central limit theorem for a random walk with random obstacles in mathrm r d for some positive constants r and ro with r ro for a probability measure v on the space t we write p ffv d q p we call the process x t p a random walk with random obstacles let t be the set of all q ee tr such that p x l e has an unbounded support and let u be a gibbs state associated with a pair potential d which is translation invariant one of the main results of this paper is the central limit theorem for x t pa where 4w is a conditional probability measure of the gibbs state u given the event t that is 4t a i1 in a previous paper 10 we considered a system of infinitely many hard balls with the same diameter r moving discontinuously in rd we constructed the markov process 4t which describes the system this process has a gibbs state as a stationary measure we showed the ergodicity of the stationary markov

variable transformation to obtain geometric ergodicity in the random walk metropolis algorithm a random walk metropolis sampler is geometrically ergodic if its equilibrium density is super exponentially light and satisfies a curvature condition stochastic process appl 85 2000 341 361 many applications including bayesian analysis with conjugate priors of logistic and poisson regression and of log linear models for categorical data result in posterior distributions that are not super exponentially light we show how to apply the change of variable formula for diffeomorphisms to obtain new densities that do satisfy the conditions for geometric ergodicity sampling the new variable and mapping the results back to the old gives a geometrically ergodic sampler for the original variable this method of obtaining geometric ergodicity has very wide applicability

efficient implementation of markov chain monte carlo when using an unbiased likelihood estimator summary when an unbiased estimator of the likelihood is used within a metropolis hastings chain it is necessary to trade off the number of monte carlo samples used to construct this estimator against the asymptotic variances of the averages computed under this chain using many monte carlo samples will typically result in metropolis hastings averages with lower asymptotic variances than the corresponding averages that use fewer samples however the computing time required to construct the likelihood estimator increases with the number of samples under the assumption that the distribution of the additive noise introduced by the loglikelihood estimator is gaussian with variance inversely proportional to the number of samples and independent of the parameter value at which it is evaluated we provide guidelines on the number of samples to select we illustrate our results by considering a stochastic volatility model applied to stock index returns

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

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

einstein relation for reversible diffusions in a random environment we consider reversible diffusions in a random environment and prove the einstein relation for this model it says that the derivative at 0 of the effective velocity under an additional local drift equals the diffusivity of the model without drift the einstein relation is conjectured to hold for a variety of models but so far it has only been proved in particular cases our proof makes use of homogenization arguments the girsanov transform and a refinement of the regeneration times introduced by shen 2011 wiley periodicals inc

a central limit theorem for gibbs measures relative to brownian motion we study a gibbs measure over brownian motion with a pair potential which depends only on the increments assuming a particular form of this pair potential we establish that in the infinite volume limit the gibbs measure can be viewed as brownian motion moving in a dynamic random environment thereby we are in a position to use the technique of kipnis and varadhan and to prove a functional central limit theorem

the conjugate heat equation and ancient solutions of the ricci flow abstract we prove gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the ricci flow as a consequence for dimension 4 and higher we show that the backward limit of type i solutions of the ricci flow must be a non flat gradient shrinking ricci soliton this extends perelman s previous result on backward limits of solutions in dimension 3 in which case the curvature operator is nonnegative it follows from hamilton ivey curvature pinching estimate as an application this also addresses an issue left in naber 2010 23 where naber proves the interesting result that there exists a type i dilation limit that converges to a gradient shrinking ricci soliton but that soliton might be flat the gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest

gibbs measures on brownian currents motivated by applications to quantum field theory we consider gibbs measures for which the reference measure is wiener measure and the interaction is given by a double stochastic integral and a pinning external potential in order properly to characterize these measures through dlr equations we are led to lift wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields we prove existence and basic properties of such gibbs measures in the small coupling regime by means of cluster expansion

invariance principle for the random conductance model with unbounded conductances we study a continuous time random walk x in an environment of i i d random conductances e 1 we obtain heat kernel bounds and prove a quenched invariance principle for x this holds even when e e

tracer diffusion in lattice gases it has been proved that a tracer particle in a reversible lattice gas converges to brownian motion however only in a few particular cases has a strictly positive self diffusion coefficientd been established here we supply the missing piece and show thatd 0 in general the exceptions are one dimensional lattice gases with nearest neighbor jumps only for whichd 0 the proof establishes a variational formula ford which could be used to obtain realistic bounds

