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

concentration of measure inequalities for markov chains and phi mixing processes we prove concentration inequalities for some classes of markov chains and f mixing processes with constants independent of the size of the sample that extend the inequalities for product measures of talagrand the method is based on information inequalities put forward by marton in case of contracting markov chains using a simple duality argument on entropy our results also include the family of logarithmic sobolev inequalities for convex functions applications to bounds on supremum of dependent empirical processes complete this work

poincare and transportation inequalities for gibbs measures under the dobrushin uniqueness condition in in this paper we establish an explicit and sharp estimate of the spectral gap poincare inequality and the transportation inequality for gibbs measures under the dobrushin uniqueness condition moreover we give a generalization of the liggett s m e theorem for interacting particle systems

a characterization of dimension free concentration in terms of transportation inequalities the aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies talagrand s t 2 transportation cost inequality this theorem permits us to give a new and very short proof of a result of otto and villani generalizations to other types of concentration are also considered in particular one shows that the poincare inequality is equivalent to a certain form of dimension free exponential concentration the proofs of these results rely on simple large deviations techniques

deviation inequalities for separately lipschitz functionals of composition of random functions abstract we consider a class of non homogeneous markov chains that contains many natural examples next using martingale methods we establish some deviation and moment inequalities for separately lipschitz functions of such a chain under moment conditions on some dominating random variables

on the rate of convergence in wasserstein distance of the empirical measure let mu n be the empirical measure associated to a n sample of a given probability distribution mu on mathbb r d we are interested in the rate of convergence of mu n to mu when measured in the wasserstein distance of order p 0 we provide some satisfying non asymptotic l p bounds and concentration inequalities for any values of p 0 and d ge 1 we extend also the non asymptotic l p bounds to stationary rho mixing sequences markov chains and to some interacting particle systems

transportation cost inequalities on path and loop groups let g be a connected lie group with the lie algebra g the action of cameron martin space h g on the path space pe g introduced by l gross illinois j math 36 1992 447 is free using this fact we define the h distance on pe g which enables us to establish a transportation cost inequality on pe g this method will be generalized to the path space over the loop group le g so that we obtain a transportation cost inequality for heat measures on le g

quantitative concentration inequalities for empirical measures on non compact spaces we establish quantitative concentration estimates for the empirical measure of many independent variables in transportation distances as an application we provide some error bounds for particle simulations in a model mean field problem the tools include coupling arguments as well as regularity and moment estimates for solutions of certain diffusive partial differential equations

intersection numbers of twisted cycles associated with the selberg integral and an application to the conformal field theory intersection numbers of twisted or loaded cycles associated with the selberg integral are studied in particular the self intersection number of the cycle which is invariant under the action of the symmetric group is expressed by the product of trigonometric functions this formula reproduces the four point correlation functions in the conformal field theory calculated by dotsenko fateev in 3 in our study a compact non singular model terada model of the configuration space of n 3 points on the real projective line and a q analogue of the chu vadermonde formula for the hypergeometric series play a crucial role intersection numbers of the corresponding cocycles are also studied

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

concentration for multidimensional diffusions and their boundary local times we prove that probability laws of certain multidimensional semimartingales which includes time inhomogenous diffusions under suitable assumptions satisfy quadratic transportation cost inequality under the uniform metric from this we derive concentration properties of lipschitz functions of process paths that depend on the entire history in particular we estimate concentration of boundary local time of reflected brownian motions on a polyhedral domain we work out explicit applications of consequences of measure concentration for the case of brownian motion with rank based drifts

transportation cost information inequalities and applications to random dynamical systems and diffusions we first give a characterization of the l1 transportation cost information inequality on a metric space and next find some appropriate sufficient condition to transportation cost information inequalities for dependent sequences applications to random dynamical systems and diffusions are studied

curvature concentration and error estimates for markov chain monte carlo we provide explicit nonasymptotic estimates for the rate of convergence of empirical means of markov chains together with a gaussian or exponential control on the deviations of empirical means these estimates hold under a positive curvature assumption expressing a kind of metric ergodicity which generalizes the ricci curvature from differential geometry and on finite graphs amounts to contraction under path coupling

information inequalities and concentration of measure we derive inequalities of the form p q h p r h q r which hold for every choice of probability measures p q r where h p r denotes the relative entropy of p with respect to r and p q stands for a coupling type distance between p and q using the chain rule for relative entropies and then specializing to q with a given support we recover some of talagrand s concentration of measure inequalities for product spaces

