ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

ergodicity of the finite dimensional approximation of the 3d navier stokes equations forced by a degenerate noise we prove ergodicity of the finite dimensional approximations of the three dimen sional navier stokes equations driven by a random force the forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity the convergence rate to the unique invariant measure is shown to be exponential the uniqueness of statistical steady states for the navier stokes equations is a less known but nevertheless important problem in the mathematical theory of turbulence the question is completely open in dimension three mainly because due to the lack of uniqueness of the equations there is no way yet to give meaning to the mathematical objects involved in the subject in the present paper the property of ergodicity is proved for the finite dimensional approxi mations of the three dimensional navier stokes equations driven by a random force the same problem has been solved in two dimensions by e and mattingly 2 such result can have a qualitative interest for the statistical behaviour of an incompressible fluid indeed if the kolmogorov theory of turbulence is taken into account one can believe that the cascade of energy responsible of the transport of the energy through the scales is effective in the inertial range so that at smaller scales only the dissipation ends up to be relevant hence the long time statistical properties of the fluid can be sufficiently depicted by the low modes of the velocity field in some sense if the ultraviolet cut off is sufficiently large in order to capture all the important modes the corresponding invariant measure gives the real behaviour of the fluid in view of these considerations the conclusions of the paper can give also both a hint and a possible starting point for the analysis of the infinite dimensional case we consider a finite dimensional truncation of the three dimensional navier stokes equations driven by a random force with periodic boundary conditions the proof of ergodicity is classical and it is developed in two steps firstly we prove that the transition probability densities are regular by checking that the diffusion operator is hypoelliptic the hormander condition then we show that the markov process is irreducible in the sense that each open set is visited with positive probability at each time for this aim we study the associated control problem see section 6 irreducibility for the infinite dimensional equations was firstly proved by flandoli 4 under the assumption that the noise acts on all modes

limit theorems for markov chains by the symmetrization method abstract let p be a markov operator with invariant probability m ergodic on l 2 s m and let w n n 0 be the markov chain with state space s and transition probability p on the space of trajectories p m with initial distribution m following wu and olla we define the symmetrized operator p s p p 2 and analyze the linear manifold h 1 i p s l 2 s m we obtain for real f h 1 an explicit forward backward martingale decomposition with a coboundary remainder for such f we also obtain some maximal inequalities for s n f k 0 n f w k related to the law of iterated logarithm we prove an almost sure central limit theorem for f h 1 when p is normal in l 2 s m or when p satisfies the sector condition we characterize the sector condition by the numerical range of p on the complex l 2 s m being in a sector with vertex at 1 we then show that if p has a real normal dilation which satisfies the sector condition then h 1 i p l 2 s m we use our approach to prove that p is l 2 uniformly ergodic if and only if it satisfies the discrete poincare s inequality

ergodic and mixing properties of the boussinesq equations with a degenerate random forcing we establish the existence uniqueness and attraction properties of an ergodic invariant measure for the boussinesq equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales the central challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system towards this aim we encounter a lie bracket structure in the associated vector fields with a complicated dependence on solutions this leads us to develop a novel hormander type condition for infinite dimensional systems demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the malliavin covariance matrix

rosenthal type inequalities for the maximum of partial sums of stationary processes and examples the aim of this paper is to propose new rosenthal type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations as in the recent results by peligrad et al proc amer math soc 135 2007 541 550 and rio j theoret probab 22 2009 146 163 the estimates of the moments are expressed in terms of the norms of projections of partial sums the proofs of the results are essentially based on a new maximal inequality generalizing the doob maximal inequality for martingales and dyadic induction various applications are also provided

ergodicity of the 2d navier stokes equations with random forcing we consider the navier stokes equation on a two dimensional torus with a random force acting at discrete times and analytic in space for arbitrarily small viscosity coefficient we prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time

ergodicity for the navier stokes equation with degenerate random forcing finite dimensional approximation we study galerkin truncations of the two dimensional navier stokes equation under degenerate large scale stochastic forcing we identify the minimal set of modes that has to be forced in order for the system to be ergodic our results rely heavily on the structure of the nonlinearity c 2001 john wiley sons inc

invariance principles under the maxwell woodroofe condition in banach spaces we prove that for adapted stationary processes the so called maxwell woodroofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense that result actually holds in the context of banach valued stationary processes including the case of lplp valued random variables with 1 p 1 p in this setting we also prove the weak invariance principle hence generalizing a result of peligrad and utev ann probab 33 2005 798 815 the proofs make use of a new maximal inequality and of approximation by martingales for which some of our results are also new

