m n coherent pairs of linear functionals and jacobi matrices abstract a pair of regular linear functionals u v in the linear space of polynomials with complex coefficients is said to be an m n coherent pair of order m if their corresponding sequences of monic orthogonal polynomials p n x n 0 and q n x n 0 satisfy a structure relation i 0 m a i n p n m i m x i 0 n b i n q n i x n 0 where m n and m are non negative integers a i n n 0 0 i m and b i n n 0 0 i n are sequences of complex numbers such that a m n 0 if n m b n n 0 if n n and a i n b i n 0 if i n when m 1 u v is called an m n coherent pair in this work we give a matrix interpretation of m n coherent pairs of linear functionals indeed an algebraic relation between the corresponding monic tridiagonal jacobi matrices associated with such linear functionals is stated as a particular situation we analyze the case when one of the linear functionals is classical finally the relation between the jacobi matrices associated with m n coherent pairs of linear functionals of order m and the hessenberg matrix associated with the multiplication operator in terms of the basis of monic polynomials orthogonal with respect to the sobolev inner product defined by the pair u v is deduced

reasonable properties for the ordering of fuzzy quantities ii as a continuation of the first part related to the first and second class of ordering approaches this paper deals with the fulfilment of reasonable properties in the third class of ordering approaches to do so we briefly introduce fuzzy relations on which the third class of approaches is based then we recall some transitivity related concepts and an ordering procedure based on a acyclic fuzzy relation acyclicity is a very weak restriction on a fuzzy relation we prove that many fuzzy relations used for the comparison of fuzzy quantities satisfy some conditions stronger than acyclicity so we give a widely applicable formulation to derive a total ranking order from a fuzzy relation with our formulation we examine all the ordering indices in the third class with respect to the proposed axioms in part i

orthogonal polynomials generated by a linear structure relation inverse problem abstract let p n n and q n n be two sequences of monic polynomials linked by a type structure relation such as q n x r n q n 1 x p n x s n p n 1 x t n p n 2 x where r n n s n n and t n n are sequences of complex numbers first we state necessary and sufficient conditions on the parameters such that the above relation becomes non degenerate when both sequences p n n and q n n are orthogonal with respect to regular moment linear functionals u and v respectively second assuming that the above relation is non degenerate and p n n is an orthogonal sequence we obtain a characterization for the orthogonality of the sequence q n n in terms of the coefficients of the polynomials and which appear in the rational transformation in the distributional sense u v some illustrative examples of the developed theory are presented

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

a matrix characterization for the dν semiclassical and dν coherent orthogonal polynomials abstract we present a new structure relation for the sequence of orthogonal polynomials associated with a d semiclassical linear functional of class s and then we use it to obtain a matrix characterization of the d semiclassical orthogonal polynomials in terms of the jacobi matrix associated with the multiplication operator in the basis of orthonormal polynomials and the nonsingular lower triangular matrix that represents the orthogonal polynomials with respect to some bases of polynomials we also provide a matrix characterization of d coherent pairs of linear functionals

statistical properties of functionals of the paths of a particle diffusing in a one dimensional random potential we present a formalism for obtaining the statistical properties of functionals and inverse functionals of the paths of a particle diffusing in a one dimensional quenched random potential we demonstrate the implementation of the formalism in two specific examples 1 where the functional corresponds to the local time spent by the particle around the origin and 2 where the functional corresponds to the occupation time spent by the particle on the positive side of the origin within an observation time window of size t we compute the disorder average distributions of the local time the inverse local time the occupation time and the inverse occupation time and show that in many cases disorder modifies the behavior drastically

approaches to multistage one shot decision making abstract in this research multistage one shot decision making under uncertainty is studied in such a decision problem a decision maker has one and only one chance to make a decision at each stage with possibilistic information based on the one shot decision theory approaches to multistage one shot decision making are proposed in the proposed approach a decision maker chooses one state amongst all the states according to his her attitude about satisfaction and possibility at each stage the payoff at each stage is associated with the focus points at the succeeding stages based on the selected states focus points the sequence of optimal decisions is determined by dynamic programming the proposed method is a fundamental alternative for multistage decision making under uncertainty because it is scenario based instead of lottery based as in the other existing methods the one shot optimal stopping problem is analyzed where a decision maker has only one chance to determine stopping or continuing at each stage the theoretical results have been obtained

