for continued fractions generated by four term recurrence relations in the study of simultaneous rational approximation of functions using rational functions with a common denominator which can be viewed as the german polynomial problem in simultaneous padeapproximation cf 1 the quest for convergence results lead to the study of generalized continued fractions a type of jacobi perron algorithm 2 3 it then becomes important to exploit the connection between the convergence of the generalized continued fraction and the solutions of the associated difference equation cf 4 5

an algebraic geometric approach to integrable maps of the plane we show that the dynamics of a birational map on an elliptic curve over a field is typically conjugate to addition by a point under the associated group law when the field is taken to be the function field of rational complex functions of one variable this amounts to an algebraic geometric version of the arnold liouville integrability theorem for planar integrable maps by products of this approach are that birational maps preserving foliations are necessarily the composition of two involutions and that relationships between birational maps preserving the same foliation can be described in terms of the respective points they add on the corresponding weierstrass curves when the result is applied to finite fields it helps explain some universal features of the periodic orbit distribution function for the reductions of integrable maps

pringsheim s theory for generalized continued fractions abstract in this paper we present a generalization to generalized continued fractions of pringsheim s theorem on the convergence of ordinary continued fractions

quantitative isoperimetric inequalities for log convex probability measures on the line abstract the purpose of this paper is to analyze the isoperimetric inequality for symmetric log convex probability measures on the line using geometric arguments we first re prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals a result due to bobkov and houdre then we give a quantitative form of the isoperimetric inequality leading to a somehow anomalous behavior indeed it could be that a set is very close to be optimal in the sense that the isoperimetric inequality is almost an equality but at the same time is very far in the sense of the symmetric difference between sets from any extremal sets from the results on sets we derive quantitative functional inequalities of weak cheeger type

reflection couplings and contraction rates for diffusions we consider contractivity for diffusion semigroups w r t kantorovich l 1 wasserstein distances based on appropriately chosen concave functions these distances are inbetween total variation and usual wasserstein distances it is shown that by appropriate explicit choices of the underlying distance contractivity with rates of close to optimal order can be obtained in several fundamental classes of examples where contractivity w r t standard wasserstein distances fails applications include overdamped langevin diffusions with locally non convex potentials products of these processes and systems of weakly interacting diffusions both of mean field and nearest neighbour type

the behavior of the spectral gap under growing drift we analyze the behavior of the spectral gap of the laplace beltrami operator on a compact riemannian manifold when a divergence free drift vector field is added we increase the drift by multiplication with a large constant c and ask the question how the spectral gap behaves as c goes to infinity it turns out that the spectral gap stays bounded if and only if the drift vector field has eigenfunctions in h 1 in that case the spectral gaps converge and we determine the limit

lie symmetries of birational maps preserving genus 0 fibrations we prove that any planar birational integrable map which preserves a fibration given by genus 0 curves has a lie symmetry and some associated invariant measures the obtained results allow to study in a systematic way the global dynamics of these maps using this approach the dynamics of several maps is described in particular we are able to give for particular examples the explicit expression of the rotation number function and the set of periods of the considered maps

poincare inequality for weighted first order sobolev spaces on loop spaces abstract let e be the loop space over a compact connected riemannian manifold with a torsion skew symmetric connection let l d be the ornstein uhlenbeck operator on a nonempty connected component d of the loop space e and let v d r be the restriction on d of the potential in the logarithmic sobolev inequality found by l gross on the loop group by s aida and by f z gong and z m ma on the loop space respectively we prove that the schrodinger operator l v l d v always has a spectral gap at the bottom 0 v of its spectrum and thus has its ground state transformed operator 1 l v 0 v where is the unique ground state of l v in particular our result proves l gross s conjecture about the existence of a spectral gap for the ground state transform of the schrodinger operator studied by him on the loop group in addition in all the above cases we identify the domain of the dirichlet forms associated with the ground state transforms as weighted first order sobolev spaces with weight given by 2 thus establishing a poincare inequality for them all these results are consequences from some new results in this paper on dirichlet forms characterizing certain classes with spectral gaps and from results by s aida and m hino

extremal of log sobolev inequality and w entropy on noncompact manifolds abstract let m be a complete connected noncompact manifold with bounded geometry under a condition near infinity we prove that the log sobolev functional 1 1 has an extremal function decaying exponentially near infinity we also prove that an extremal function may not exist if the condition is violated this result has the following consequences 1 it seems to give the first example of connected complete manifolds with bounded geometry where a standard log sobolev inequality does not have an extremal 2 it gives a negative answer to the open question on the existence of extremal of perelman s w entropy in the noncompact case which was stipulated by perelman 2002 22 p 9 3 2 remark 3 it helps to prove in some cases that noncompact shrinking breathers of ricci flow are gradient shrinking solitons

