invariance of white noise for kdv on the line we consider the korteweg de vries equation with white noise initial data posed on the whole real line and prove the almost sure existence of solutions moreover we show that the solutions obey the group property and follow a white noise law at all times past or future as an offshoot of our methods we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting

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

periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

statistical mechanics of nonlinear wave equations 4 cubic schrodinger the cubic schrodinger equation is considered on the circle both in the de focussing and the focussing case the existence of the flow is proved together with the invariance of the appropriate gibbsian measure namely the petit canonical measure in the defocussing case and the micro canonical measure in the focussing case

a metric between probability distributions on finite sets of different cardinalities and applications to order reduction in this paper we define a metric distance between probability distributions on two distinct finite sets of possibly different cardinalities the metric is defined in terms of a joint distribution on the product of the two sets which has the two given distributions as its marginals and has minimum entropy computing the metric exactly turns out to be np hard therefore an efficient greedy algorithm is presented for finding an upper bound on the distance we then study the problem of optimal order reduction in the metric defined here it is shown that every optimal reduced order approximation must be an aggregation of the original distribution and that optimal reduced order approximation is equivalent to finding an aggregation with maximum entropy this problem also turns out to be np hard so again a greedy algorithm is constructed for finding a suboptimal reduced order approximation taken together all the results presented here permit the approximation of an independent and identically distributed i i d process over a set of large cardinality by another i i d process over a set of smaller cardinality in future work attempts will be made to extend this work to markov processes over finite sets

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

three dimensional navier stokes equations driven by space time white noise abstract in this paper we prove existence and uniqueness of local solutions to the three dimensional 3d navier stokes n s equation driven by space time white noise using two methods first the theory of regularity structures introduced by martin hairer in 16 and second the paracontrolled distribution proposed by gubinelli imkeller perkowski in 12 we also compare the two approaches

asynchronous mappings and asynchronous cellular automata the aim of this paper is the study of asynchronous automata a special kind of automata which encode the independency relation between actions and which enable their concurrent execution these automata introduced by zielonka rairo inform theor appl 21 99 135 1987 constitute a natural extension of finite automata to the case of asynchronous parallelism their behaviour is described by trace languages subsets of partially commutative monoids the main result concerning this class of automata states that they accept exactly all recognizable trace languages in this paper we give new improved constructions of asynchronous automata in the final part of the paper we present a distributed system of messages with bounded time stamps based on asynchronous automata

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

an extension of kleene s and ochman ski s theorems to infinite traces abstract as was noted by mazurkiewicz traces constitute a convenient tool for describing finite behaviour of concurrent systems extending in a natural way mazurkiewicz s original definition infinite traces have recently been introduced enabling one to deal with infinite behaviour of nonterminating concurrent systems in this paper we examine the basic families of recognizable sets and of rational sets of infinite traces the seminal kleene characterization of recognizable subsets of the free monoid and its subsequent extensions to infinite words due to buchi and to finite traces due to ochmanski are the cornerstones of the corresponding theories the main result of our paper is an extension of these characterizations to the domain of infinite traces using recognizing and weakly recognizing morphisms as well as a generalization of the schutzenberger product of monoids we prove various closure properties of recognizable trace languages moreover we establish normal form representations for recognizable and rational sets of infinite traces

aggregation of graph models and markov chains by deterministic annealing we consider the problem of simplifying large weighted directed graphs by aggregating nodes and edges this problem is recast as a clustering resource allocation problem and a solution method that incorporates features of the deterministic annealing da algorithm is proposed the novelty in our method is a quantitive measure of dissimilarity that allows us to compare directed graphs of possibly different sizes i e the original and the aggregated graphs the approach we propose is insensitive to initial conditions and less likely to converge to poor local minima than lloyd type algorithms we apply our graph aggregation clustering method to markov chains where low order markov chains that approximate high order chains are obtained through appropriate aggregation of state transition matrices we further develop a decentralized computational scheme to improve tractability of the algorithm

unique ergodicity for a class of stochastic hyperbolic equations with additive space time white noise in this paper we consider a certain class of second order nonlinear pdes with damping and space time white noise forcing posed on the d dimensional torus this class includes the wave equation for d 1 and the beam equation for d le 3 we show that the gibbs measure is the unique invariant measure for this system since the flow does not satisfy the strong feller property we introduce a new technique for showing unique ergodicity this approach may be also useful in situations in which finite time blowup is possible

mixed product and asynchronous automata abstract we show the interest of the mixed or synchronous product for studying traces and trace languages in the free partially commutative monoids we use this product to construct particular asynchronous automata and we show that each asynchronous automaton is the image by a strictly alphabetic morphism of a mixed product of automata then we use this result and zielonka s theorem 1984 to obtain a result similar to mezei s theorem in the free partially commutative monoids by showing that each recognizable trace language is the homomorphic image of some finite union of synchronous products of recognizable languages

on recognizable subsets of free partially commutative monoids on montre que dans un mono de partiellement commutatif libre engendre par un alphabet fini a le sous ensemble x de a forme des mots equivalents a un des mots de x est reconnaissable si x est reconnaissable et si tout facteur iterant de x a un graphe de non commutation connexe

