homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity this article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices under polynomial growth assumptions we prove that the energy functionals f varepsilon stored in the deformation of an varepsilon scaling of a stochastic lattice converge to a continuous energy functional when varepsilon goes to zero in particular the limiting energy functional is of integral type and deterministic if the lattice is ergodic we also generalize to systems and nonlinear settings well known results on stochastic homogenization of discrete elliptic equations as an application of the main result we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics we finally address some mechanical properties of the limiting models such as frame invariance isotropy and natural states

solvable groups acting on the line actions of discrete groups on the real line are considered when the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line in the process of obtaining these results sufficient conditions are also determined for the existence of invariant quasi invariant measures for abelian solvable groups acting on the line it turns out for example that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi invariant measure and that any abelian group of c diffeomorphisms has an invariant measure an example is given to show that some restrictions are necessary in order to obtain such conclusions some applications to the study of codimension one foliations are indicated introduction if g is a group of homeomorphisms of the line it is reasonable to ask for a dynamic description of how g acts when reasonable restrictions are placed on g for example in 9 it is shown that if g is finitely generated and nilpotent then there is a g invariant borel measure on r which is finite on compact sets this already says much about g the proof is quite different from classical arguments which guarantee the existence of a finite invariant measure when an amenable group acts on a compact hausdorff space 6 in that one uses pseudogroup properties of g rather than group properties specifically nonexponential growth of orbits rather than nilpotence of g it is reasonable to determine what can be said when g is solvable but as pointed out in 9 solvable groups need not have invariant measures e g subgroups of the affine group which contain nontrivial translations and nontrivial dilations such examples suggest consideration of a more general invariance property a measure is called quasi invariant for g if each element of g multiplies the measure by a nonzero constant which depends on the particular element of g usual lebesgue measure is quasi invariant for the affine group so it is reasonable to ask if every finitely generated solvable g has a quasi invariant measure it turns out that a large class of solvable groups including polycyclic groups acting on the line must have quasi invariant measures but there are solvable groups which do not have such measures an important step in the investigation of solvable groups is the consideration of infinitely generated abelian groups if differentiability is assumed stronger received by the editors october 30 1980 and in revised form august 3 1982 1980 mathematics subject classification primary 57e25 secondary 58f11 58f15 58f18

constitutive equations of elastic polycrystalline materials where is the reference local placement of the material element f is the gradient of some transplacement which brings the material element from placement to another placement and t is the cauchy stress in the configuration if is an undistorted placement the material symmetries of an elastic element are formulated by means of members of the rotation group orth precisely we say thatq orth is a symmetry transformation relative to the placement if

rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media we establish a logarithmic type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second order pde in strongly mixing media with similar i e logarithmic decorrelation rate the proof consists of two major steps the first which is actually the only place in the paper where probability plays a role establishes the rate for special quadratic data using the methodology developed by the authors and wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media the second is a general argument based on the new notion of viscosity solutions which is introduced in this paper that shows that rates known for quadratic can be extended to general data as an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments the former is algebraic while the latter depends on the particular equation

regulating flows topology of foliations and rigidity a flow transverse to a foliation is regulating if in the universal cover an arbitrary orbit of the flow intersects every leaf of the lifted foliation this implies that the foliation is r covered that is its leaf space in the universal cover is homeomorphic to the reals we analyse the converse of this implication to study the topology of the leaf space of certain foliations we prove that if a pseudo anosov flow is transverse to an r covered foliation and the flow is not an r covered anosov flow then the flow is regulating for the foliation using this we show that several interesting classes of foliations are not r covered finally we show a rigidity result if an r covered anosov flow is transverse to a foliation but is not regulating then the foliation blows down to one topologically conjugate to the stable or unstable foliations of the transverse flow

random parking euclidean functionals and rubber elasticity we study subadditive functions of the random parking model previously analyzed by the second author in particular we consider local functions s of subsets of mathbb r d and of point sets that are almost subadditive in their first variable denoting by the random parking measure in mathbb r d and by r the random parking measure in the cube qr r r d we show under some natural assumptions on s that there exists a constant overline s in mathbb r such that lim r to infty frac s q r xi q r lim r to infty frac s q r xi r q r overline s almost surely if zeta mapsto s q r zeta is the counting measure of zeta in qr then we retrieve the result by the second author on the existence of the jamming limit the present work generalizes this result to a wide class of almost subadditive functions in particular classical euclidean optimization problems as well as the discrete model for rubber previously studied by alicandro cicalese and the first author enter this class of functions in the case of rubber elasticity this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit as well as a generalization to stochastic networks generated on bounded sets