on eigenfunctions of markov processes on trees we begin by studying the eigenvectors associated to irreducible finite birth and death processes showing that the i nontrivial eigenvector i admits a succession of i decreasing or increasing stages each of them crossing zero imbedding naturally the finite state space into a continuous segment one can unequivocally define the zeros of i which are interlaced with those of i 1 these kind of results are deduced from a general investigation of minimax multi sets dirichlet eigenproblems which leads to a direct construction of the eigenvectors associated to birth and death processes this approach can be generically extended to eigenvectors of markov processes living on trees this enables to reinterpret the eigenvalues and the eigenvectors in terms of the previous dirichlet eigenproblems and a more general conjecture is presented about related higher order cheeger inequalities finally we carefully study the geometric structure of the eigenspace associated to the spectral gap on trees

spectral gap for spherically symmetric log concave probability measures and beyond let mu be a probability measure on rr n n geq 2 with lebesgue density proportional to e v vert x vert where v rr to rr is a smooth convex potential we show that the associated spectral gap in l 2 mu lies between n 1 int rr n vert x vert 2 mu dx and n int rr n vert x vert 2 mu dx improving a well known two sided estimate due to bobkov our markovian approach is remarkably simple and is sufficiently robust to be extended beyond the log concave case at the price of potentially modifying the underlying dynamics in the energy leading to weighted poincare inequalities all our results are illustrated by some classical and less classical examples

logarithmic sobolev inequalities for pinned loop groups abstract letgbe a connected compact type lie group equipped with anadg invariant inner product on the lie algebra g ofg given this data there is a well known left invariant h1 riemannian structure on l l g the infinite dimensional group of continuous based loops ing using this riemannian structure we define and construct a heat kernel t g0 associated to the laplace beltrami operator on l g heret 0 g0 l g and t g0 is a certain probability measure on l g for fixedg0 l g andt 0 we use the measure t g0 and the riemannian structure on l g to construct a classical pre dirichlet form the main theorem of this paper asserts that this pre dirichlet form admits a logarithmic sobolev inequality

an isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in gauss space we prove an isoperimetric inequality on the discrete cube which is the precise analog of a logarithmic inequality due to talagrand as a consequence the gaussian isoperimetric inequality is derived

weak poincare inequalities and l2 convergence rates of markov semigroups abstract in order to describe l 2 convergence rates slower than exponential the weak poincare inequality is introduced it is shown that the convergence rate of a markov semigroup and the corresponding weak poincare inequality can be determined by each other conditions for the weak poincare inequality to hold are presented which are easy to check and which hold in many applications the weak poincare inequality is also studied by using isoperimetric inequalities for diffusion and jump processes some typical examples are given to illustrate the general results in particular our results are applied to the stochastic quantization of field theory in finite volume moreover a sharp criterion of weak poincare inequalities is presented for poisson measures on configuration spaces

isoperimetric inequality for radial probability measures on euclidean spaces abstract we generalize the poincare limit which asserts that the n dimensional gaussian measure is approximated by the projections of the uniform probability measure on the euclidean sphere of appropriate radius to the first n coordinates as the dimension diverges to infinity the generalization is done by replacing the projections with certain maps using this generalization we derive a gaussian isoperimetric inequality for an absolutely continuous probability measure on euclidean spaces with respect to the lebesgue measure whose density is a radial function

criteria of spectral gap for markov operators let e f be a probability space and let p be a markov operator on l2 with 1 a simple eigenvalue such that p i e is an invariant probability measure of p then p 12 p p has a spectral gap i e 1 is isolated in the spectrum of p if and only if p limr sup f2 1 f pf r 1 this strengthens a conjecture of simon and hoegh krohn on the spectral gap for hyperbounded operators solved recently by l miclo in 10 consequently for a symmetric conservative irreducible dirichlet form on l2 a poincare log sobolev type inequality holds if and only if so does the corresponding defective inequality extensions to sub markov operators and non conservative dirichlet forms are also presented

functional inequalities and spectrum estimates the infinite measure case abstract as a continuation of 13 where a poincare type inequality was introduced to study the essential spectrum on the l 2 space of a probability measure this paper provides a modification of this inequality so that the infimum of the essential spectrum is well described even if the reference measure is infinite high order eigenvalues as well as the corresponding semigroup are estimated by using this new inequality criteria of the inequality and estimates of the inequality constants are presented finally some concrete examples are considered to illustrate the main results in particular estimates of high order eigenvalues obtained in this paper are sharp as checked by two examples on the euclidean space