simple random walk on long range percolation clusters ii scaling limits supported in part at the technion by a landau fellowship supported in part by an alfred sloan fellowship in mathematics

self diffusion in simple models systems with long range jumps we review some exact results for the morion of a tagged particle in simple models then we study the density dependence of the sill diffusion coefficient d n in lattice systems with simple symmetric exclusion in which the particles can jump with equal rates to a set of n neighboring sites we obtain positive upper and lower bounds on f n n 1 d n d n 0 1 for is an element of 0 1 computer simulations for the square triangular and one dimensional lattices suggest that fn becomes effectively independent of n for n greater than or equal to 20

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

quenched large deviations for simple random walks on percolation clusters including long range correlations we prove a quenched large deviation principle ldp for a simple random walk on a supercritical percolation cluster srwpc on mathbb z d d geq 2 the models under interest include classical bernoulli bond and site percolation as well as models that exhibit long range correlations like the random cluster model the random interlacement and the vacant set of random interlacements for d geq 3 and the level sets of the gaussian free field d geq 3 inspired by the methods developed by kosygina et al commun pure appl math 59 1489 1521 2006 for proving quenched ldp for elliptic diffusions with a random drift and by yilmaz commun pure appl math 62 8 1033 1075 2009 and rosenbluth quenched large deviations for multidimensional random walks in a random environment a variational formula ph d thesis nyu arxiv 0804 1444v1 for similar results regarding elliptic random walks in random environment we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures of the environment markov chain in the non elliptic case of srwpc via a contraction principle this reduces easily to a quenched ldp for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas the main difficulty in our set up lies in the inherent non ellipticity as well as the lack of translation invariance stemming from conditioning on the fact that the origin belongs to the infinite cluster we develop a unifying approach for proving quenched large deviations for srwpc based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

efficient markov chain monte carlo methods for decoding neural spike trains stimulus reconstruction or decoding methods provide an important tool for understanding how sensory and motor information is represented in neural activity we discuss bayesian decoding methods based on an encoding generalized linear model glm that accurately describes how stimuli are transformed into the spike trains of a group of neurons the form of the glm likelihood ensures that the posterior distribution over the stimuli that caused an observed set of spike trains is log concave so long as the prior is this allows the maximum a posteriori map stimulus estimate to be obtained using efficient optimization algorithms unfortunately the map estimate can have a relatively large average error when the posterior is highly nongaussian here we compare several markov chain monte carlo mcmc algorithms that allow for the calculation of general bayesian estimators involving posterior expectations conditional on model parameters an efficient version of the hybrid monte carlo hmc algorithm was significantly superior to other mcmc methods for gaussian priors when the prior distribution has sharp edges and corners on the other hand the hit and run algorithm performed better than other mcmc methods using these algorithms we show that for this latter class of priors the posterior mean estimate can have a considerably lower average error than map whereas for gaussian priors the two estimators have roughly equal efficiency we also address the application of mcmc methods for extracting nonmarginal properties of the posterior distribution for example by using mcmc to calculate the mutual information between the stimulus and response we verify the validity of a computationally efficient laplace approximation to this quantity for gaussian priors in a wide range of model parameters this makes direct model based computation of the mutual information tractable even in the case of large observed neural populations where methods based on binning the spike train fail finally we consider the effect of uncertainty in the glm parameters on the posterior estimators

fixed width output analysis for markov chain monte carlo markov chain monte carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages a fundamental question is when sampling should stop that is at what point the ergodic averages are good estimates of the desired quantities we consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user specified value hence calculating a monte carlo standard error is a critical step in assessing the simulation output we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution we give sufficient conditions for the strong consistency of both methods and investigate their finite sample properties in various examples

