uniqueness of gradient gibbs measures with disorder we consider in a uniformly strictly convex potential regime two versions of random gradient models with disorder in model a the interface feels a bulk term of random fields while in model b the disorder enters through the potential acting on the gradients we assume a general distribution on the disorder with uniformly bounded finite second moments it is well known that for gradient models without disorder there are no gibbs measures in infinite volume in dimension d 2 while there are shift invariant gradient gibbs measures describing an infinite volume distribution for the gradients of the field as was shown by funaki and spohn commun math phys 185 1 36 1997 van enter and kulske proved in ann appl probab 18 1 109 119 2008 that adding a disorder term as in model a prohibits the existence of such gradient gibbs measures for general interaction potentials in d 2 in cotar and kulske ann appl probab 22 5 1650 1692 2012 we proved the existence of shift covariant random gradient gibbs measures for model a when d ge 3 the disorder is i i d and has mean zero and for model b when d ge 1 and the disorder has a stationary distribution in the present paper we prove existence and uniqueness of shift covariant random gradient gibbs measures with a given expected tilt u in mathbb r d and with the corresponding annealed measure being ergodic for model a when d ge 3 and the disordered random fields are i i d and symmetrically distributed and for model b when d ge 1 and for any stationary disorder dependence structure we also compute for both models for any gradient gibbs measure constructed as in cotar and kulske ann appl probab 22 5 1650 1692 2012 when the disorder is i i d and its distribution satisfies a poincare inequality assumption the optimal decay of covariances with respect to the averaged over the disorder gradient gibbs measure

time reversal for infinite dimensional diffusions in this paper we analize the reversibility of the diffusion property for the solution of certain infinite dimensional systems of stochastic differential equations necessary and sufficient conditions ensuring this reversibility are given the proofs use the techniques of the stochastic calculus of variations

delocalization of two dimensional random surfaces with hard core constraints we study the fluctuations of random surfaces on a two dimensional discrete torus the random surfaces we consider are defined via a nearest neighbor pair potential which we require to be twice continuously differentiable on a possibly infinite interval and infinity outside of this interval no convexity assumption is made and we include the case of the so called hammock potential when the random surface is uniformly chosen from the set of all surfaces satisfying a lipschitz constraint our main result is that these surfaces delocalize having fluctuations whose variance is at least of order log n where n is the side length of the torus we also show that the expected maximum of such surfaces is of order at least log n the main tool in our analysis is an adaptation to the lattice setting of an algorithm of richthammer who developed a variant of a mermin wagner type argument applicable to hard core constraints we rely also on the reflection positivity of the random surface model the result answers a question mentioned by brascamp et al on the hammock potential and a question of velenik

persistence of gaussian processes non summable correlations suppose the auto correlations of real valued centered gaussian process z cdot are non negative and decay as rho s t for some rho cdot regularly varying at infinity of order alpha in 1 0 with i rho t int 0 t rho s ds its primitive we show that the persistence probabilities decay rate of log mathbb p sup t in 0 t z t 0 is precisely of order t i rho t log i rho t thereby closing the gap between the lower and upper bounds of newell and rosenblatt ann math stat 33 1306 1313 1962 which stood as such for over fifty years we demonstrate its usefulness by sharpening recent results of sakagawa adv appl probab 47 146 163 2015 about the dependence on d of such persistence decay for the langevin dynamics of certain nabla phi interface models on mathbb z d

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

a new class of solvable dynamical systems a new class of dynamical systems are presented together with their solutions some of these models are isochronous namely their generic solutions are all completely periodic with the same period others are characterized by friction all solutions vanishing in the remote future and still others are asymptotically isochronous approaching an isochronous behavior in the remote future

supersymmetry and algebraic darboux transformations we describe a class of algebraically solvable susy models by considering the deformation of invariant polynomial flags by means of the darboux transformation the algebraic deformations corresponding to the addition of a bound state to a shape invariant potential are particularly interesting the polynomial flags in question are indexed by a deformation parameter m 1 2 and lead to new algebraically solvable models we illustrate these ideas by considering deformations of the hyperbolic p schl teller potential