analysis and geometry on configuration spaces abstract in this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds namely configuration spaces more precisely a differential geometry is introduced on the configuration space x over a riemannian manifold x this geometry is non flat even if x r d it is obtained as a natural lifting of the riemannian structure on x in particular a corresponding gradient divergence div and laplace beltrami operator h div gamma are constructed the associated volume elements i e all measures on x w r t which and div gamma become dual operators on l 2 x are identified as exactly the mixed poisson measures with mean measure equal to a multiple of the volume element dx on x in particular all these measures obey an integration by parts formula w r t vector fields on x the corresponding dirichlet forms e on l 2 x are therefore defined each is shown to be associated with a diffusion process which is thus the brownian motion on x and which is sub sequently identified as the usual independent infinite particle process on x the associated heat semigroup t t t 0 is calculated explicitly it is also proved that the diffusion process when started with is time ergodic or equivalently e gamma is irreducible or equivalently t t t 0 is ergodic if and only if is poisson measure z dx with intensity z dx for some z 0 furthermore it is shown that the laplace beltrami operator h div gamma gamma on l 2 x z dx is unitary equivalent to the second quantization of the laplacian x on x on the corresponding fock space n 0 l 2 x z dx n as another direct consequence of our results we obtain a representation of the lie algebra of compactly supported vector fields on x on poisson space finally generalizations to the case where dx is replaced by an absolutely continuous measure and also to interacting particle systems on x are described in particular the case where the mixed poisson measures are replaced by gibbs measures of ruelle type on x

on recurrence equations associated with invariant varieties of periodic points a recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed we show that infinitely many recurrence equations can be derived from the information about invariant varieties of periodic points of higher dimensional integrable maps

on the role of convexity in isoperimetry spectral gap and concentration we show that for convex domains in euclidean space cheeger s isoperimetric inequality spectral gap of the neumann laplacian exponential concentration of lipschitz functions and the a priori weakest requirement that lipschitz functions have arbitrarily slow uniform tail decay are all quantitatively equivalent to within universal constants independent of the dimension this substantially extends previous results of maz ya cheeger gromov milman buser and ledoux as an application we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume up to constants and under maps which are on average lipschitz we also provide a new characterization up to constants of the spectral gap of a convex domain as one over the square of the average distance from the worst subset having half the measure of the domain in addition we easily recover and extend many previously known lower bounds on the spectral gap of convex domains due to payne weinberger li yau kannan lovasz simonovits bobkov and sodin the proof involves estimates on the diffusion semi group following bakry ledoux and a result from riemannian geometry on the concavity of the isoperimetric profile our results extend to the more general setting of riemannian manifolds with density which satisfy the cd 0 curvature dimension condition of bakry emery

functional inequalities for empty essential spectrum abstract in terms of the equivalence of poincare inequality and the existence of spectral gap the super poincare inequality is suggested in the paper for the study of essential spectrum it is proved for symmetric diffusions that such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator this inequality recovers known sobolev and nash type ones it is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below some results are also proved for a more general setting including symmetric jump processes moreover estimates of inequality constants are also presented which lead to a proof of a result on ultracontractivity suggested recently by d stroock finally concentration of reference measures for super poincare inequalities is studied the resulting estimates extend previous ones for poincare and log sobolev inequalities

heat kernel analysis on infinite dimensional heisenberg groups abstract we introduce a class of non commutative heisenberg like infinite dimensional lie groups based on an abstract wiener space the ricci curvature tensor for these groups is computed and shown to be bounded brownian motion and the corresponding heat kernel measures t t 0 are also studied we show that these heat kernel measures admit 1 gaussian like upper bounds 2 cameron martin type quasi invariance results 3 good l p bounds on the corresponding radon nikodym derivatives 4 integration by parts formulas and 5 logarithmic sobolev inequalities the last three results heavily rely on the boundedness of the ricci tensor

probability distance inequalities on riemannian manifolds and path spaces abstract we construct otto villani s coupling for general reversible diffusion processes on a riemannian manifold as an application some new estimates are obtained for wasserstein distances by using a sobolev poincare type inequality introduced by latala and oleszkiewicz the corresponding concentration estimates of the measure are presented finally our main result is applied to obtain the transportation cost inequalities on the path space with respect to both of the l2 distance and the intrinsic distance in particular talagrand s inequality holds on the path space over a compact manifold

