equivalences between nonuniform exponential dichotomy and admissibility abstract relationship between exponential dichotomies and admissibility of function classes is a significant problem for hyperbolic dynamical systems it was proved that a nonuniform exponential dichotomy implies several admissible pairs of function classes and conversely some admissible pairs were found to imply a nonuniform exponential dichotomy in this paper we find an appropriate admissible pair of classes of lyapunov bounded functions which is equivalent to the existence of nonuniform exponential dichotomy on half lines r separately on both half lines r simultaneously and on the whole line r additionally the maximal admissibility is proved in the case on both half lines r simultaneously

a grobman hartman theorem for nonuniformly hyperbolic dynamics abstract we establish a version of the grobman hartman theorem in banach spaces for nonuniformly hyperbolic dynamics we also consider the case of sequences of maps which corresponds to a nonautonomous dynamics with discrete time more precisely we consider sequences of lipschitz maps a m f m such that the linear parts a m admit a nonuniform exponential dichotomy and we establish the existence of a unique sequence of topological conjugacies between the maps a m f m and a m furthermore we show that the conjugacies are holder continuous with holder exponent determined by the ratios of lyapunov exponents with the same sign to the best of our knowledge this statement appeared nowhere before in the published literature even in the particular case of uniform exponential dichotomies although some experts claim that it is well known in this case we are also interested in the dependence of the conjugacies on the perturbations f m we show that it is holder continuous with the same holder exponent as the one for the conjugacies we emphasize that the additional work required to consider the case of nonuniform exponential dichotomies is substantial in particular we need to consider several additional lyapunov norms

existence and stability for fokker planck equations with log concave reference measure we study markov processes associated with stochastic differential equations whose non linearities are gradients of convex functionals we prove a general result of existence of such markov processes and a priori estimates on the transition probabilities the main result is the following stability property if the associated invariant measures converge weakly then the markov processes converge in law the proofs are based on the interpretation of a fokker planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures endowed with the wasserstein distance

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

white noise driven quasilinear spdes with reflection we study reflected solutions of the heat equation on the spatial interval 0 1 with dirichlet boundary conditions driven by an additive space time white noise roughly speaking at any point x t where the solutionu x t is strictly positive it obeys the equation and at a point x t whereu x t is zero we add a force in order to prevent it from becoming negative this can be viewed as an extension both of one dimensional sdes reflected at 0 and of deterministic variational inequalities an existence and uniqueness result is proved which relies heavily on new results for a deterministic variational inequality

motion by mean curvature from the ginzburg landau interface model we consider the scalar field t with a reversible stochastic dynamics which is defined by the standard dirichlet form relative to the gibbs measure with formal energy the potential v is even and strictly convex we prove that under a suitable large scale limit the t field becomes deterministic such that locally its normal velocity is proportional to its mean curvature except for some anisotropy effects as an essential input we prove that for every tilt there is a unique shift invariant ergodic gibbs measure for the field

decomposing constraint satisfaction problems using database techniques abstract there is a very close relationship between constraint satisfaction problems and the satisfaction of join dependencies in a relational database which is due to a common underlying structure namely a hypergraph by making that relationship explicit we are able to adapt techniques previously developed for the study of relational databases to obtain new results for constraint satisfaction problems in particular we prove that a constraint satisfaction problem may be decomposed into a number of subproblems precisely when the corresponding hypergraph satisfies a simple condition we show that combining this decomposition approach with existing algorithms can lead to a significant improvement in efficiency

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

integration by parts formulae on convex sets of paths and applications to spdes with reflection we consider the law of the bessel bridge of dimension 3 on the convex set k 0 of continuous non negative paths on 0 1 we prove an integration by parts formula on k 0 w r t to where an explicit infinite dimensional boundary measure appears we apply this to the solution u of a white noise driven stochastic partial differential equation with reflection introduced by nualart and pardoux where u 0 0 1 is a random non negative function and is a random positive measure on 0 0 1 indeed we prove that u is the radial part in the sense of dirichlet forms of the 3 valued solution of a linear stochastic heat equation and that has the following structure s 2 0 s 0 1 is the additive functional of u with revuz measure for ds 0 1 a e s there exists a unique r s in 0 1 s t u s r s 0 and ds d r s d ds 0 1 where a is the dirac mass at a 0 1 this gives a complete description of u as solution of a skorohod problem in the infinite dimensional non smooth convex set k 0