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

equilibrium fluctuations for interacting brownian particles we consider an infinitely extended system of brownian particles interacting by a pair force gradv their initial distribution is stationary and given by the gibbs measure associated with the potentialv with fugacityz we assume thatv is symmetric finite range three times continuously differentiable superstable and positive and that the fugacity is small in the sense that 0 z 0 28 eedq 1 e v q in addition a certain essential self adjointness property is assumed we prove then that the time dependent fluctuations in the density on a spatial scale of order 1 and on a time scale of order 2 converge as 0 to a gaussian field with covariance edqg q e 2 t f q withp the density and the compressibility

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

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

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

homogenization of parabolic equations with a continuum of space and time scales this paper addresses the issue of the homogenization of linear divergence form parabolic operators in situations where no ergodicity and no scale separation in time or space are available namely we consider divergence form linear parabolic operators in omega subset mathbb r n with l infty omega times 0 t coefficients it appears that the inverse operator maps the unit ball of l 2 omega times 0 t into a space of functions which at small time and space scales are close in h 1 norm to a functional space of dimension n it follows that once one has solved these equations at least n times it is possible to homogenize them both in space and in time reducing the number of operation counts necessary to obtain further solutions in practice we show under a cordes type condition that the first order time derivatives and second order space derivatives of the solution of these operators with respect to caloric coordinates are in l 2 instead of h 1 with euclidean coordinates if the medium is time independent then it is sufficient to solve n times the associated elliptic equation in order to homogenize the parabolic equation

limit theory for random walks in degenerate time dependent random environments abstract we study continuous time variable speed random walks in random environments on z d d 2 where at time t the walk at x jumps across edge x y at time dependent rate a t x y the rates which we assume stationary and ergodic with respect to space time shifts are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time we formulate conditions on the environment under which the law of diffusively scaled random walk path tends to brownian motion for almost every sample of the rates the proofs invoke moser iteration to prove sublinearity of the corrector in pointwise sense a key additional input is a conversion of certain weighted energy norms to ordinary ones our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the helffer sjostrand representation of gradient models with certain non strictly convex potentials

policy iteration for customer average performance optimization of closed queueing systems we consider the optimization of queueing systems with service rates depending on system states the optimization criterion is the long run customer average performance which is an important performance metric different from the traditional time average performance we first establish with perturbation analysis a difference equation of the customer average performance in closed networks with exponentially distributed service times and state dependent service rates then we propose a policy iteration optimization algorithm based on this difference equation this algorithm can be implemented on line with a single sample path and does not require knowing the routing probabilities of queueing systems finally we give numerical experiments which demonstrate the efficiency of our algorithm this paper gives a new direction to efficiently optimize the customer centric performance in queueing systems

comparison of quenched and annealed invariance principles for random conductance model we show that there exists an ergodic conductance environment such that the weak annealed invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle does not hold

rank based inference in the proportional hazards model for interval censored data summary a marginal likelihood approach to fitting the proportional hazards model to interval censored or grouped data is proposed this approach maximises a likelihood that is the sum over all rankings of the data that are consistent with the observed censoring intervals as in the usual proportional hazards model the method does not require specification of the baseline hazard function the score equations determining the maximum marginal likelihood estimator can be written as the expected value of the score of the usual proportional hazards model with respect to a certain distribution of rankings a gibbs sampling scheme is given to generate rankings from this distribution and stochastic approximation is used to solve the score equations simulation results under various censoring schemes give point estimates that are close to estimates obtained using actual failure times

t e harris contributions to interacting particle systems and percolation this is a bird s eye view of t e harris work on interacting particle systems and percolation and of its impact on later work in probability theory and mathematical physics nal illness and death was a great loss to me personally and professionally as well as to probability theory in southern california and beyond for three decades he and i were the primary probabilists at usc and ucla respectively our approach to mathematics and our mathematical interests were very similar my wife chris and i enjoyed many wonderful social occasions at the harris home in beverly hills with ted and connie they were great hosts the conversation was always stimulating they were interested in so many things ted initiated the southern california probability symposium in about 1970 it is probably the oldest meeting of its type in the us and continues to provide an exciting place for interactions among southern california probabilists to this day in two of the years 1989 and 2007 the meeting was dedicated to ted the rst was on the occasion of his 70th birthday and the second was in his memory even after his retirement ted attended the usc probability seminar regularly always sitting in the front row and asking perceptive questions few mathematicians have had a greater ratio of number of ideas to number of papers ted wrote many fewer papers 31 than most prominent mathematicians but each is a jewel in each of the several areas in which he worked he was among the

