transference principles for log sobolev and spectral gap with applications to conservative spin systems we obtain new principles for transferring log sobolev and spectral gap inequalities from a source metric measure space to a target one when the curvature of the target space is bounded from below as our main application we obtain explicit estimates for the log sobolev and spectral gap constants of various conservative spin system models consisting of non interacting and weakly interacting particles constrained to conserve the mean spin when the self interaction is a perturbation of a strongly convex potential this partially recovers and partially extends previous results of caputo chafai grunewald landim lu menz otto panizo villani westdickenberg and yau when the self interaction is only assumed to be non strongly convex as in the case of the two sided exponential measure we obtain sharp estimates on the system s spectral gap as a function of the mean spin independently of the size of the system

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

on asymptotic stabilization of driftless systems a simple approach is proposed to investigate the stabilization of driftless systems by taking the control input as a function of system states we can transform the stabilization design into a stability problem the known stability results on the nonlinear autonomous system are then employed to derive the asymptotically stabilizing controllers by solving an algebraic equation existence conditions of the asymptotic stabilizing feedback controller for the driftless systems by employing the stability criteria on linear systems and a class of third order homogeneous systems are obtained to demonstrate the application of the approach

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

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

isoperimetric and analytic inequalities for log concave probability measures we discuss an approach based on the brunn minkowski inequality to isoperimetric and analytic inequalities for probability measures on euclidean space with logarithmically concave densities in particular we show that such measures have positive isoperimetric constants in the sense of cheeger and thus always share poincare type inequalities we then describe those log concave measures which satisfy isoperimetric inequalities of gaussian type the results are precised in dimension 1

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

numerical approximations of algebraic riccati equations for abstract systems modelled by analytic semigroups and applications this paper provides a numerical approximation theory of algebraic riccati operator equations with unbounded coefficient operators a and b such as arise in the study of optimal quadratic cost problems over the time interval 0 oo for the abstract dynamics j ay bu here a is the generator of a strongly continuous analytic semigroup and b is an unbounded operator with any degree of unboundedness less than that of a convergence results are provided for the riccati operators as well as for all the other relevant quantities which enter into the dynamic optimization problem the present numerical theory is the counterpart of a known continuous theory several examples of partial differential equations with boundary point control where all the required assumptions are verified illustrate the theory they include parabolic equations with l2 dirichlet control as well as plate equations with a strong degree of damping and point control

a two sided estimate for the gaussian noise stability deficit the gaussian noise stability of a set a subset mathbb r n is defined by begin aligned mathcal s rho a mathbb p left x in a y in a right end aligned where x y are standard jointly gaussian vectors satisfying mathbb e x i y j delta ij rho borell s inequality states that for all 0 rho 1 among all sets a subset mathbb r n with a given gaussian measure the quantity mathcal s rho a is maximized when a is a half space we give a novel short proof of this fact based on stochastic calculus moreover we prove an almost tight two sided dimension free robustness estimate for this inequality by introducing a new metric to measure the distance between the set a and its corresponding half space h namely the distance between the two centroids we show that the deficit mathcal s rho h mathcal s rho a can be controlled from both below and above by essentially the same function of the distance up to logarithmic factors as a consequence we also establish the conjectured exponent in the robustness estimate proven by mossel neeman which uses the total variation distance as a metric in the limit rho rightarrow 1 we obtain an improved dimension free robustness bound for the gaussian isoperimetric inequality our estimates are also valid for a generalized version of stability where more than two correlated vectors are considered

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

convergence rates for the approximations of the solutions to algebraic riccati equations with unbounded coefficients case of analytic semigroups approximations schemes for the solutions of the algebraic riccati equations will be considered we shall concentrate on the case when the input operator is unbounded and the dynamics of the system is described by an analytic semigroup the main goal of the paper is to establish the optimal rates of convergence of the underlying approximations by optimal we mean such approximations which would reconstruct the optimal regularity of the original solutions as well as the best approximation properties of the finite dimensional subspaces it turns out that if one aims to obtain the optimal rates in the case of unbounded input operators the choice of the approximations to the generator as well as to the control operator is very critical while the convergence results hold with any consistent approximations the optimal rates require a careful selection of the approximating schemes our theoretical results will be illustrated by several examples of boundary point control problems where the optimal rates of convergence are achieved with the appropriate numerical algorithms