a concrete estimate for the weak poincare inequality on loop space the aim of the paper is to study the pinned wiener measure on the loop space over a simply connected compact riemannian manifold together with a hilbert space structure and the ornstein uhlenbeck operator d d we give a concrete estimate for the weak poincare inequality assuming positivity of the ricci curvature of the underlying manifold the order of the rate function is s for any 0

weak logarithmic sobolev inequalities and entropic convergence in this paper we introduce and study a weakened form of logarithmic sobolev inequalities in connection with various others functional inequalities weak poincare inequalities general beckner inequalities we also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi group in particular we exhibit an example where poincare inequality can not be used for deriving entropic convergence whence weak logarithmic sobolev inequality ensures the result

g continued fractions for basic hypergeometric functions ii in this paper we apply a modification of a generalized pringsheim s theorem to obtain a gcontinued fraction expansion for the quotient of two contiguous basic hypergeometric functions in arbitrarily many variables as an application we obtain a g continued fraction extension of the rogers ramanujan continued fraction 2003 elsevier inc all rights reserved

asymptotic statistical equivalence for ergodic diffusions the multidimensional case asymptotic local equivalence in the sense of le cam is established for inference on the drift in multidimensional ergodic diffusions and an accompanying sequence of gaussian shift experiments the nonparametric local neighbourhoods can be attained for any dimension provided the regularity of the drift is sufficiently large in addition a heteroskedastic gaussian regression experiment is given which is also locally asymptotically equivalent and which does not depend on the centre of localisation for one direction of the equivalence an explicit markov kernel is constructed

nonlinearity and temporal dependence nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models we study this link using three measures of temporal dependence mixing mixing and mixing stationary diffusions that are mixing have mixing coefficients that decay exponentially to zero when they fail to be mixing they are still mixing and mixing but coefficient decay is slower than exponential for such processes we find transformations of the markov states that have finite variances but infinite spectral densities at frequency zero the resulting spectral densities behave like those of stochastic processes with long memory finally we show how state dependent poisson sampling alters the temporal dependence

spectral gap for measure valued diffusion processes abstract the spectral gap is estimated for some measure valued processes which are induced by the intrinsic extrinsic derivatives on the space of finite measures over a riemannian manifold these processes are symmetric with respect to the dirichlet and gamma distributions arising from population genetics in addition to the evolution of allelic frequencies investigated in the literature they also describe stochastic movements of individuals

isoperimetric rigidity and distributions of 1 lipschitz functions abstract we prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal then the space is foliated by minimal geodesics where the observable variance is defined to be the supremum of the variance of 1 lipschitz functions on the space our result can be considered as a variant of cheeger gromoll s splitting theorem and also of cheng s maximal diameter theorem as an application we obtain a new isometric splitting theorem for a complete weighted riemannian manifold with a positive bakry emery ricci curvature

stability of the bakry emery theorem on rn abstract we establish quantitative stability estimates for the bakry emery bound on logarithmic sobolev and poincare constants of uniformly log concave measures more specifically we show that if a 1 uniformly log concave measure has almost the same logarithmic sobolev or poincare constant as the standard gaussian measure then it almost splits off a gaussian factor our results are dimension free leading to dimension free stability estimates for gaussian concentration of lipschitz functions the proofs are based on stein s method optimal transport and an approximate integration by parts identity relating measures and near extremals in the associated functional inequality

log sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality let m be a smooth connected manifold endowed with a smooth measure mu and a smooth locally subelliptic diffusion operator l which is symmetric with respect to mu we assume that l satisfies a generalized curvature dimension inequality as introduced by baudoin garofalo cite bg1 our goal is to discuss functional inequalities for mu like the poincare inequality the log sobolev inequality or the gaussian logarithmic isoperimetric inequality

estimates of logarithmic sobolev constant an improvement of bakry emery criterion this paper is mainly devoted to estimate the logarithmic sobolev abbrev l s constant for diffusion operators on manifold or in rd in most cases we study the lower bounds but a generalization to a korzeniowski j funct anal 71 1987 363 370 theorem 1 for the upper bound is also presented theorem 1 5 based on a simple observation due to j d deuschel and d w stroock j funct anal 92 1990 30 48 of the comparison between the l s constants for different potentials the powerful bakry emery criterion for the l s inequality is improved considerably in the paper especially for the manifolds with non positive sectional curvatures theorem 1 3 1 in terms of our notation r inf x p r infx tx m x 1 ricc hessv x x where x p is the distance betweenxand an arbitrary fixed pointp m the improvement can be roughly stated as follows the condition infr 0 r 0 for which the criterion is available is now replaced by supr 0 r 0