free transport entropy inequalities for non convex potentials and application to concentration for random matrices talagrand s inequalities provide a link between two fundamentals concepts of probability transportation and entropy the study of the counterpart of these inequalities in the context of free probability has been initiated by biane and voiculescu and later extended by hiai petz and ueda for convex potentials in this work we prove a free analogue of a result of bobkov and gotze in the classical setting thus providing free transport entropy inequalities for a very natural class of measures appearing in random matrix theory these inequalities are weaker than the ones of hiai petz and ueda but still hold beyond the convex case we then use this result to get a concentration estimate for beta ensembles under mild assumptions on the potential

measure concentration for a class of random processes let x x i i be a stationary random process with a countable alphabet and distribution q let q x k 0 denote the conditional distribution of x x 1 x 2 x n given the k length past write d 1 x 1 0 if 1 x 1 and d 1 x 1 1 otherwise we say that the process x admits a joining with finite distance u if for any two past sequences k 0 k 1 0 and x k 0 x k 1 x 0 there is a joining of q k 0 and q x k 0 say dist 0 x 0 k 0 x k 0 such that the main result of this paper is the following inequality for processes that admit a joining with finite distance

a family of functional inequalities łojasiewicz inequalities and displacement convex functions abstract for displacement convex functionals in the probability space equipped with the monge kantorovich metric we prove the equivalence between the gradient and functional type ojasiewicz inequalities we also discuss the more general case of convex functions and we provide a general convergence theorem for the corresponding gradient dynamics specialising our results to the boltzmann entropy we recover otto villani s theorem asserting the equivalence between logarithmic sobolev and talagrand s inequalities the choice of power type entropies shows a new equivalence between gagliardo nirenberg inequality and a nonlinear talagrand inequality some nonconvex results and other types of equivalences are discussed

ricci curvature of markov chains on metric spaces abstract we define the coarse ricci curvature of metric spaces in terms of how much small balls are closer in wasserstein transportation distance than their centers are this definition naturally extends to any markov chain on a metric space for a riemannian manifold this gives back after scaling the value of ricci curvature of a tangent vector examples of positively curved spaces for this definition include the discrete cube and discrete versions of the ornstein uhlenbeck process moreover this generalization is consistent with the bakry emery ricci curvature for brownian motion with a drift on a riemannian manifold positive ricci curvature is shown to imply a spectral gap a levy gromov like gaussian concentration theorem and a kind of modified logarithmic sobolev inequality the bounds obtained are sharp in a variety of examples

characterization of talagrand s like transportation cost inequalities on the real line abstract in this paper we give necessary and sufficient conditions for talagrand s like transportation cost inequalities on the real line this brings a new wide class of examples of probability measures enjoying a dimension free concentration of measure property another byproduct is the characterization of modified log sobolev inequalities for log concave probability measures on r

the heat flow on metric random walk spaces abstract in this paper we study the heat flow on metric random walk spaces which unifies into a broad framework the heat flow on locally finite weighted connected graphs the heat flow determined by finite markov chains and some nonlocal evolution problems we give different characterizations of the ergodicity and prove that a metric random walk space with positive ollivier ricci curvature is ergodic furthermore we prove a cheeger inequality and as a consequence we show that a poincare inequality holds if and only if an isoperimetric inequality holds we also study the bakry emery curvature dimension condition and its relation with functional inequalities like the poincare inequality and the transport information inequalities

a modified graded mesh and higher order finite element approximation for singular perturbation problems abstract a singularly perturbed convection diffusion problem is solved numerically using finite element method based on higher order polynomials more precisely we introduce a modified graded mesh generated using some implicitly defined functions higher order parameter uniform convergence is obtained in weighted energy norm moreover the error estimates obtained are optimal in the sense that they are free from logarithmic factor a number of test examples are taken into account and a rigorous comparative analysis is presented moreover we compare the proposed method with others found in the literature

a singularly perturbed problem with two parameters on a bakhvalov type mesh a singularly perturbed problem with two small parameters is considered on a bakhvalov type mesh we prove uniform convergence of a galerkin finite element method with piecewise linear functions arguments in the error analysis include interpolation error bounds for a clement quasi interpolant as well as discretization error estimates in an energy norm numerical experiments support theoretical findings

hitting times functional inequalities lyapunov conditions and uniform ergodicity the use of lyapunov conditions for proving functional inequalities was initiated in 5 it was shown in 4 30 that there is an equivalence between a poincare inequality the existence of some lyapunov function and the exponential integrability of hitting times in the present paper we close the scheme of the interplay between lyapunov conditions and functional inequalities by showing that strong functional inequalities are equivalent to lyapunov type conditions showing that these lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times we also give some complement concerning the link between lyapunov conditions and in tegrability property of the invariant probability measure and as such transportation inequalities and we show that some unbounded lyapunov conditions can lead to uniform ergodicity and coming down from infinity property

