cores for feller semigroups with an invariant measure abstract let a i j 1 n a i j x d i j i 1 n b i x d i be an elliptic differential operator with unbounded coefficients on r n and assume that the associated feller semigroup t t t 0 has an invariant measure then t t t 0 extends to a strongly continuous semigroup t p t t 0 on l p l p r n for every 1 p we prove that under mild conditions on the coefficients of a the space of test functions c c r n is a core for the generator a p d p of t p t t 0 in l p for 1 p

kernel estimates for nonautonomous kolmogorov equations abstract using time dependent lyapunov functions we prove pointwise upper bounds for the heat kernels of some nonautonomous kolmogorov operators with possibly unbounded drift and diffusion coefficients

elliptic operators on rd with unbounded coefficients u t t x 1 2 tr c x du t x f x du t x 1 2 u 0 x x if the coefficients c and f are continuous and bounded and c is nondegenerate then a satisfactory theory of both equations is available see 18 15 the situation changes dramatically when the coefficients f and c are unbounded in this case even uniqueness of solutions becomes problematic as shown by the example 2 xxx see 20 the main goal of this paper is to study equations 1 1 and 1 2 in case of unbounded f and c we will consider well posedness of problems 1 1 doi 10 1006 jdeq 2000 3866 available online at http www idealibrary com on

regularity of invariant measures on finite and infinite dimensional spaces and applications abstract in this paper we prove new results on the regularity i e smoothness of measures solving the equation l 0 for operators of type l b on finite and infinite dimensional state spaces e in particular we settle a conjecture of i shigekawa in the situation where h is the gross laplacian e h is an abstract wiener space and b id e v where v takes values in the cameron martin space h using gross logarithmic sobolev inequality in an essential way we show that is always absolutely continuous w r t the gaussian measure and that the square root of the density is in the malliavin test function space of order 1 in l 2 furthermore we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for l 0 these include results on the inverse problem i e we give conditions ensuring that b is the vector logarithmic derivative of a measure we also prove necessary and sufficient conditions for to be symmetrizing i e l is symmetric on l 2 finally a substantial part of this work is devoted to the uniqueness of symmetrizing measures for l we characterize the cases where we have uniqueness by the irreducibility of the associated classical dirichlet forms

invariant measure for the stochastic navier stokes equations in unbounded 2d domains building upon a recent work by two of the authors and j seidler on bwbw feller property for stochastic nonlinear beam and wave equations we prove the existence of an invariant measure to stochastic 2 d navier stokes with multiplicative noise equations in unbounded domains this answers an open question left after the first author and y li proved a corresponding result in the case of an additive noise

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

the navier stokes limit of the boltzmann equation for bounded collision kernels the present work establishes a navier stokes limit for the boltzmann equation considered over the infinite spatial domain r3 appropriately scaled families of diperna lions renormalized solutions are shown to have fluctuations whose limit points in the w l1 topology are governed by leray solutions of the limiting navier stokes equations this completes the arguments in bardos golse levermore commun pure appl math 46 5 667 753 1993 for the steady case and in lions masmoudi arch ration mech anal 158 3 173 193 2001 for the time dependent case

cores for parabolic operators with unbounded coefficients abstract let a i j 1 n q i j s x d i j i 1 n b i s x d i be a family of elliptic differential operators with unbounded coefficients defined in r n 1 in m kunze l lorenzi a lunardi nonautonomous kolmogorov parabolic equations with unbounded coefficients trans amer math soc in press under suitable assumptions it has been proved that the operator g a d s generates a semigroup of positive contractions t p t in l p r n 1 for every 1 p where is an infinitesimally invariant measure of t p t here under some additional conditions on the growth of the coefficients of a which cover also some growths with an exponential rate at we provide two different cores for the infinitesimal generator g p of t p t in l p r n 1 for p 1 and we also give a partial characterization of d g p finally we extend the results so far obtained to the case when the coefficients of the operator a are t periodic with respect to the variable s for some t 0

a pregenerator for burgers equation forced by conservative noise we consider the stationarity of a burgers equation with an external random force of gradient type in one space dimension the expected stationary measure is the white noise measure on the space of tempered distributions as a consequence the nonlinearity of the formal equation u t u u x u xx x is ill defined introducing a pregenerator we can formulate a generalized martingale problem leading to a meaningful version of the formal equation which was an open problem