random walk in markovian environment we prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on z d we assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is markovian with strong spatial and temporal mixing properties

central limit theorems for additive functionals of markov chains central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic markov chain say s n g x 1 g x n where e g x 1 0 and e g x 1 2 the conditions imposed restrict the moments of g and the growth of the conditional means e s n x 1 no other restrictions on the dependence structure of the chain are required when specialized to shift processes the conditions are implied by simple integral tests involving g

malliavin calculus for infinite dimensional systems with additive noise abstract we consider an infinite dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of wiener processes by studying how randomness is spread by the dynamics we develop in this setting a partial counterpart of hormander s classical theory of hypoelliptic operators we study the distributions of finite dimensional projections of the solutions and give conditions that provide existence and smoothness of densities of these distributions with respect to the lebesgue measure we also apply our results to concrete spdes such as a stochastic reaction diffusion equation and the stochastic 2d navier stokes system

persistent random walks in random environment weak convergence of a class of functionals of prwre is proved as a consequence clt is obtained for the normed trajectory

law of the iterated logarithm for stationary processes there has been recent interest in the conditional central limit question for strictly stationary ergodic processes x 1 x 0 x 1 whose partial sums s n x 1 x n are of the form s n m n r n where m n is a square integrable martingale with stationary increments and r n is a remainder term for which e r n 2 o n here we explore the law of the iterated logarithm lil for the same class of processes letting denote the norm in l 2 p a sufficient condition for the partial sums of a stationary process to have the form s n m n r n is that n 3 2 e sn x 0 x 1 be summable a sufficient condition for the lil is only slightly stronger requiring n 3 2 log 3 2 n e sn x 0 x 1 to be summable as a by product of our main result we obtain an improved statement of the conditional central limit theorem invariance principles are obtained as well

distortion of effects caused by indirect confounding summary undetected confounding may severely distort the effect of an explanatory variable on a response variable as defined by a stepwise data generating process the best known type of distortion which we call direct confounding arises from an unobserved explanatory variable common to a response and its main explanatory variable of interest it is relevant mainly for observational studies since it is avoided by successful randomization by contrast indirect confounding which we identify in this paper is an issue also for intervention studies for general stepwise generating processes we provide matrix and graphical criteria to decide which types of distortion may be present when they are absent and how they are avoided we then turn to linear systems without other types of distortion but with indirect confounding for such systems the magnitude of distortion in a least squares regression coefficient is derived and shown to be estimable so that it becomes possible to recover the effect of the generating process from the distorted coefficient

central limit theorem for stationary linear processes we establish the central limit theorem for linear processes with dependent innovations including martingales and mixingale type of assumptions as defined in mcleish ann probab 5 1977 616 621 and motivated by gordin soviet math dokl 10 1969 1174 1176 in doing so we shall preserve the generality of the coefficients including the long range dependence case and we shall express the variance of partial sums in a form easy to apply ergodicity is not required

lie backlund transformations for the massive thirring model second and third order lie backlund transformations of the massive thirring model are computed by symbolic integration

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

an almost sure invariance principle for random walks in a space time random environment we consider a discrete time random walk in a space time i i d random environment we use a martingale approach to show that the walk is diffusive in almost every fixed environment we improve on existing results by proving an invariance principle and considering environments with an l2 averaged drift we also state an a s invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean under an ergodic invariant measure for the environment chain

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

quenched invariance principle for multidimensional ballistic random walk in a random environment with a forbidden direction we consider a ballistic random walk in an i i d random environment that does not allow retreating in a certain fixed direction we prove an invariance principle functional central limit theorem under almost every fixed environment the assumptions are nonnestling at least two spatial dimensions and a 2 e moment for the step of the walk uniformly in the environment the main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively

heavy tail phenomenon and convergence to stable laws for iterated lipschitz maps we consider the markov chain x n x n 0 infty on mathbb r d defined by the stochastic recursion x n x psi theta n x n 1 x starting at x in mathbb r d where 1 2 are i i d random variables taking their values in a metric space theta mathfrak r and psi theta n mathbb r d mapsto mathbb r d are lipschitz maps assume that the markov chain has a unique stationary measure under appropriate assumptions on psi theta n we will show that the measure has a heavy tail with the exponent 0 i e nu x in mathbb r d x t asymp t alpha using this result we show that properly normalized birkhoff sums s n x sum k 1 n x k x converge in law to an stable law for alpha in 0 2