the malliavin calculus a functional analytic approach in his groundbreaking articles s 91 malliavin introduced a technique for obtaining elliptic regularity results using function space calculus in his formulation the function space calculus on which his theory rests is the martingale calculus coming from the ornstein uhlenbeck process on wiener space cf 121 for a detailed discussion of malliavin s theory from this point of view although this formulation has a great deal to recommend it and is particularly pleasing to afficionados of the so called brownian sheet most analysts including those who are reasonably facile with the machinery of probability theory are unlikely to find the technical difficulties inherent in this approach outweighed by the eventual rewards shigekawa 111 discovered that there is an alternative formulation of malliavin s theory in shigekawa s formulation no mention need be made of the ornstein uhlenbeck process instead what he relies on is a sobolev type extension of the frechet dervative with wiener measure playing the role played by lebesgue measure in the finite dimensional context a third formulation of malliavin s theory was recently provided by bismut l bismut s idea is to exploit the quasi invariance of wiener measure as it is manifested in a beautiful relation due to haussmann 4 the present article has two aims in the first place section 1 2 and 3 are devoted to yet another formulation of malliavin s calculus although the end results are precisely the same as those obtained in 121 no reference is made to the ornstein uhlenbeck process on which the original theory depended instead the machinery is entirely that of elementary functional analysis and is devoid of any sophisticated martingale calculus it has been my intent that these three sections should be readily accessible to any analyst who has had some acquintance with wiener measure

on the asymptotic behaviour of spitzer s model for evolution of one dimensional point systems a nearest neighbor gradient dynamics of one dimensional infinite particle systems is considered the model admits a two parameter family of stationary configurations some domains of attraction of stationary configurations are described and the continuum hydrodynamical limit of the system is investigated it is shown that the mean density of points satisfies a nonlinear diffusion equation in the hydrodynamical limit

random walks among time increasing conductances heat kernel estimates for any graph having a suitable uniform poincare inequality and volume growth regularity we establish two sided gaussian transition density estimates and parabolic harnack inequality for constant speed continuous time random walks evolving via time varying uniformly elliptic conductances provided the vertex conductances i e reversing measures increase in time such transition density upper bounds apply for discrete time uniformly lazy walks with the matching lower bounds holding once the parabolic harnack inequality is proved

worst case analysis of the perceptron and exponentiated update algorithms abstract the absolute loss is the absolute difference between the desired and predicted outcome this paper demonstrates worst case upper bounds on the absolute loss for the perception learning algorithm and the exponentiated update learning algorithm which is related to the weighted majority algorithm the bounds characterize the behavior of the algorithms over any sequence of trials where each trial consists of an example and a desired outcome interval any value in the interval is an acceptable outcome the worst case absolute loss of both algorithms is bounded by the absolute loss of the best linear function in a comparison class plus a constant dependent on the initial weight vector plus a per trial loss the per trial loss can be eliminated if the learning algorithm is allowed a tolerance from the desired outcome for concept learning the worst case bounds lead to mistake bounds that are comparable to past results

pde with random coefficients and euclidean field theory in this paper a new proof of an identity of giacomin olla and spohn is given the identity relates the 2 point correlation function of a euclidean field theory to the expectation of the green s function for a pde with random coefficients the euclidean field theory is assumed to have convex potential an inequality of brascamp and lieb therefore implies gaussian bounds on the fourier transform of the 2 point correlation function by an application of results from random pde the previously mentioned identity implies pointwise gaussian bounds on the 2 point correlation function

one particle subspaces in the stochastic xy model we study the spectrum of the generatorh of the glauber dynamics for a model of planar rotators on a lattice in the case of a high temperature 1 we construct two so called one particle subspacesh forh and describe the spectrum of the generator in these subspaces as a consequence we find time asymptotics of the correlations for the glauber dynamics

infinite lattice systems of interacting diffusion processes existence and regularity properties generalized stochastic gradient systems for infinite lattice models are investigated the allowed strength of the interaction depends on the dimension of the lattice the semigroup of transition probabilities is constructed and its regularity properties are also discussed some results of doss and royer 2 are improved

goldfishing a new solvable many body problem a recent technique allows one to identify and investigate solvable dynamical systems naturally interpretable as classical many body problems being characterized by equations of motion of newtonian type generally in two dimensional space in this paper we tersely review results previously obtained in this manner and present novel findings of this kind mainly solvable variants of the goldfish many body model including models that feature isochronous classes of completely periodic solutions different formulations of these models are presented the behavior of one of these isochronous dynamical systems in the neighborhood of its equilibrium configuration is investigated and in this manner some remarkable diophantine findings are obtained

