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

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

schubert polynomials for the affine grassmannian of the symplectic group we study the schubert calculus of the affine grassmannian gr of the symplectic group the integral homology and cohomology rings of gr are identified with dual hopf algebras of symmetric functions defined in terms of schur s p and q functions an explicit combinatorial description is obtained for the schubert basis of the cohomology of gr and this is extended to a definition of the affine type c stanley symmetric functions a homology pieri rule is also given for the product of a special schubert class with an arbitrary one

solving monotone stochastic variational inequalities and complementarity problems by progressive hedging the concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover in particular the optimality conditions for a multistage stochastic programming problem one of the long standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm that approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity thus increasing the range of applications for progressive hedging stochastic complementarity problems as a special case are explored numerically in a linear two stage formulation

on strong anomalous diffusion superdiffusive behavior i e x2 t t2 with 1 2 is in general not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusive regime i e q const 1 2 occurring in random shear flows strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this is due to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling p x t t f x t of the probability density this implies that the effective equation at large scale and long time for p x t obeys neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

stochastic variational inequalities residual minimization smoothing sample average approximations the stochastic variational inequality vi has been used widely in engineering and economics as an effective mathematical model for a number of equilibrium problems involving uncertain data this paper presents a new expected residual minimization erm formulation for a class of stochastic vi the objective of the erm formulation is lipschitz continuous and semismooth which helps us guarantee the existence of a solution and convergence of approximation methods we propose a globally convergent a s smoothing sample average approximation ssaa method to minimize the residual function this minimization problem is convex for the linear stochastic vi if the expected matrix is positive semidefinite we show that the erm problem and its ssaa problems have minimizers in a compact set and any cluster point of minimizers and stationary points of the ssaa problems is a minimizer and a stationary point of the erm problem a s our examples come from applications involving traffic flow problems we show that t

affine stanley symmetric functions we define a new family inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of generating functions for w inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i which are affine analogues of stanley symmetric functions we establish basic properties of these functions including symmetry dominance and conjugation we conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine schur functions as applications we show how affine stanley symmetric functions generalize the dual of the k schur functions of lapointe lascoux and morse as well as the cylindric schur functions of postnikov conjecturally affine stanley symmetric functions should be related to the cohomology of the affine flag variety

the terminal velocity of sedimenting particles in a flowing fluid the influence of an underlying carrier flow on the terminal velocity of sedimenting particles is investigated both analytically and numerically our theoretical framework works for a general class of laminar or turbulent velocity fields and by means of an ordinary perturbation expansion at small stokes number leads to closed partial differential equations pde whose solutions contain all relevant information on the sedimentation process the set of pde is solved by means of direct numerical simulations for a class of 2d cellular flows static and time dependent and the resulting phenomenology is analysed and discussed

pieri rule for the affine flag variety abstract we prove the affine pieri rule for the cohomology of the affine flag variety conjectured by lam lapointe morse and shimozono we study the cap operator on the affine nilhecke ring that is motivated by kostant and kumar s work on the equivariant cohomology of the affine flag variety we show that the cap operators for pieri elements are the same as pieri operators defined by berg saliola and serrano this establishes the affine pieri rule we also discuss properties of cap operators which are not necessarily affine a type

heavy particles in incompressible flows the large stokes number asymptotics abstract the dynamics of very heavy particles suspended in incompressible flows is studied in the asymptotics in which their response time is much larger than any characteristic time of fluid motion in this limit of very large stokes numbers particles behave as if suspended in a correlated in time gaussian flow at those spatial scales where the fluid velocity field is smooth following piterbarg l i piterbarg the top lyapunov exponent for stochastic flow modeling the upper ocean turbulence siam j appl math 62 2002 777 and mehlig et al b mehlig m wilkinson k duncan t weber m ljunggren aggregation of inertial particles in random flows phys rev e 72 2005 051104 the two particle dynamics is reduced to a nonlinear system of three stochastic differential equations with additive noise this model is used to single out the mechanisms leading to the preferential concentration of particles scaling arguments are used to predict the large stokes number behavior of the distribution of the stretching rate and of the probability distribution function of the longitudinal velocity difference between two particles as for the fractal character of the particle distribution strong numerical evidence is obtained in favor of saturation of the correlation dimension to the space dimension at large stokes numbers numerical results at finite stokes number values reveal that this model catches some important qualitative features of particle clustering observed in more realistic flows