max min optimality of service rate control in closed queueing networks in this technical note we discuss the optimality properties of service rate control in closed jackson networks we prove that when the cost function is linear to a particular service rate the system performance is monotonic w r t with respect to that service rate and the optimal value of that service rate can be either maximum or minimum we call it max min optimality when the second order derivative of the cost function w r t a particular service rate is always positive negative which makes the cost function strictly convex concave the optimal value of such service rate for the performance maximization minimization problem can be either maximum or minimum to the best of our knowledge this is the most general result for the optimality of service rates in closed jackson networks and all the previous works only involve the first conclusion moreover our result is also valid for both the state dependent and load dependent service rates under both the time average and customer average performance criteria

positivity of the self diffusion matrix of interacting brownian particles with hard core we prove the positivity of the self diffusion matrix of interacting brownian particles with hard core when the dimension of the space is greater than or equal to 2 here the self diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle we will do this for all activities z 0 of gibbs measures in particular for large z the case of high density particles a typical example of such a particle system is an infinite amount of hard core brownian balls

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

on homogenization of time dependent random flows we study a diffusion with a random time dependent drift we prove the invariance principle when the spectral measure of the drift satisfies a certain integrability condition this result generalizes the results of 13 7

asymptotic efficiency of the sample mean in markov chain monte carlo schemes in many cases a univariate time series derived from a time reversible stationary markov chain has the property that there is a distribution function on 1 1 0 1 such that for all non negative integers l its lth moment is the ith lag autocorrelation of the time series we discuss some consequences of this property effects of the asymptotic efficiency of the sample mean on the efficiency of subsampling a stationary markov chain are discussed the stationarity assumption is then relaxed we show that under certain regularity conditions the sample mean of a strongly asymptotically stationary process is asymptotically efficient in that it has the smallest mean square error asymptotically in a broad class of linear estimators

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

large deviations for the symmetric simple exclusion process in dimensions d 3 we consider symmetric simple exclusion processes with l macr nd particles in a periodic d dimensional lattice of width n we perform the diffusive hydrodynamic scaling of space and time the initial condition is arbitrary and is typically far away form equilibrium it specifies in the scaling limit a density profile on the d dimensional torus we are interested in the large deviations of the empirical process n d l1 xi as random variables taking values in the space of measures on d 0 1 we prove a large deviation principle with a rate function that is more or less universal involving explicity besides the initial profile only such canonical objects as bulk and self diffusion coefficients

scaling limit for a class of gradient fields with nonconvex potentials we consider gradient fields x x d whose law takes the gibbs boltzmann form z 1 exp x y v v x where the sum runs over nearest neighbors we assume that the potential v admits the representation v log d exp 1 2 2 where is a positive measure with compact support in 0 hence the potential v is symmetric but nonconvex in general while for strictly convex v s the translation invariant ergodic gradient gibbs measures are completely characterized by their tilt a nonconvex potential as above may lead to several ergodic gradient gibbs measures with zero tilt still every ergodic zero tilt gradient gibbs measure for the potential v above scales to a gaussian free field

towards a unified dynamical theory of the brownian particle in an ideal gas we consider the trajectoryqm t of a brownian particle of massm in an ideal gas of identical particles of mass 1 and of density 1 in equilibrium at inverse temperature 1 the dynamics is uniform motion plus elastic collisions with the brownian particle our theory in dimension one describes a variety of limiting processes containing the wiener process and the ornstein uhlenbeck process fora 1 2qm a at depending on the asymptotic behaviour ofm a part of the theory is hypothetical while another part relies upon known results we also prove that ifa1 2 e m a a thena 1 2qm a at converges to a wiener process whose variance is known from papers of sinai soloveichik and of the present authors