almost sure enhanced dissipation and uniform in diffusivity exponential mixing for advection diffusion by stochastic navier stokes we study the mixing and dissipation properties of the advection diffusion equation with diffusivity 0 kappa ll 1 and advection by a class of random velocity fields on mathbb t d d 2 3 including solutions of the 2d navier stokes equations forced by sufficiently regular in space non degenerate white in time noise we prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity kappa namely that there is a deterministic exponential rate independent of kappa such that all mean zero h 1 initial data decays exponentially fast in h 1 at this rate with probability one this implies almost sure enhanced dissipation in l 2 specifically that there is a deterministic uniform in kappa exponential decay in l 2 after time t rsim left log kappa right both the mathcal o left log kappa right time scale and the uniform in kappa exponential mixing are optimal for lipschitz velocity fields this work is also a major step in our program on scalar mixing and lagrangian chaos necessary for a rigorous proof of the batchelor power spectrum of passive scalar turbulence

ergodicity of 2d navier stokes equations with random forcing and large viscosity the stochastically forced two dimensional incompressable navier stokes equations are shown to possess an unique invariant measure if the viscosity is taken large enough this result follows from a stronger result showing that at high viscosity there is a unique stationary solution which attracts solutions started from arbitrary initial conditions that is to say the system has a trivial random attractor along the way results controling the expectation and averaging time of the energy and enstrophy are given

the emergence of large scale coherent structure under small scale random bombardments we provide mathematical justification of the emergence of large scale coherent structure in a two dimensional fluid system under small scale random bombardments with small forcing and appropriate scaling assumptions the analysis shows that the large scale structure emerging out of the small scale random forcing is not the one predicted by equilibrium statistical mechanics but the error is very small which explains earlier successful prediction of the large scale structure based on equilibrium statistical mechanics 2005 wiley periodicals inc

uniqueness of the invariant measure for a stochastic pde driven by degenerate noise we consider the stochastic ginzburg landau equation in a bounded domain we assume the stochastic forcing acts only on high spatial frequencies the low lying frequencies are then only connected to this forcing through the non linear cubic term of the ginzburg landau equation under these assumptions we show that the stochastic pde has a unique invariant measure the techniques of proof combine a controllability argument for thelow lying frequencies with an infinite dimensional version of the malliavin calculus to show positivity and regularity of the invariant measure this then implies the uniqueness of that measure

symmetries of the massive thirring model for a hamiltonian system every non hamiltonian symmetry gives rise to a recursion operator for symmetries using this method two recursion operators for symmetries of the massive thirring model are constructed the structure of the lie algebra of symmetries generated by these operators is given

an invariance principle for maps with polynomial decay of correlations we give a general method of deriving statistical limit theorems such as the central limit theorem and its functional version in the setting of ergodic measure preserving transformations this method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations

martingale approximations for sums of stationary processes approximations to sums of stationary and ergodic sequences by martingales are investigated necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained it is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating martingales satisfy a lindeberg feller condition using the explicit construction of the approximating martingales a central limit theorem is derived for the sample means of linear processes the conditions are not sufficient for the functional version of the central limit theorem this is shown by an example and a slightly stronger sufficient condition is given

a dynamical polynomial chaos approach for long time evolution of spdes abstract we propose a dynamical generalized polynomial chaos dgpc method to solve time dependent stochastic partial differential equations spdes with white noise forcing the long time simulation of spde solutions by polynomial chaos pc methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time exploiting the markovian property of white noise dgpc 1 implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise the dimension of the representation is kept minimal by application of a karhunen loeve kl expansion using frequent restarts and low degree polynomials on sparse multi index sets the method allows us to perform long time simulations including the calculation of invariant measures for systems which possess one we apply the method to the numerical simulation of stochastic burgers and navier stokes equations with white noise forcing our method also allows us to incorporate time independent random coefficients such as a random viscosity we propose several numerical simulations and show that the algorithm compares favorably with standard monte carlo methods

matrix representations and independencies in directed acyclic graphs for a directed acyclic graph there are two known criteria to decide whether any specific conditional independence statement is implied for all distributions factorized according to the given graph both criteria are based on special types of path in graphs they are called separation criteria because independence holds whenever the conditioning set is a separating set in a graph theoretical sense we introduce and discuss an alternative approach using binary matrix representations of graphs in which zeros indicate independence statements a matrix condition is shown to give a new path criterion for separation and to be equivalent to each of the previous two path criteria institute of mathematical statistics 2009

creating and annihilating lie backlund transformations of the federbush model using software developed for symbolic integration infinitesimal symmetries conserved currents and first and second order lie backlund transformations for the federbush model are established moreover four x t dependent lie backlund transformations are constructed leading to infinite hierarchies of lie backlund transformations