homogenization and enhancement of the g equation in random environments we study the homogenization of a g equation which is advected by a divergence free stationary vector field in a general ergodic random environment we prove that the averaged equation is an anisotropic deterministic g equation and we give necessary and sufficient conditions in order to have enhancement since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions in addition as we show the associated minimal first passage time function does not satisfy in general the uniform integrability condition which is necessary to apply the sub additive ergodic theorem we overcome these obstacles by i establishing a new reachability controllability estimate for the minimal function and ii constructing for each direction and almost surely a random sequence which has both a long time averaged limit due to the sub additive ergodic theorem and stays in the same sense asymptotically close to the minimal time

realizable average stress average strain pairs in a plate with holes here a complete characterization is given of the set of all possible average stress average strain pairs that can exist in a plate containing a fixed volume fraction f of holes specifically for a given average stress the range of values the average strain takes as the microgeometry is varied while keeping f fixed is determined it is shown that multiple rank laminate materials suffice to generate all possible values of the average strain when the microgeometry is restricted to be a periodic array of holes with only one hole per unit cell the average strain takes a smaller range of values as the hole shape is varied a certain necessary condition for optimality must be satisfied if the hole is such that the average strain is on the boundary of this range numerical results are obtained for the range in the limit where the holes are well separated and occupy a small volume fraction optimal hole shapes associated with average strains on the boundary of the range of admissible values are identif

stochastic homogenization of the bending plate model abstract we use the notion of stochastic two scale convergence introduced in 32 to solve the problem of stochastic homogenization of the elastic plate in the bending regime

higher order linearization and regularity in nonlinear homogenization we prove large scale c regularity for solutions of nonlinear elliptic equations with random coefficients thereby obtaining a version of the statement of hilbert s 19th problem in the context of homogenization the analysis proceeds by iteratively improving three statements together i the regularity of the homogenized lagrangian l ii the commutation of higher order linearization and homogenization and iii large scale c 0 1 type regularity for higher order linearization errors we consequently obtain a quantitative estimate on the scaling of linearization errors a liouville type theorem describing the polynomially growing solutions of the system of higher order linearized equations and an explicit heterogenous analogue of the taylor series for an arbitrary solution of the nonlinear equations with the remainder term optimally controlled these results give a complete generalization to the nonlinear setting of the large scale regularity theory in homogenization for linear elliptic equations

hypocoercivity in metastable settings and kinetic simulated annealing combining classical arguments for the analysis of the simulated annealing algorithm with the more recent hypocoercive method of distorted entropy we prove the convergence for large time of the kinetic langevin annealing with logarithmic cooling schedule

quasigeodesic pseudo anosov flows in hyperbolic 3 manifolds and connections with large scale geometry abstract in this article we obtain a simple topological and dynamical systems condition which is necessary and sufficient for an arbitrary pseudo anosov flow in a closed hyperbolic three manifold to be quasigeodesic quasigeodesic means that orbits are efficient in measuring length up to a bounded multiplicative distortion when lifted to the universal cover we prove that such flows are quasigeodesic if and only if there is an upper bound depending only on the flow on the number of orbits which are freely homotopic to an arbitrary closed orbit of the flow the main ingredient is a proof that under the boundedness condition the fundamental group of the manifold acts as a uniform convergence group on a flow ideal boundary of the universal cover we also construct a flow ideal compactification of the universal cover and prove that it is equivariantly homeomorphic to the gromov compactification this implies the quasigeodesic behavior of the flow the flow ideal boundary and flow ideal compactification are constructed using only the structure of the flow

stochastic homogenization of nonconvex unbounded integral functionals with convex growth we consider the well trodden ground of the problem of the homogenization of random integral functionals when the integrand has standard growth conditions the qualitative theory is well understood when it comes to unbounded functionals that is when the domain of the integrand is not the whole space and may depend on the space variable there is no satisfactory theory in this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth we first prove that if the integrand is convex and has p growth from below with p d the dimension then it admits homogenization regardless of growth conditions from above this result that crucially relies on the existence and sublinearity at infinity of correctors is also new in the periodic case in the case of nonconvex integrands we prove that a similar homogenization result holds provided that the nonconvex integrand admits a two sided estimate by a convex integrand the domain of which may depend on the space variable that itself admits homogenization this result is of interest to the rigorous derivation of rubber elasticity from polymer physics which involves the stochastic homogenization of such unbounded functionals

