a mixed variational method for linear coupled thermoelastic analysis abstract a variational method for linear coupled quasi static thermoelastic analysis is presented the variational support is a statement in terms of displacement temperature stress and heat flux the statement is based on the hybrid stress formulation for the elastic part and on the mixed flux temperature formulation for the thermal one and includes the rate dependent terms of the energy balance equations and the initial condition a finite element model for the semidiscrete analysis is developed within this variational framework and a guideline for implementing a family of thermoelastic finite elements is given some test cases enlighten the effectiveness and reliability of the approach proposed

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

diffusive limit of a tagged particle in asymmetric simple exclusion processes invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d 3 the method of proof is by martingale techniques which rely on the fact that symmetric random walks are transient in high dimensions c 2000 john wiley sons inc

homogenization of an advection equation with locally stationary random coefficients abstract in the paper we consider the solution of an advection equation with rapidly changing coefficients t u e 1 e v t e 2 x e x u e 0 for t t and u e t x u 0 x x r d here e 0 is some small parameter and the drift term v t x t x r 1 d is assumed to be a d dimensional vector valued random field with incompressible spatial realizations we prove that when the field is gaussian locally stationary quasi periodic in the x variable and strongly mixing in time the solutions u e t x converge in law as e 0 to u 0 x t t x where x s t x s t is a diffusion satisfying x t t x x the averages of u e t x converge then to the solution of the corresponding kolmogorov backward equation

diffusion approximation for the advection of particles in a strongly turbulent random environment in this paper we prove several theorems concerning the motion of a particle in a random environment the trajectory of a particle is the solution of the differential equation dx t dt v x t where v x v e 1 f x 0 1 v is a constant vector f is a mean zero fluctuation field and e 1 is a parameter measuring the size of the fluctuations we show that both in case of a motion of a single particle and of a particle system considered in the macroscopic coordinate system moving along with velocity v i e x x vt e t t e 2 the diffusion approximation holds provided that f is divergence free moreover we show how to renormalize trajectories to obtain a similar result for non divergence free fields these results generalize theorems due to khasminskii and to kesten and papanicolaou

on the superdiffusive behavior of passive tracer with a gaussian drift in the present article we consider a motion of a passive tracer particle whose trajectory satisfies the ito stochastic differential equation d x t v t x t dt sqrt 2 kappa d w t where w is a brownian motion v is a stationary gaussian random field with incompressible realizations independent of w and 0 we prove the superdiffusive character of the motion under certain conditions on the energy spectrum of the velocity field the result is shown both for steady time independent and time dependent and markovian velocity fields in addition we provide explicit upper and lower bounds for the hurst exponent of the trajectory all previous rigorous results concerned explicitely solvable shear flows cases

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

equidistribution on the sphere a concept of generalized discrepancy which involves pseudodifferential operators to give a criterion of equidistributed pointsets is developed on the sphere a simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy with the help of this formula five kinds of point systems on the sphere namely lattices in polar coordinates transformed two dimensional sequences rotations on the sphere triangulations and sum of three squares sequence are investigated quantitative tests are done and the results are compared with one another our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

some variational principles for linear coupled thermoelasticity guided by the principle of virtual work the governing equations describing the physical behavior of a thermoelastic continuum were expressed as the euler lagrange equations of certain variational principles the differential variational principles were formulated for the thermoelastic continuum with or without an internal surface of discontinuity by introducing the dislocation potentials and lagrange undetermined multipliers these principles were shown to recover some of the earlier variational principles as special cases and their reciprocals were also recorded

deterministic homogenization for fast slow systems with chaotic noise consider a fast slow system of ordinary differential equations of the form x a x y e 1b x y y e 2g y where it is assumed that b averages to zero under the fast flow generated by g we give conditions under which solutions x to the slow equations converge weakly to an it o diffusion x as e 0 the drift and diffusion coefficients of the limiting stochastic differential equation satisfied by x are given explicitly our theory applies when the fast flow is anosov or axiom a as well as to a large class of nonuniformly hyperbolic fast flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on the fast flow