time dependent polynomials with one multiple root and new solvable dynamical systems a time dependent monic polynomial in the z variable with n distinct roots such that exactly one root has multiplicity m 2 is considered for k 1 2 the kth derivatives of the n roots are expressed in terms of the derivatives of order j k of the first n coefficients of the polynomial and of the derivatives of order j k 1 of the roots themselves these relations are utilized to construct new classes of algebraically solvable first order systems of ordinary differential equations as well as n body problems multiple examples of solvable isochronous all solutions are periodic with the same period 2 and 3 body problems are provided a time dependent monic polynomial in the z variable with n distinct roots such that exactly one root has multiplicity m 2 is considered for k 1 2 the kth derivatives of the n roots are expressed in terms of the derivatives of order j k of the first n coefficients of the polynomial and of the derivatives of order j k 1 of the roots themselves these relations are utilized to construct new classes of algebraically solvable first order systems of ordinary differential equations as well as n body problems multiple examples of solvable isochronous all solutions are periodic with the same period 2 and 3 body problems are provided

on the correlation for kac like models in the convex case the aim of this paper is to stu the behavior asm tends to of a family of measures exp m x dx m on m where m is a potential on m which is a perturbation in a suitable sense of the harmonic potential j x j 2

exponentiated gradient versus gradient descent for linear predictors we consider two algorithm for on line prediction based on a linear model the algorithms are the well known gradient descent gd algorithm and a new algorithm which we call eg they both maintain a weight vector using simple updates for the gd algorithm the update is based on subtracting the gradient of the squared error made on a prediction the eg algorithm uses the components of the gradient in the exponents of factors that are used in updating the weight vector multiplicatively we present worst case loss bounds for eg and compare them to previously known bounds for the gd algorithm the bounds suggest that the losses of the algorithms are in general incomparable but eg has a much smaller loss if only a few components of the input are relevant for the predictions we have performed experiments which show that our worst case upper bounds are quite tight already on simple artificial data

existence of a smooth density for the filter in nonlinear filtering with infinite dimensional noise abstract in this paper we prove by means of malliavin calculus that under a local hormander condition the unnormalized filter associated with a nonlinear filtering problem in which the system process depends on an infinite dimensional noise admits a smooth density with respect to the lebesgue measure

central limit theorem for an infinite lattice system of interacting diffusion processes on donne une demonstration basee sur une representation par integrale stochastique de dimension infinie des fonctionnelles lisses des processus de diffusion

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

strict convexity of the free energy for a class of non convex gradient models we consider a gradient interface model on the lattice with interaction potential which is a non convex perturbation of a convex potential we show using a one step multiple scale analysis the strict convexity of the surface tension at high temperature this is an extension of funaki and spohn s result 8 where the strict convexity of potential was crucial in their proof

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

a priori estimates for symmetrizing measures and their applications to gibbs states abstract we prove existence and uniform a priori estimates for tempered gibbs states of certain classical lattice systems with unbounded spins nonharmonic pair potentials and infinite radius of interaction we use an alternative characterization of gibbs measures in terms of their radon nikodym derivatives w r t local shifts of the configuration space and corresponding integration by parts formulas

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

some remarks on nonequilibrium dynamics of infinite particle systems classical mechanics of infinitely many particles in dimensions one and two is considered particles interacting by a superstable pair potential of finite range the group of motion generated by newton s equations is constructed in the space of locally finite configurations with a logarithmic order of energy fluctuations at infinity a core of the liouville operator is also described results of dobrushin and the author and of marchioro pellegrinotti pulvirenti are improved

absolute continuity of the law of an infinite dimensional wiener functional with respect to the wiener probability in this paper we study conditions ensuring that the law of ac 0 1 valued functional defined on an abstract wiener space is absolutely continuous with respect to the wiener measure onc 0 1 these conditions extend those established byp malliavin 12 13 for finite dimensional wiener functionals and those of 15 for hilbert valued functionals

malliavin calculus for the stochastic 2d navier stokes equation we consider the incompressible two dimensional navier stokes equation with periodic boundary conditions under the effect of an additive white in time stochastic forcing under mild restrictions on the geometry of the scales forced we show that any finite dimensional projection of the solution possesses a smooth strictly positive density with respect to lebesgue measure in particular our conditions are viscosity independent we are mainly interested in forcing that excites a very small number of modes all of the results rely on proving the nondegeneracy of the infinite dimensional malliavin matrix c 2006 wiley periodicals inc