gradient flow formulation and longtime behaviour of a constrained fokker planck equation abstract we consider a fokker planck equation which is coupled to an externally given time dependent constraint on its first moment this constraint introduces a lagrange multiplier which renders the equation nonlocal and nonlinear in this paper we exploit an interpretation of this equation as a wasserstein gradient flow of a free energy f on a time constrained manifold first we prove existence of solutions by passing to the limit in an implicit euler scheme obtained by minimizing h f w 2 2 0 among all satisfying the constraint for some 0 and time step h 0 second we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant the proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint the rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic sobolev inequalities

from molecular models to continuum mechanics we present here a limiting process allowing us to write some continuum mechanics models as a natural asymptotic of molecular models the approach is based on the hypothesis that the macroscopic displacement is equal to the microscopic one we carry out the corresponding calculations in the case of two body energies including higher order terms and also in the case of thomas fermi type models

foliations with good geometry the goal of this article is to show that there is a large class of closed hyperbolic 3 manifolds admitting codimension one foliations with good large scale geometric properties we obtain results in two directions first there is an internal result a possibly singular foliation in a manifold is quasi isometric if when lifted to the universal cover distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover this means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior we previously proved that nonsingular codimension one foliations in closed hyperbolic 3 manifolds can never be quasi isometric in this article we produce a large class of singular quasi isometric codimension one foliations in closed hyperbolic 3 manifolds the foliations are stable and unstable foliations of pseudo anosov flows our second result is an external result relating nonsingular foliations in hyperbolic 3 manifolds with their limit sets in the universal cover that is showing that leaves in the universal cover have good asymptotic behavior let g be a reebless finite depth foliation in a closed hyperbolic 3 manifold then g is not quasi isometric but suppose that g is transverse to a quasigeodesic pseudo anosov flow with quasi isometric stable and unstable foliations which are given by the internal result we then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis the main tools used to prove these results are first a link between geometric properties of stable unstable foliations of pseudo anosov flows and the topology of these foliations in the tuniversal cover and second a topological theory of the joint structure of the pseudo anosov foliations in the universal cover reebless codimension one foliations are extremely useful for understanding the topology of 3 manifolds for instance they imply that the manifold is irreducible ro its universal cover is homeomorphic to r3 pa leaves are 7rl injective no and transversals are never null homotopic no hence they reflect topological properties of the manifold as for which manifolds have reebless foliations gabai gal ga2 ga3 proved that any compact oriented irreducible 3 manifold with nonzero first betti number has many reebless finite depth foliations roughly a

stochastic homogenisation of free discontinuity problems in this paper we study the stochastic homogenisation of free discontinuity functionals assuming stationarity for the random volume and surface integrands we prove the existence of a homogenised random free discontinuity functional which is deterministic in the ergodic case moreover by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise subadditive ergodic theorem by akcoglou and krengel we characterise the limit volume and surface integrands in terms of asymptotic cell formulas

homogenization of multiparameter integrals where is a bounded domain in rn u rm 0 and w rn rmn 0 is supposed to be 0 1 periodic with respect to the 7rst variable x rn the distinguishing feature of 1 here is that the integrand is permitted to depend on a vector of parameters 1 k r with k 1 we are interested in the asymptotic behavior of 1 as 0 and 0 integral functionals of this type appear in the mathematical modeling of cellular composite materials when n m 3 1 can be interpreted as the stored strain energy of an elastic and heterogeneous material u being a deformation or displacement 7eld when m 1 u may be a di erence of potential in a condenser in any case the medium under consideration is composed of several materials which are periodically distributed at the microscopic scale given by in the applications we often deal with

nonlinear stochastic homogenization and ergodic theory on etudie l homogeneisation stochastique non lineaire d une classe de fonctionnelles integrales on relie le probleme a la theorie ergodique

promoting essential laminations we show that a co orientable taut foliation of a closed orientable algebraically atoroidal 3 manifold is either the weak stable foliation of an anosov flow or else there are a pair of very full genuine laminations transverse to the foliation