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

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

joint response graphs and separation induced by triangular systems summary we consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions the independences are captured by missing edges in a directed graph a matrix form of such a graph called the generating edge matrix is triangular so the distributions that are generated over such graphs are called triangular systems we study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models i e for alternative recursive factorizations of the given density using joint conditional distributions for this we introduce families of linear triangular equations which do not require assumptions of distributional form the strength of the associations that are implied by such linear families for chain graph models is derived the edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix it is shown how induced independences and dependences can be studied by graphs by edge matrix calculations and via the properties of densities some ways of using the results are illustrated

limits of random differential equations on manifolds consider a family of random ordinary differential equations on a manifold driven by vector fields of the form sum ky k alpha k z t epsilon omega where y k are vector fields epsilon is a positive number z t epsilon is a 1 over epsilon mathcal l 0 diffusion process taking values in possibly a different manifold alpha k are annihilators of mathrm ker mathcal l 0 under hormander type conditions on mathcal l 0 we prove that as epsilon approaches zero the stochastic processes y t over epsilon epsilon converge weakly and in the wasserstein topologies we describe this limit and give an upper bound for the rate of the convergence

exponential mixing properties of stochastic pdes through asymptotic coupling we consider parabolic stochastic partial differential equations driven by white noise in time we prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions these conditions amount essentially to the fact that the equation transmits the noise to all its determining modes several examples are investigated including some where the noise does not act on every determining mode directly

on ergodicity of some markov processes we formulate a criterion for the existence and uniqueness of an invariant measure for a markov process taking values in a polish phase space in addition weak ergodicity that is the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution is established the principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity the latter is called the e property the general result is applied to solutions of some stochastic evolution equations in hilbert spaces as an example we consider an evolution equation whose solution describes the lagrangian observations of the velocity field in the passive tracer model the weak mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer

generalized couplings and convergence of transition probabilities we provide sufficient conditions for the uniqueness of an invariant measure of a markov process as well as for the weak convergence of transition probabilities to the invariant measure our conditions are formulated in terms of generalized couplings we apply our results to several spdes for which unique ergodicity has been proven in a recent paper by glatt holtz mattingly and richards and show that under essentially the same assumptions the weak convergence of transition probabilities actually holds true

numerical simulation of polynomial speed convergence phenomenon we provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for markov processes the hybrid method that we introduce is based on the coupling technique and renewal theory we propose to replace some estimates in classical results about the ergodicity of markov processes by numerical simulations when the corresponding analytical proof is difficult after that all remaining conclusions can be derived from rigorous analysis then we apply our results to seek numerical justification for the ergodicity of two 1d microscopic heat conduction models the mixing rate of these two models are expected to be polynomial but very difficult to prove in both examples our numerical results match the expected polynomial mixing rate well

a coupling approach to randomly forced nonlinear pde s ii we consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure the result obtained applies in particular to the 2d navier stokes system and multidimensional complex ginzburg landau equation with random kick force

convergence of numerical time averaging and stationary measures via poisson equations numerical approximation of the long time behavior of a stochastic differential equation sde is considered error estimates for time averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the sde the error analysis is based on using an associated poisson equation for the underlying sde the main advantages of this approach are its simplicity and universality it works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables and for hypoelliptic sdes to simplify the exposition we consider only the case where the state space of the sde is a torus and we study only smooth test functions however we anticipate that the approach can be applied more widely an analogy between our approach and stein s method is indicated some practical implications of the results are discussed