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

dynamics of excitation in systems with a randomly modulated decay channel abstract we have investigated the kinetics of excitation in the frame of the conventional pauli master equation for a molecular aggregate with a decay channel attached to one distinguished molecule or site as a principally new feature we assume the rate of probability flow through the decay channel to be a randomly modulated function of time the modulation is described by a markoff stochastic process and we do not invoke the white noise assumption an exact calculation is given for a broad family of stochastic processes which bridges and includes the asymmetric random telegraph signal and the gaussian process the general method is then applied in three physically relevant situations one molecule aggregate cyclic antenna system and infinite linear chain in these models we discuss the site occupation probabilities the aggregate excitation function and the mean de excitation time

the theory of fuzzy stochastic processes abstract in this paper we deal with the general theory of fuzzy stochastic processes we give the suitable definitions of fuzzy random function fuzzy stochastic process and their fall shadow distribution functions the concepts of separability and measurability of a fuzzy random function are introduced and some properties of these new concepts are derived we investigate the continuity of fuzzy sample function of a fuzzy stochastic process

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

mapping multiplicative to additive noise the langevin formulation of a number of well known stochastic processes involves multiplicative noise in this work we present a systematic mapping of a process with multiplicative noise to a related process with additive noise which may often be easier to analyse the mapping is easily understood in the example of the branching process in a second example we study the random neighbour or infinite range contact process which is mapped to an ornstein uhlenbeck process with absorbing wall the present work might shed some light on absorbing state phase transitions in general such as the role of conditional expectation values and finite size scaling and elucidate the meaning of the noise amplitude while we focus on the physical interpretation of the mapping we also provide a mathematical derivation

homogenization of linear and semilinear second order parabolic pdes with periodic coefficients a probabilistic approach we study the limit of the solution of linear and semilinear second order pdes of parabolic type with rapidly oscillating periodic coefficients singular drift and singular coefficient of the zeroth order term our method of proof is fully probabilistic and builds upon the arguments in earlier work in the linear case we use the feynman kac formula to represent the solution of the parabolic pde and in the semilinear case we use an associated backward stochastic differential equation

zeros of sobolev orthogonal polynomials following from coherent pairs let sn denote the monic orthogonal polynomial sequence with respect to the sobolev inner product f g s fgd 0 f g d 1 where d 0 d 1 is a so called coherent pair and 0 then sn has n different real zeros the position of these zeros with respect to the zeros of other orthogonal polynomials in particular laguerre and jacobi polynomials is investigated coherent pairs are found where the zeros of sn 1 separte the zeros of sn

einstein s relation between diffusion constant and mobility for a diffusion model an ornstein uhlenbeck process in a periodic potential inr d is considered it has been shown previously that this process satisfies a central limit theorem in the sense that by rescaling space and time in a suitable way the distribution of the process converges to that of a wiener process with nonsingular diffusion matrix here a rigorous proof is given of a version of einstein s formula for this model relating the diffusion constant to the mobility of the system

random walk on a disordered directed bethe lattice the random walk of a particle on a directed bethe lattice of constant coordinancez is examined in the case of random hopping rates as a result the higher the coordinance the narrower the regions of anomalous drift and diffusion the annealed and quenched mean square dispersions are calculated in all dynamical phases in opposition to the one dimensional z 2 case the annealed and quenched mean quadratic dispersions are shown to be identical in all phases

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

homogenization of ornstein uhlenbeck process in random environment we consider a tracer particle moving in a random environment the velocity of the tracer is modelled by an ornstein uhlenbeck process which takes into account inertia and friction the medium results in a possibly unbounded random potential we prove an invariance principle for this kind of motion the method used is generalized in order to obtain a central limit theorem for a large class of process the most interesting application being a tagged particle in a medium of infinitely many ornstein uhlenbeck particles

on rational transformations of linear functionals direct problem abstract let u be a quasi definite linear functional we find necessary and sufficient conditions in order to the linear functional v satisfying x a u x a v be a quasi definite one also we analyze some linear relations linking the polynomials orthogonal with respect to u and v

