multiscale expansion of invariant measures for spdes we derive the first two terms in an expansion for the invariant measure of a class of semilinear parabolic spdes near a change of stability when the noise strength and the linear instability are of comparable order 2 this result gives insight into the stochastic bifurcation and allows to rigorously approximate correlation functions the error between the approximate and the true invariant measure is bounded in both the wasserstein and the total variation distance

fick s law and fractality of nonequilibrium stationary states in a reversible multibaker map nonequilibrium stationary states are studied for a multibaker map a simple reversible chaotic dynamical system the probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function the equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of derham type functional equations to select the physical states the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type this system corresponds to a diffusive flow experiment through a slab of material for long times any initial distribution approaches the stationary one obeying fick s law at stationarity the intracell distribution is singular in the stable direction and expressed by the takagi function which is continuous but has no finite derivatives the result suggests that singular measures play an important role in the dynamical description of non equilibrium states

regularity of laws and ergodicity of hypoelliptic sdes driven by rough paths we consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior in particular we focus on the case when the driving noise is a rough path valued fractional brownian motion with hurst parameter h 2 1 1 our contribution in this work is twofold first when the driving vector fields satisfy hormander s c elebrated lie bracket condition we derive explicit quantitative bounds on the i nverse of the malliavin ma trix en route to this we provide a novel deterministic version of norris s lemma for differential equations driven by rough paths this result with the added assumption that the linearized equation has moments will then yield that the transition laws have a smooth density with respect to lebesgue measure our second main result states that under hormander s condi tion the solutions to rough differential equations driven by fractional brownian motion with h 2 1 1 enjoy a suitable version of the strong feller property under a standard controllability condition this implies that they admit a unique stationary solution that is physical in the sense that it does not look into the future

periodic homogenization for hypoelliptic diffusions we study the long time behavior of an ornstein uhlenbeck process under the influence of a periodic drift we prove that under the standard diffusive rescaling the law of the particle position converges weakly to the law of a brownian motion whose covariance can be expressed in terms of the solution of a poisson equation we also derive upper bounds on the convergence rate in several metrics

isotropic hypoellipticity and trend to equilibrium for the fokker planck equation with a high degree potential we consider the fokker planck equation with a confining or anti confining potential which behaves at infinity like a possibly high degree homogeneous function hypoellipticity techniques provide the well posedness of the weak cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding witten laplacian with detailed applications

asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators we consider a model of heat conduction introduced in 6 which consists of a finite nonlinear chain coupled to two heat reservoirs at different temperatures we study the low temperature asymptotic behavior of the invariant measure we show that in this limit the invariant measure is characterized by a variational principle the main technical ingredients are some control theoretic arguments to extend the freidlin wentzell theory of large deviations to a class of degenerate diffusions

hamiltonian treatment of time dispersive and dissipative media within the linear response theory we develop a hamiltonian theory for a time dispersive and dissipative tdd inhomogeneous medium as described by a linear response equation respecting causality and power dissipation the canonical hamiltonian constructed here exactly reproduces the original dissipative evolution after integrating out auxiliary fields in particular for a dielectric medium we obtain a simple formula for the hamiltonian and closed form expressions for the energy density and energy flux involving the auxiliary fields the developed approach also allows to treat a long standing problem of scattering from a lossy non spherical obstacle and more generally wave propagation in tdd media

spectral properties of hypoelliptic operators we study hypoelliptic operators with polynomially bounded coefficients that are of the form k i 1 m x i t x i x 0 f where the x j denote first order differential operators f is a function with at most polynomial growth and x i t denotes the formal adjoint of x i in l 2 for any 0 we show that an inequality of the form u c u 0 k iy u 0 0 holds for suitable and c which are independent of y r in weighted sobolev spaces the first index is the derivative and the second the growth we apply this result to the fokker planck operator for an anharmonic chain of oscillators coupled to two heat baths using a method of herau and nier hn02 we conclude that its spectrum lies in a cusp x iy x y c 0 1 c r

ergodic systems of n balls in a billiard table we consider the motion ofn balls in billiard tables of a special form and we prove that the resulting dynamical systems are ergodic on a constant energy surface in fact they enjoy thek property these are the first systems of interacting particles proven to be ergodic for an arbitrary number of particles

sufficient burn in for gibbs samplers for a hierarchical random effects model we consider gibbs and block gibbs samplers for a bayesian hierarchical version of the one way random effects model drift and minorization conditions are established for the underlying markov chains the drift and minorization are used in conjunction with results from j s rosenthal j amer statist assoc 90 1995 558 566 and g o roberts and r l tweedie stochastic process appl 80 1999 211 229 to construct analytical upper bounds on the distance to stationarity these lead to upper bounds on the amount of burn in that is required to get the chain within a prespecified total variation distance of the stationary distribution the results are illustrated with a numerical example 1 introduction we consider a bayesian hierarchical version of the standard normal theory one way random effects model the posterior density for this model is intractable in the sense that the integrals required for making inferences cannot be computed in closed form hobert and geyer 1998 analyzed a gibbs sampler and a block gibbs sampler for this problem and showed that the markov chains underlying these algorithms converge to the stationary i e posterior distribution at a geometric rate however hobert and geyer stopped short of constructing analytical upper bounds on the total variation distance to stationarity in this article we construct such upper bounds and this leads to a method for determining a sufficient burn in our results are useful from a practical standpoint because they obviate troublesome ad hoc convergence diagnostics cowles and carlin 1996 and