an invariance principle for non symmetric markov processes and reflecting diffusions in random domains we study an invariance principle for additive functionals of nonsymmetric markov processes with singular mean forward velocities we generalize results of kipnis and varadhan kv and de masi et al de in two directions markov processes are non symmetric and mean forward velocities are distributions we study continuous time markov processes we use our result to homogenize non symmetric reflecting diffusions in random domains

on an alleged philosophers stone concerning variational principles with subsidiary conditions in general the lagrange multiplier method lmm is used to incorporate subsidiary conditions into variational principles the lagrange multipliers represent an additional field of independent variables the attempt to satisfy subsidiary conditions without employing additional independent unknowns has led to the development of simplified variational principles svp they are characterized by expressing lagrange multipliers in terms of original field variables by means of the euler lagrange equations for the multipliers providing their physical interpretation in the first part of the theoretical investigation systems with infinitely many degrees of freedom are studied it is shown that the euler lagrange equations of a lmm based on a modification of the principle of minimum of potential energy pmipe do not hold unconditionally that is for arbitrary subsidiary conditions for the corresponding svp the second part of the theoretical investigation is concerned with systems with a finite number of degrees of freedom the finite element method fem is employed to discuss and compare the characteristics of the lmm and of the corresponding svp contrary to the former the latter are found to be problem dependent several shortcomings of the svp are listed including the possibility of obtaining an infinite sequence of singular coefficient matrices in the process of a systematic mesh refinement consequently convergence of finite element solutions to the true solution in the limit of finite element representations is not guaranteed the theoretical findings are corroborated by the results of a detailed numerical study

diffusion in turbulence we prove long time diffusive behavior homogenization for convection diffusion in a turbulent flow that it incompressible and has a stationary and square integrable stream matrix simple shear flow examples show that this result is sharp for flows that have stationary stream matrices

on superdiffusive behavior of a passive tracer in a random flow abstract in this note we consider a passive tracer model describing particle dispersion in a turbulent flow the trajectory of the particle is given by the solution of an ordinary differential equation x t f x t x 0 x 0 where f x is a divergence free random vector field that is spatially homogeneous and isotropic we show that trajectories of the tracer display superdiffusive behavior when the random velocity f x decorrelates at large distances but does it not rapidly but rather at some moderate rate the main tools used in the proofs are variational principles and tauberian type theorems

central limit theorem for random walks in doubly stochastic random environment mathscr h _ 1 suffices we prove a central limit theorem under diffusive scaling for the displacement of a random walk on zdzd in stationary and ergodic doubly stochastic random environment under the h 1h 1 condition imposed on the drift field the condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable this improves the best existing result fluctuations in markov processes time symmetry and martingale approximation 2012 springer where it is assumed that the stream tensor is in lmax 2 d lmax 2 d with 0 0 our proof relies on an extension of the relaxed sector condition of bull inst math acad sin n s 7 2012 463 476 and is technically rather simpler than existing earlier proofs of similar results by oelschlager ann probab 16 1988 1084 1126 and komorowski landim and olla fluctuations in markov processes time symmetry and martingale approximation 2012 springer

diffusive and nondiffusive limits of transport in nonmixing flows we study the passive scalar transport in a class of nonmixing markovian flows with power law spectra and correlation times we establish a new diffusion regime under an optimal condition convergent kubo formula on the spatial temporal structure of this family of flows under such a condition the peclet number of the problem may be infinite we propose a general criterion for the diffusion regime that takes into account of both the effect of molecular diffusion and the spatial temporal structure of the velocity field we conjecture the criterion to be applicable in general for temporally ergodic reversible markovian flows we show heuristically that the violation of this criterion may lead to a nondiffusive scaling limit

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

spherical designs discrepancy and numerical integration a spherical design is a point configuration on the sphere which yields exact equal weight quadrature formulae for polynomials up to a given de gree until now only very specific constructions for spherical designs are known we establish connections to spherical cap discrepancy and show some general discrepancy bounds furthermore we reformulate the problem of construct ing designs as an optimization problem and develop an algorithm for finding practical designs

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

polynomial discrepancy of sequences generalizing e hlawka s concept of polynomial discrepancy we introduce a similar concept for sequences in the unit cube and on the sphere we investigate the relation of this polynomial discrepancy to the usual discrepancy and obtain lower and upper bounds in a final section some computational results are established

the sector constants of continuous state branching processes with immigration abstract continuous state branching processes with immigration are studied we are particularly concerned with the associated non symmetric dirichlet form after observing that gamma distributions are only reversible distributions for this class of models we prove that every generalized gamma convolution is a stationary distribution of the process with suitably chosen branching mechanism and with continuous immigration for such non reversible processes the strong sector condition is discussed in terms of a characteristic called the thorin measure in addition some connections with notion from non commutative probability theory will be pointed out through calculations involving the stieltjes transform

diffusive limits for true or myopic self avoiding random walks and self repellent brownian polymers in d 3 the problems considered in the present paper have their roots in two different cultures the true or myopic self avoiding walk model tsaw was introduced in the physics literature by amit et al phys rev b 27 1635 1645 1983 this is a nearest neighbor non markovian random walk in mathbb z d which prefers to jump to those neighbors which were less visited in the past the self repelling brownian polymer model srbp initiated in the probabilistic literature by durrett and rogers probab theory relat fields 92 337 349 1992 independently of the physics community is the continuous space time counterpart a diffusion in mathbb r d pushed by the negative gradient of the mollified occupation time measure of the process in both cases similar long memory effects are caused by a path wise self repellency of the trajectories due to a push by the negative gradient of softened local time we investigate the asymptotic behaviour of tsaw and srbp in the non recurrent dimensions first we identify a natural stationary in time and ergodic distribution of the environment the local time profile as seen from the moving particle the main results are diffusive limits in the case of tsaw for a wide class of self interaction functions we establish diffusive lower and upper bounds for the displacement and for a particular more restricted class of interactions we prove full clt for the finite dimensional distributions of the displacement in the case of srbp we prove full clt without restrictions on the interaction functions these results settle part of the conjectures based on non rigorous renormalization group arguments equally valid for the tsaw and srbp cases in amit et al 1983 the proof of the clt follows the non reversible version of kipnis varadhan theory on the way to the proof we slightly weaken the so called graded sector condition

homogenization of a diffusion process in a divergence free random field on etudie le probleme d homogeneisation pour un processus de diffusion dans un champ de derive aleatoire independant de la divergence dans le cas particulier d un petit champ gaussien on derive un developpement asymptotique de la matrice de diffusion efficace en termes de sa densite spectrale

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

on the poisson equation and diffusion approximation 3 three different results are established which turn out to be closely connected so that the first one implies the second one which in turn implies the third one the first one states the smoothness of an invariant diffusion density with respect to a parameter the second establishes a similar smoothness of the solution of the poisson equation in rd the third one states a diffusion approximation result or in other words an averaging of singularly perturbed diffusion for fully coupled sde systems or sde systems with complete dependence

computation of the effective diffusivity tensor for transport of a passive scalar in a turbulent incompressible flow we consider the passive scalar transport in an incompressible random flow our basic result is a proof of the convergence of a certain numerical scheme for the computation of the eddy diffusivity tensor the scheme leads to the formula for the diffusivity expressed in terms of an infinite series we give a rigorous proof of the geometric bounds on the magnitude of the terms of the series provided that the eulerian field is markovian and gaussian and its temporal dynamics has a sufficiently large spectral gap the principal tools used in the proofs are the decomposition of the space of square integrable fields formed over the possible realizations of the eulerian velocity field in the gaussian chaos and the hypercontractivity properties of gaussian measures

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

on the sector condition and homogenization of diffusions with a gaussian drift abstract in this paper we present the functional central limit theorem for a class of markov processes whose l 2 generator satisfies the so called graded sector condition we apply the result to obtain homogenization theorems for certain classes of diffusions with a random gaussian drift additionally we present a result concerning the regularity of the effective diffusivity tensor with respect to the parameters related to the statistics of the drift the abstract central limit theorem see theorem 2 2 is obtained by applying the technique used in sethuraman et al comm pure appl math 53 2000 972 to the case of infinite particle systems