global properties of invariant measures abstract we study global regularity properties of invariant measures associated with second order differential operators in r n under suitable conditions we prove global boundedness of the density sobolev regularity a harnack inequality and pointwise upper and lower bounds

singular stochastic equations on hilbert spaces harnack inequalities for their transition semigroups we consider stochastic equations in hilbert spaces with singular drift in the framework of g da prato m rockner singular dissipative stochastic equations in hilbert spaces probab theory related fields 124 2 2002 261 303 we prove a harnack inequality in the sense of f y wang logarithmic sobolev inequalities on noncompact riemannian manifolds probab theory related fields 109 1997 417 424 for its transition semigroup and exploit its consequences in particular we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding kolmogorov operator has at most one infinitesimally invariant measure satisfying some mild integrability conditions finally we prove existence of such a measure for noncontinuous drifts

euclidean gibbs measures on loop lattices existence and a priori estimates we present a new method to prove existence and uniform a priori estimates for euclidean gibbs measures corresponding to quantum anharmonic crystals it is based first on the alternative characterization of gibbs measures in terms of their logarithmic derivatives through integration by parts formulas and second on the choice of appropriate lyapunov functionals

some properties of invariant measures of non symmetric dissipative stochastic systems we consider transition semigroups generated by stochastic partial differential equations with dissipative nonlinear terms we prove an integration by part formula and a logarithmic sobolev inequality for the invariant measure no symmetry or reversibility assumptions are made furthemore we prove some compactness results on the transition semigroup and on the embedding of the sobolev spaces based on the invariant measure we use these results to derive asymptotic properties for a stochastic reaction diffusion equation

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

l p uniqueness for elliptic operators with unbounded coefficients in r n let a be an elliptic operator with unbounded and sufficiently smooth coefficients and let be a sub invariant measure of the operator a in this paper we give sufficient conditions guaranteeing that the closure of the operator a cc rn generates a sub markovian strongly continuous semigroup of contractions in lp rn applications are given in the case when a is a generalized schrodinger operator

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

symmetry classes of variable coefficient nonlinear schrodinger equations a variable coefficient nonlinear schrodinger vcnls equation involving three arbitrary complex functions of space time in 1 1 dimensions is analysed from the point of view of its symmetries all equations of the type studied having non trivial lie point symmetry groups g are identified the symmetry group is shown to be at most five dimensional and only when the equation is equivalent to the nls equation itself or to a rather special complex ginzburg landau equation lie point transformations are used to obtain solutions of specific vcnls equations that should be of interest in nonlinear optics or other branches of physics

ergodicity ofl2 semigroups and extremality of gibbs states we extend the classical results of holley stroock on the characterization of extreme gibbs states for the ising model in terms of the irreducibility resp ergodicity of the corresponding glauber dynamics to the case of lattice systems with unbounded linear spin spaces we first develop a general framework to discuss questions of this type using classical dirichlet forms on infinite dimensional state spaces and their associated diffusions we then describe concrete applications to lattice models with polynomial interactions i e the discretep d models of euclidean quantum field theory in addition we prove the equivalence of extremality and shift ergodicity for tempered gibbs states of these models and also discuss this question in the general framework

weak solutions to the stochastic porous media equation via kolmogorov equations the degenerate case a stochastic version of the porous medium equation with coloured noise is studied the corresponding kolmogorov equation is solved in the space l 2 h where is an infinitesimally excessive measure then a weak solution is constructed

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

analysis and geometry on configuration spaces the gibbsian case using a natural riemannian geometry like structure on the configuration space over rd we prove that for a large class of potentials the corresponding canonical gibbs measures on 1 can be completely characterized by an integration by parts formula that is if is the gradient of the riemannian structure on one can define a corresponding divergence div such that the canonical gibbs measures are exactly those measures for which and div are dual operators onl2 one consequence is that for such the corresponding dirichlet forms e are defined in addition each of them is shown to be associated with a conservative diffusion process on with invariant measure the corresponding generators are extensions of the operator div the diffusions can be characterized in terms of a martingale problem and they can be considered as a brownian motion on perturbed by a singular drift another main result of this paper is the following if is a canonical gibbs measure then it is extreme or a pure phase if and only if the corresponding weak sobolev spacew1 2 on is irreducible as a consequence we prove that for extreme canonical gibbs measures the above mentioned diffusions are time ergodic in particular this holds for tempered grand canonical gibbs measures ruelle measures provided that the activity constant is small enough we also include a complete discussion of the free case i e 0 where the underlying space rdis even replaced by a riemannian manifoldx

group invariant solutions of the 2 1 dimensional cubic schrodinger equation we use lie point symmetries of the 2 1 dimensional cubic schrodinger equation to obtain new analytic solutions in a systematic manner we present an analysis of the reduced odes and in particular show that although the original equation is not integrable they typically can belong to the class of painleve type equations

absolutely continuous flows generated by sobolev class vector fields in finite and infinite dimensions abstract we prove the existence of the global flow ut generated by a vector field a from a sobolev class w1 1 on a finite or infinite dimensional space x with a measure provided is sufficiently smooth and that a a and a where a is the divergence with respect to are exponentially integrable in addition the measure is shown to be quasi invariant under ut in the case x r n and p dx where p w1 1loc r n is a locally uniformly positive probability density a sufficient condition is exp c a exp c a p p l1 for all c in the infinite dimensional case we get analogous results for measures differentiable along sufficiently many directions examples of measures which fit our framework important for applications are symmetric invariant measures of infinite dimensional diffusions and gibbs measures typically in both cases such measures are essentially non gaussian our result in infinite dimensions significantly extends previously studied cases where was a gaussian measure finally we study flows generated by vector fields whose values are not necessarily in the cameron martin space

from the q tensor flow for the liquid crystal to the harmonic map flow in this paper we consider the solutions of the relaxed q tensor flow in mathbb r 3 with small parameter epsilon we show that the limiting map is the so called harmonic map flow as a consequence we present a new proof for the global existence of a weak solution for the harmonic map flow in three dimensions as in 18 23 where the ginzburg landau approximation approach was used

kolmogorov equations in infinite dimensions well posedness and regularity of solutions with applications to stochastic generalized burgers equations we develop a new method to uniquely solve a large class of heat equations so called kolmogorov equations in infinitely many variables the equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper lyapunov type functions this way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally lipschitz drifts apart from general analytic interest the main motivation is to apply this to uniquely solve martingale problems in the sense of stroock varadhan given by stochastic partial differential equations from hydrodynamics such as the stochastic navier stokes equations in this paper this is done in the case of the stochastic generalized burgers equation uniqueness is shown in the sense of markov flows

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

ito type measure valued stochastic differential equations a class of ito type measure valued stochastic differential equations is studied on a locally compact polish space the sdes are driven by countably many brownian motions with interactions caused by the diffusion and the drift coefficients through countably many continuous functions explicit conditions are presented for the existence uniqueness and ergodicity of the solution

on nonuniqueness of solutions to elliptic equations for probability measures we consider the set of all probability measures on rd satisfying an elliptic equation l 0 in the weak worm we give sufficient conditions in order that this set contains at least two different elements we also construct new examples of nonuniqueness for such equations

large deviations for the invariant measure of a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the invariant measure of the non gaussian stochastic partial differential equation spde t v e v e f x v e e x v e ddot w tx here is a strongly elliptic second order operator with constant coefficients h dh xx h and the space variablex takes values on the unit circles 1 the functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic pde and in particular we require that 0 m m wherem andm are some finite positive constants the perturbationw is a brownian sheet it is well known that under some simple assumptions the solutionv 2 is ac k s 1 valued markov process for each 0 1 2 wherec s 1 is the banach space of real valued continuous functions ons 1 which are holder continuous of exponent we prove under some further natural assumptions onf and which imply that the zero element ofc s 1 is a globally exponentially stable critical point of the unperturbed equation t 0 0 f x 0 that e has a unique stationary distributionv k on c s 1 c k s 1 when the perturbation parameter e is small enough some further calculations show that as e tends to zero v k tends tov k 0 the point mass centered on the zero element ofc s 1 the main goal of this paper is to show that in factv k is governed by a large deviations principle ldp our starting point in establishing the ldp forv k is the ldp for the process e which has been shown in an earlier paper our methods of deriving the ldp forv k based on the ldp for e are slightly non standard compared to the corresponding proofs for finite dimensional stochastic differential equations since the state spacec s 1 is inherently infinite dimensional