non local effects by homogenization or 3d 1d dimension reduction in elastic materials reinforced by stiff fibers abstract we first consider an elastic thin heterogeneous cylinder of radius of order e the interior of the cylinder is occupied by a stiff material fiber that is surrounded by a soft material matrix by assuming that the elasticity tensor of the fiber does not scale with e and that of the matrix scales with e 2 we prove that the one dimensional model is a nonlocal system we then consider a reference configuration domain filled out by periodically distributed rods similar to those described above we prove that the homogenized model is a second order nonlocal problem in particular we show that the homogenization problem is directly connected to the 3d 1d dimensional reduction problem

on l spaces and left orderable fundamental groups examples suggest that there is a correspondence between l spaces and three manifolds whose fundamental groups cannot be left ordered in this paper we establish the equivalence of these conditions for several large classes of manifolds in particular we prove that they are equivalent for any closed connected orientable geometric three manifold that is non hyperbolic a family which includes all closed connected orientable seifert fibred spaces we also show that they are equivalent for the twofold branched covers of non split alternating links to do this we prove that the fundamental group of the twofold branched cover of an alternating link is left orderable if and only if it is a trivial link with two or more components we also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in text homeo s 1

homogenization of a hele shaw problem in periodic and random media we investigate the homogenization limit of a free boundary problem with space dependent free boundary velocities the problem under consideration has a well known obstacle problem transformation formally obtained by integrating with respect to the time variable by making rigorous the link between these two problems we are able to derive an explicit formula for the homogenized free boundary velocity and we establish the uniform convergence of the free boundaries

γ convergence and homogenisation for a class of degenerate functionals abstract this paper is on convergence for degenerate integral functionals related to homogenisation problems in the heisenberg group here both the rescaling and the notion of invariance or periodicity are chosen in a way motivated by the geometry of the heisenberg group without using special geometric features these functionals would be neither coercive nor periodic so classic results do not apply all the results apply to the more general case of carnot groups

regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity we prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form the equation is required to be strictly elliptic but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite d th moment where d is the dimension in the general stationary ergodic framework we show that the equation homogenizes to a deterministic uniformly elliptic equation and we obtain an explicit estimate of the effective ellipticity which is new even in the uniformly elliptic context showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory we prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity which are controlled in the macroscopic limit by the ergodic theorem we show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite p th moment for every p less than d but for which regularity and homogenization break down while these results are new even for linear equations we prove them in the fully nonlinear context in probabilistic terms our homogenization results correspond to quenched invariance principles for controlled diffusion processes in random media including linear diffusions as well as diffusions controlled by one controller or two competing players

stochastic reinforcement problems and ergodic theory we study the asymptotic behaviour of dirichlet problems in domains of r 2 bounded by thin layers whose thickness is given by means of an assigned ergodic random function using a capacitary method together with ergodic theorems for additive and superadditive processes we are able to characterise the limit problem precisely

quasistatic droplets in randomly perforated domains we consider the hele shaw problem in a randomly perforated domain with zero neumann boundary conditions a homogenization limit is obtained as the characteristic scale of the domain goes to zero specifically we prove that the solutions as well as their free boundaries converge uniformly to those corresponding to a homogeneous and anisotropic hele shaw problem set in mathbb r d the main challenge when deriving the limit lies in controlling the oscillations of the free boundary this is overcome first by extending de giorgi nash moser type estimates to perforated domains and second by proving the almost sure non degenerate growth of the solution near its free boundary

homogenization of polycrystalline aggregates the effective elastic properties of a polycrystalline material depend on the single crystal elastic constants of the crystallites comprising the polycrystal and on the manner in which the crystallites are arranged in this paper we apply the techniques of homogenization to put the problem of determining effective elastic constants in a precise mathematical framework that permits us to derive an expression for the effective elasticity tensor we also study how the homogenized elasticity tensor changes as the probability characterizing the ensemble changes under the assumption that the field of orientations of the crystallographic axes of the crystallites is an independent random field we show that our theory is compatible with the formulation used in texture analysis in particular we are able to prove that the physical assumption made by 10 in his study of weakly textured polycrystals holds true in addition we establish some elementary bounds on the material constants that characterize the effective elasticity tensor of weakly textured orthorhombic aggregates of cubic crystallites

hypercontractivity and the bakry emery criterion for compact lie groups on etudie l ingredient essentiel du critere de bakry emery on utilise ce critere pour obtenir des estimations ameliorees de la constante hypercontractive

homogenization of one phase stefan type problems in periodic and random media we investigate the homogenization of stefan type problems with oscillating diusion coecients both cases of periodic and random stationary ergodic medium are considered the proof relies on the coincidence of viscosity solutions and weak solutions which are the time derivatives of the solutions of an obstacle problem for the stefan problem this coincidence result is of independent interest

the boundary of negatively curved groups gromov s article gr contains fundamental properties of negatively curved groups several sets of seminar notes are available frn swn usn that contain more detailed accounts of and further expand on gromov s article a sequence xi e r i 1 2 is convergent at infinity provided xi x t 00 as i j t 00 two sequences xi yj convergent at infinityare equivalent if xi yi t 00 as i t 00 a point in the boundary ar of r is an equivalence class of sequences convergent at infinity now r u ar has a natural topology in which r is a discrete subspace and a typical not necessarily open neighborhood of a point a e ar represented by a sequence xj is given by y e r u arj a y n where a y lim xi y if y e r and a y lim xi yi if year is represented by a sequence yj ar is a compact metrizable finite dimensional space gr swn the important tool that relates ar with the cohomological properties of r is the rips complex pd r for every d 0 pd r is the simplicial complex whose vertices are elements of r and a collection xi xk e r spans a simplex if d xi x d for all i j the natural group action of r on itself by left translations gives rise to an action on pd r the key observation of rips is that when d is sufficiently large pd r is contractible and therefore when r is torsion free provides a model for er more precisely we have proposition 1 1 gr lemma 1 7 a swn 4 2 proposition 9 suppose that r is a negatively curved group and let t5 be as in the definition choose an integer

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 additive structure of elliptic homogenization one of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions since the latter are nonlocal functions of the former in this paper we address this problem in a new way in the context of linear elliptic equations in divergence form by showing that certain quantities associated to the energy density of solutions are essentially additive as a result we are able to prove quantitative estimates on the weak convergence of the gradients fluxes and energy densities of the first order correctors under blow down which are optimal in both scaling and stochastic integrability the proof of the additivity is a bootstrap argument completing the program initiated in armstrong et al commun math phys 347 2 315 361 2016 using the regularity theory recently developed for stochastic homogenization we reduce the error in additivity as we pass to larger and larger length scales in the second part of the paper we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables in particular we prove that the first order correctors converge in the large scale limit to a variant of the gaussian free field

mesoscopic higher regularity and subadditivity in elliptic homogenization we introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form unlike previous works on the topic our method does not use concentration inequalities such as poincare or logarithmic sobolev inequalities in the probability space and relies instead on a higher ck k 1 regularity theory for solutions of the heterogeneous equation which is valid on length scales larger than a certain specified mesoscopic scale this regularity theory which is of independent interest allows us to in effect localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument the fluctuations of the energy are then tightly controlled using subadditivity the convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes that is it quantifies the weak convergence of these quantities which yields by a new multiscale poincare inequality quantitative estimates on the sublinearity of the corrector

nil anosov actions we consider anosov actions of a lie group g of dimension k on a closed manifold of dimension k n we introduce the notion of nil anosov action of g which includes the case where g is nilpotent and establishes the invariance by the entire group g of the associated stable and unstable foliations we then prove a spectral decomposition theorem for such an action when the group g is nilpotent finally we focus on the case where g is nilpotent and the unstable bundle has codimension one we prove that in this case the action is a nil extension over an anosov action of an abelian lie group in particular if n ge 3 then the action is topologically transitive if n 2 then the action is a nil extension over an anosov flow

homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions a microscopic heterogeneous system under random influence is considered the randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium this system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes obstacles or heterogeneities together with random dynamical boundary conditions on the boundaries of these small holes

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

poincare and logarithmic sobolev inequalities by decomposition of the energy landscape we consider a diffusion on a potential landscape which is given by a smooth hamiltonian h rn r in the regime of low temperature e we proof the eyring kramers formula for the optimal constant in the poincare pi and logarithmic sobolev inequality lsi for the associated generator l e h of the diffusion the proof is based on a refinement of the two scale approach introduced by grunewald otto westdickenberg and villani govw09 and of the mean difference estimate introduced by chafai and malrieu cm10 the eyring kramers formula follows as a simple corollary from two main ingredients the first one shows that the pi and lsi constant of the diffusion restricted to a basin of attraction of a local minimum scales well in e this mimics the fast convergence of the diffusion to metastable states the second ingredient is the estimation of a mean difference by a weighted transport distance it contains the main contribution to the pi and lsi constant resulting from exponentially long waiting times of jumps between metastable states of the diffusion

a spectral like decomposition for transitive anosov flows in dimension three given a transitive or non transitive anosov vector field x on a closed three dimensional manifold m one may try to decompose m x by cutting m along tori and klein bottles transverse to x we prove that one can find a finite collection s 1 dots s n of pairwise disjoint pairwise non parallel tori and klein bottles transverse to x such that the maximal invariant sets lambda 1 dots lambda m of the connected components v 1 dots v m of m s 1 cup dots cup s n satisfy the following properties each lambda i is a compact invariant locally maximal transitive set for x the collection lambda 1 dots lambda m is canonically attached to the pair m x i e it can be defined independently of the collection of tori and klein bottles s 1 dots s n the lambda i s are the smallest possible for every possibly infinite collection s i i in i of tori and klein bottles transverse to x the lambda i s are contained in the maximal invariant set of m cup i s i to a certain extent the sets lambda 1 dots lambda m are analogs for anosov vector field in dimension 3 of the basic pieces which appear in the spectral decomposition of a non transitive axiom a vector field then we discuss the uniqueness of such a decomposition we prove that the pieces of the decomposition v 1 dots v m equipped with the restriction of the anosov vector field x are almost unique up to topological equivalence

stochastic homogenization of the keller segel chemotaxis system abstract in this paper we focus on the keller segel chemotaxis system in a random heterogeneous domain we assume that the corresponding diffusion and chemotaxis coefficients are given by stationary ergodic random fields and apply stochastic two scale convergence methods to derive the homogenized macroscopic equations in establishing our results we also derive a priori estimates for the keller segel system that rely only on the boundedness of the coefficients in particular no differentiability assumption on the diffusion and chemotaxis coefficients for the chemotactic species is required finally we prove the convergence of a periodization procedure for approximating the homogenized macroscopic coefficients

homology of closed geodesics in a negatively curved manifold soit m une variete de riemann compacte dont le flot geodesique sur le fibre tangent unite est de type anosov si on note par n x le nombre de geodesiques premieres p dans m dont la classe d homologie est un donne et de longueur l p x alors le taux de croissance exponentiel lim x x a1 log n x est egal a l entropie topologique h du flot geodesique 0

homogenization of the poisson equation with dirichlet conditions in random perforated domains we consider a sequence of open sets o e contained in a fixed bounded open set o of r n n 3 which vary randomly with e 0 the corresponding distribution function is given by an ergodic measure preserving dynamical system in such a way that o o e is a union of closed sets of size e n n 2 and the distance between them of order e for this sequence o e we study the asymptotic behavior of the solutions of the poisson equation with dirichlet conditions on o e similarly to the classical cioranescu murat result for the deterministic problem we show the existence of a new term of zero order in the limit equation we emphasize the fact that this new term is deterministic

a functional version of the birkhoff ergodic theorem for a normal integrand a variational approach in this paper we prove a new version of the birkhoff ergodic theorem bet for random variables depending on a parameter alias integrands this involves variational convergences namely epigraphical hypographical and uniform convergence and requires a suitable definition of the conditional expectation of integrands we also have to establish the measurability of the epigraphical lower and upper limits with respect to the field of invariant subsets from the main result applications to uniform versions of the bet to sequences of random sets and to the strong consistency of estimators are briefly derived

homogenization of the free boundary velocity in this paper we investigate some free boundary problems with space dependent free boundary velocities based on maximum principle type arguments we show the uniform convergence of the solutions in the homogenization limit the main step is to show the uniqueness of the limiting free boundary velocity which turns out to be a continuous function of the normal direction of the free boundary

the two scale convergence method applied to generalized besicovitch spaces the twoscale convergence method has proved to be a very useful tool for dealing with periodic homogenization problems in the present paper we develop this theory to generalized besicovitch spaces

lipschitz regularity for elliptic equations with random coefficients we develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients the main result is a large scale l type estimate for the gradient of a solution the estimate is proved with optimal stochastic integrability under a one parameter family of mixing assumptions allowing for very weak mixing with non integrable correlations to very strong mixing for example finite range of dependence we also prove a quenched l2 estimate for the error in homogenization of dirichlet problems the approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence form equations

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media