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

an efficient second order in time scheme for approximating long time statistical properties of the two dimensional navier stokes equations we investigate the long time behavior of the following efficient second order in time scheme for the 2d navier stokes equations in a periodic box begin array ll frac 3 omega n 1 4 omega n omega n 1 2k nabla perp 2 psi n psi n 1 cdot nabla 2 omega n omega n 1 nu delta omega n 1 f n 1 quad delta psi n omega n end array the scheme is a combination of a 2nd order in time backward differentiation and a particular explicit adams bashforth treatment of the advection term therefore only a linear constant coefficient poisson solver is needed at each time step we prove uniform in time bounds on this scheme in dot l 2 dot h 1 per and dot h 2 per provided that the time step is sufficiently small these time uniform estimates further lead to the convergence of long time statistics stationary statistical properties of the scheme to that of the nse itself at vanishing time step

ergodicity and exponential mixing of the real ginzburg landau equation with a degenerate noise abstract in this paper we establish the existence uniqueness and attraction properties of an invariant measure for the real ginzburg landau equation in the presence of a degenerate stochastic forcing acting only in four directions the main challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system to achieve this we propose a condition which only requires four noises

homogenization of periodic linear degenerate pdes it is well known under the name of periodic homogenization that under a centering condition of the drift a periodic diffusion process on r d converges under diffusive rescaling to a d dimensional brownian motion existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion in this paper we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set as a consequence it is no longer the case that the effective diffusivity matrix is necessarily non degenerate it turns out that provided that some very weak regularity conditions are met the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion in particular this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank we also discuss the application of our results to the homogenization of a class of elliptic and parabolic pdes c 2008 elsevier inc all rights reserved

a new maximal inequality and invariance principle for stationary sequences we derive a new maximal inequality for stationary sequences under a martingale type condition introduced by maxwell and woodroofe ann probab 28 2000 713 724 then we apply it to establish the donsker invariance principle for this class of stationary sequences a markov chain example is given in order to show the optimality of the conditions imposed

a strong pair correlation bound implies the clt for sinai billiards we investigate the possibility of proving the central limit theorem clt for dynamical systems using only information on pair correlations a strong bound on multiple correlations is known to imply the clt chernov and markarian in chaotic billiards 2006 in chernov s paper j stat phys 122 6 2006 such a bound is derived for dynamically holder continuous observables of dispersing billiards here we weaken the regularity assumption and subsequently show that the bound on multiple correlations follows directly from the bound on pair correlations thus a strong bound on pair correlations alone implies the clt for a wider class of observables the result is extended to anosov diffeomorphisms in any dimension some non invertible maps are also considered

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

central limit theorems for mixing sequences of random variables under minimal conditions soit xj j 1 une suite strictement stationnaire de variables aleatoires de moyenne nulle de variance finie et satisfaisant une condition de melange forte soit s n la nieme somme partielle et supposons var s n variant regulierement et d ordre 1 on demontre que si s n var s n 1 2 ne converge pas vers 0 en l 1 alors xj j 1 est dans le domaine d attraction partielle d une loi gaussienne

a useful version of the central limit theorem for a general class of markov chains abstract in the paper we propose certain conditions relatively easy to verify which ensure the central limit theorem for some general class of markov chains to justify the usefulness of our criterion we further verify it for a particular discrete time markov dynamical system from the application point of view the examined system provides a useful tool in analysing the stochastic dynamics of gene expression in prokaryotes

the central limit theorem for markov chains started at a point the aim of this paper is to prove a central limit theorem and an invariance principle for an additive functional of an ergodic markov chain on a general state space with respect to the law of the chain started at a point no irreducibility assumption nor mixing conditions are imposed the only assumption bears on the growth of the l 2 norms of the ergodic sums for the function generating the additive functional which must be with the result holds almost surely with respect to the invariant probability of the chain

ergodic theory of infinite dimensional systems with applications to dissipative parabolic pdes we consider a class of randomly perturbed dynamical systems satisfying conditions which reflect the properties of general nonlinear dissipative parabolic pdes results on invariant measures and their exponential mixing properties are proved and applications to 2d navier stokes systems are included

stability of random attractors under perturbation and approximation abstract the comparison of the long time behaviour of dynamical systems and their numerical approximations is not straightforward since in general such methods only converge on bounded time intervals however one can still compare their asymptotic behaviour using the global attractor and this is now standard in the deterministic autonomous case for random dynamical systems there is an additional problem since the convergence of numerical methods for such systems is usually given only on average in this paper the deterministic approach is extended to cover stochastic differential equations giving necessary and sufficient conditions for the random attractor arising from a random dynamical system to be upper semi continuous with respect to a given family of perturbations or approximations

multiplicative ergodic theorem for a non irreducible random dynamical system we study the asymptotic properties of the trajectories of a discrete time random dynamical system in an infinite dimensional hilbert space under some natural assumptions on the model we establish a multiplica tive ergodic theorem with an exponential rate of convergence the assumptions are satisfied for a large class of parabolic pdes including the 2d navier stokes and complex ginzburg landau equations perturbed by a non degenerate bounded random kick force as a consequence of this er godic theorem we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system in particular we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible the proof relies on a refined version of the uniform feller property combined with some contraction and bootstrap arguments

on the convergence of stationary solutions in the smoluchowski kramers approximation of infinite dimensional systems abstract we prove the convergence in the small mass limit of statistically invariant states for a class of semi linear damped wave equations perturbed by an additive gaussian noise both with lipschitz continuous and with polynomial non linearities in particular we prove that the first marginals of any sequence of invariant measures for the stochastic wave equation converge in a suitable wasserstein metric to the unique invariant measure of the limiting stochastic semi linear parabolic equation obtained in the smoluchowski kramers approximation the wasserstein metric is associated to a suitable distance on the space of square integrable functions that is chosen in such a way that the dynamics of the limiting stochastic parabolic equation is contractive with respect to such a wasserstein metric this implies that the limiting result is a consequence of the validity of a generalized smoluchowski kramers limit at fixed times the proof of such a generalized limit requires new delicate bounds for the solutions of the stochastic wave equation that must be uniform with respect to the size of the mass

asymptotic coupling and a general form of harris theorem with applications to stochastic delay equations there are many markov chains on infinite dimensional spaces whose one step transition kernels are mutually singular when starting from different initial conditions we give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of harris theorem the first uses the existence of couplings which draw the solutions together as time goes to infinity such asymptotic couplings were central to mattingly and sinai in comm math phys 219 3 523 565 2001 mattingly in comm math phys 230 3 461 462 2002 hairer in prob theory relat field 124 345 380 2002 bakhtin and mattingly in commun contemp math 7 553 582 2005 on which this work builds as in bakhtin and mattingly 2005 the emphasis here is on stochastic differential delay equations harris celebrated theorem states that if a markov chain admits a lyapunov function whose level sets are small in the sense that transition probabilities are uniformly bounded from below then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed this convergence takes place in a total variation norm weighted by the lyapunov function a second aim of this article is to replace the notion of a small set by the much weaker notion of a d small set which takes the topology of the underlying space into account via a distance like function d with this notion at hand we prove an analogue to harris theorem where the convergence takes place in a wasserstein like distance weighted again by the lyapunov function this abstract result is then applied to the framework of stochastic delay equations in this framework the usual theory of harris chains does not apply since there are natural examples for which there exist no small sets except for sets consisting of only one point this gives a solution to the long standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it

the central limit theorem for random dynamical systems we consider random dynamical systems with randomly chosen jumps the choice of deterministic dynamical system and jumps depends on a position the central limit theorem for random dynamical systems is established

exponential mixing of the 2d stochastic navier stokes dynamics we consider the navier stokes equation on a two dimensional torus with a random force which is white noise in time and excites only a finite number of modes the number of excited modes depends on the viscosity v and grows like v 3 when v goes to zero we prove that this markov process has a unique invariant measure and is exponentially mixing in time

limit theorems for some markov chains abstract the exponential rate of convergence and the central limit theorem for some markov operators are established these operators were efficiently used in some biological models see hille horbacz szarek 2015 8 which generalize the cell cycle model given by lasota mackey 1999 12

ergodicity and stability of a dynamical system perturbed by impulsive random interventions abstract we determine all ergodic measures and their stability properties of a markov operator that is associated to a markov chain which ensues from impulsive random interventions in a one dimensional deterministic dynamical system at equally spaced time points this setting is inspired by a biological application in population dynamics where samples catches are drawn regularly from a growing population or part of a bacterial population is eradicated e g through antibiotics on the way we formulate and prove a version of orey s convergence theorem and exponential ergodicity using essentially banach lattice arguments and banach s fixed point theorem valid in the generality of a polish state space we use the krylov bogoliubov beboutov yosida decomposition to show that we found all ergodic measures finally we prove that the extinction probability is a continuous function of the initial population size that is strictly positive on part of the state space

exponential ergodicity for stochastic burgers and 2d navier stokes equations abstract it is shown that transition measures of the stochastic navier stokes equation in 2 d converge exponentially fast to the corresponding invariant measures in the distance of total variation as a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state transformation analogous results are proved for the stochastic burgers equation