control lyapunov razumikhin functions and robust stabilization of time delay systems motivated by control lyapunov functions and razumikhin theorems on stability of time delay systems we introduce the concept of control lyapunov razumikhin functions clrf the main reason for considering clrfs is construction of robust stabilizing control laws for time delay systems most existing universal formulas that apply to clfs are not applicable to clrfs it turns out that the domination redesign control law applies achieving global practical stability and under an additional assumption global asymptotic stability this additional assumption is satisfied in the practically important case when the quadratic part of a clrf is itself a clrf for the jacobian linearization of the system the clrf based domination redesign possesses robustness to input unmodeled dynamics including an infinite gain margin while in general construction of clrfs is an open problem we show that for several classes of time delay systems a clrf can be constructed in a systematic way

control of an unstable reaction diffusion pde with long input delay abstract a smith predictor like design for compensation of arbitrarily long input delays is available for general controllable possibly unstable lti finite dimensional systems such a design has not been proposed previously for problems where the plant is a pde we present a design and stability analysis for a prototype problem where the plant is a reaction diffusion parabolic pde with boundary control the plant has an arbitrary number of unstable eigenvalues and arbitrarily long delay with an unbounded input operator the predictor based feedback design extends fairly routinely within the framework of infinite dimensional backstepping however the stability analysis contains interesting features that do not arise in predictor problems when the plant is an ode the unbounded character of the input operator requires that the stability be characterized in terms of the h 1 rather than the usual l 2 norm of the actuator state the analysis involves an interesting structure of interconnected pdes of parabolic and first order hyperbolic types where the feedback gain kernel for the undelayed problem becomes an initial condition in a pde arising in the compensator design for the problem with input delay space and time variables swap their roles in an interesting manner throughout the analysis

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

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

global stabilization of the generalized korteweg de vries equation posed on a finite domain this paper is concerned with the internal stabilization of the generalized korteweg de vries equation on a bounded domain the global well posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval 1 4 the global exponential stability is obtained whatever the location where the damping is active confirming positively a conjecture of perla menzala vasconcellos and zuazua quart appl math 60 2002 pp 111 129

equilibrium fluctuations for a system of harmonic oscillators with conservative noise we investigate the harmonic chain forced by a multiplicative noise the evolution is given by an infinite system of stochastic differential equations total energy and deformation are preserved the conservation of momentum is destroyed by the noise gaussian product measures are the extremal stationary states of this model equilibrium fluctuations of the conserved fields at a diffusive scaling are described by a couple of generalized ornstein uhlenbeck processes

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

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

output feedback stabilization of the korteweg de vries equation this paper presents an output feedback control law for the korteweg de vries equation the control design is based on the backstepping method and the introduction of an appropriate observer the local exponential stability of the closed loop system is proven some numerical simulations are shown to illustrate this theoretical result

game of singular stochastic control and strategic exit we investigate a game of singular control and strategic exit in a model of competitive market share control in the model each player can make irreversible investments to increase his market share which is modeled as a diffusion process in addition each player has an option to exit the market at any point in time we formulate a verification theorem for best responses of the game and characterize markov perfect equilibria mpe under a set of verifiable assumptions we find a class of mpes with a rich structure in particular each player maintains up to two disconnected intervals of singular control regions one of which plays a defensive role and the other plays an offensive role we also identify a set of conditions under which the outcome of the game may be unique despite the multiplicity of the equilibria

structure of the condensed phase in the inclusion process we establish a complete picture of condensation in the inclusion process in the thermodynamic limit with vanishing diffusion covering all scaling regimes of the diffusion parameter and including large deviation results for the maximum occupation number we make use of size biased sampling to study the structure of the condensed phase which can extend over more than one lattice site and exhibit an interesting hierarchical structure characterized by the poisson dirichlet distribution while this approach is established in other areas including population genetics or random permutations we show that it also provides a powerful tool to analyse homogeneous condensation in stochastic particle systems with stationary product distributions we discuss the main mechanisms beyond inclusion processes that lead to the interesting structure of the condensed phase and the connection to other generic particle systems our results are exact and we present monte carlo simulation data and recursive numerics for partition functions to illustrate the main points