a quasivariational principle for fluid solid interaction discussion it was shown that the straightforward application of the ritz procedure with piecewise cubic hermitian interpolation functions in order to produce time finite element equations will result in an unstable algorithm this is a rather disappointing result if one refers e g to the second order static equilibrium problem of a rod for which the same interpolation model works so well in the authors opinion in order to get a better understanding of this phenomenon one must refer to the parent matrix k before the initial or the boundary conditions are imposed the degeneracy of k represents the need of the number of the essential conditions that must be imposed in order to make the formulation well defined for the given problem the parent matrix of eqs 4 and 5 is singular of degeneracy 1 this is exactly the single rigid body motion that must be removed in order to make the formulation for the static rod problem well defined however in the dynamic problem both the initial displacement and velocity are essential conditions that must be imposed in order to make the formulation well defined indeed the parent matrix of the modified elements eqs 12 and 13 is singular of degeneracy 2 and both the initial displacement and velocity must be imposed in order to solve the equations

symmetric simple exclusion process regularity of the self diffusion coefficient we prove that the self diffusion coefficient of a tagged particle in the symmetric exclusion process in zd which is in equilibrium at density is of class c as a function of in the closed interval 0 1 the proof provides also a recursive method to compute the taylor expansion at the boundaries

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

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 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

diffusion approximation for the convection diffusion equation with random drift we consider the asymptotic behavior of the solutions ofscaled convection diffusion equations t u t x x t x 1 v t 2 x x u t x with the initial condition u 0 x u 0 x as the parameter 0 under the assumptions that 0 and v t x t x r d is a d dimensional stationary zero mean incompressible gaussian random field markovian and mixing in t we show that the laws of u t t 0 in an appropriate functional space converge weakly as 0 to a type measureconcentrated on a solution of a certain constant coefficient heat equation

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

transport by time dependent stationary flow we consider transport properties for gaussian stationary divergence free random velocity fields in 2 which are markov in time we prove the existence of effective diffusivity we also obtain its full asymptotics in the case of short time correlations on a fully rigorous level the main regularity assumption is that almost every realization of the random velocity field should be continuous in space and time and lipschitz continuous in space

coupled thermoelastic shell equations with second sound for high frequency vibrations of temperature dependent materials guided by the three dimensional theory of coupled thermoelasticity with second sound a system of shear deformable shell equations is consistently derived in invariant differential and variational forms for the high frequency vibrations of temperature dependent materials the first part of the paper is concerned with a unified variational principle describing the fundamental equations of thermoelasticity the differential type of variational principle is presented by expressing hamilton s principle for the thermal part of a thermoelastic region and then combining it with its mechanical part in the second part the hierarchic system of non isothermal shell equations is systematically established by use of the variational principle together with mindlin s kinematic hypothesis for shells the system of two dimensional approximate equations which may take account of all the significant mechanical and thermal effects including the temperature dependency of material governs the extensional thickness shear flexural and torsional as well as coupled vibrations of shells of uniform thickness lastly in the third part emphasis is placed on certain cases involving special material geometry and kinematics besides a theorem is devised so as to enumerate the initial and boundary conditions sufficient for the uniqueness in solutions of the system of non isothermal shell equations

statistical estimates for the navier stokes equations and the kraichnan theory of 2 d fully developed turbulence a mathematical formulation of the kraichnan theory for 2 d fully developed turbulence is given in terms of ensemble averages of solutions to the navier stokes equations a simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber up to a logarithmic correction this is followed by partial rigorous support for kraichnan s eddy breakup mechanism a rigorous estimate for the total energy is found to be consistent with kraichnan s theory finally it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case

a martingale approach to homogenization of unbounded random flows we study the asymptotic behavior of brownian motion in steady unbounded incompressible random flows we prove an invariance principle for almost all realizations of random flows the key compactness result is obtained by moser s iterative scheme in pde theory

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

the dissipative scale of the stochastics navier stokes equation regularization and analyticity we prove spatial analyticity for solutions of the stochastically forced navier stokes equation provided that the forcing is sufficiently smooth spatially we also give estimates which extend to the stationary regime providing strong control of both of the expected rate of dissipation and fluctuations about this mean surprisingly we could not obtain non random estimates of the exponential decay rate of the spatial fourier spectra