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

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

optimal kullback leibler aggregation via information bottleneck in this paper we present a method for reducing a regular discrete time markov chain dtmc to another dtmc with a given typically much smaller number of states the cost of reduction is defined as the kullback leibler divergence rate between a projection of the original process through a partition function and a dtmc on the correspondingly partitioned state space finding the reduced model with minimal cost is computationally expensive as it requires an exhaustive search among all state space partitions and an exact evaluation of the reduction cost for each candidate partition our approach deals with the latter problem by minimizing an upper bound on the reduction cost instead of minimizing the exact cost the proposed upper bound is easy to compute and it is tight if the original chain is lumpable with respect to the partition then we express the problem in the form of information bottleneck optimization and propose using the agglomerative information bottleneck algorithm for searching a suboptimal partition greedily rather than exhaustively the theory is illustrated with examples and one application scenario in the context of modeling bio molecular interactions

strong solutions to the stochastic quantization equations we prove the existence and uniqueness of a strong solution of the stochastic quantization equation in dimension 2 for almost all initial data with respect to the invariant measure the method is based on a fixed point result in suitable besov spaces

the kullback leibler divergence rate between markov sources in this work we provide a computable expression for the kullback leibler divergence rate lim sub n spl rarr spl infin 1 nd p sup n spl par q sup n between two time invariant finite alphabet markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p sup n and q sup n respectively we illustrate it numerically and examine its rate of convergence the main tools used to obtain the kullback leibler divergence rate and its rate of convergence are the theory of nonnegative matrices and perron frobenius theory similarly we provide a formula for the shannon entropy rate lim sub n spl rarr spl infin 1 nh p sup n of markov sources and examine its rate of convergence

weighted graph cuts without eigenvectors a multilevel approach a variety of clustering algorithms have recently been proposed to handle data that is not linearly separable spectral clustering and kernel k means are two of the main methods in this paper we discuss an equivalence between the objective functions used in these seemingly different methods in particular a general weighted kernel k means objective is mathematically equivalent to a weighted graph clustering objective we exploit this equivalence to develop a fast high quality multilevel algorithm that directly optimizes various weighted graph clustering objectives such as the popular ratio cut normalized cut and ratio association criteria this eliminates the need for any eigenvector computation for graph clustering problems which can be prohibitive for very large graphs previous multilevel graph partitioning methods such as metis have suffered from the restriction of equal sized clusters our multilevel algorithm removes this restriction by using kernel k means to optimize weighted graph cuts experimental results show that our multilevel algorithm outperforms a state of the art spectral clustering algorithm in terms of speed memory usage and quality we demonstrate that our algorithm is applicable to large scale clustering tasks such as image segmentation social network analysis and gene network analysis

statistical mechanics of nonlinear wave equations 3 metric transitivity for hyperbolic sine gordon mckean and vaninsky proved that the canonical measuree hd q d p based upon the hamiltonian h smallint tfrac 1 2 p 2 tfrac 1 2 q 2 f q dx of the wave equation 2q t2 2q x2 f q 0 with restoring forcef q f q is preserved by the associated flow ofq andp q and they conjectured that metric transitivity prevails always on the whole line and likewise on the circleunless f q q orf q shq here the metric transitivity is proved for the whole line in the second case the proof employs the beautiful d alembert formula of krichever

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

randomly forced cgl equation stationary measures and the inviscid limit we study a complex ginzburg landau cgl equation perturbed by a random force which is white in time and smooth in the space variable x assuming that dim x 4 we prove that this equation has a unique solution and discuss its asymptotic in time properties next we consider the case when the random force is proportional to the square root of the viscosity and study the behaviour of stationary solutions as the viscosity goes to zero we show that under this limit a subsequence of solutions in question converges to a nontrivial stationary process formed by global strong solutions of the nonlinear schrodinger equation

new results on the star problem in trace monoids we prove that if it is decidable whether x is recognizable for a recognizable subset x of a free partially commutative monoid then it is decidable whether a recognizable subset of a free partially commutative monoid possesses the finite power property we prove that if any trace of a set x is connected we can decide whether x possesses the finite power property finally we also show that if x is a finite set containing at most four traces or at most two connected traces it is decidable whether x is recognizable

optimal kullback leibler aggregation via spectral theory of markov chains this paper is concerned with model reduction for complex markov chain models the kullback leibler divergence rate is employed as a metric to measure the difference between the markov model and its approximation for a certain relaxation of the bi partition model reduction problem the solution is shown to be characterized by an associated eigenvalue problem the form of the eigenvalue problem is closely related to the markov spectral theory for model reduction this result is the basis of a heuristic proposed for the m ary partition problem resulting in a practical recursive algorithm the results are illustrated with examples

two dimensional navier stokes equations driven by a space time white noise abstract we study the two dimensional navier stokes equations with periodic boundary conditions perturbed by a space time white noise it is shown that although the solution is not expected to be smooth the nonlinear term can be defined without changing the equation we first construct a stationary martingale solution then we prove that for almost every initial data with respect to a measure supported by negative spaces there exists a unique global solution in the strong probabilistic sense

invariant gibbs measure evolution for the radial nonlinear wave equation on the 3d ball abstract we establish new global well posedness results along gibbs measure evolution for the nonlinear wave equation posed on the unit ball in r 3 via two distinct approaches the first approach invokes the method established in the works bourgain 1994 1996 3 4 5 based on a contraction mapping principle and applies to a certain range of nonlinearities the second approach allows to cover the full range of nonlinearities admissible to treatment by gibbs measure working instead with a delicate analysis of convergence properties of solutions the method of the second approach is quite general and we shall give applications to the nonlinear schrodinger equation on the unit ball in subsequent works bourgain and bulut 2013 10 11

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness

maximal serializability of iterated transactions abstract the serializability condition is usually considered in order to maintain the consistency of a database in the presence of conflicting accesses to the database performed by concurrent transactions this serializability condition is considered herein as a general synchronization problem among transactions or processes which can be iterated infinitely often the behaviour of such a system of transactions is represented by an infinite word over the alphabet of the operations performed by the transactions then a characterization of the prefixes of such behaviours is given and it is shown that satisfying the serializability condition so called correct behaviours is given and it is shown that the set of all correct behaviours can be controlled by a finite automaton as an example the classical dining philosophers problem is shown to be entirely modelled and solved as a serializability problem