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

functional inequalities on arbitrary riemannian manifolds abstract for any connected not necessarily complete riemannian manifold we construct a probability measure of type e v x d x where d x is the riemannian volume measure and v is a function c smooth outside a closed set of zero volume satisfying poincare sobolev type functional inequalities in particular v is c smooth on the whole manifold when the poincare and the super poincare inequalities are considered the sobolev inequality for infinite measures are also studied

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

on the space of bv functions and a related stochastic calculus in infinite dimensions functions of bounded variation bv functions are defined on an abstract wiener space e h in a way similar to that in finite dimensions some characterizations are given which justify describing a bv function as a function in l log l 1 2 with the first order derivative being an h valued measure it is also shown that the space of bv functions is obtained by a natural extension of the sobolev space d1 1 moreover some stochastic formulae related to bv functions are investigated

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

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

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

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

the linear regulator problem for parabolic systems we present an approximation framework for computation in finite dimensional spaces of riccati operators that can be guaranteed to converge to the riccati operator in feedback controls for abstract evolution systems in a hilbert space it is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems

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

asymptotic stabilization of driftless systems issues of asymptotic stabilization of a class of non linear driftless systems are presented in addition to the necessary and sufficient condition for the existence of a smooth time invariant asymptotic stabilizer sufficient condition for the existence of a quadratic type lyapunov function candidate is also proposed herein to alleviate the construction of stabilizing control laws following the deduction of the equivalence of the sufficient condition and the determination of the local definiteness of a defined scalar function the stabilizability checking conditions are then derived in terms of system dynamics and its derivatives at the origin only these are achieved by taking taylor s series expansion on system dynamics the derived conditions are shown to be consistent with those obtained by brockett comparative results of liaw and liang are also included finally examples are given to demonstrate the use of the main results

geometric random edge we show that a variant of the random edge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs max c tx x in mathbb r n ax leqslant b max ctx x rn ax b whose constraint matrix a satisfies a geometric property introduced by brunsch and roglin the sine of the angle of a row of a to a hyperplane spanned by n 1 n 1 other rows of a is at least delta this property is a geometric generalization of a being integral and each sub determinant of a being bounded by delta in absolute value in this case delta geqslant 1 delta 2 n 1 2n in particular linear programs defined by totally unimodular matrices are captured in this framework here delta geqslant 1 n 1 n and dyer and frieze previously described a strongly polynomial time randomized simplex algorithm for linear programs with a totally unimodular the expected number of pivots of the simplex algorithm is polynomial in the dimension and 1 delta 1 and independent of the number of constraints of the linear program our main result can be viewed as an algorithmic realization of the proof of small diameter for such polytopes by bonifas et al using the ideas of dyer and frieze

riccati equation arising in a boundary control problem with distributed parameters global existence is proved for the solution of a riccati differential equation connected with the synthesis of a boundary control problem governed by parabolic partial differential equations

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

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

higher order sobolev embeddings and isoperimetric inequalities abstract optimal higher order sobolev type embeddings are shown to follow via isoperimetric inequalities this establishes a higher order analogue of a well known link between first order sobolev embeddings and isoperimetric inequalities sobolev type inequalities of any order involving arbitrary rearrangement invariant norms on open sets in r n possibly endowed with a measure density are reduced to much simpler one dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets as a consequence the optimal target space in the relevant sobolev embeddings can be determined both in standard and in non standard classes of function spaces and underlying measure spaces in particular our results are applied to any order sobolev embeddings in regular john domains of the euclidean space in maz ya classes of possibly irregular euclidean domains described in terms of their isoperimetric function and in families of product probability spaces of which the gauss space is a classical instance

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

galerkin approximations of abstract parabolic boundary value problems with rough boundary data l p theory etude des approximations de galerkin d un probleme aux valeurs limites parabolique abstrait avec des donnees aux limites grossieres determination des vitesses de convergence optimales dans les normes lp

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

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

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

tail estimates for markovian rough paths we work in the context of markovian rough paths associated to a class of uniformly subelliptic dirichlet forms 26 and prove a better than exponential tail estimate for the accumulated local p variation functional which has been introduced and studied in 17 we comment on the significance of these estimates to a range of currently studied problems including the recent results of ni hao 32 and chevyrev and lyons 18

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

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

pointwise symmetrization inequalities for sobolev functions and applications we develop a technique to obtain new symmetrization inequalities that provide a unified framework to study sobolev inequalities concentration inequalities and sharp integrability of solutions of elliptic equations

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

bounds on the deficit in the logarithmic sobolev inequality abstract the deficit in the logarithmic sobolev inequality for the gaussian measure is considered and estimated by means of transport and information theoretic distances