coexistence of stable limit cycles in a generalized curie weiss model with dissipation in this paper we modify the langevin dynamics associated to the generalized curie weiss model by introducing noisy and dissipative evolution in the interaction potential we show that when a zero mean gaussian is taken as single site distribution the dynamics in the thermodynamic limit can be described by a finite set of odes depending on the form of the interaction function the system can have several phase transitions at different critical temperatures because of the dissipation effect not only the magnetization of the systems displays a self sustained periodic behavior at sufficiently low temperature but in certain regimes any finite number of stable limit cycles can exist we explore some of these peculiarities with explicit examples

on the diffusive nature of entropy flow in infinite systems remarks to a paper by guo papanicolau varadhan the hydrodynamic behaviour of interacting diffusion processes is investigated by means of entropy free energy arguments the methods of 13 are simplified and extended to infinite systems including a case of anharmonic oscillators in a degenerate thermal noise following 14 15 and 3 5 we derive a priori bounds for the rate of entropy production in finite volumes as the size of the whole system is infinitely extended the flow of entropy through the boundary is controlled in much the same way as energy flow in diffusive systems 4

two novel classes of solvable many body problems of goldfish type with constraints two novel classes of many body models with nonlinear interactions of goldfish type are introduced they are solvable provided the initial data satisfy a single constraint in one case in the other two constraints i e for such initial data the solution of their initial value problem can be achieved via algebraic operations such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials entirely isochronous versions of some of these models are also exhibited i e versions of these models whose nonsingular solutions are all completely periodic with the same period

green s functions for elliptic and parabolic equations with random coefficients ii this paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues it is assumed that the coefficients of the equation are stationary random variables random in both space and time the green s functions for the equations are then random variables regularity properties for expectation values of green s functions are obtained in particular it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the green s function for the heat equation similar results are obtained for the related finite difference equations this paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space

approximation of the effective conductivity of ergodic media by periodization this paper is concerned with the approximation of the effective conductivity a associated to an elliptic operator xa x x where for x d d 1 a x is a bounded elliptic random symmetric d d matrix and takes value in an ergodic probability space x writing an x the periodization of a x on the torus tdn of dimension d and side n we prove that for almost all we extend this result to non symmetric operators x a e x x corresponding to diffusions in ergodic divergence free flows a is d d elliptic symmetric matrix and e x an ergodic skew symmetric matrix and to discrete operators corresponding to random walks on d with ergodic jump rates

local stability and hydrodynamical limit of spitzer s one dimensional lattice model an infinite system of ordinary differential equations is considered the right hand side is just the negative gradient of potential energy of a one dimensional system of unbounded spins interacting by a symmetric and convex pair potential constant configurations are stationary points and the mean spin is conserved it is shown that each of these stationary points has its own domain of attraction the initial distribution need not be translation invariant as a consequence we obtain that the mean spin satisfies the heat equation in the hydrodynamical limit

low temperature asymptotics of integrable systems in an external field an asymptotic low temperature expansion is performed for an integrable bosonic lattice model and for the critical spin 1 2 heisenberg chain in a magnetic field the results apply to the integrable bose gas as well we also comment on a high temperature expansion of the bosonic lattice model

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

the weighted majority algorithm we study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials with a prediction to be made in each and the goal of the learner is to make few mistakes we are interested in the case where the learner has reason to believe that one of some pool of known algorithms will perform well but the learner does not know which one a simple and effective method based on weighted voting is introduced for constructing a compound algorithm in such a circumstance we call this method the weighted majority algorithm we show that this algorithm is robust in the presence of errors in the data we discuss various versions of the weighted majority algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool for example given a sequence of trials if there is an algorithm in the pool a that makes at most m mistakes then the weighted majority algorithm will make at most c log a m mistakes on that sequence where c is fixed constant

on the hydrodynamic limit of a ginzburg landau lattice model the law of large numbers in arbitrary dimensions a lattice system of interacting diffusion processes is investigated the evolution is attractive and time reversible the spin satisfies a conservation law it is shown that the rescaled spin field converges in probability to the corresponding solution to a nonlinear diffusion equation