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

variable coefficient nonlinear schrodinger equations with four dimensional symmetry groups and analysis of their solutions analytical solutions of variable coefficient nonlinear schrodinger equations having four dimensional symmetry groups which are in fact the next closest to the integrable ones occurring only when the lie symmetry group is five dimensional are obtained using two different tools the first tool is to use one dimensional subgroups of the full symmetry group to generate solutions from those of the reduced ordinary differential equations namely group invariant solutions the other is by truncation in their painleve expansions

relative entropy and mixing properties of infinite dimensional diffusions let be a diffusion process taking values on the infinite dimensional space t z where t is the circle and with components satisfying the equations d i i dw i b i dt for some coefficients i and b i i z suppose we have an initial distribution and a sequence of times t n such that lim n s tn exists where s t is the semi group of the process we prove that if i and b i are bounded of finite range have uniformly bounded second order partial derivatives and inf i i 0 then is invariant

on the ergodic properties of glauber dynamics we show that if there is an infinite volume gibbs measure which satisfies a logarithmic sobolev inequality with local coefficients of moderate growth then the corresponding stochastic dynamics decays to equilibrium exponentially fast in the uniform norm

hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic ising models abstract we describe some results relating the spectral gap and logarithmic sobolev constants we work initially in the context of symmetric diffusions on a finite dimensional manifold and later apply our results to the analysis of certain infinite systems of mildly interacting diffusions

global existence of weak solutions of the nematic liquid crystal flow in dimension three for any bounded smooth domain 3 or 3 we establish the global existence of a weak solution u d 0 3 s2 of the initial boundary value or the cauchy problem of the simplified ericksen leslie system llf modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary or cauchy data u0 d0 h h1 s2 with d0 s 2 the upper hemisphere furthermore u d satisfies the global energy inequality 2016 wiley periodicals inc

a comparative analysis of painleve lax pair and similarity transformation methods in obtaining the integrability conditions of nonlinear schrodinger equations we derive the integrability conditions of nonautonomous nonlinear schrodinger equations using the lax pair and similarity transformation methods we present a comparative analysis of these integrability conditions with those of the painleve method we show that while the painleve integrability conditions restrict the dispersion nonlinearity and dissipation gain coefficients to be space independent and the external potential to be only a quadratic function of position the lax pair and the similarity transformation methods allow for space dependent coefficients and an external potential that is not restricted to the quadratic form the integrability conditions of the painleve method are retrieved as a special case of our general integrability conditions we also derive the integrability conditions of nonautonomous nonlinear schrodinger equations for two and three spacial dimensions

elliptic equations for measures on infinite dimensional spaces and applications we introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces this concept allows one to consider equations whose coefficients are not globally integrable by using a suitably extended lyapunov function technique we derive a priori estimates for the solutions of such equations and prove new existence results as an application we consider stochastic burgers reaction diffusion and navier stokes equations and investigate the elliptic equations for the corresponding invariant measures our general theorems yield a priori estimates and existence results for such elliptic equations we also obtain moment estimates for gibbs distributions and prove an existence result applicable to a wide class of models

uniqueness of generalized schrodinger operators part ii abstract we prove that for h 1 2 loc r d dx 0 dx a e the generalized schrodinger operator s 2 1 dom s c 0 r d has exactly one self adjoint extension on l 2 r d 2 dx which generates a sub markovian semigroup on l 2 r d 2 dx this is based on our previous work where a necessary and sufficient condition on for this to hold was proved but which was only verified to always hold if d 1 we also prove a corresponding result where r d is replaced by an infinite dimensional space and the lebesgue measure by some gaussian measure whose covariance operator has a discrete spectrum

the logarithmic sobolev inequality for continuous spin systems on a lattice abstract in this article we give cf theorem 1 6 a sufficient condition on a local specification for the associated gibbs state to satisfy a logarithmic sobolev inequality in the final section cf theorem 3 10 we relate our condition to the type of mixing condition introduced by dobrushin and shlosman in connection with their work on complete analyticity

sharp upper bounds for the density of some invariant measures we prove sharp upper bounds for invariant measures of markov processes in n associated with second order elliptic differential operators with unbounded coefficients

invariant measures for stochastic nonlinear beam and wave equations existence of an invariant measure for a stochastic extensible beam equation and for a stochastic damped wave equation with polynomial nonlinearities is proved it is shown first that the corresponding transition semigroups map the space of all bounded sequentially weakly continuous functions on the state space into itself and then by a lyapunov functions approach solutions bounded in probability are found

exact adaptive pointwise drift estimation for multidimensional ergodic diffusions the problem of pointwise adaptive estimation of the drift coefficient of a multivariate diffusion process is investigated we propose an estimator which is sharp adaptive on scales of sobolev smoothness classes the analysis of the exact risk asymptotics allows to identify the impact of the dimension and other influencing values such as the geometry of the diffusion coefficient of the prototypical drift estimation problem for a large class of multidimensional diffusion processes we further sketch generalizations of our results to arbitrary diffusions satisfying suitable bernstein type inequalities