monge kantorovitch measure transportation and monge ampere equation on wiener space let w h be an abstract wiener space assume two i i 1 2 probabilities on w w we give some conditions for the wasserstein distance between 1 and 2 with respect to the cameron martin space to be finite where the infimum is taken on the set of probability measures on w w whose first and second marginals are 1 and 2 in this case we prove the existence of a unique cyclically monotone map t iw with w h such that t maps 1 to 2 moreover if 2 then t is stochastically invertible i e there exists s w w such that s t iw 1 a s and t s iw 2 a s if in addition 1 then there exists a 1 convex function in the gaussian sobolev space such that these results imply that the quasi invariant transformations of the wiener space with finite wasserstein distance from can be written as the composition of a transport map t and a rotation i e a measure preserving map we give also 1 convex sub solutions and ito type solutions of the monge ampere equation on w

probabilistic approach for granular media equations in the non uniformly convex case we use here a particle system to prove both a convergence result with convergence rate and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex the proof of convergence is simpler than the one in carrillo mccann villani rev mat iberoamericana 19 971 1018 2003 arch rat mech anal 179 217 263 2006 all the results complete former results of malrieu ann appl probab 13 540 560 2003 in the uniformly convex case the main tool is an uniform propagation of chaos property and a direct control in wasserstein distance of solutions starting with different initial measures the deviation inequality is obtained via a t 1 transportation cost inequality replacing the logarithmic sobolev inequality which is no more clearly dimension free

properties of isoperimetric functional and transport entropy inequalities via concentration various properties of isoperimetric functional transport entropy and concentration inequalities are studied on a riemannian manifold equipped with a measure whose generalized ricci curvature is bounded from below first stability of these inequalities with respect to perturbation of the measure is obtained the extent of the perturbation is measured using several different distances between perturbed and original measure such as a one sided l bound on the ratio between their densities wasserstein distances and kullback leibler divergence in particular an extension of the holley stroock perturbation lemma for the log sobolev inequality is obtained and the dependence on the perturbation parameter is improved from linear to logarithmic second the equivalence of transport entropy inequalities with different cost functions is verified by obtaining a reverse jensen type inequality the main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting of independent interest is a new dimension independent characterization of transport entropy inequalities with respect to the 1 wasserstein distance which does not assume any curvature lower bound

a large deviation approach to some transportation cost inequalities new transportation cost inequalities are derived by means of elementary large deviation reasonings their dual characterization is proved this provides an extension of a well known result of s bobkov and f gotze their tensorization properties are investigated sufficient conditions and necessary conditions too for these inequalities are stated in terms of the integrability of the reference measure applying these results leads to new deviation results concentration of measure and deviations of empirical processes

transportation information inequalities for markov processes in this paper one investigates the transportation information tci inequalities tc i for all probability measures on a metric space mathcal x d where is a given probability measure tc is the transportation cost from to with respect to the cost function c x y on mathcal x 2 i is the fisher donsker varadhan information of with respect to and 0 0 is a left continuous increasing function using large deviation techniques it is shown that tci is equivalent to some concentration inequality for the occupation measure of a reversible ergodic markov process related to i the tensorization property of tci and comparisons of tci with poincare and log sobolev inequalities are investigated several easy to check sufficient conditions are provided for special important cases of tci and several examples are worked out

transportation cost inequalities for diffusions with jumps and its application to regime switching processes abstract by using the malliavin calculus we establish the t 1 transportation cost inequalities for stochastic differential equations with jumps which generalizes the results in 14 as an application we establish the t 1 transportation cost inequalities for regime switching diffusion processes

a new characterization of quadratic transportation information inequalities it is known that a quadratic transportation information inequality mathrm w 2 mathrm i interpolates between the talagrand s inequality mathrm w 2 mathrm h and the log sobolev inequality lsi for short the aim of this paper is threefold 1 to prove the equivalence of mathrm w 2 mathrm i and the lyapunov condition which gives a new characterization inspired by cattiaux et al probab theory relat fields 148 1 2 285 304 2010 2 to prove the stability of mathrm w 2 mathrm i under bounded perturbations which gives a transference principle in the sense of holley stroock 3 to prove mathrm w 2 mathrm h through a restricted mathrm w 2 mathrm i which gives a counterpart of the restricted lsi presented by gozlan et al ann probab 39 3 857 880 2011