une approche gibbsienne des diffusions browniennes infini dimensionnelles nous montrons qu il y a correspondance biunivoque entre les lois de diffusions browniennes infini dimensionnelles et les etats de gibbs sur l espace des trajectoires omega c 0 1 mathbb z d ce resultat est ensuite applique aux problemes de transition de phases du retournement du temps et a l etude des mesures reversibles le principal outil est une caracterisation des etats de gibbs par une version infini dimensionnelle de la formule d integration par parties du calcul de malliavin

construction of non gaussian self similar random fields with hierarchical structure in the present work we construct non gaussian self similar random fields with hierarchical structure the construction is based on non gaussian solutions of the main nonlinear equation of the hierarchical models theory the existence of such solutions was proved originally by sinai and the author and later by another method by collet and eckmann next we establish the uniqueness of a gibbs state for the constructed self similar field finally for a class of hierarchical models we prove the convergence of renormalization transformations of a random field at the critical point to the self similar field

worst case quadratic loss bounds for prediction using linear functions and gradient descent studies the performance of gradient descent gd when applied to the problem of online linear prediction in arbitrary inner product spaces we prove worst case bounds on the sum of the squared prediction errors under various assumptions concerning the amount of a priori information about the sequence to predict the algorithms we use are variants and extensions of online gd whereas our algorithms always predict using linear functions as hypotheses none of our results requires the data to be linearly related in fact the bounds proved on the total prediction loss are typically expressed as a function of the total loss of the best fixed linear predictor with bounded norm all the upper bounds are tight to within constants matching lower bounds are provided in some cases finally we apply our results to the problem of online prediction for classes of smooth functions

stochastic analysis on product manifolds dirichlet operators on differential forms abstract we define a de rham complex over a product manifold infinite product of compact manifolds and dirichlet operators on differential forms associated with differentiable measures in particular gibbs measures which generalize the notions of bochner and de rham laplacians we give probabilistic representations for corresponding semigroups and study properties of the corresponding stochastic dynamics

how the unbounded drift shapes the dirichlet semigroups behaviour of non gaussian gibbs measures abstract we investigate the dirichlet operator semigroups of non gaussian gibbs measures on linear spin lattice systems we prove that dirichlet semigroups preserve spaces of l 2 differentiable functions we show how the non linearity and unboundedness of drift term in dirichlet operator affects the structure of the spaces involved

scaling limit for a class of gradient fields with nonconvex potentials we consider gradient fields x x d whose law takes the gibbs boltzmann form z 1 exp x y v v x where the sum runs over nearest neighbors we assume that the potential v admits the representation v log d exp 1 2 2 where is a positive measure with compact support in 0 hence the potential v is symmetric but nonconvex in general while for strictly convex v s the translation invariant ergodic gradient gibbs measures are completely characterized by their tilt a nonconvex potential as above may lead to several ergodic gradient gibbs measures with zero tilt still every ergodic zero tilt gradient gibbs measure for the potential v above scales to a gaussian free field

a characterisation of the gaussian free field we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain markov property is a multiple of the gaussian free field this result holds subject only to a fourth moment assumption

central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions we prove a functional central limit theorem for additive functionals of stationary reversible ergodic markov chains under virtually no assumptions other than the necessary ones we use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk

log sobolev inequalities for infinite one dimensional lattice systems it is shown that a unique gibbs measure of infinite spin system with short range interaction on one dimensional lattice satisfies log sobolev inequality

monotonicity of the free energy in the stochastic heisenberg model the specific free energy of the state at timet of the stochastic heisenberg model is shown to be non increasing witht and to strictly decrease whenever this state is not a gibbs state of the hamiltonian the initial state is assumed to be translation invariant and suitably smooth for such states a convergence theorem is obtained

ergodic and mixing properties of equilibrium measures for markov processes let s t be the semigroup corresponding to a markov process on a metric space x suppose s t commutes with a homeomorphism t of x we prove that under certain conditions an equilibrium measure for the process is ergodic under t we also show that under stronger conditions this measure must be mixing under t several applications of these results are given