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

non abelian symmetries of stochastic processes derivation of correlation functions for random vertex models and disordered interacting particle systems we consider systems of particles hopping stochastically on d dimensional lattices with space dependent probabilities we map the master equation onto an evolution equation in a fock space where the dynamics are given by a quantum hamiltonian continuous time or a transfer matrix discrete time using non abelian symmetries of these operators we derive duality relations expressing the time evolution of a given initial configuration in terms of correlation functions of simpler dual processes particularly simple results are obtained for the time evolution of the density profile as a special case we show that for any su 2 symmetric system the two point and three point density correlation functions in the n particle steady state can be computed from the probability distribution of a single particle moving in the same environment we apply our results to various models among them partial exclusion a simple diffusion reaction system and the two dimensional six vertex model with space dependent vertex weights for a random distribution of the vertex weights one obtains a version of the random barrier model describing diffusion of particles in disordered media we derive exact expressions for the averaged two point density correlation functions in the presence of weak correlated disorder

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

self duality for the two component asymmetric simple exclusion process we study a two component asymmetric simple exclusion process asep that is equivalent to the asep with second class particles we prove self duality with respect to a family of duality functions which are shown to arise from the reversible measures of the process and the symmetry of the generator under the quantum algebra uq x1d524 x1d529 3 we construct all invariant measures in explicit form and discuss some of their properties we also prove a sum rule for the duality functions

variational inequalities for combined control and stopping we study the variational inequality associated with the combined stochastic control problem and establish the existence of a unique viscosity solution without uniform ellipticity an application to the quasi variational inequality is given

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

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

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

feedback stabilization of a 1 d linear reaction diffusion equation with delay boundary control the goal of this paper is to design a stabilizing feedback boundary control for a reaction diffusion partial differential equation pde where the boundary control is subject to a constant delay while the equation may be unstable without any control for this system which is equivalent to a parabolic equation coupled with a transport equation a prediction based control is explicitly computed by splitting the infinite dimensional system into two parts a finite dimensional unstable part and a stable infinite dimensional part a finite dimensional delayed controller is computed for the unstable part and it is shown that this controller stabilizes the whole pde the proof is based on an explicit expression of the classical artstein transformation combined with an adequately designed lyapunov function a numerical simulation illustrates the constructive feedback design method

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

brief paper lyapunov tools for predictor feedbacks for delay systems inverse optimality and robustness to delay mismatch we consider lti finite dimensional completely controllable but possibly open loop unstable plants with arbitrarily long actuator delay and the corresponding predictor based feedback for delay compensation we study the problem of inverse optimal re design of the predictor based feedback law we obtain a simple modification of the basic predictor based controller which employs a low pass filter and has been proposed previously by mondie and michiels for achieving robustness to discretization of the integral term in the predictor feedback law the key element in our work is the employment of an infinite dimensional backstepping transformation and the resulting lyapunov function for the infinite dimensional systems consisting of the state of the ode plant and the delay state the lyapunov function allows us to quantify the lyapunov stability properties under the modified feedback the inverse optimality of the feedback and its disturbance attenuation properties for the basic predictor feedback the availability of the lyapunov function also allows us to prove robustness to small delay mismatch in both positive and negative directions

on the robust optimal stopping problem we study a robust optimal stopping problem with respect to a set cal p of mutually singular probabilities this can be interpreted as a zero sum controller stopper game in which the stopper is trying to maximize its payoff while an adverse player wants to minimize this payoff by choosing an evaluation criteria from cal p we show that the upper snell envelope overline z of the reward process y is a supermartingale with respect to an appropriately defined nonlinear expectation underline mathscr e and overline z is further an underline mathscr e martingale up to the first time tau when overline z meets y consequently tau is the optimal stopping time for the robust optimal stopping problem and the corresponding zero sum game has a value although the result seems similar to the one obtained in the classical optimal stopping theory the mutual singularity of probabilities and the game aspect of the problem give rise to major technical hurdles which we circum

not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks we present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls we show that in a general case when the controls are distributed stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks we also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems

correlation inequalities for interacting particle systems with duality we prove a comparison inequality between a system of independent random walkers and a system of random walkers which either interact by attracting each other a process which we call here the symmetric inclusion process sip or repel each other a generalized version of the well known symmetric exclusion process as an application new correlation inequalities are obtained for the sip as well as for some interacting diffusions which are used as models of heat conduction the so called brownian momentum process and the brownian energy process these inequalities are counterparts of the inequalities in the opposite direction for the symmetric exclusion process showing that the sip is a natural bosonic analogue of the symmetric exclusion process which is fermionic finally we consider a boundary driven version of the sip for which we prove duality and then obtain correlation inequalities

on the existence of optimal controls the optimal control problem where the state is governed by an ito stochastic differential equation possibly just an ordinary differential equation is formulated in martingale terms under a coercivity condition which is weaker than compactness of the control set a convexity condition and mild continuity hypotheses on the data it is shown by the direct method that optimal controls exist hard and soft constraints are allowed in the absence of soft constraints it is shown that there exists an optimal control that is a function only of the present time and state i e the synthesis problem has a solution the main tool here is krylov s markovian selection theorem

utility maximization with discretionary stopping utility maximization problems of mixed optimal stopping control type are considered which can be solved by reduction to a family of related pure optimal stopping problems sufficient conditions for the existence of optimal strategies are provided in the context of continuous time ito process models for complete markets the mathematical tools used are those of optimal stopping theory continuous time martingales convex analysis and duality theory several examples are solved explicitly including one which demonstrates that optimal strategies need not always exist

martingale approach to stochastic differential games of control and stopping we develop a martingale approach for studying continuous time stochastic differential games of control and stopping in a non markovian framework and with the control affecting only the drift term of the state process under appropriate conditions we show that the game has a value and construct a saddle pair of optimal control and stopping strategies crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities

large deviations for a stochastic model of heat flow we investigate a one dimensional chain of 2n harmonic oscillators in which neighboring sites have their energies redistributed randomly the sites n and n are in contact with thermal reservoirs at different temperature and kipnis et al j statist phys 27 65 74 1982 proved that this model satisfies fourier s law and that in the hydrodynamical scaling limit when n the stationary state has a linear energy density profile bar theta u u 1 1 we derive the large deviation function s u for the probability of finding in the stationary state a profile u different from bar theta u the function s has striking similarities to but also large differences from the corresponding one of the symmetric exclusion process like the latter it is nonlocal and satisfies a variational equation unlike the latter it is not convex and the gaussian normal fluctuations are enhanced rather than suppressed compared to the local equilibrium state we also briefly discuss more general models and find the features common in these two and other models whose s is known

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

rapid stabilization for a korteweg de vries equation from the left dirichlet boundary condition this paper deals with the stabilization problem for the korteweg de vries equation posed on a bounded interval the control acts on the left dirichlet boundary condition at the right end point dirichlet and neumann homogeneous boundary conditions are considered the proposed feedback law forces the exponential decay of the system under a smallness condition on the initial data moreover the decay rate can be tuned to be as large as desired the feedback control law is designed by using the backstepping method

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

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

on dualities for ssep and asep with open boundary conditions duality relations for simple exclusion processes with general open boundaries are discussed it is shown that a combination of spin operators and bosonic operators enables us to have a unified discussion about duality relations with open boundaries as for the symmetric simple exclusion process ssep more general results than those from previous studies are obtained it is clarified that not only the absorbing sites but also additional sites called copying sites are needed for the boundaries in the dual process for the ssep the role of the copying sites is to conserve information about the particle states on the boundary sites similar discussions are applied to the asymmetric simple exclusion process asep in which the q analogues are employed and it is clarified that the asep with open boundaries has a complicated dual process on the boundaries

rapid exponential stabilisation of linear kdv equation with long input delay on the left boundary in this paper we consider the one dimensional linear kdv korteweg de vries equation posed in bounded interval with delay boundary control acting on the left boundary by dirichlet condition where

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

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

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

optimal investment for worst case crash scenarios a martingale approach we investigate the optimal portfolio problem under the threat of a financial market crash in a multidimensional jump diffusion framework we set up a nonprobabilistic crash model and consider an investor that seeks to maximize crra utility in the worst possible crash scenario we recast the problem as a stochastic differential game with the help of the fundamental notion of indifference strategies we completely solve the portfolio problem using martingale arguments

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

the controller and stopper game for a linear diffusion consider a process x x t 0 t 1 with values in the interval i 0 1 absorption at the boundary points of i and dynamics dx t t dt t dw t x 0 x the values t t are selected by a controller from a subset of 0 1 that depends on the current position x t for every t 0 at any stopping rule of his choice a second player called stopper can halt the evolution of the process x upon which he receives from the controller the amount e u x here 2 0 1 is a discount factor and u 0 1 is a continuous reward function under appropriate conditions on this function and on the contoller s set of choices it is shown that the two players have a saddle point of optimal strategies these can be described fairly explicitly by reduction to a suitable problem of optimal stopping whose maximal expected reward v coincides with the value of the game v sup inf x e e u x inf x sup e e u x

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

linear systems with delayed controls a reduction linear systems with delayed control action are transformed into systems without delays under an absolute continuity condition the new system is an ordinary differential control equation in the general case the new system is a measure differential control system it is shown how the controllability stabilization and various optimization problems can be analyzed via the reduced systems

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

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

orthogonal stochastic duality functions from lie algebra representations we obtain stochastic duality functions for specific markov processes using representation theory of lie algebras the duality functions come from the kernel of a unitary intertwiner between representations which provides generalized orthogonality relations for the duality functions in particular we consider representations of the heisenberg algebra and mathfrak su 1 1 both cases lead to orthogonal self duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes

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

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

portfolio risk minimization and differential games abstract a risk minimization problem is considered in a continuous time markovian regime switching financial model modulated by a continuous time finite state markov chain we interpret the states of the chain as different states of an economy a particular form of convex risk measure which includes the entropic risk measure as a particular case as a measure of risk and an optimal portfolio is determined by minimizing the convex risk measure of the terminal wealth we explore the state of the art of the stochastic differential game to formulate the problem as a markovian regime switching version of a two player zero sum stochastic differential game a novel feature of our model is that we provide the flexibility of controlling both the diffusion risk and the regime switching risk a verification theorem for the hamilton jacobi bellman hjb solution of the game is provided

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

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

new formulation of restricted growth processes we present a new formulation of a class of growth models those which evolve according to an exclusion process this formulation is based upon a transformation of the probability distribution function which involves grassmann variables this method is very general and enables one to derive an exact stochastic differential equation for the model of interest we describe this method using the traffic model as an example

exact and approximate nash equilibria in discounted markov stopping games with terminal redemption abstract this work is concerned with a class of discrete time zero sum games with two players and markov transitions on a denumerable space the general structure of the game is as follows before the game starts player i receives a bond from player ii and at each decision time player ii can stop the system paying a terminal bond to player i or if the game is not halted player i selects an action to drive the system and receives a running reward bond from player ii at termination player i redeems the accumulated bonds and the performance of a pair of decision strategies is measured by the expectation of the value at time zero of the bonds accumulated by player i under standard continuity compactness conditions the following conclusions are established i it is shown that this stopping game has a value function which is characterized by an equilibrium equation and ii such a result is used to obtain the existence of a nash equilibrium also iii the method of successive approximations is used to construct approximate nash equilibria for the game

brief paper dynamic stabilization of an euler bernoulli beam equation with time delay in boundary observation an euler bernoulli beam equation under boundary control and delayed observation is considered an observer predictor based scheme is developed to stabilize the equation on a time interval where the observation is available the state is tracked by the observer when the observation is not available due to the delay the state is estimated by the predictor the estimation is then used in proportional feedback it is shown that the state of closed loop system decays exponentially for smooth initial values

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

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