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

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

second fundamental form and gradient of neumann semigroups abstract the second fundamental form for the boundary of a compact riemannian manifold is described by a short time behavior for the gradient of neumann semigroups as an application we prove that the manifold is convex if and only if the neumann heat semigroup p t satisfies the gradient estimate p t f e k t p t f for some constant k r and all t 0 f c 1

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

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

entropy bounds for gibbs measures with non gaussian tails abstract we introduce and study relative entropy bounds which under some mixing conditions allow us to classify the distribution tails in the infinite dimensional setting

robust control of non linear affine systems issue of robust control for non linear affine systems with uncertainties appearing in both drift and non drift terms are presented based on lyapunov function approach control laws are proposed to guarantee uniformly asymptotic stability of the equilibrium point to facilitate the design and simplify the checking procedure stabilization control for the uncertain systems possess asymptotic stabilizable nominal time invariant driftness terms are proposed for the demonstration of robust design

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

control of a driftless bilinear vector field on n sphere in this paper we consider a multi input driftless bilinear system evolving on the n dimensional sphere s n we first provide examples drawn from rigid body mechanics that provide the motivation for the control of bilinear systems on s n for the general framework we establish the global controllability on s n and propose two linear control laws on s n that achieve asymptotic stabilization of an equilibrium point with an almost global domain of attraction further the asymptotically stable closed loop system trajectories are shown to be arcs on the geodesics of s n for a particular choice of the equilibrium point next we propose two linear time varying control laws to achieve trajectory tracking on s n and show the asymptotic stability of the tracking error a distributed control is designed for the consensus of multiagent bilinear systems on s n with an undirected tree as the communication graph the consensus manifold is shown to have an almost global domain of attraction

algebraic riccati equation arising in boundary control problems an algebraic riccati equation is studied with application to the optimal control of deterministic and stochastic parabolic systems with boundary control the control function can act on the boundary through dirichlet or neumann conditions

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

backstepping observer design for parabolic pdes with measurement of weighted spatial averages this paper is concerned with the observer design for one dimensional linear parabolic partial differential equations whose output is a weighted spatial average of the state over the entire spatial domain we focus on the backstepping approach which provides a systematic procedure to design an observer gain for systems with boundary measurement if the output is not a boundary value of the state the backstepping approach is not directly applicable to obtaining an observer gain that stabilizes the error dynamics therefore we attempt to convert the error system into another system to which backstepping is applicable the conversion is successfully achieved for a class of weighting functions and the resultant observer realizes exponential convergence of the estimation error with an arbitrary decay rate in terms of the l 2 norm in addition an explicit expression of the observer gain is available in a special case the effectiveness of the proposed observer is also confirmed by numerical simulations

curvature dimension inequalities on sub riemannian manifolds obtained from riemannian foliations part ii using the curvature dimension inequality proved in part i we look at consequences of this inequality in terms of the interaction between the sub riemannian geometry and the heat semigroup p t corresponding to the sub laplacian we give bounds for the gradient entropy a poincare inequality and a li yau type inequality these results require that the gradient of p t f remains uniformly bounded whenever the gradient of f is bounded and we give several sufficient conditions for this to hold

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

dirichlet boundary control problems for parabolic equations with quadratic cost analyticity and riccati s feedback synthesis riccati type feedback synthesis of optimal controls for dirichlet boundary parabolic equations is considered the functional cost penalizes the l2 energy over 0 t of state and control action u and also final state y t at t t this latter fact makes the functional cost discontinuous on the space of admissible controls l2 0t after overcoming some technical difficulties related to the above mentioned discontinuity we prove that the optimal control uo can be written in the desired feedback form uo t cp t yo t for all 0 t t

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

characterization of domains of fractional powers of certain operators arising in elastic systems and applications with x a hilbert space where a the elastic operator and b the damping operator are closed operators in x with dense domains q a and 9 b respectively moreover a is throughout assumed to be self adjoint and strictly positive stimulated by two conjectures raised in crl the most general of which referred to the case lx f described below we have already studied the operators and sb defined respectively by

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

convergence estimates for semidiscrete approximations of nonselfadjoint parabolic equations semidiscrete galerkin approximations of the second order nonselfadjoint parabolic equations are considered l 2 error estimates for l 2 initial conditions are derived by use of semigroup methods and of analyticity of the underlined semigroup the scheme here presented can be also freely applied to nonconforming elements

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