on stable oscillations and equilibriums induced by small noise we consider the motion of a light particle when the force field is perturbed by a small nose if a certain relation between the mass of the particle and the noise intensity holds the motion of the particle will be close to periodic oscillations or to a stable equilibrium which do not exist without the noise we study various classes of random perturbations in particular we consider the question of computer simulation of these effects and calculate the correction term which appears when the gaussian perturbations are replaced by the simple random walk these are the stochastic resonance type effects and their mathematical description is based on the large deviation theory

note on the kaplan yorke dimension and linear transport coefficients a number of new relations between the kaplan yorke dimension phase space contraction transport coefficients and the maximal lyapunov exponents are given for dissipative thermostatted systems subject to a small but non zero external field in a nonequilibrium stationary state a condition for the extensivity of phase space dimension reduction is given a new expression for the linear transport coefficients in terms of the kaplan yorke dimension is derived alternatively the kaplan yorke dimension for a dissipative macroscopic system can be expressed in terms of the linear transport coefficients of the system the agreement with computer simulations for an atomic fluid at small shear rates is very good

fractional laplacian in bounded domains the fractional laplacian operator ensuremath ensuremath ensuremath delta ensuremath alpha 2 appears in a wide class of physical systems including l evy flights and stochastic interfaces in this paper we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval the implementation of boundary conditions is justified by appealing to two physical models namely hopping particles and elastic springs the eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions some analytical results concerning the structure of the eigenvalue spectrum are also obtained

short and long time behavior of the fokker planck equation in a confining potential and applications we consider the linear fokker planck equation in a confining po tential in space dimension d 3 using spectral methods we prove bounds on the derivatives of the solution for short and long time and give some applica tions

the effect of classical noise on a quantum two level system we consider a quantum two level system perturbed by classical noise the noise is implemented as a stationary diffusion process in the off diagonal matrix elements of the hamiltonian representing a transverse magnetic field we determine the invariant measure of the system and prove its uniqueness in the case of ornstein uhlenbeck noise we determine the speed of convergence to the invariant measure finally we determine an approximate one dimensional diffusion equation for the transition probabilities the proofs use both spectral theoretic and probabilistic methods

nonequilibrium statistical mechanics of hamiltonian rotators with alternated spins we consider a finite region of a d dimensional lattice of nonlinear hamiltonian rotators where neighbouring rotators have opposite alternated spins and are coupled by a small potential of size varepsilon a a ge 1 2 we weakly stochastically perturb the system in such a way that each rotator interacts with its own stochastic thermostat with a force of order varepsilon then we introduce action angle variables for the system of uncoupled rotators varepsilon 0 and note that the sum of actions over all nodes is conserved by the purely hamiltonian dynamics of the system with varepsilon 0 we investigate the limiting as varepsilon rightarrow 0 dynamics of actions for solutions of the varepsilon perturbed system on time intervals of order varepsilon 1 it turns out that the limiting dynamics is governed by a certain autonomous stochastic equation for the vector of actions this equation has a completely non hamiltonian nature this is a consequence of the fact that the system of rotators with alternated spins do not have resonances of the first order the varepsilon perturbed system has a unique stationary measure widetilde mu varepsilon and is mixing any limiting point of the family widetilde mu varepsilon of stationary measures as varepsilon rightarrow 0 is an invariant measure of the system of uncoupled integrable rotators there are plenty of such measures however it turns out that only one of them describes the limiting dynamics of the varepsilon perturbed system we prove that a limiting point of widetilde mu varepsilon is unique its projection to the space of actions is the unique stationary measure of the autonomous equation above which turns out to be mixing and its projection to the space of angles is the normalized lebesque measure on the torus mathbb t n the results and convergences which concern the behaviour of actions on long time intervals are uniform in the number n of rotators those concerning the stationary measures are uniform in n in some natural cases

comments on the entropy of nonequilibrium steady states we discuss the entropy of nonequilibrium steady states we analyze the so called spontaneous production of entropy in certain reversible deterministic nonequilibrium system and its link with the collapse of such systems towards an attractor that is of lower dimension than the dimension of phase space this means that in the steady state limit the gibbs entropy diverges to negative infinity we argue that if the gibbs entropy is expanded in a series involving 1 2 body terms the divergence of the gibbs entropy is manifest only in terms involving integrals whose dimension is higher than approximately the kaplan yorke dimension of the steady state attractor all the low order terms are finite and sum in the weak field limit to the local equilibrium entropy of linear irreversible thermodynamics

interpolation of polynomials given by straight line programs we give an algorithm for the interpolation of a polynomial a given by a straight line program its complexity is polynomial in t log d l n where t is an input bound on the number of terms in a d is a bound on its partial degree in all variables l is the length of the given straight line program and n is the number of variables

duality and exact correlations for a model of heat conduction we study a model of heat conduction with stochastic diffusion of energy we obtain a dual particle process which describes the evolution of all the correlation functions an exact expression for the covariance of the energy exhibits long range correlations in the presence of a current we discuss the formal connection of this model with the simple symmetric exclusion process