large deviations and concentration properties for ϕ interface models we consider the massless field with zero boundary conditions outside d n d d n n d a suitable subset of d i e the continuous spin gibbs measure n on d n with hamiltonian given by h x y x y 1 v x y and x 0 for x d n c the interaction v is taken to be strictly convex and with bounded second derivative this is a standard effective model for a d 1 dimensional interface represents the height of the interface over the base d n due to the choice of scaling of the base we scale the height with the same factor by setting n n we study various concentration and relaxation properties of the family of random surfaces n and of the induced family of gradient fields n n as the discretization step 1 n tends to zero n in particular we prove a large deviation principle for n and show that the corresponding rate function is given by d u x dx where is the surface tension of the model this is a multidimensional version of the sample path large deviation principle we use this result to study the concentration properties of n under the volume constraint i e the constraint that 1 n d x dn n x stays in a neighborhood of a fixed volume v 0 and the hard wall constraint i e n x 0 for all x this is therefore a model for a droplet of volume v lying above a hard wall we prove that under these constraints the field n of rescaled heights concentrates around the solution of a variational problem involving the surface tension as it would be predicted by the phenomenological theory of phase boundaries our principal result however asserts local relaxation properties of the gradient field n n to the corresponding extremal gibbs states thus our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments the proofs have both probabilistic and analytic aspects essential analytic tools are p estimates for elliptic equations and the theory of young measures on the side of probability tools a central role is played by the helffer sjostrand 31 pde representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of t funaki and h spohn 25 on the structure of gradient fields

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

quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics we study quantitatively the effective large scale behavior of discrete elliptic equations on the lattice mathbb z d with random coefficients the theory of stochastic homogenization relates the random stationary and ergodic field of coefficients with a deterministic matrix of effective coefficients this is done via the corrector problem which can be viewed as a highly degenerate elliptic equation on the infinite dimensional space of admissible coefficient fields in this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a spectral gap estimate w r t a glauber dynamics on coefficient fields as it is the case for independent and identically distributed coefficients as a main result we prove an optimal decay in time of the semigroup associated with the corrector problem i e of the generator of the process called random environment as seen from the particle as a corollary we recover existence of stationary correctors in dimensions d 2 and prove new optimal estimates for regularized versions of the corrector in dimensions d ge 2 we also give a self contained proof of a new estimate on the gradient of the parabolic variable coefficient green s function which is a crucial analytic ingredient in our approach as an application of these results we prove the first and optimal estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients

dobrushin uniqueness theorem and logarithmic sobolev inequalities we formulate a condition on a local specification e on a countable product space m m being a riemannian manifold or a discrete set 1 1 assuring that the corresponding set of gibbs measures consists of a unique measure satisfying a logarithmic sobolev inequality

pattern densities in non frozen planar dimer models in this paper we introduce a family of observables for the dimer model on a bi periodic bipartite planar graph called pattern density fields we study the scaling limit of these objects for non frozen gibbs measures of the dimer model and prove that they converge to a linear combination of a derivative of the gaussian massless free field and an independent white noise

a new class of solvable many body problems with constraints associated with an exceptional polynomial subspace of codimension 2 a new class of many body models is identified and investigated just as those we recently discovered these many body problems are solvable provided the initial data satisfy certain constraints for such data the solution of the initial value problem can be achieved via algebraic operations such as finding the zeros of known polynomials entirely isochronous subclasses of these models are also exhibited whose generic solutions are completely periodic with the same fixed period in their entire phase space

convection diffusion equation with space time ergodic random flow we prove the homogenization of convection diffusion in a time dependent ergodic incompressible random flow which has a bounded stream matrix and a constant mean drift we also prove two variational formulas for the effective diffusivity as a consequence we obtain both upper and lower bounds on the effective diffusivity

ergodicity for generalized kawasaki dynamics we give a necessary and sufficient condition for a gibbs measure on the product space s1 d to satisfy the spectral gap or the logarithmic sobolev inequality with the following quadratic form where y is a finite set and al are integers as a consequence we prove that the generalized kawasaki dynamics decays exponentially to equilibrium in the supremum norm in a strong mixing region

stationary states and their stability of the stepping stone model involving mutation and selection on considere un modele de diffusion qui est un processus de diffusion de dimension infinie on etudie la regularite des distributions marginales de dimension finie des etats stationnaires

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

correlation structure of the corrector in stochastic homogenization recently the quantification of errors in the stochastic homogenization of divergence form operators has witnessed important progress our aim now is to go beyond error bounds and give precise descriptions of the effect of the randomness in the large scale limit this paper is a first step in this direction our main result is to identify the correlation structure of the corrector in dimension 33 and higher this correlation structure is similar to but different from that of a gaussian free field

learning linear threshold approximations using perceptrons we demonstrate sufficient conditions for polynomial learnability of suboptimal linear threshold functions using perceptrons the central result is as follows suppose there exists a vector w of n weights including the threshold with accuracy 1 average error and balancing separation i e with probability 1 w correctly classifies an example x over examples incorrectly classified by w the expected value of w x is source of inaccuracy does not matter and over a certain portion of correctly classified examples the expected value of w x is then with probability 1 the perceptron achieves accuracy at least 1 1 after o n 2 2 ln 1 examples

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

non explosion boundedness and ergodicity for stochastic semilinear equations abstract sufficient conditions for non explosion boundedness and ergodicity of solutions of a class of semilinear stochastic equations are given applications of the results are based on a new regularity theorem for solutions of the linear problem

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

a cluster expansion for stochastic lattice fields a langevin equation of landau ginzburg type for the stochastic dynamics of a scalar field on a lattice is studied a cluster expansion is developed for this problem which converges for large mass as a consequence one establishes uniformly in the volume a exponential decay of correlations in space and time and b exponential approach to equilibrium for a class of nearby initial distributions

convergence to equilibrium for classical and quantum spin systems the paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on euclidean spaces existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted hilbert and banach spaces this paper is an enlarged and rewritten version of the paper 7

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

invariance implies gibbsian some new results we investigate stationary distributions of stochastic gradient systems in riemannian manifolds and prove that under certain assumptions such distributions are symmetric these results are extended to countable products of finite dimensional manifolds and applied to gibbs distributions in the case where the single spin spaces are riemannian manifolds in particular we obtain a new result concerning the question whether all invariant measures are gibbsian actually we consider a more general object weak elliptic equations for measures which on the one hand yields the results obtained stronger than the above mentioned statements and on the other hand enables us to give simpler proofs of more general than previously known facts applications to concrete models of lattice systems over d with not necessarily compact spin space are presented also in the case d 3 under certain assumptions of decay of interaction

asymptotic ergodicity of the process of conditional law in some problem of non linear filtering abstract we consider a sample of a trajectory of an ergodic process m x through a stationary noisy observation process we study the asymptotic ergodicity of the phase process

gradient dynamics of infinite point systems nonequilibrium gradient dynamics of d dimensional particle systems is investigated the interaction is given by a superstable pair potential of finite range solutions are constructed in the well defined set of locally finite configurations with a logarithmic order of energy fluctuations if the system is deterministic and d 2 then singular potentials are also allowed for stochastic models with a smooth interaction we need d 4 in order to develop some prerequisites for the theory of hydrodynamical fluctuations in equilibrium we investigate smoothness of the markov semigroup and describe some properties of its generator 0 introduction the purpose of this paper is to study existence and regularity properties of solutions to the following infinite system of stochastic differential equations consider a countable set s of d dimensional particles suspended in a fluid where the interaction is given by a pair potential u rgd 00 oo in a quasi microscopic description of such systems the effect of collisions with the particles of the fluid can be represented by uncorrelated stochastic forces and the soft resistance of the liquid medium reduces the order of the equations of motion from two to one see 13 23 and 24 for some further references configurations of these systems are countable subsets of rd such that any bounded domain contains a finite number of points only particles of a configuration will be identified by labelling points by elements of s thus a

scaling limits of weakly asymmetric interfaces we consider three models of evolving interfaces intimately related to the weakly asymmetric simple exclusion process with n particles on a finite lattice of 2n sites our model 1 defines an evolving bridge on 0 1 our model 1 w an evolving excursion on 0 1 while our model 2 consists of an evolving pair of non crossing bridges on 0 1 based on the observation that the invariant measures of the dynamics depend on the area under or between the interface s we characterise the scaling limits of the invariant measures when the asymmetry of the exclusion process scales like n frac 3 2 then we show that the scaling limits of the dynamics themselves are expressed in terms of variants of the stochastic heat equation in particular in model 1 w we obtain the well studied reflected stochastic heat equation introduced by nualart and pardoux probab theory relat fields 93 1 77 89 1992