the equivalence of the logarithmic sobolev inequality and the dobrushin shlosman mixing condition given a finite range lattice gas with a compact continuous spin space it is shown cf theorem 1 2 that a uniform logarithmic sobolev inequality cf 1 4 holds if and only if the dobrushin shlosman mixing condition cf 1 5 holds as a consequence of our considerations we also show cf theorems 3 2 and 3 6 that these conditions are equivalent to a statement about the uniform rate at which the associated glauber dynamics tends to equilibrium in this same direction we show cf theorem 3 19 that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the glauber dynamics

entropy reversible diffusion processes and markov uniqueness abstract consider a symmetric bilinear form e defined on c c r d by formula in this paper we study the stochastic process associated with the smallest closed markovian extension of e c c and give a new proof of markov uniqueness i e the uniqueness of a closed markovian extension based on purely probabilistic arguments we also give another purely analytic one as a consequence we show that all invariant measures are reversible provided they are of finite energy the problem of uniqueness of such measures is also partially solved

the small deborah number limit of the doi onsager equation to the ericksen leslie equation we present a rigorous derivation of the ericksen leslie equation starting from the doi onsager equation as in the fluid dynamic limit of the boltzmann equation we first make the hilbert expansion for the solution of the doi onsager equation the existence of the hilbert expansion is connected to an open question whether the energy of the ericksen leslie equation is dissipated we show that the energy is dissipated for the ericksen leslie equation derived from the doi onsager equation the most difficult step is to prove a uniform bound for the remainder in the hilbert expansion this question is connected to the spectral stability of the linearized doi onsager operator around a critical point by introducing two important auxiliary operators the detailed spectral information is obtained for the linearized operator around all critical points however these are not enough to justify the small deborah number limit for the inhomogeneous doi onsager equation since the elastic stress in the velocity equation is also strongly singular for this we need to establish a precise lower bound for a bilinear form associated with the linearized operator in the bilinear form the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated we find a coordinate transform and introduce a five dimensional space called the maier saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square however the lower bound is very weak for the part inside the maier saupe space in order to apply them to the error estimates we have to analyze the structure of the singular terms and introduce a suitable energy functional

compactness and the maximal gibbs state for random gibbs fields on a lattice we prove for a general class of gibbsian random field on that the set of tempered gibbs states is compact this class contains the euclidean random fields moreover if the interaction is attractive there is a unique minimal and maximal gibbs state and are unique translation invariant ant and have the global markov property we also prove that uniqueness of the tempered gibbs state is equivalent to the magnetizationsm qx being equal which is true if the pressure is differentiable

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

the log sobolev inequality for unbounded spin systems abstract in this note we would like to give a proof of log sobolev inequality for un bounded spin systems with weaker assumptions on the potentials than previously obtained

first order perturbations of dirichlet operators existence and uniqueness we study perturbations of typeb of dirichlet operators l0 d l0 associated with dirichlet forms of typee0 u v 12 u v hd onl2 e whereeis a finite or infinite dimensional banach space herehdenotes a hilbert space densely and continuously embedded ine assuming quasi regularity of e0 d e0 we show that there always exists a closed extension oflu l0u b u hthat generates a sub markovianc0 semigroup of contractions onl2 e resp l1 e ifb l2 e h and b u hd 0 u 0 ifdis an appropriate core for l0 d l0 we show that there is only one closed extension of l d inl1 e generating a strongly continuous semigroup in particular we apply our results to operators of type h b where hdenotes the gross laplacian on an abstract wiener space e h andb ide v wherevtakes values in the cameron martin spaceh

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

random perturbations of reaction diffusion equations the quasi deterministic approximation on considere des champs aleatoires u e t x u 1 e t x u n e t x definis comme les solutions d un systeme d equations aux derivees partielles u k e t l k u k e f k x u 1 e u n e e k t x les l k sont des operateurs elliptiques lineaires d ordre 2 k sont des champs gaussiens bruit blanc