numerical study of strong anomalous diffusion the superdiffusion behavior i e x2 t t2 with 1 2 in general is not completely characterized by a unique exponent we study some systems exhibiting strong anomalous diffusion i e x t q tq q where 2 1 2 and q q is not a linear function of q this feature is different from the weak superdiffusion regime i e q const 1 2 as in random shear flows the strong anomalous diffusion can be generated by nontrivial chaotic dynamics e g lagrangian motion in 2d time dependent incompressible velocity fields 2d symplectic maps and 1d intermittent maps typically the function q q is piecewise linear this corresponds to two mechanisms a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions in order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem this happens only in a very narrow region of the control parameters space in the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling of the probability distribution i e p x t t f x t this implies that the effective equation at large scale and long time for p x t can obey neither the usual fick equation nor other linear equations involving temporal and or spatial fractional derivatives

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

a k tableau characterization of k schur functions we study k schur functions characterized by k tableaux proving combinatorial properties such as a k pieri rule and a k conjugation this new approach relies on developing the theory of k tableaux and includes the introduction of a weight permuting involution on these tableaux that generalizes the bender knuth involution this work lays the groundwork needed to prove that the set of k schur littlewood richardson coefficients contains the 3 point gromov witten invariants structure constants for the quantum cohomology ring

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

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

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

schubert polynomials for the affine grassmannian let g be a complex simply connected simple group and k a maximal compact subgroup let f c t denote the field of formal laurent series and o c t the ring of formal power series the quotient qr g f g 0 is known as the affine grassmannian of g it is known that qr is homotopy equivalent to the space of based loops k into the compact group k see 9 30 the multiplication on qk endows the homology h qr and cohomology h qr with structures of dual hopf algebras these hopf algebras were first identified by bott 4 another descrip tion is given by ginzburg 10 which we shall not use the bruhat decomposition of g f induces a stratification of qr by schubert cells ilw indexed by grassman nian elements w of the affine weyl group waffwe denote by aw g h qr and aw g h qr the schubert classes in homology and cohomology when g sl n c bott identifies h qr and h qr with a subring an and a quotient an of the ring of symmetric functions our main theorem theorem 7 1 identifies the schubert classes aw g qr and aw g h qr as explicit symmet ric functions under this isomorphism these symmetric functions are combinatori ally defined in homology the aw g h qr are represented by lascoux lapointe morse s fc schur functions sw x g an and in cohomology the aw g h qr are represented by the dual fc schur functions or affine schur functions fw x g an see 21 24 17 our theorem was originally conjectured by mark shimozono the conjecture was made explicit in the cohomology case by jennifer morse kostant and kumar 14 studied the topology of homogeneous spaces for arbi trary kac moody groups and in particular calculated the structure constants of h qr in the schubert basis using the algebraic construction of the nilhecke ring our connection between the topology and the combinatorics proceeds via the study of three subalgebras of the nilhecke ring the first subalgebra is the nilcoxeter al gebra ao which is the algebra generated by the divided difference operators the second algebra is a certain centralizer algebra aaff s which we call the peterson subalgebra peterson 28 constructs an isomorphism j from the t equivariant ho mology hr qr of the affine grassmannian to z ff s the third algebra b is a combinatorially defined subalgebra of a0 which we call the affine fomin stanley

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

two stage quadratic games under uncertainty and their solution by progressive hedging algorithms a model of a two stage n person noncooperative game under uncertainty is studied in which at the first stage each player solves a quadratic program parameterized by other players decisions and