entropy production from open volume preserving to dissipative systems we generalize gaspard s method for computing the e entropy production rate in hamiltonian systems to dissipative systems with attractors considered earlier by tel vollmer and breymann this approach leads to a natural definition of a coarse grained gibbs entropy which is extensive and which can be expressed in terms of the srb measures and volumes of the coarse graining sets which cover the attractor one can also study the entropy and entropy production as functions of the degree of resolution of the coarse graining process and examine the limit as the coarse graining size approaches zero we show that this definition of the gibbs entropy leads to a positive rate of irreversible entropy production for reversible dissipative systems we apply the method to the case of a two dimensional map based upon a model considered by vollmer tel and breymann that is a deterministic version of a biased random walk we treat both volume preserving and dissipative versions of the basic map and make a comparison between the two cases we discuss the e entropy production rate as a function of the size of the coarse graining cells for these biased random walks and for an open system with flux boundary conditions show regions of exponential growth and decay of the rate of entropy production as the size of the cells decreases this work describes in some detail the relation between the results of gaspard those of of tel vollmer and breymann and those of ruelle on entropy production in various systems described by anosov or anosov like maps

green kubo formula for weakly coupled systems with noise we study the green kubo formula kappa varepsilon varsigma for the heat conductivity of an infinite chain of d dimensional finite systems cells coupled by a smooth nearest neighbor potential varepsilon v the uncoupled systems evolve according to hamiltonian dynamics perturbed stochastically by an energy conserving noise of strength varsigma noting that kappa varepsilon varsigma exists and is finite whenever varsigma 0 we are interested in what happens when the strength of the noise varsigma to 0 for this we start in this work by formally expanding kappa varepsilon varsigma in a power series in varepsilon kappa varepsilon varsigma varepsilon 2 sum n geq 2 varepsilon n 2 kappa n varsigma and investigating the formal equations satisfied by kappa n varsigma we show in particular that kappa 2 varsigma is well defined when no pinning potential is present and coincides formally with the heat conductivity obtained in the weak coupling van hove limit where time is rescaled as varepsilon 2 t for the cases where the latter has been established liverani and olla in jams 25 555 583 2012 dolgopyat and liverani in commun math phys 308 201 225 2011 for one dimensional systems we investigate kappa 2 varsigma as varsigma to 0 in three cases the disordered harmonic chain the rotor chain and a chain of strongly an harmonic oscillators moreover we formally identify kappa 2 varsigma with the conductivity obtained by having the chain between two reservoirs at temperature t and t delta t in the limit delta t to 0 n to infty varepsilon to 0

smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics this paper reviews various applications of the theory of smooth dynamical systems to conceptual problems of nonequilibrium statistical mecanics we adopt a new point of view which has emerged progressively in recent years and which takes seriously into account the chaotic character of the microscopic time evolution the emphasis is on nonequilibrium steady states rather than the traditional approach to equilibrium point of view of boltzmann the nonequilibrium steady states in presence of a gaussian thermostat are described by srb measures in terms of these one can prove the gallavotti cohen fluctuation theorem one can also prove a general linear response formula and study its consequences which are not restricted to near equilibrium situations at equilibrium one recovers in particular the onsager reciprocity relations under suitable conditions the nonequilibrium steady states satisfy the pairing theorem of dettmann and morriss the results just mentioned hold so far only for classical systems they do not involve large size i e they hold without a thermodynamic limit

towards a derivation of fourier s law for coupled anharmonic oscillators we consider a hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice mathbb z 2n times mathbb z 2 and subjected to stochastic forcing mimicking heat baths of temperatures t1 and t2 on the hyperplanes at 0 and n we introduce a truncation of the hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two point stationary correlation functions we prove that these equations have a unique solution which for n large is approximately a local equilibrium state satisfying fourier law that relates the heat current to a local temperature gradient the temperature exhibits a nonlinear profile

nonequilibrium dynamics of a stochastic model of anomalous heat transport we study the dynamics of covariances in a chain of harmonic oscillators with conservative noise in contact with two stochastic langevin heat baths the noise amounts to random collisions between nearest neighbour oscillators that exchange their momenta in a recent paper lepri et al 2009 j phys a math theor 42 025001 we have studied the stationary state of this system with fixed boundary conditions finding analytical exact expressions for the temperature profile and the heat current in the thermodynamic continuum limit in this paper we extend the analysis to the evolution of the covariance matrix and to generic boundary conditions our main purpose is to construct a hydrodynamic description of the relaxation to the stationary state starting from the exact equations governing the evolution of the correlation matrix we identify and adiabatically eliminate the fast variables arriving at a continuity equation for the temperature profile t y t complemented by an ordinary equation that accounts for the evolution in the bulk altogether we find that the evolution of t y t is the result of fractional diffusion

randomized interpolation and approximationof sparse polynomials we present a randomized algorithm that interpolates a sparse polynomial in polynomial time in the bit complexity model the algorithm can be also applied to approximate polynomials that can be approximated by sparse polynomials the approximation is in the l 2 norm