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

on some canonical classes of cubic quintic nonlinear schrodinger equations abstract in this paper we bring into attention variable coefficient cubic quintic nonlinear schrodinger equations which admit lie symmetry algebras of dimension four within this family we obtain the reductions of canonical equations of nonequivalent classes to ordinary differential equations using tools of lie theory painleve integrability of these reduced equations is investigated exact solutions through truncated painleve expansions are achieved in some cases one of these solutions a conformal group invariant one exhibits blow up behavior in finite time in l p l norm and in distributional sense

ergodicity for the stochastic dynamics of quasi invariant measures with applications to gibbs states the convex set maof quasi invariant measures on a locally convex spaceewith given shift radon nikodym derivatives i e cocycles a atk k k0 t ris analyzed the extreme points of maare characterized and proved to be non empty a specification of lattice type is constructed so that macoincides with the set of the corresponding gibbs states as a consequence via a well known method due to dynkin and follmer a unique representation of an arbitrary element in main terms of extreme ones is derived furthermore the corresponding classical dirichlet forms e d e and their associated semigroups t t t 0onl2 e are discussed under a mild positivity condition it is shown that mais extreme if and only if e d e is irreducible or equivalently t t t 0is ergodic this implies time ergodicity of associated diffusions applications to gibbs states of classical and quantum lattice models as well as those occuring in euclidean quantum field theory are presented in particular it is proved that the stochastic quantization of a guerra rosen simon gibbs state on d r2 ininfinite volumewith polynomial interaction is ergodic if the gibbs state is extreme i e is a pure phase which solves a long standing open problem

small noise asymptotics for invariant densities for a class of diffusions a control theoretic view abstract we consider multi dimensional nondegenerate diffusions with invariant densities with the diffusion matrix scaled by a small 0 the o d e limit corresponding to 0 is assumed to have the origin as its unique globally asymptotically stable equilibrium using control theoretic methods we show that in the 0 limit the invariant density has the form exp w x 2 where the w is characterized as the optimal cost of a deterministic control problem this generalizes an earlier work of sheu extension to multiple equilibria is also given

regularity of invariant measures the case of non constant diffusion part we prove regularity i e smoothness of measures on rdsatisfying the equationl 0 wherelis an operator of typelu tr au b u hereais a lipschitz continuous uniformly elliptic matrix valued map andbis merely square integrable we also treat a class of corresponding infinite dimensional cases where rdis replaced by a locally convex topological vector spacex in this cases is proved to be absolutely continuous w r t a gaussian measure onxand the square root of the radon nikodym density belongs to the malliavin test function space d2 1

the small deborah number limit of the doi onsager equation without hydrodynamics abstract we study the small deborah number limit of the doi onsager equation in the case when hydrodynamic effects are neglected this is a smoluchowski type equation that describes the dynamics of nematic liquid crystals at a molecular level by characterizing the evolution of a number density function depending upon both particle position x r d d 2 3 and orientation vector m s 2 the unit sphere we prove that when the deborah number tends to zero the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into s 2 this flow is a special case of the gradient flow of the well known oseen frank energy functional for nematic liquid crystals the key ingredient is to show the strong compactness of the family of number density functions the proof relies on the strong compactness of the corresponding q tensor namely the second moment a detailed analysis of the linearized operator near the limit local equilibrium distribution and an energy dissipation estimate

estimates for invariant probability measures of degenerate spdes with singular and path dependent drifts in terms of a nice reference probability measure integrability conditions on the path dependent drift are presented for infinite dimensional degenerate pdes to have regular positive solutions to this end the corresponding stochastic partial differential equations are proved to possess the weak existence and uniqueness of solutions as well as the existence uniqueness and entropy estimates of invariant probability measures when the reference measure satisfies the log sobolev inequality sobolev estimates are derived for the density of invariant probability measures some results are new even for non degenerate sdes with path independent drifts the main results are applied to nonlinear functional spdes and degenerate functional sdes spdes

reversibility in infinite hamiltonian systems with conservative noise the set of stationary measures of an infinite hamiltonian system with noise is investigated the model consists of particles moving in r 3 with bounded velocities and subject to a noise that does not violate the classical laws of conservation see ovy following lo we assume that the noise has also a finite radius of interaction and prove that translation invariant stationary states of finite specific entropy are reversible with respect to the stochastic component of the evolution therefore the results of lo imply that such invariant measures are superpositions of gibbs states

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

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

an analytic approach to existence and uniqueness for martingale problems in infinite dimensions we prove existence and uniqueness for a class of martingale problems in a hilbert space we solve the associated kolmogorov equation and prove that the corresponding semigroup is determined by a kernel of measures if a schauder type regularity is satisfied