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

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

periodic homogenization for inertial particles we study the problem of homogenization for inertial particles moving in a periodic velocity field and subject to molecular diffusion we show that under appropriate assumptions on the velocity field the large scale long time behavior of the inertial particles is governed by an effective diffusion equation for the position variable alone to achieve this we use a formal multiple scale expansion in the scale parameter this expansion relies on the hypo ellipticity of the underlying diffusion an expression for the diffusivity tensor is found and various of its properties studied in particular an expansion in terms of the non dimensional particle relaxation time the stokes number is shown to co incide with the known result for passive non inertial tracers in the singular limit 0 this requires the solution of a singular perturbation problem achieved by means of a formal multiple scales expansion in incompressible and potential fields are studied as well as fields which are neither and theoretical findings are supported by numerical simulations

slowdown and neutral pockets for a random walk in random environment we consider a d dimensional random walk in random environment for which transition probabilities at each site are either neutral or present an effective drift pointing to the right we obtain large deviation estimates on the probability that the walk moves in a too slow ballistic fashion both under the annealed and quenched measures these estimates underline the key role of large neutral pockets of the medium in the occurrence of slowdowns of the walk

nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps we prove an invariance principle for a tagged particle in a simple exclusion process out of equilibrium the scaling limit is a time inhomogeneous process of independent increments related to the solution of a fractional heat equation

asymptotic variance and convergence rates of nearly periodic markov chain monte carlo algorithms this article considers nearly periodic markov chains that may have excellent functional estimation properties but poor distributional convergence rate it shows how simple modifications of the chain involving using a random number of iterations can greatly improve the distributional convergence of the chain various theoretical results about convergence rates of the modified chains are proven a number of examples including a transdimensional markov chain monte carlo example a card shuffling example and several antithetic metropolis algorithms are considered

equilibrium fluctuations for nabla_ varphi interface model we study the large scale space time fluctuations of an interface which is modeled by a massless scalar field with reversible langevin dynamics for a strictly convex interaction potential we prove that on a large space time scale these fluctuations are governed by an infinite dimensional ornstein uhlenbeck process its effective diffusion type covariance matrix is characterized through a variational formula

finite dimensional approximation of the self diffusion coefficient for the exclusion process we show that for the symmetric simple exclusion process on z d the self diffusion coefficient of a tagged particle is stable when approximated by simple exclusion processes on large periodic lattices the proof depends on a similar stability property of the asymptotic variance of additive functionals of mean 0 this requires establishing a property for the dirichlet space known as the liouville d property

noise can speed markov chain monte carlo estimation and quantum annealing carefully injected noise can speed the average convergence of markov chain monte carlo mcmc estimates and simulated annealing optimization this includes quantum annealing and the mcmc special case of the metropolis hastings algorithm mcmc seeks the solution to a computational problem as the equilibrium probability density of a reversible markov chain the algorithm must cycle through a long burn in phase until it reaches equilibrium because the markov samples are statistically correlated the special injected noise reduces this burn in period in mcmc a related theorem shows that it reduces the cooling time in simulated annealing simulations showed that optimal noise gave a 76 speed up in finding the global minimum in the schwefel optimization benchmark the noise boosted simulations found the global minimum in 99 8 of trials compared with only 95 4 of trials in noiseless simulated annealing simulations also showed that the noise boost is robust to accelerated cooling schedules and that noise decreased convergence times by more than 32 under aggressive geometric cooling molecular dynamics simulations showed that optimal noise gave a 42 speed up in finding the minimum potential energy configuration of an eight argon atom gas system with a lennard jones 12 6 potential the annealing speed up also extends to quantum monte carlo implementations of quantum annealing noise improved ground state energy estimates in a 1024 spin simulated quantum annealing simulation by 25 6 the quantum noise flips spins along a trotter ring the noisy mcmc algorithm brings each markov step closer on average to equilibrium if an inequality holds between two expectations gaussian or cauchy jump probabilities reduce the noise benefit inequality to a simple quadratic inequality simulations show that noise boosted simulated annealing is more likely than noiseless annealing to sample high probability regions of the search space and to accept solutions that increase the search breadth

gibbs measures for brownian paths under the effect of an external and a small pair potential we consider brownian motion in the presence of an external and a weakly coupled pair interaction potential and show that its stationary measure is a gibbs measure uniqueness of the gibbs measure for two cases is shown also the typical path behaviour the degree of mixing and some further properties are derived we use cluster expansion in the small coupling parameter