m n coherent pairs of order m k and sobolev orthogonal polynomials a pair of regular linear functionals u v is said to be a m n coherent pair of order m k if their corresponding sequences of monic orthogonal polynomials p n x n 0 and q n x n 0 satisfy a structure relation such as i 0ma i np n m i m x i 0nb i nq n k i k x n 0 where a i n and b i n are complex numbers such that a m n 0 if n m b n n 0 if n n and a i n b i n 0 when i n in the first part of this work we focus our attention on the algebraic properties of an m n coherent pair of order m k to be more precise let us assume that m k if m k then u and v are related by a rational factor in the distributional sense if m k then u and v are semiclassical and they are again related by a rational factor in the second part of this work we deal with a sobolev type inner product defined in the linear space of polynomials with real coefficients p as l rp x q x d m 0 x l rp m x q m x d m 1 x p q p where l is a positive real number m is a positive integer number and m 0 m 1 is a m n coherent pair of order m of positive borel measures supported on an infinite subset of the real line meaning that the sequences of monic orthogonal polynomials p n x n 0 and q n x n 0 with respect to m 0 and m 1 respectively satisfy a structure relation as above with k 0 a i n and b i n being real numbers fulfilling the above mentioned conditions we generalize several recent results known in the literature in the framework of sobolev orthogonal polynomials and their connections with coherent pairs introduced in a iserles p e koch s p norsett j m sanz serna on polynomials orthogonal with respect to certain sobolev inner products j approx theory 65 2 1991 151 175 and their extensions in particular we show how to compute the coefficients of the fourier expansion of functions on an appropriate sobolev space defined by the above inner product in terms of the sequence of sobolev orthogonal polynomials s n x l n 0

intermittency in random maps abstract the escape probability for a random walk on a one dimensional lattice is discussed in terms of random maps the global dynamics is found to be intermittent the laminar regions with long sojourn near the upper fixed point are broken by irregular bursts intermittency emerges even if the sinai condition is not satisfied it is shown that if p is the probability of choosing the upper map the average laminar length diverges as 1 p

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

scaling limit for interacting ornstein uhlenbeck processes the problem of describing the bulk behavior of an interacting system consisting of a large number of particles comes up in different contexts see for example 1 for a recent exposition in 4 one of the authors considered the case of interacting diffusions on a circle and proved that the density of particles evolves according to a nonlinear diffusion equation the interacting particles evolved according to a generator that was symmetric in equilibrium in this article we consider interacting ornstein uhlenbeck processes here the diffusion generator is not symmetric relative to the equilibrium and the earlier methods have to be modified considerably we use some ideas that were employed in 3 to extend the central limit theorem from the symmetric to nonsymmetric cases

on linearly related orthogonal polynomials and their functionals let pn be a sequence of polynomials orthogonal with respect a linear functional u and qn a sequence of polynomials defined by pn x snpn 1 x qn x tnqn 1 x we find necessary and sufficient conditions in order to qn be a sequence of polynomials orthogonal with respect to a linear functional v furthermore we prove that the relation between these linear functionals is x a u x a v even more if u and v are linked in this way we get that pn and qn satisfy a formula as above

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

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

diffusion in a one dimensional lattice with random asymmetric transition rates the authors study one dimensional continuous time random walks for which the pairs wn wn 1 of nearest neighbour transition rates are assumed to be independent equally distributed random variables the long time asymptotic behaviour of the mean displacement x t is determined exactly for a specific model system in which diodes u 0 and two way bonds lambda v v occur with probabilities p and 1 p respectively for lambda 1 p we find that x t approximately tnu f beta 1ln t where nu ln 1 p ln lambda and beta ln lambda and where f is a periodic function with period 1 the mean displacement thus not only increases slower than linearly in time but exhibits superimposed non decaying oscillations

on the smoluchowski kramers approximation for a system with an infinite number of degrees of freedom according to the smolukowski kramers approximation we show that the solution of the semi linear stochastic damped wave equations utt t x u t x ut t x b x u t x qopen image in new window t u 0 u0 ut 0 v0 endowed with dirichlet boundary conditions converges as goes to zero to the solution of the semi linear stochastic heat equation ut t x u t x b x u t x qopen image in new window t u 0 u0 endowed with dirichlet boundary conditions moreover we consider relations between asymptotics for the heat and for the wave equation more precisely we show that in the gradient case the invariant measure of the heat equation coincides with the stationary distributions of the wave equation for any 0

a limit theorem in some dynamic fuzzy systems abstract in this paper using a fuzzy relation we define a dynamic fuzzy system with a compact state space by introducing a contraction property we study the limit of a sequence of fuzzy states and obtain a theorem for the existence and uniqueness of the solution of a fuzzy relational equation as an application we deal with a dynamic fuzzy system with a terminal fuzzy gain and give some characterizations for the fuzzy expected gain a numerical example is given to comprehend our idea in this paper

a matrix approach for the semiclassical and coherent orthogonal polynomials we obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials we also provide a matrix characterization for coherent pairs of linear functionals

when do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials given p n n 0 a sequence of monic orthogonal polynomials we analyze their linear combinations with constant coefficients and fixed length i e q n x p n x a 1p n 1 x a kp n k a k 0 n k necessary and sufficient conditions are given for the orthogonality of the sequence q n n 0 an interesting interpretation in terms of the jacobi matrices associated with p n n 0 and q n n 0 is shown moreover in the case k 2 we characterize the families p n n 0 such that the corresponding polynomials q n n 0 are also orthogonal

on the asymptotic behavior of an ornstein uhlenbeck process with random forcing we study the asymptotic behavior of an inertial tracer particle in a random force field we show that there exists a probability measure under which the process describing the velocity and environment seen from the vantage point of the moving particle is stationary and ergodic this measure is equivalent to the underlying probability for the eulerian flow as a consequence of the above we obtain the law of large numbers for the trajectory of the tracer moreover we prove also some decorrelation properties of the velocity of the particle which lead to the existence of a non degenerate asymptotic covariance tensor

ranking and defuzzification methods based on area compensation abstract we present some interesting properties related to the area compensation procedure to compare fuzzy numbers it has been proved that this method produces more than a fuzzy interval order it induces a ranking of fuzzy numbers some further results are given about the transitivity property and about computational aspects extensions to non normal fuzzy numbers and fuzzy quantities are also proposed

dynamical properties of a random walk in a one dimensional random medium abstract the random motion of a particle in a one dimensional continuous random medium with random forces is investigated by making use of the large scale equivalence of the properties of the walk with those of a directed random walk on a discrete lattice when the disorder is weak enough a normal drift diffusion regime takes place whereas for strong disorder anomalous behaviours occur it is shown that these results may be obtained by introducing a renormalized lattice with a finite spacing in the normal phase this spacing appears explicitly in the result while it turns out to be irrelevant in the anomalous ones the dynamical exponents as well as the prefactors of the power laws of the particle position and of its average dispresion are explicitly calculated in the anomalous phases

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

ito versus stratonovich a survey is given of the facts and fancies concerning the nonlinear langevin or ito equation actually it is merely a pre equation which becomes an equation when an interpretation rule is added the rules of ito and stratonovich differ but both are mathematically consistent and therefore equally admissible conventions the reason why they seem to lead to physical differences is that the langevin approach used to arrive at the equation involves a tacit assumption for systems with external noise this assumption can be justified and it is then clear that the stratonovich rule applies systems with internal noise however can only be properly described by a master equation and the ito stratonovich controversy never enters afterward one is free to model the resulting fluctuations either with an ito or a stratonovich scheme but that does not lead to any new information

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

optimal stopping problems in a stochastic and fuzzy system in a stochastic and fuzzy environment two kinds of stopping models are discussed and compared the optimal fuzzy stopping times are given under the assumptions of monotonicity and regularity for stopping rules also we find that fuzzy stopping times are favored in a comparison between fuzzy and classical stopping models

limits of random differential equations on manifolds consider a family of random ordinary differential equations on a manifold driven by vector fields of the form sum ky k alpha k z t epsilon omega where y k are vector fields epsilon is a positive number z t epsilon is a 1 over epsilon mathcal l 0 diffusion process taking values in possibly a different manifold alpha k are annihilators of mathrm ker mathcal l 0 under hormander type conditions on mathcal l 0 we prove that as epsilon approaches zero the stochastic processes y t over epsilon epsilon converge weakly and in the wasserstein topologies we describe this limit and give an upper bound for the rate of the convergence

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

sobolev orthogonal polynomials and m n coherent pairs of measures we introduce the notion of m n coherent pair of measures as a generalization of the concept of coherent pair of measures introduced by iserles et al a iserles p e koch s p norsett j m sanz serna on polynomials orthogonal with respect to certain sobolev inner products j approx theory 65 2 1991 151 175 and subsequently generalized by several authors a pair of measures email protected 0 email protected 1 is called m n coherent if the corresponding orthogonal polynomial sequences p n n and q n n resp satisfy a non zero structure relation such as i 0nr i np n i x email protected i 0ms i nq n i 1 x for all n 0 1 2 where m and n are fixed non negative integer numbers and r i n and s i n are given real parameters satisfying some natural conditions we prove that the regular moment linear functionals associated to an m n coherent pair are semiclassical and they are related by a rational modification in the usual sense of distribution theory we also discuss the converse statement under the assumption that email protected 0 email protected 1 form an m n coherent pair we study the sequence s n l n of the monic orthogonal polynomials with respect to the sobolev inner product l email protected email protected email protected f g email protected 1 defined in the space of all polynomials with real coefficients where l 0 an efficient algorithm is stated to compute the coefficients in the fourier sobolev type series f x email protected n 0 c n ls n l x with respect to l for suitable smooth functions f such that email protected l m 0 2 r and f l m 1 2 r finally some illustrative computational examples are presented

on linearly related sequences of difference derivatives of discrete orthogonal polynomials let be either c 0 or q c 0 1 and let d be the corresponding difference operator defined in the usual way either by d p x p x p x or d q p x p q x p x q 1 x let u and v be two moment regular linear functionals and let p n x n 0 and q n x n 0 be their corresponding orthogonal polynomial sequences ops we discuss an inverse problem in the theory of discrete orthogonal polynomials involving the two ops p n x n 0 and q n x n 0 assuming that their difference derivatives d of higher orders m and k resp are connected by a linear algebraic structure relation such as i 0 m a i n d m p n m i x i 0 n b i n d k q n k i x n 0 where m n m k n 0 a m n 0 for n m b n n 0 for n n and a i n b i n 0 for i n under certain conditions we prove that u and v are related by a rational factor in the distributional sense moreover when m k then both u and v are d semiclassical functionals this leads us to the concept of m n d coherent pair of order m k extending to the discrete case several previous works as an application we consider the ops with respect to the following sobolev type inner product p x r x u p x r x v d m p x d m r x 0 assuming that u and v which eventually may be represented by discrete measures supported either on a uniform lattice if or on a q lattice if q constitute a m n d coherent pair of order m that is an m n d coherent pair of order m 0 m n being fixed

orthogonal polynomials associated with an inverse quadratic spectral transform let p n n 0 be a sequence of monic orthogonal polynomials with respect to a quasi definite linear functional u and q n n 0 a sequence of polynomials defined by q n x p n x s np n 1 x t np n 2 x n 1 with t n 0 for n 2 we obtain a new characterization of the orthogonality of the sequence q n n 0 with respect to a linear functional v in terms of the coefficients of a quadratic polynomial h such that h x v u we also study some cases in which the parameters s n and t n can be computed more easily and give several examples finally the interpretation of such a perturbation in terms of the jacobi matrices associated with p n n 0 and q n n 0 is presented

fuzzy stopping problems in continuous time fuzzy stochastic systems in a continuous time fuzzy stochastic system a stopping model with fuzzy stopping times is presented the optimal fuzzy stopping times are given under an assumption of regularity for stopping rules also the optimal fuzzy reward is characterized as a unique solution of an optimality equation under a differentiability condition an example in the markov models is discussed

ohrnstein uhlenbeck process in a random potential on considere le processus d ohmstein uhlenbeck sur r d r d dans le potentiel aleatoire q x on montre que le processus d echelle x e t ex t e2 obeit au theoreme de la limite centrale en probabilite par rapport a

the effect of boundaries on the spectrum of a one dimensional random mass dirac hamiltonian the average density of states dos of the one dimensional dirac hamiltonian with a random mass on a finite interval 0 l is derived our method relies on the eigenvalue distributions extreme value statistics problem which are obtained explicitly the well known dyson singularity is recovered above the crossover energy below c we find a log normal suppression of the average dos

convection diffusion equation with space time ergodic random flow we prove the homogenization of convection diffusion in a time dependent ergodic incompressible random flow which has a bounded stream matrix and a constant mean drift we also prove two variational formulas for the effective diffusivity as a consequence we obtain both upper and lower bounds on the effective diffusivity

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

microscopic dynamical exponents for random random directed walk on a one dimensional lattice with quenched disorder we demonstrate that the dynamical exponent for the time dependence of the coordinate previously found for an average over disorder is already present in any realization of a given sample this ergodicity comes from the existence of a scaling law for the probability distribution of the parameter defining the asymptotic dynamical regime the self averaging or non self averaging properties of the normal or anomalous phases are direct consequences of this result

generalized coherent pairs abstract a pair of quasi definite moment functionals u0 u1 is a generalized coherent pair if monic orthogonal polynomials pn x n 0 and rn x n 0 relative to u0 and u1 respectively satisfy a relation r n x 1 n 1 p n 1 x n n p n x n 1 n 1 p n 1 x n 2 where n and n are arbitrary constants which may be zero if u0 u1 is a generalized coherent pair then u0 and u1 must be semiclassical we find conditions under which either u0 or u1 is classical in such a case we also determine the types of the companion moment functionals also some illustrating examples and two ways of generating generalized coherent pairs are given we also discuss the corresponding sobolev orthogonal polynomials

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

exact results on sinai s diffusion we study the continuum version of sinai s problem of a random walker in a random force field in one dimension a method of stochastic representation is used to represent various probability distributions in this problem mean probability density function and first passage time distributions this method reproduces already known rigorous results and also confirms directly some recent results derived using approximation schemes we demonstrate clearly in the sinai scaling regime that the disorder dominates the problem and that the thermal distributions tend to zero one laws

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

on linearly related sequences of derivatives of orthogonal polynomials abstract we discuss an inverse problem in the theory of standard orthogonal polynomials involving two orthogonal polynomial families p n n and q n n whose derivatives of higher orders m and k resp are connected by a linear algebraic structure relation such as i 0 n r i n p n i m m x i 0 m s i n q n i k k x for all n 0 1 2 where m and n are fixed nonnegative integer numbers and r i n and s i n are given complex parameters satisfying some natural conditions let u and v be the moment regular functionals associated with p n n and q n n resp assuming 0 m k we prove the existence of four polynomials m m i and n k i of degrees m m i and n k i resp such that d k m m m i u n k i v i 0 1 the k m th derivative as well as the left product of a functional by a polynomial being defined in the usual sense of the theory of distributions if k m then u and v are connected by a rational modification if k m 1 then both u and v are semiclassical linear functionals which are also connected by a rational modification when k m the stieltjes transform associated with u satisfies a non homogeneous linear ordinary differential equation of order k m with polynomial coefficients

an approach to stopping problems of a dynamic fuzzy system we formulate a stopping problem for dynamic fuzzy systems concerning with fuzzy decision environment it could be regarded as a natural fuzzification of non fuzzy stopping problem with a deterministic dynamic system the validity of the approach by cuts of fuzzy sets will be discussed in constructing one step look ahead policy of an optimal fuzzy stopping time a numerical example is given to illustrate the theoretical results and to show a merit to use fuzzy stopping times in dynamic fuzzy systems

dynamical exponents for one dimensional random random directed walks the dynamical exponents of the coordinate and of the mean square displacement are explicitly calculated in the case of a directed random walk on a one dimensional random lattice moreover it is shown that in the dynamical phase where the coordinate increases slower thant the latter is not a self averaging quantity

on an extension of symmetric coherent pairs of orthogonal polynomials given two symmetric and positive definite linear functionals w and v we study the coefficients in the recurrence relation for the system of monic polynomials orthogonal with respect to the second linear functional assuming that the first one is classical and that there exists an algebraic differential relation between these two families of polynomials moreover we determine this companion linear functional as a rational modification of the classical one

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

on the linear functionals associated to linearly related sequences of orthogonal polynomials abstract an inverse problem is solved by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials p n n and q n n respectively in the sense p n x i 1 n r i n p n i x q n x i 1 m s i n q n i x for all n 0 1 2 where r i n and s i n are complex numbers satisfying some natural conditions are connected by a rational modification i e there exist polynomials and with degrees m and n respectively such that u v we also make some remarks concerning the corresponding direct problem stating a characterization theorem in the case n 1 and m 2 as an example we give a linear relation of the above type involving jacobi polynomials with distinct parameters

metrics for probability distributions and the trend to equilibrium for solutions of the boltzmann equation this paper deals with the trend to equilibrium of solutions to the spacehomogeneous boltzmann equation for maxwellian molecules with angular cutoff as well as with infinite range forces the solutions are considered as densities of probability distributions the tanaka functional is a metric for the space of probability distributions which has previously been used in connection with the boltzmann equation our main result is that if the initial distribution possesses moments of order 2 e then the convergence to equilibrium in his metric is exponential in time in the proof we study the relation between several metrics for spaces of probability distributions and relate this to the boltzmann equation by proving that the fourier transformed solutions are at least as regular as the fourier transform of the initial data this is also used to prove that even if the initial data only possess a second moment then v r f v t v 2 dv 0 asr and this convergence is uniform in time

parameter estimation for multiscale diffusions we study the problem of parameter estimation for time series possessing two widely separated characteristic time scales the aim is to understand situations where it is desirable to fit a homogenized single scale model to such multiscale data we demonstrate numerically and analytically that if the data is sampled too finely then the parameter fit will fail in that the correct parameters in the homogenized model are not identified we also show numerically and analytically that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly the ideas are studied in the context of thermally activated motion in a two scale potential however the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data

one dimensional diffusion in a semiinfinite poisson random force we consider the one dimensional diffusion of a particle on a semiinfinite line and in a piecewise linear random potential we first present a new formalism which yields an analytical expression for the green function of the fokker planck equation valid for any deterministic construction of the potential profile the force is then taken to be an asymmetric dichotomic process solving the corresponding energy dependent stochastic riccati equation in the space asymptotic regime we give an exact probabilistic description of returns to the origin this method allows for a time asymptotic characterization of the underlying dynamical phases when the two values taken by the dichotomic force are of different signs there occur trapping potential wells with a broad distribution of trapping times and dynamical phases may appear depending on the man force if both values are negative the time asymptotic mean value of the probability density at the origin is proportional to the absolute value of the mean force if they are both positive traps no longer exist and the dynamics is always normal problems with a shot noise force and with a gaussian white noise force are solved as appropriate limiting cases

random walk in a one dimensional random medium abstract the random walk of a particle in a one dimensional random medium is examined by means of the equivalent transfer rates technique in the discrete as well as in the continuous version of the model the probability distributions of the energy dependent equivalent transfer rates are found analytically either by a matching procedure in the discrete case or exactly in the continuous model both discrete and continuous models are shown to belong to the same universality class the average probability of presence of the particle at its initial point is then computed for a non zero global bias it decreases at large times according to a negative power law with an exponent depending on disorder and bias when there is no global bias sinai s model the decay at large times follows a logarithmic law

anomalous diffusion localization aging and subaging effects in trap models at very low temperature we study in detail the dynamics of the one dimensional symmetric trap model via a real space renormalization procedure which becomes exact in the limit of zero temperature in this limit the diffusion front in each sample consists of two delta peaks which are completely out of equilibrium with each other the statistics of the positions and weights of these delta peaks over the samples allows to obtain explicit results for all observables in the limit t 0 we first compute disorder averages of one time observables such as the diffusion front the thermal width the localization parameters the two particle correlation function and the generating function of thermal cumulants of the position we then study aging and subaging effects our approach reproduces very simply the two different aging exponents and yields explicit forms for scaling functions of the various two time correlations we also extend the real space renormalization group method to include systematic corrections to the previous zero temperature procedure via a series expansion in t we then consider the generalized trap model with parameter alpha in 0 1 and obtain that the large scale effective model at low temperature does not depend on alpha in any dimension so that the only observables sensitive to alpha are those that measure the local persistence such as the probability to remain exactly in the same trap during a time interval finally we extend our approach at a scaling level for the trap model in d 2 and obtain the two relevant time scales for aging properties

exact results and self averaging properties for random random walks on a one dimensional infinite lattice we present new exact results for a one dimensional asymmetric disordered hopping model the lattice is taken infinite from the start and we do not resort to the periodization scheme used by derrida an explicit resummation allows for the calculation of the velocityv and the diffusion constantd which are found to coincide with those given by derrida and for demonstrating thatv is indeed a self averaging quantity the same property is established ford in the limiting case of a directed walk