We show how the Clark-Ocone-Haussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the Ornstein-Uhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space. By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diffusion measure as long as the torsion of the corresponding Riemannian connection satisfies Driver's total antisymmetry condition. 1 Introduction Let ! = (! t ) t0 be a standard Brownian motion starting from the origin with values in IR n and denote by W 0 (IR n ) the path space of continuous functions from [0; 1] to IR n starting from the origin. We denote further by IE expectation with respect to the law ¯ (the Wiener measure) of ! on W 0 (IR n ). Gross [G] proved the following logarithmic Sobolev inequality holds: ...

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We study some discrete isoperimetric and Poincar'e-type inequalities for product probability measures ¯ n on the discrete cube f0; 1g n and on the lattice Z n . In particular we prove sharp lower estimates for the product measures of 'boundaries ' of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions ¯ on Z which satisfy these inequalities on Z n . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincar'e inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. 1 Introduction For a number p 2 (0; 1), let ¯ p denote the Bernoulli measure on f0; 1g with probabilities q = 1 \Gamma p and p, respectively, and let ¯ n p denote the corresponding product measure on f0; 1g n . Denote by s i (x) the 'neighbour' of x 2 f0; 1g n obtained b...

We show that a covariance representation in Gauss space implies large deviation inequalities for smooth functions of Gauss random vectors. This covariance representation characterizes Gaussian measures. of the Gaussian. This representation has an analogue for Bernoulli measures and implies sharp large deviation inequalities for Bernoulli sums as well.

We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We examine similarly dimension free isoperimetric bounds using these parameters.

We give a martingale proof of Gaussian isoperimetry, which also contains Bobkov’s inequality on the two-point space and its extension to non symmetric Bernoulli measures. We derive the equivalence of different forms of Gaussian type isoperimetry. This allows us to prove a sharp form of Bobkov’s inequality for the sphere and to get new isoperimetric estimates for the unit cube.

We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem. 1

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Extremal properties of central half-spaces for product measures F. Barthe\Lambda Abstract We deal with the isoperimetric and the shift problem for subsets of measure one half 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 `p-sums of Minkowski spaces.

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– Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semigroup 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-Émery. 1