explicit constants for gaussian upper bounds on heat kernels on utilise des inegalites logarithmiques de sobolev pour obtenir des bornes superieures gaussiennes sur les noyaux de la chaleur associes a des operateurs d ordre 2 elliptiques et hypoelliptiques

exponential convergence to equilibrium for a class of random walk models we prove exponential convergence to equilibrium l 2 geometric ergodicity for a random walk with inward drift on a sub cayley rooted tree this randomwalk model generalizes a monte carlo algorithm for the self avoiding walk proposed by berretti and sokal if the number of vertices of leveln in the tree grows asc n n n 1 we prove that the autocorrelation time satisfies n 2 n 1

convergence to fractional kinetics for random walks associated with unbounded conductances we consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail we prove that the scaling limit of this process is a fractional kinetics process that is the time change of a d dimensional brownian motion by the inverse of an independent stable subordinator we further show that the same process appears in the scaling limit of the non symmetric bouchaud s trap model

quantification of ergodicity in stochastic homogenization optimal bounds via spectral gap on glauber dynamics we study quantitatively the effective large scale behavior of discrete elliptic equations on the lattice mathbb z d with random coefficients the theory of stochastic homogenization relates the random stationary and ergodic field of coefficients with a deterministic matrix of effective coefficients this is done via the corrector problem which can be viewed as a highly degenerate elliptic equation on the infinite dimensional space of admissible coefficient fields in this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a spectral gap estimate w r t a glauber dynamics on coefficient fields as it is the case for independent and identically distributed coefficients as a main result we prove an optimal decay in time of the semigroup associated with the corrector problem i e of the generator of the process called random environment as seen from the particle as a corollary we recover existence of stationary correctors in dimensions d 2 and prove new optimal estimates for regularized versions of the corrector in dimensions d ge 2 we also give a self contained proof of a new estimate on the gradient of the parabolic variable coefficient green s function which is a crucial analytic ingredient in our approach as an application of these results we prove the first and optimal estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients

harmonic functions h transform and large deviations for random walks in random environments in dimensions four and higher we consider large deviations for nearest neighbor random walk in a uniformly elliptic i i d environment on d there exist variational formulae for the quenched and averaged rate functions i q and i a obtained by rosenbluth and varadhan respectively i q and i a are not identically equal however when d 4 and the walk satisfies the so called t condition of sznitman they have been previously shown to be equal on an open set a eq for every a eq we prove the existence of a positive solution to a laplace like equation involving and the original transition kernel of the walk we then use this solution to define a new transition kernel via the h transform technique of doob this new kernel corresponds to the unique minimizer of varadhan s variational formula at it also corresponds to the unique minimizer of rosenbluth s variational formula provided that the latter is slightly modified

decentralized control of service rates in a closed jackson network the authors consider a closed jackson network with m nodes the service rate at each node is controllable in a decentralized manner i e it will be controlled based on local information extracted from that node only for each node there is a holding cost and an operating cost it is assumed that both costs are time homogeneous and that the operating cost is a linear function of the service rate however both costs are arbitrary functions of the number of jobs at the node the objective is to minimize the time average expected total cost it is shown that there exists an optimal control characterized by a set of thresholds one for each node such that it is optimal for each node to serve at zero rates if the number of jobs there is below or at the threshold and serve at maximum allowed rates when the number of jobs exceeds the threshold this structure of the optimal control is identified by using the product form solution of the jackson network and the duality theory of linear programming

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

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

einstein relation and steady states for the random conductance model we consider random walk among i i d uniformly elliptic conductances on zdzd and prove the einstein relation see theorem 1 it says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium for fixed bias we show that there is an invariant measure for the environment seen from the particle these invariant measures are often called steady states the einstein relation follows at least for d 3d 3 from an expansion of the steady states as a function of the bias see theorem 2 which can be considered our main result this expansion is proved for d 3d 3 in contrast to guo ann probab 44 2016 324 359 we need not only convergence of the steady states but an estimate on the rate of convergence see theorem 4

perturbation realization potentials and sensitivity analysis of markov processes two fundamental concepts and quantities realization factors and performance potentials are introduced for markov processes the relations among these two quantities and the group inverse of the infinitesimal generator are studied it is shown that the sensitivity of the steady state performance with respect to the change of the infinitesimal generator can be easily calculated by using either of these three quantities and that these quantities can be estimated by analyzing a single sample path of a markov process based on these results algorithms for estimating performance sensitivities on a single sample path of a markov process can be proposed the potentials in this paper are defined through realization factors and are shown to be the same as those defined by poisson equations the results provide a uniform framework of perturbation realization for infinitesimal perturbation analysis ipa and non ipa approaches to the sensitivity analysis of steady state performance they also provide a theoretical background for the pa algorithms developed in recent years