thermodynamic entropy production fluctuation in a two dimensional shear flow model we investigate fluctuations in the momentum flux across a surface perpendicular to the velocity gradient in a stationary shear flow maintained by either thermostated deterministic or by stochastic boundary conditions in the deterministic system the fluctuation relation for the probability of large deviations which holds for the phase space volume contraction giving the gibbs ensemble entropy production never seems to hold for the flux which gives the hydrodynamic entropy production in the stochastic case the fluctuation relation is found to hold for the total flux as predicted by various exact results but not for the flux across part of the surface the latter appear to satisfy a modified fluctuation relation similar results are obtained for the heat flux in a steady state produced by stochastic boundaries at different temperatures

stationary states for a mechanical system with stochastic boundary conditions we consider a system of newtonian particles with a long range repulsive pair potential moving in a cavity whose surface temperature is spatially varying when a particle hits the surface it is thermalized at the temperature of the collision point we prove that this system has a unique stationary ensemble to which any initial distribution converges for large times we show that this stationary ensemble depends continuously on the surface temperature profile

the long time behavior of brownian motion in tilted periodic potentials a b s t r a c t a variety of phenomena in physics and other fields can be modeled as brownian motion in a heat bath under tilted periodic potentials we are interested in the long time average velocity considered as a function of the external force that is the tilt of the potential in many cases the long time behavior pinning and de pinning phenomenon has been observed we use the method of stochastic differential equation to study the langevin equation describing such diffusion in the over damped limit we show the convergence of the long time average velocity to that of the smoluchowski kramers approximation and carry out asymptotic analysis based on risken s and reimann et al s formula in the under damped limit applying freidlin et al s theory we first show the existence of three pinning and de pinning thresholds of the normalized tilt corresponding to the bi stability phenomenon and second as noise approaches zero derive formulas of the mean transition times between the pinning and running states

parabolic h measures abstract classical h measures introduced by tartar 1990 and independently by gerard 1991 are not well suited for the study of parabolic equations recently several parabolic variants have been proposed together with a number of applications we introduce a new parabolic variant and call it the parabolic h measure which is suitable for these known applications moreover for this variant we prove the localisation and propagation principle establishing a basis for more demanding applications of parabolic h measures similarly as it was the case with classical h measures in particular the propagation principle enables us to write down a transport equation satisfied by the parabolic h measure associated to a sequence of solutions of a schrodinger type equation some applications to specific equations are presented illustrating the possible use of this new tool a comparison to similar results for classical h measures has been made as well

a direct search algorithm for optimization with noisy function evaluations we consider the unconstrained optimization of a function when each function evaluation is subject to a random noise we assume that there is some control over the variance of the noise term in the sense that additional computational effort will reduce the amount of noise this situation may occur when function evaluations involve simulation or the approximate solution of a numerical problem it also occurs in an experimental setting when averaging repeated observations at the same point can lead to a better estimate of the underlying function value we describe a new direct search algorithm for this type of problem we prove convergence of the new algorithm when the noise is controlled so that the standard deviation of the noise approaches zero faster than the step size we also report some numerical results on the performance of the new algorithm

stationary random metrics on hierarchical graphs via min type recursive distributional equations this paper is inspired by the problem of understanding in a mathematical sense the liouville quantum gravity on surfaces here we show how to define a stationary random metric on self similar spaces which are the limit of nice finite graphs these are the so called hierarchical graphs they possess a well defined level structure and any level is built using a simple recursion stopping the construction at any finite level we have a discrete random metric space when we set the edges to have random length using a multiplicative cascade with fixed law m we introduce a tool the cut off process by means of which one finds that renormalizing the sequence of metrics by an exponential factor they converge in law to a non trivial metric on the limit space such limit law is stationary in the sense that glueing together a certain number of copies of the random limit space according to the combinatorics of the brick graph the obtained random metric has the same law when rescaled by a random factor of law m in other words the stationary random metric is the solution of a distributional equation when the measure m has continuous positive density on mathbf r the stationary law is unique up to rescaling and any other distribution tends to a rescaled stationary law under the iterations of the hierarchical transformation we also investigate topological and geometric properties of the random space when m is log normal detecting a phase transition influenced by the branching random walk associated to the multiplicative cascade

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

dynamical system models of transport chaos characteristics the macroscopic limit and irreversibility the escape rate formalism and the thermostating algorithm describe relaxation towards a decaying state with absorbing boundaries and a steady state of periodic systems respectively it has been shown that the key features of the transport properties of both approaches if modeled by low dimensional dynamical systems can conveniently be described in the framework of multibaker maps in the present paper we discuss in detail the steps required to reach a meaningful macroscopic limit the limit involves a sequence of coarser and coarser descriptions projections until one reaches the level of irreversible macroscopic advection diffusion equations the influence of boundary conditions is studied in detail only a few of the chaos characteristics possess a meaningful macroscopic limit but none of these is sufficient to determine the entropy production in a general non equilibrium state 2003 elsevier b v all rights reserved

existence of nonequilibrium steady state for a simple model of heat conduction this paper contains rigorous results for a simple stochastic model of heat conduction similar to the kmp knipnis marchiori presutti model but with possibly energy dependent interaction rates we prove the existence and uniqueness of nonequilibrium steady states their relation to lebesgue measure and exponential convergence to steady states from suitable initial conditions

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

on convergence to equilibrium distribution i the klein gordon equation with mixing consider the klein gordon equation kge in n n 2 with constant or variable coefficients we study the distribution t of the random solution at time t we assume that the initial probability measure 0 has zero mean a translation invariant covariance and a finite mean energy density we also assume that 0 satisfies a rosenblatt or ibragimov linnik type mixing condition the main result is the convergence of t to a gaussian probability measure as t which gives a central limit theorem for the kge the proof for the case of constant coefficients is based on an analysis of long time asymptotics of the solution in the fourier representation and bernstein s room corridor argument the case of variable coefficients is treated by using an averaged version ofthe scattering theory for infinite energy solutions based on vainberg s results on local energy decay

homogenization of linear and nonlinear transport equations we study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields two types of homogenized equations are derived for general n dimensional linear and nonlinear problems we derive homogenized equations by introducing additional independent variables to represent the small scales for the two dimensional linear transport equations we derive effective equations for the averaged quantities such equations take the form of either a degenerate non local diffusion equation with memory or a higher order hyperbolic equation to study the nonlinear transport equations we introduce the concept of two scale young measure and extend diperna s method to prove that it reduces to a family of dirac measures

greatest common divisors of polynomials given by straight line programs algorithms on multivariate polynomials represented by straight line programs are developed first it is shown that most algebraic algorithms can be probabilistically applied to data that are given by a straight line computation testing such rational numeric data for zero for instance is facilitated by random evaluations modulo random prime numbers then auxiliary algorithms that determine the coefficients of a multivariate polynomial in a single variable are constructed the first main result is an algorithm that produces the greatest common divisor of the input polynomials all in straight line representation the second result shows how to find a straight line program for the reduced numerator and denominator from one for the corresponding rational function both the algorithm for that construction and the greatest common divisor algorithm are in random polynomial time for the usual coefficient fields and output a straight line program which with controllably high probability correctly determines the requested answer the running times are polynomial functions in the binary input size the input degrees as unary numbers and the logarithm of the inverse of the failure probability the algorithm for straight line programs for the numerators and denominators of rational functions implies that every degree bounded rational function can be computed fast in parallel that is in polynomial size and polylogarithmic depth

large deviation of the density profile in the steady state of the open symmetric simple exclusion process we consider an open one dimensional lattice gas on sites i 1 n with particles jumping independently with rate 1 to neighboring interior empty sites the simple symmetric exclusion process the particle fluxes at the left and right boundaries corresponding to exchanges with reservoirs at different chemical potentials create a stationary nonequilibrium state sns with a steady flux of particles through the system the mean density profile in this state which is linear describes the typical behavior of a macroscopic system i e this profile occurs with probability 1 when n the probability of microscopic configurations corresponding to some other profile x x i n has the asymptotic form exp n f f is the large deviation functional in contrast to equilibrium systems for which f eq is just the integral of the appropriately normalized local free energy density the f we find here for the nonequilibrium system is a nonlocal function of this gives rise to the long range correlations in the sns predicted by fluctuating hydrodynamics and suggests similar non local behavior of f in general sns where the long range correlations have been observed experimentally

probability metrics and uniqueness of the solution to the boltzmann equation for a maxwell gas we consider a metric for probability densities with finite variance on d and compare it with other metrics we use it for several applications both in probability and in kinetic theory the main application in kinetic theory is a uniqueness result for the solution of the spatially homogeneous boltzmann equation for a gas of true maxwell molecules

billiards in a general domain with random reflections we study stochastic billiards on general tables a particle moves according to its constant velocity inside some domain fancyscript d subset mathbb r d until it hits the boundary and bounces randomly inside according to some reflection law we assume that the boundary of the domain is locally lipschitz and almost everywhere continuously differentiable the angle of the outgoing velocity with the inner normal vector has a specified absolutely continuous density we construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location velocity we mainly focus on the case of bounded domains then we prove exponential ergodicity of these two markov processes we study their invariant distribution and their normal gaussian fluctuations of particular interest is the case of the cosine reflection law the stationary distributions for the two processes are uniform in this case the discrete time chain is reversible though the continuous time process is quasi reversible also in this case we give a natural construction of a chord picked at random in fancyscript d and we study the angle of intersection of the process with a d 1 dimensional manifold contained in fancyscript d

equivalence of nonequilibrium ensembles for simple maps abstract in this paper we discuss the equivalence of different approaches to the same transport problems with particular attention to recent works by breyman et al these suggest a singular equivalence between the stationary states of models from gaussian dynamics and those from chaotic scattering theory although the mentioned equivalence may concern but a limited set of the properties of those systems it is very important from a theoretical standpoint we argue that a special form of dissipativity in the evolution equations of the phase space distributions is to be invoked to understand those facts we discuss one application of periodic orbit theory to illustrate how the stationary states of thermostatted systems can be investigated and some qualitative features obtained

random perturbations of nonlinear oscillators degenerate white noise perturbations of hamiltonian systems in r 2 are studied in particular perturbations of a nonlinear oscillator with 1 degree of freedom are considered if the oscillator has more than one stable equilibrium the long time behavior of the system is defined by a diffusion process on a graph inside the edges the process is defined by a standard averaging procedure but to define the process for all t 0 one should add gluing conditions at the vertices calculation of the gluing conditions is based on delicate hormander type estimates

simple maps with fractal diffusion coefficients we consider chains of one dimensional piecewise linear chaotic maps with uniform slope we study the diffusive behavior of an initially nonuniform distribution of points as a function of the slope of the map by solving the frobenius perron equation for markov partition values of the slope we relate the diffusion coefficient to eigenvalues of the topological transition matrix the diffusion coefficient obtained shows a fractal structure as a function of the slope of the map this result may be typical for a wide class of maps such as two dimensional sawtooth maps

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

ergodicity of stochastic differential equations driven by fractional brownian motion we study the ergodic properties of finite dimensional syste ms of sdes driven by non degenerate additive fractional brownian motion with arbitrary hurst parameter h 2 0 1 a general framework is constructed to make precise the notions of invariant measure and stationary state for such a system we then prove under rather weak dissipativity conditions that such an sde possesses a unique stationary solution and that the convergence rate of an arbitrary solut ion towards the stationary one is at least algebraic a lower bound on the exponent is also given

ordinary differential equations transport theory and sobolev spaces we obtain some new existence uniqueness and stability results for ordinary differential equations with coefficients in sobolev spaces these results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions

pseudo free energies and large deviations for non gibbsian fkg measures a large deviation theorem for the invariant measures of translation invariant attractive interacting particle systems on 0 1 zd is proven in this way a pseudo free energy and pressure is defined for ergodic systems the large deviations property holds with the usual scaling the case of non ergodic systems is also discussed a similar result holds for occupation times the perturbation by an external field is treated

from ballistic to diffusive behavior in periodic potentials the long time large scale small friction asymptotic for the one dimensional langevin equation with a periodic potential is studied in this paper it is shown that the freidlin wentzell and central limit theorem homogenization limits commute we prove that in the combined small friction long time large scale limit the particle position converges weakly to a brownian motion with a singular diffusion coefficient which we compute explicitly we show that the same result is valid for a whole one parameter family of space time rescalings the proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator

diffusion effusion and chaotic scattering an exactly solvable liouvillian dynamics we study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area preserving mapping of an infinitely extended strip onto itself this exactly solvable mapping sustains chaotic behaviors and diffusion processes the relationship between the diffusion coefficient the lyapunov exponent and the entropy per unit time is derived the long lived classical resonances of the liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation in this sense there is a quasi isomorphism between the resonance spectrum of the liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation furthermore we show that a fractal repeller is associated to each non equilibrium state in the isolated and finite multibaker chain the nonequilibrium states are all unstable with respect to the equilibrium validating a weak form of the second principle of thermodynamics for the present dynamical system consequences of nonequilibrium fractals on classical measurements are discussed we then describe the open multibaker chain as a scattering system fractal properties of chaotic scattering are here shown to be related to diffusion in the chain

rates of convergence of the hastings and metropolis algorithms we apply recent results in markov chain theory to hastings and metropolis algorithms with either independent or symmetric candidate distributions and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution pi in the independence case in mathbb r k these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi in the symmetric case in mathbb r only we show geometric convergence essentially occurs if and only if pi has geometric tails we also evaluate recently developed computable bounds on the rates of convergence in this context examples show that these theoretical bounds can be inherently extremely conservative although when the chain is stochastically monotone the bounds may well be effective

ergodic properties of classical dissipative systems i we consider a class of models in which a hamiltonian system a with a finite number of degrees of freedom is brought into contact with an infinite heat reservoir b we develop the formalism required to describe these models near thermal equilibrium using a combination of abstract spectral techniques and harmonic analysis we investigate the singular spectrum of the liouvillean l of the coupled system a b we provide a natural set of conditions which ensure that the spectrum of l is purely absolutely continuous except for a simple eigenvalue at zero it then follows from the spectral theory of dynamical systems koopmanism that the system a b is strongly mixing from a probabilistic point of view we study a new class of random processes on finite dimensional manifolds non markovian ornstein uhlenbeck processes the paths of such a process are solutions of a random integro differential equation with gaussian noise which is a natural generalization of the well known langevin equation in this context we establish that under appropriate conditions the ou process is strongly mixing even when the langevin equation has memory and is driven by non white noise

ergodicity of some open systems with particle disk interactions we consider steady states for a class of mechanical systems with particle disk interactions coupled to two possibly unequal heat baths we show that any steady state that satisfies some natural assumptions is ergodic and absolutely continuous with respect to a lebesgue type reference measure and conclude that there exists at most one absolutely continuous steady state

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

large time asymptotics of solutions to the kramers fokker planck equation with a short range potential in this work we use a scattering method to study the kramers fokker planck equation with a potential whose gradient tends to zero at the infinity for short range potentials in dimension three we show that complex eigenvalues do not accumulate at low energies and obtain the low energy resolvent asymptotics this combined with high energy pseudospectral estimates valid in more general situations gives the large time asymptotics of solutions in weighted l 2 spaces

entropy production in open volume preserving systems we describe a mechanism leading to positive entropy production in volume preserving systems under nonequilibrium conditions we consider volume preserving systems sustaining a diffusion process like the multibaker map or the lorentz gas a continuous flux of particles is imposed across the system resulting in a steady gradient of concentration in the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient the invariant measure becomes singular for instance in the multibaker map the limit invariant measure has a cumulative function given in terms of the nondifferentiable takagi function because of this singularity of the invariant measure the entropy must be defined as a coarse grained entropy instead of the fined grained gibbs entropy which would require the existence of a regular measure with a density the coarse grained entropy production is then shown to be asymptotically positive and moreover given by the entropy production expected from irreversible thermodynamics

perturbative analysis of anharmonic chains of oscillators out of equilibrium we compute the first order correction to the correlation functions of the stationary state of a stochastically forced harmonic chain out of equilibrium when a small on site anharmonic potential is added this is achieved by deriving a suitable formula for the covariance matrix of the invariant state we find that the first order correction of the heat current does not depend on the size of the system second the temperature profile is linear when the harmonic part of the on site potential is zero the sign of the gradient of the profile however is opposite to the sign of the temperature difference of the two heat baths

approximative analytical method for some langevin dynamical systems abstract we develop an approach which leads to an analytically treatable integral representation for the steady state correlation functions of some langevin dynamical systems which are related to the microscopic hamiltonian description of the heat mechanism for the simpler case of harmonic interactions where rigorous results exist and a comparison with our method is possible we perform the computations for the thermal conductivity given by two point functions and show that our approximative scheme works perfectly well

exponential convergence to non equilibrium stationary states in classical statistical mechanics we continue the study of a model for heat conduction 6 consisting of a chain of non linear oscillators coupled to two hamiltonian heat reservoirs at different temperatures we establish existence of a liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators

controlling general polynomial networks we consider networks of massive particles connected by non linear springs some particles interact with heat baths at different temperatures which are modeled as stochastic driving forces the structure of the network is arbitrary but the motion of each particle is 1d for polynomial interactions we give sufficient conditions for hormander s bracket condition to hold which implies the uniqueness of the steady state if it exists as well as the controllability of the associated system in control theory these conditions are constructive they are formulated in terms of inequivalence of the forces modulo translations and or conditions on the topology of the connections we illustrate our results with examples including conducting chains of variable cross section this then extends the results for a simple chain obtained in eckmann et al in commun math phys 201 657 697 1999

on the convergence of the markov chain simulation method the markov chain simulation method has been successfully used in many problems including some that arise in bayesian statistics we give a self contained proof of the convergence of this method in general state spaces under conditions that are easy to verify

microscopic chaos and reaction diffusion processes in the periodic lorentz gas we apply the hypothesis of microscopic chaos to diffusion controlled reaction which we study in a reactive periodic lorentz gas the relaxation rate of the reactive eigenmodes is obtained as eigenvalue of the frobenius perron operator which determines the reaction rate the cumulative functions of the eigenstates of the frobenius perron operator are shown to be generalizations of lebesgue s singular continuous functions for small enough densities of catalysts the reaction is controlled by the diffusion a random walk model of this diffusion controlled reaction process is presented which is used to study the dependence of the reaction rate on the density of catalysts

exponential return to equilibrium for hypoelliptic quadratic systems we study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators quadratic operators are non selfadjoint differential operators with complex valued quadratic symbols under appropriate assumptions a complete description of the spectrum of such operators is given and the exponential return to equilibrium with sharp estimates on the rate of convergence is proven some applications to the study of chains of oscillators and to the generalized langevin equation are given

density estimates for a random noise propagating through a chain of differential equations we here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d the first one being forced by a non degenerate random noise and the n 1 other ones being degenerate the system formed by the n equations satisfies a suitable hormander condition the second equation feels the noise plugged into the first equation the third equation feels the noise transmitted from the first to the second equation and so on so that the noise propagates one way through the system when the coefficients of the system are lipschitz continuous we show that the density of the solution satisfies gaussian bounds with non diffusive time scales the proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem

on two temperature problem for harmonic crystals we consider the dynamics of a harmonic crystal in d dimensions with n components d n 1 the initial date is a random function with finite mean density of the energy which also satisfies a rosenblatt or ibragimov linnik type mixing condition the random function is translation invariant in x1 xd 1 and converges to different translation invariant processes as xd with the distributions we study the distribution t of the solution at time t in mathbb b the main result is the convergence of t to a gaussian translation invariant measure as t the proof is based on the long time asymptotics of the green function and on bernstein s room corridor argument the application to the case of the gibbs measures g with two different temperatures t is given limiting mean energy current density is 0 0 c t t with some positive constant c 0 what corresponds to second law

long time behaviour of heat flow global estimates and exact asymptotics we obtain global upper and lower bounds on the heat kernel of an elliptic second order differential operator which become sharp in certain long time and large space asymptotics we prove a generalization of aronson s gaussian bounds which identifies correctly an effective drift for heat flow in the case of periodic coefficients we give variational characterizations of the effective conductivity which is then made to appear in heat kernel bounds these results are for heat kernels with measurable coefficients for differentiable coefficients we prove tighter estimates in which the rate of homogenization is known to be optimal

non equilibrium statistical mechanics of strongly anharmonic chains of oscillators we study the model of a strongly non linear chain of particles coupled to two heat baths at different temperatures our main result is the existence and uniqueness of a stationary state at all temperatures this result extends those of eckmann pillet rey bellet epr99a epr99b to potentials with essentially arbitrary growth at infinity this extension is possible by introducing a stronger version of hormander s theorem for kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains

change of order for regular chains in positive dimension we discuss changing the variable order for a regular chain in positive dimension this quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations we propose a modular method reducing the problem to computations in dimension zero and one the problems raised by the choice of the specialization points and the lack of the crucial information of what are the free and algebraic variables for the new order are discussed strong but not unusual hypotheses for the initial regular chain are required the main required subroutines are change of order in dimension zero and a formal newton iteration

noise can speed convergence in markov chains a new theorem shows that noise can speed convergence to equilibrium in discrete finite state markov chains the noise applies to the state density and helps the markov chain explore improbable regions of the state space the theorem ensures that a stochastic resonance noise benefit exists for states that obey a vector norm inequality such noise leads to faster convergence because the noise reduces the norm components a corollary shows that a noise benefit still occurs if the system states obey an alternate norm inequality this leads to a noise benefit algorithm that requires knowledge of the steady state an alternative blind algorithm uses only past state information to achieve a weaker noise benefit simulations illustrate the predicted noise benefits in three well known markov models the first model is a two parameter ehrenfest diffusion model that shows how noise benefits can occur in the class of birth death processes the second model is a wright fisher model of genotype drift in population genetics the third model is a chemical reaction network of zeolite crystallization a fourth simulation shows a convergence rate increase of 64 for states that satisfy the theorem and an increase of 53 for states that satisfy the corollary a final simulation shows that even suboptimal noise can speed convergence if the noise applies over successive time cycles noise benefits tend to be sharpest in markov models that do not converge quickly and that do not have strong absorbing states

heat flow in an exactly solvable model a chain of one dimensional oscillators is considered they are mechanically uncoupled and interact via a stochastic process which redistributes the energy between nearest neighbors the total energy is kept constant except for the interactions of the extremal oscillators with reservoirs at different temperatures the stationary measures are obtained when the chain is finite the thermodynamic limit is then considered approach to the gibbs distribution is proven and a linear temperature profile is obtained

fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields the authors consider the problem of reconstructing i e interpolating a t sparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments it is shown that if the polynomial has coefficients in a finite field gf q and the black box can evaluate the polynomial in the field gf q ulcorner 2 log q nt 3 urcorner where n is the number of variables then there is an algorithm to interpolate the polynomial in o log 3 nt boolean parallel time and o n 2 t 6 log 2 nt processors this algorithm yields the first efficient deterministic polynomial time algorithm and moreover boolean nc algorithm for interpolating t sparse polynomials over finite fields and should be contrasted with the fact that efficient interpolation using a black box that only evaluates the polynomial at points in gf q is not possible cf m clausen a dress j grabmeier and m karpinski theoret comput sci 1990 to appear this algorithm tog

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

non equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures we study the statistical mechanics of a finite dimensional non linear hamiltonian system a chain of anharmonic oscillators coupled to two heat baths described by wave equations assuming that the initial conditions of the heat baths are distributed according to the gibbs measures at two different temperatures we study the dynamics of the oscillators under suitable assumptions on the potential and on the coupling between the chain and the heat baths we prove the existence of an invariant measure for any temperature difference i e we prove the existence of steady states furthermore if the temperature difference is sufficiently small we prove that the invariant measure is unique and mixing in particular we develop new techniques for proving the existence of invariant measures for random processes on a non compact phase space these techniques are based on an extension of the commutator method of hormander used in the study of hypoelliptic differential operators

microscopic stationary states for stochastic systems with particle flux let t be the symmetric nearest neighbor simple exclusion process on 3 modified to allow the spontaneous creation and destruction of particles at the origin we study the invariant measures for the system in this non equilibrium situation for which a net current of particles flows from the origin we analyze the n point correlations of the invariant measures which for n 2 behave like max x1x2 4

chapman enskog as an application of the method for eliminating fast variables although the chapman enskog treatment of the boltzmann equation is one of the first examples of the elimination of fast variables it is not usually presented consistently from that point of view here it is developed systematically as a special case of the general method for eliminating fast variables from nonlinear equations as a result certain ambiguities can be remedied first it is inconsistent with the separation of time scales to extend the phenomenological description of the gas by including some of the higher moments of the distribution function such as the heat flow in the application to the relativistic boltzmann equation the dilemma concerning the choice of the lowest order approximation is resolved in the final section it is demonstrated that unsystematic elimination of fast variables leads to secular terms

non equilibrium current fluctuations in stochastic lattice gases we study current fluctuations in lattice gases in the macroscopic limit extending the dynamic approach for density fluctuations developed in previous articles more precisely we establish a large deviation principle for a space time fluctuation j of the empirical current with a rate functional i j we then estimate the probability of a fluctuation of the average current over a large time interval this probability can be obtained by solving a variational problem for the functional i we discuss several possible scenarios interpreted as dynamical phase transitions for this variational problem they actually occur in specific models we finally discuss the time reversal properties of i and derive a fluctuation relationship akin to the gallavotti cohen theorem for the entropy production