fluctuations for a φ interface model with repulsion from a wall we consider a interface model on a one dimensional lattice with repulsion from a hard wall we suppose that the repulsion is of the form c 1 where c 0 and denotes the height of the interface from the wall we prove convergence of the equilibrium fluctuations around the hydrodynamic limit to the solution of a spde with singular drift if c 0 the system becomes the funaki olla interface model with reflection at the wall whose equilibrium fluctuations converge to the solution of a spde with reflection we give a new proof of this result using the characterization of such solution as the diffusion generated by an infinite dimensional dirichlet form obtained in a previous paper our method is based on a study of integration by parts formulae w r t the equilibrium measure of the interface model and allows to avoid the proof of the so called boltzmann gibbs principle we also obtain convergence of finite dimensional distributions of non equilibrium fluctuations around the stationary hydrodynamic limit 0

on minimal constraint networks in a minimal binary constraint network every tuple of a constraint relation can be extended to a solution the tractability or intractability of computing a solution to such a minimal network was a long standing open question dechter conjectured this computation problem to be np hard we prove this conjecture we also prove a conjecture by dechter and pearl stating that for k 2 it is np hard to decide whether a single constraint can be decomposed into an equivalent k ary constraint network we show that this holds even in case of bi valued constraints where k 3 which proves another conjecture of dechter and pearl finally we establish the tractability frontier for this problem with respect to the domain cardinality and the parameter k

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

stable invariant manifolds for parabolic dynamics abstract we consider nonautonomous equations v a t v in a banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates e c t for some function t this corresponds to the existence of a generalized exponential dichotomy which is known to be robust when t t this behavior can be described as a type of parabolic dynamics we consider the general case of nonuniform exponential dichotomies for which the lyapunov stability is not uniform we show that for any sufficiently small perturbation f of a generalized exponential dichotomy there is a stable invariant manifold for the perturbed equation v a t v f t v we also consider the case of exponential contractions which allow a simpler treatment and we show that they persist under sufficiently small nonlinear perturbations

probleme de skorohod multivoque on prouve un resultat d existence et d unicite pour une generalisation par introduction d un operateur maximal monotone multivoque du probleme de skorohod avec reflexion normale deterministe associe a un convexe ferme d de r d la formulation a partir des operateurs maximaux monotones multivoques permet de considerer des drifts singuliers explosant au bord du domaine cette approche multivoque clarifie la connexion entre probleme de skorohod et semigroupes nonlineaires en guise d application on considere ensuite le cas stochastique les equations differentielles stochastiques multivoques sont ainsi revisitees par consequent cette contribution fournit une nouvelle methode de construction de diffusions reflechies normalement possedant un coefficient de derive discontinu explosif

integration by parts on bolds delta bessel bridges bolds delta 3 and related spdes we study a white noise driven semilinear partial differential equation on the spatial interval 0 1 with dirichlet boundary condition and with a singular drift of the form cu 3 c 0 we prove existence and uniqueness of a non negative continuous adapted solution u on 0 oo x 0 1 for every nonnegative continuous initial datum x satisfying x 0 x 1 0 we prove that the law of the bessel bridge on 0 1 of dimension 3 is the unique invariant probability measure of the process x u with c 1 3 8 and if n that u is the radial part in the sense of dirichlet forms of the r valued solution of a linear stochastic heat equation an explicit integration by parts formula w r t is given for all 3

stability in delay difference equations with nonuniform exponential behavior abstract we study the stability under perturbations for delay difference equations in banach spaces namely we establish the nonuniform stability of linear nonuniform exponential contractions under sufficiently small perturbations we also obtain a stable manifold theorem for perturbations of linear delay difference equations admitting a nonuniform exponential dichotomy and show that the stable manifolds are lipschitz in the perturbation

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

equilibrium fluctuations for nabla_ varphi interface model we study the large scale space time fluctuations of an interface which is modeled by a massless scalar field with reversible langevin dynamics for a strictly convex interaction potential we prove that on a large space time scale these fluctuations are governed by an infinite dimensional ornstein uhlenbeck process its effective diffusion type covariance matrix is characterized through a variational formula

nonuniform mean square exponential dichotomies and mean square exponential stability abstract in this paper the existence conditions of nonuniform mean square exponential dichotomy nms ed for a linear stochastic differential equation sde are established the difference of the conditions for the existence of a nonuniform dichotomy between an sde and an ordinary differential equation ode is that the first one needs an additional assumption nonuniform lyapunov matrix to guarantee that the linear sde can be transformed into a decoupled one while the second does not therefore the first main novelty of our work is that we establish some preliminary results to tackle the stochasticity this paper is also concerned with the mean square exponential stability of nonlinear perturbation of a linear sde under the condition of nonuniform mean square exponential contraction nms ec for this purpose the concept of second moment regularity coefficient is introduced this concept is essential in determining the stability of the perturbed equation and hence we deduce the lower and upper bounds of this coefficient our results imply that the lower and upper bounds of the second moment regularity coefficient can be expressed solely by the drift term of the linear sde

smoothness of invariant manifolds for nonautonomous equations for semiflows generated by ordinary differential equations v a t v admitting a nonuniform exponential dichotomy we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v a t v f t v as an application we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow in particular we obtain smooth invariant manifolds for a class of vector fields that need not be c1 for any 0 1 to the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting we emphasize that we do not need to assume the existence of an exponential dichotomy but only the existence of a nonuniform exponential dichotomy with sufficiently small nonuniformity when compared to the lyapunov exponents of the original linear equation furthermore for example in the case of stable manifolds we only need to assume that there exist negative lyapunov exponents while we also allow zero exponents our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones

robustness of nonuniform exponential dichotomies in banach spaces we give conditions for the robustness of nonuniform exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation x a t x persists under a sufficiently small linear perturbation we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the angles between the stable and unstable subspaces our proofs exhibit implicitly the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions we emphasize that we do not need the notion of admissibility of bounded nonlinear perturbations we also obtain related robustness results in the case of nonuniform exponential contractions in addition we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time

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

stable manifolds for nonautonomous equations without exponential dichotomy abstract we consider nonautonomous ordinary differential equations v a t v in banach spaces and under fairly general assumptions we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v a t v f t v which corresponds to the set of negative lyapunov exponents of the original linear equation the main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity i e a small deviation from the classical notion of uniform exponential dichotomy in fact we showed that essentially any linear equation v a t v admits a nonuniform exponential dichotomy and thus the above assumption only concerns the smallness of the nonuniformity of the dichotomy this smallness is a rather common phenomenon at least from the point of view of ergodic theory almost all linear variational equations obtained from a measure preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity we emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small only sufficiently small our approach is related to the notion of lyapunov regularity which goes back to lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

conservative stochastic cahn hilliard equation with reflection we consider a stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average the equation is driven by the derivative in space of a space time white noise and contains a double laplacian in the drift due to the lack of the maximum principle for the double laplacian the standard techniques based on the penalization method do not yield existence of a solution we propose a method based on infinite dimensional integration by parts formulae obtaining existence and uniqueness of a strong solution for all continuous nonnegative initial conditions and detailed information on the associated invariant measure and dirichlet form

nonuniform μ ν dichotomies and local dynamics of difference equations abstract we obtain a local stable manifold theorem for perturbations of nonautonomous linear difference equations possessing a very general type of nonuniform dichotomy possibly with different growth rates in the uniform and nonuniform parts note that we consider situations where the classical lyapunov exponents can be zero additionally we study how the manifolds decay along the orbit of a point as well as the behavior under perturbations and give examples of nonautonomous linear difference equations that admit the dichotomies considered

stable manifolds for nonuniform polynomial dichotomies we establish the existence of smooth stable manifolds in banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy this new dichotomy is more restrictive in the nonuniform part but allow the uniform part to obey a polynomial law instead of an exponential more restrictive law we consider two families of perturbations for one of the families we obtain local lipschitz stable manifolds and for the other family assuming more restrictive conditions on the perturbations and its derivatives we obtain c1 global stable manifolds finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family

evolution problems in spaces of probability measures abstract this is a short survey paper on the application of monge s theory of optimal mass transportation to evolution problems of a gradient type in the case when the natural state space is the space of probability measures this occurs for instance for fokker planck equations and optimal mass transportation methods provide in this setting very general existence and stability results

parameter dependence of stable manifolds under nonuniform hyperbolicity abstract for a nonautonomous linear equation v a t v in a banach space with a nonuniform exponential dichotomy we show that the nonlinear equation v a t v f t v has stable invariant manifolds v which are lipschitz in the parameter provided that f is a sufficiently small lipschitz perturbation since any linear equation with nonzero lyapunov exponents has a nonuniform exponential dichotomy the above assumption is very general we emphasize that passing from a classical uniform exponential dichotomy to a general nonuniform exponential dichotomy requires a substantially new approach

lyapunov regularity of impulsive differential equations abstract for linear impulsive differential equations we give a simple criterion for the existence of a nonuniform exponential dichotomy which includes uniform exponential dichotomies as a very special case for this we introduce the notion of lyapunov regularity for a linear impulsive differential equation in terms of the so called regularity coefficient the theory is then used to show that if the lyapunov exponents are nonzero then there is a nonuniform exponential behavior which can be expressed in terms of the lyapunov exponents of the differential equation and of the regularity coefficient we also consider the particular case of nonuniform exponential contractions when there are only negative lyapunov exponents having this relation in mind it is also of interest to provide alternative characterizations of lyapunov regularity and particularly to obtain sharp lower and upper bound for the regularity coefficient in particular we obtain bounds expressed in terms of the matrices defining the impulsive linear system and we obtain characterizations in terms of the exponential growth rate of volumes in addition we establish the persistence of the stability of a linear impulsive differential equation under sufficiently small nonlinear perturbations