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

heat kernel bounds ancient κ solutions and the poincare conjecture abstract we establish certain gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient solutions to the ricci flow as an application using the w entropy associated with the heat kernel we give a different and much shorter proof of perelman s classification of backward limits of these ancient solutions the method is partly motivated by cao 2007 1 and sesum 2006 27 the current paper or chow and lu 2004 6 combined with chen and zhu 2006 4 and zhang 2009 31 lead to a simplified proof of the poincare conjecture without using reduced distance and reduced volume

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

transience of second class particles and diffusive bounds for additive functionals in one dimensional asymmetric exclusion processes consider a one dimensional exclusion process with finite range translation invariant jump rates with nonzero drift let the process be stationary with product bernoulli invariant distribution at density place a second class particle initially at the origin for the case 1 2 we show that the time spent by the second class particle at the origin has finite expectation this strong transience is then used to prove that variances of additive functionals of local mean zero functions are diffusive when 1 2 as a corollary to previous work we deduce the invariance principle for these functionals the main arguments are comparisons of h 1 norms a large deviation estimate for second class particles and a relation between occupation times of second class particles and additive functional variances

random dirichlet environment viewed from the particle in dimension d ge 3 we consider random walks in random dirichlet environment rwde which is a special type of random walks in random environment where the exit probabilities at each site are i i d dirichlet random variables on zd rwde are parameterized by a 2d tuple of positive reals called weights in this paper we characterize for d 3 the weights for which there exists an absolutely continuous invariant probability distribution for the process viewed from the particle we can deduce from this result and from ann inst henri poincare probab stat 47 2011 1 8 a complete description of the ballistic regime for d 3

central limit theorems for additive functionals of the simple exclusion process definition 2 1 in the above paper is incorrectly stated in the proof of theo rem 2 1 which gives an invariance principle for certain processes satisfying defi nition 2 1 conditions in definition 2 1 are sufficient to deduce finite dimensional convergence but not enough to apply a maximal inequality for demimartingales to obtain tightness the problem is definition 2 1 as stated only considers pair increment associations and not more general associations needed for the demi martingale property we slightly strengthen the definition here in this correction so that the proof of tightness in theorem 2 1 holds details of how this is accom plished are given below by substituting the corrected definition 2 1 for the previous one all results in the article hold as written in particular proposition 2 1 which is the link between theorem 2 1 and the main results and which states certain additive processes sat isfy definition 2 1 holds with the same argument

occupation times of long range exclusion and connections to kpz class exponents with respect to a class of long range exclusion processes on mathbb z d with single particle transition rates of order cdot d alpha starting under bernoulli invariant measure nu rho with density rho we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals part of our motivation is to investigate the dependence on alpha d and rho with respect to the variance of these functionals and associated scaling limits in the case the rates are symmetric among other results we find the scaling limits exhaust a range of fractional brownian motions with hurst parameter h in 1 2 3 4 however in the asymmetric case we study the asymptotics of the variances which when d 1 and rho 1 2 points to a curious dichotomy between long range strength parameters 0 alpha le 3 2 and alpha 3 2 in the former case the order of the occupation time variance is the same as under the process with symmetrized transition rates which are calculated exactly in the latter situation we provide consistent lower and upper bounds and other motivations that this variance order is the same as under the asymmetric short range model which is connected to kpz class scalings of the space time bulk mass density fluctuations

spectral measure and approximation of homogenized coefficients this article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations the originality of this work is the use of a well known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients in particular we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error and conversely combined with results by otto and the first author in low dimension and results by the second author in high dimension this allows us to prove that for any dimension d 2 there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem