on simultaneous linearization of diffeomorphisms of the sphere let r1 r2 rm be rotations generating sod 1 d 2 and f1 f2 fm be their small smooth perturbations we show that f can be simultaneously linearized if and only if the associated random walk has zero lyapunov exponents as a consequence we obtain stable ergodicity of actions of random rotations in even dimensions 1 main results let f1 f2 fm be diffeomorphisms of s d 2 let i i n be a sequence of independent random variables uniformly distributed on 1 m consider the markov process on s 1 xn f nxn 1 if is an invariant measure for this process let 1 2 d be the lyapunov exponents of denote

coupling control variates for markov chain monte carlo we show that markov couplings can be used to improve the accuracy of markov chain monte carlo calculations in some situations where the steady state probability distribution is not explicitly known the technique generalizes the notion of control variates from classical monte carlo integration we illustrate it using two models of nonequilibrium transport

information entropy of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy e n of the classical orthogonal polynomials e email protected p 2 n x logp 2 n x w x dx here an explicit formula for the entropy of general orthogonal polynomials is obtained in terms of the eigenvalues of jacobi matrix associated with the orthogonal polynomials p n x and the weighted moments of p 2 n x m n email protected x kp 2 n x w x dx k 1 2

transition functions paths and path integrals the main theme of this expository paper is the relation between analysis and probability in the context of diffusion theory section 1 discusses in rather heuristic fashion the very satisfying solution to the problem of describing diffusion processes which kolmogorov achieved via pde theory the theory of partial differential equations and his criterion for path continuity section 2 describes how ito calculus totally transformed the subject by allowing us to construct the sample paths of a diffusion process x by solving an sde stochastic differential equation driven by brownian motion of course sdes have great intrinsic importance too as noisy perturbations of nonlinear dynamical systems though 2 begins heuristically the mathematics is then tightened up this paper is after all a tribute to the man whose greatest contribution to science is his setting probability theory on a rigorous foundation once kolmogorov s precise language is available 2 then takes a quick sight seeing trip through some of the great developments by doob ito and their successors in an age in which so many do simulations of ito equations i have explained precisely in the briefest possible fashion what the exact theory is it is easy and usable you will just have time for a snapshot of how brownian motion on the orthonormal frame bundle is linked to index theorems and of what the malliavin calculus is about you will however be advised on guide books on these and other areas including physicists favourites large deviations measure valued diffusions etc so that you can later explore at your leisure confession the paper consists very largely of selected tracks remixed from the album rogers williams 1987 diffusions markov processes and martingales chichester wiley tributes to kolmogorov s work in probability and statistics have appeared in every book ever written on probability and in ann probability 17 1989 815 964 ann statist 18 1990 987 1031 bull lond math soc 22 1990 31 100 teor veroyatnost primenen 34 1989 no 1 usp mat nauk 43 1988 no 6 the official biography by shiryaev will be a wonderful volume for me this paper is a further expression of my thanks to kolmogorov and as he would have wished to levy doob and ito too

where does the norm of a weighted polynomial live a characterization is given of the sets supporting the uniform norms of weighted polynomials w x npn x wherepn is any polynomial of degree at mostn the closed support ofw x may be bounded or unbounded of special interest is the case whenw x has a nonempty zero setz the treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of z one main result of this paper states that there is a unique compact set independent ofn andpn maximizing this functional that contains the points where the norms of weighted polynomials are attained the distribution of the zeros of chebyshev polynomials corresponding to the weights w x n is also studied the main theorems give a unified method of investigating many particular examples applications to weighted approximation on the real line with respect to a fixed weight are included

convergence rate of pade type approximants for stieltjes functions for a wide class of stieltjes functions we estimate the rate of convergence of pade type approximants when the number of fixed poles represents a fixed proportion with respect to the order of the rational approximant

entropic integrals of hyperspherical harmonics and spatial entropy of d dimensional central potentials the information entropy of a single particle in a quantum mechanical d dimensional central potential is separated in two parts one depends only on the specific form of the potential radial entropy and the other depends on the angular distribution spatial entropy the latter is given by an entropic like integral of the hyperspherical harmonics which is expressed in terms of the entropy of the gegenbauer polynomials this entropy is expressed in terms of the values of the quadratic logarithmic potential of gegenbauer polynomials cn t at the zeros of these polynomials then this potential for integer is given as a finite expansion of chebyshev polynomials of even order whose coefficients are shown to be wilson polynomials

polymer transport in random flow the dynamics of polymers in a random smooth flow is investigated in the framework of the hookean dumbbell model the analytical expression of the time dependent probability density function of polymer elongation is derived explicitly for a gaussian rapidly changing flow when polymers are in the coiled state the pdf reaches a stationary state characterized by power law tails both for small and large arguments compared to the equilibrium length the characteristic relaxation time is computed as a function of the weissenberg number in the stretched state the pdf is unstationary and exhibits multiscaling umerical simulations for the two dimensional navier stokes flow confirm the relevance of theoretical results obtained for the correlated model

entropy formula for random transformations we exhibit random strange attractors with random sinai bowen ruelle measures for the composition of independent random diffeomorphisms

integration of brownian vector fields summary using the wiener chaos decomposition we show that strong solutions of non lipschitzian s d e s are given by random markovian kernels the example of sobolev flows is studied in some detail exhibiting interesting phase transitions

superprocesses of stochastic flows we construct a continuous superprocess x on m r d which is the unique weak feller extension of the empirical process of consistent k point motions generated by a family of differential operators the process x differs from known dawson watanabe type fleming viot type and ornstein uhlenbeck type superprocesses this new type of superprocess provides a connection between stochastic flows and measure valued processes and determines a stochastic coalescence which is similar to those of smoluchowski moreover the support of x describes how an initial measure on r d is transported under the flow as an example the process realizes a viewpoint of darling about the isotropic stochastic flows under certain conditions

dimension formula for random transformations we consider compositions of random diffeomorphisms and show that the dimension of sample measures equals lyapunov dimension as conjectured in the nonrandom case by yorke et al

multiple regimes of diffusion we consider the diffusion of independent particles experiencing random accelerations by a space and time dependent force as well as viscous damping this model can exhibit several asymptotic behaviors depending upon the limiting cases which are considered some of which have been discussed in earlier work here we explore the full space of dimensionless parameters and present an asymptotic phase diagram which delineates the limiting regimes

information entropy of gegenbauer polynomials the information entropy of gegenbauer polynomials is relevant since this is related to the angular part of the information entropies of certain quantum mechanical systems such as the harmonic oscillator and the hydrogen atom in d dimensions we give an effective method to compute the entropy for gegenbauer polynomials with an integer parameter and obtain the first few terms in the asymptotic expansion as the degree of the polynomial tends to infinity

ergodicity of a measure valued markov chain induced by random transformations let z1 z2 be a sequence of i i d random transformations possibly discontinuous of a compact metric space m and let e denote the space of normalized mass distributions on m given in e let n denote the random measure z n z1 1 when well defined we construct the transition probability p of the e valued markov chain n and give a necessary and sufficient condition for p to have a unique invariant measure concentrated on the degenerate mass distributions convergence to statistical equilibrium of the associated discrete time stochastic flow is investigated

modulus of continuity of the canonic brownian motion on the group of diffeomorphisms of the circle abstract an explicit modulus of holder continuity is given for the flow associated to the canonic brownian motion on the diffeomorphism group of the circle

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

sample path properties of the stochastic flows we consider a stochastic flow driven by a finite dimensional brownian motion we show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the equilibrium state and the central limit theorem the proof uses new estimates of the mixing rates of the multi point motion

kraichnan flow in a square an example of integrable chaos the kraichnan flow provides an example of a random dynamical system accessible to an exact analysis we study the evolution of the infinitesimal separation between two lagrangian trajectories of the flow its long time asymptotics is reflected in the large deviation regime of the statistics of stretching exponents whereas in the flow that is isotropic at small scales the distribution of such multiplicative large deviations is gaussian this does not have to be the case in the presence of an anisotropy we analyze in detail the flow in a two dimensional periodic square where the anisotropy generally persists at small scales the calculation of the large deviation rate function of the stretching exponents reduces in this case to the study of the ground state energy of an integrable periodic schrodinger operator of the lame type the underlying integrability permits to explicitly exhibit the non gaussianity of the multiplicative large deviations and to analyze the time scales at which the large deviation regime sets in in particular we indicate how the divergence of some of those time scales when the two lyapunov exponents become close allows a discontinuity of the large deviation rate function in the parameters of the flow the analysis of the two dimensional anisotropic flow permits to identify the general scenario for the appearance of multiplicative large deviations together with the restrictions on its applicability

isotropic stochastic flows on considere les flots stochastiques isotropes dans un espace euclidien de d dimensions d 2 on relie la tendance des distances entre deux points et des vecteurs tangents a se borner ou s etendre a la dimension et a la proportion du flot solenoidal ou potentiel

on sobolev orthogonal polynomials with coherent pairs the laguerre case type 1 we study the asymptotic properties of orthogonal polynomials with sobolev inner product formula the pair d d is called a coherent pair if there exists nonzero constantsdnsuch that formula wherepn x andqn x are thenth monic orthogonal polynomials with respect tod andd respectively one can divide the coherent pairs into two cases the jacobi case and the laguerre case there are two types in each case we consider thenth root asymptotics and the zero distribution for the laguerre case type 1

t e harris s contributions to recurrent markov processes and stochastic flows this is a brief survey of t e harris s work on recurrent markov processes and on stochastic flows and of some more recent work in these fields

asymptotic behaviour of stochastic flows of diffeomorphisms two case studies the two stochastic flows studied are i the canonical stochastic flow on the orthonormal frame bundle of hyperbolic space which gives stochastic parallel translation along brownian paths in hyperbolic space and ii a stochastic flow on the spheres n 1 arising from its embedding as the unit sphere in n both flows are controlled by the same stochastic differential equation in a finite dimensional lie group in each case the lyapunov exponents are computed and a complete description is given of the local and global stability of the flow

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

entropic integrals of orthogonal hypergeometric polynomials with general supports the boltzmann shannon information entropy of probability measures which involve the continuous hypergeometric type polynomials fpn x g orthogonal with respect to a general weight function x is determined by two integral quantities one with kernel p 2 x x lnp 2 x called as entropy of the polynomial pn x and another one with kernel p 2 x x ln x here an explicit expression for the latter quantity and for a broader family of related integrals is obtained in terms only of the second order dierential equation satised by the involved polynomials for illustration the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type hermite laguerre and jacobi c 2000 elsevier science b v all rights reserved

clustering and coalescence from multiplicative noise the kraichnan ensemble we study the dynamics of the two point statistics of the kraichnan ensemble which describes the transport of a passive pollutant by a stochastic turbulent flow characterized by scale invariant structure functions the fundamental equation of this problem consists of the fokker planck equation for the two point correlation function of the density of particles performing spatially correlated brownian motions with scale invariant correlations this problem is equivalent to the stochastic motion of an effective particle driven by a generic multiplicative noise in this paper we propose an alternative and more intuitive approach to the problem than the original one gawedzki and vergassola 2000 physica d 138 63 leading to the same conclusions the general features of this new approach make possible to fit it to other more complex contexts

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

a mathematical framework for stochastic climate models there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction here a systematic mathematical strategy for stochastic climate modeling is developed and some of the new phenomena in the resulting equations for the climate variables alone are explored the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances all of these phenomena are derived from a systematic self consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit the theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated the feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography explicit concrete examples with the new phenomena are presented for the stochastically forced three mode interaction equations the conjecture of smith and waleffe phys fluids 11 1999 1608 1622 for stochastically forced three wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self consistency of the mathematical approach with the predictions of equilibrium statistical mechanics in particular for these examples the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables c 2001 john wiley sons inc

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

existence and uniqueness for stochastic 2d euler flows with bounded vorticity the strong existence and the pathwise uniqueness of solutions with l infty vorticity of the 2d stochastic euler equations are proved the noise is multiplicative and it involves the first derivatives a lagrangian approach is implemented where a stochastic flow solving a nonlinear flow equation is constructed the stability under regularizations is also proved

asymptotic support theorem for planar isotropic brownian flows it has been shown by various authors that the diameter of a given nontrivial bounded connected set x grows linearly in time under the action of an isotropic brownian flow ibf which has a nonnegative top lyapunov exponent in case of a planar ibf with a positive top lyapunov exponent the precise deterministic linear growth rate k of the diameter is known to exist in this paper we will extend this result to an asymptotic support theorem for the time scaled trajectories of a planar ibf which has a positive top lyapunov exponent starting in a nontrivial compact connected set x r2 that is we will show convergence in probability of the set of time scaled trajectories in the hausdorff distance to the set of lipschitz continuous functions on 0 1 starting in 0 with lipschitz constant k

intermittency of the tracer gradient the problem of stirring a passive scalar tracer by a random velocity field is considered for gaussian velocity fields with infinitely small time and space scales it is shown that the tracer gradient is concentrated in a discrete set of points the points that form a random point process independent of the initial tracer distribution a complete description of this point process is given if the initial tracer field is a random function with homogeneous increments a full statistics of the jumps is also given

isotropic brownian motions over complex fields as a solvable model for may wigner stability analysis we consider matrix valued stochastic processes known as isotropic brownian motions and show that these can be solved exactly over complex fields while these processes appear in a variety of questions in mathematical physics our main motivation is their relation to a may wigner like stability analysis for which we obtain a stability phase diagram the exact results establish the full joint probability distribution of the finite time lyapunov exponents and may be used as a starting point for a more detailed analysis of the stability instability phase transition our derivations rest on an explicit formulation of a fokker planck equation for the lyapunov exponents this formulation happens to coincide with an exactly solvable class of models of the calgero sutherland type originally encountered for a model of phase coherent transport the exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices and is also closely related to hermitian matrix models with an external source

logarithmic potential of hermite polynomials and information entropies of the harmonic oscillator eigenstates the problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential vn t hn x 2 ln x t e x2 dx at the zeros of the hermite polynomial hn x here a closed analytical expression for vn t is obtained which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of hn x is known an inequality for the values of vn t at the zeros of hn x is conjectured which leads to a new nonvariational upper bound for the entropies finally the exact formula for vn t is written in an alternative way which allows the entropies to be expressed in terms of the even order spectral moments of the hermite polynomials the asymptotic n 1 limit of this alternative expression for the entropies is discussed and the conjectured upper bound for the entropies is proved to be asymptotically valid

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

brownian motions on the homeomorphisms of the plane let z denote a point of r 2 we study random flows z st z in r 2 0 leq s leq t infty z in r 2 z tu z st z z su z for s leq t leq u such flows are called brownian if z is continuous in s t z and has appropriate spatial and temporal homogeneity properties and if z st z uv cdots are independent homeomorphisms of r 2 onto r 2 when s leq t leq u leq v leq cdots for a brownian flow the coordinates of any k points are a 2k dimensional continuous markov process k 1 2 cdots if these processes are diffusions whose diffusion matrices have bounded continuous derivatives of order leq 2 i e are c 2 bounded then the diffusion matrices are necessarily obtained in a certain way from the covariance tensor of the field of infinitesimal displacements a converse is given in the incompressible isotropic case given a c 2 bounded covariance tensor of an isotropic solenoidal r 2 valued field in r 2 there exists a corresponding incompressible isotropic brownian flow

the weighted lp norms of orthonormal polynomials for erdos weights let w e q where q r r is even and smooth and of faster than polynomial growth at infinity for example we consider q x exp l x 1 where exp exp exp exp denotes the kth iterated exponential weights of the form w2 for such w are often called erdos weights we compute the growth of the lp norms 0 p of the weighted orthonormal polynomials pn w2 x w x for a large class of erdos weights based on recent work of the author with levin and mthembu on the l norm of pn w2 x w x as an auxiliary result we obtain bounds on the fundamental polynomials of lagrange interpolation at the zeros of pn w2 x and as a corollary we deduce finer spacing for the zeros of pn w2 the growth of the lp norms of orthonormal polynomials is a key factor in investigating convergence of orthogonal expansions and lagrange interpolation in lp norms

infinite products of random isotropically distributed matrices statistical properties of infinite products of random isotropically distributed matrices are investigated both for continuous processes with finite correlation time and discrete sequences of independent matrices a formalism that allows to calculate easily the lyapunov spectrum and generalized lyapunov exponents is developed this problem is of interest to probability theory statistical characteristics of matrix t exponentials are also needed for turbulent transport problems dynamical chaos and other parts of statistical physics

on isotropic brownian motions this paper includes a detailed study of isotropic brownian motion on matrices and brownian flows associated with isotropic gaussian fields characteristic exponents stability asymptotic behaviour statistical equilibrium are the topics on which the main results are obtained

distribution of velocity gradients and rate of caustic formation in turbulent aerosols at finite kubo numbers in a one dimensional model for a turbulent aerosol inertial particles suspended in a random flow we compute the distributions of particle velocity gradients and the rate of caustic formation at finite but small kubo numbers ku for arbitrary stokes numbers st our results are consistent with those obtained earlier in the limit ku 0 and st such that ku 2 st remains constant we show how finite time correlations and nonergodic effects influence the inertial particle dynamics at finite but small kubo numbers

entropy of orthogonal polynomials with freud weights and information entropies of the harmonic oscillator potential the information entropy of the harmonic oscillator potential v x 1 2 x2 in both position and momentum spaces can be expressed in terms of the so called entropy of hermite polynomials i e the quantity sn h h2n x log h2n x e x2dx these polynomials are instances of the polynomials orthogonal with respect to the freud weights w x exp x m m 0 here a very precise and general result of the entropy of freud polynomials recently established by aptekarev et al j math phys 35 4423 4428 1994 specialized to the hermite kernel case m 2 leads to an important refined asymptotic expression for the information entropies of very excited states i e for large n in both position and momentum spaces to be denoted by s and s respectively briefly it is shown that for large values of n s 1 2log log 2n e o 1 and s 1 2log log 2n e o 1 so that s s log 2 2n e2 o 1 in agreement with the generalized indetermination relation of byalinicki birula and mycielski commun m

on geometric properties of stochastic flows related to the lyapunov spectrum we study the geometric properties of two stochastic flows on spheres in euclidean space the underlying one point motion in both cases is brownian both flows arise from the action of a lie group valued brownian motion on a quotient for both flows the curvature of a curve moving under the flow is shown to be a diffusion null recurrent in one case and transient in the other

polymer stretching in the inertial range of turbulence we study the deformation of flexible polymers whose contour length lies in the inertial range of a homogeneous and isotropic turbulent flow by using the elastic dumbbell model and a stochastic velocity field with nonsmooth spatial correlations we obtain the probability density function of the extension as a function of the weissenberg number and of the scaling exponent of the velocity structure functions in a spatially rough flow as in the inertial range of turbulence the statistics of polymer stretching differs from that observed in laminar flows or in smooth chaotic flows in particular the probability distribution of polymer extensions decays as a stretched exponential and the most probable extension grows as a power law of the weissenberg number furthermore the ability of the flow to stretch polymers weakens as the flow becomes rougher in space

generalized ornstein uhlenbeck processes we solve a physically significant extension of a classic problem in the theory of diffusion namely the ornstein uhlenbeck process ornstein and uhlenbeck phys rev 36 823 1930 our generalized ornstein uhlenbeck systems include a force which depends upon the position of the particle as well as upon time they exhibit anomalous diffusion at short times and non maxwellian velocity distributions in equilibrium two approaches are used some statistics are obtained from a closed form expression for the propagator of the fokker planck equation for the case where the particle is initially at rest in the general case we use spectral decomposition of a fokker planck equation employing nonlinear creation and annihilation operators to generate the spectrum which consists of two staggered ladders

chasing balls through martingale fields we consider the way sets are dispersed by the action of stochastic flows derived from martingale fields under fairly general continuity and ellipticity conditions the following dichotomy result is shown any nontrivial connected set x either contracts to a point under the action of the flow or its diameter grows linearly in time with speed at least a positive deterministic constant a the linear growth may further be identified again almost surely with a much stronger behavior which we call ball chasing if is any path with lipschitz constant smaller than a the ball of radius e around t contains points of the image of x for an asymptotically positive fraction of times t if the ball grows as the logarithm of time there are individual points in x whose images eventually remain in the ball

quadrature formulas based on rational interpolation we consider quadrature formulas based on interpolation using the ba sis functions 1 1 tkx k 1 2 3 on 1 1 where tk are parameters on the interval 1 1 we investigate two types of quadratures quadrature formulas of maximum accuracy which correctly integrate as many basis functions as possible gaussian quadrature and quadrature formulas whose nodes are the zeros of the orthogonal functions obtained by orthogonalizing the system of basis functions or thogonal quadrature we show that both approaches involve orthogonal polynomials with modified or varying weights which depend on the number of quadrature nodes the asymptotic distribution of the nodes is obtained as well as various interlacing properties and monotonicity results for the nodes

lyapunov exponent for small particles in smooth one dimensional flows this paper discusses the lyapunov exponent for small particles in a spatially and temporally smooth flow in one dimension the lyapunov exponent is obtained as a series expansion in the stokes number st which is a dimensionless measure of the importance of inertial effects the approach described here can be extended to calculations of the lyapunov exponents and of the correlation dimension for inertial particles in higher dimensions it is shown that there is a correction to this theory which arises because the particles do not sample the velocity field ergodically using this non ergodic correction it is found that contrary to expectations the first order term in the expansion of does not vanish

tate shafarevich groups and k3 surfaces this paper explores a topic taken up recently by logan and van luijk finding nontrivial 2 torsion elements of the tate shafarevich group of the jacobian of a genus 2 curve by exhibiting brauer manin obstructions to rational points on certain quotients of principal homogeneous spaces of the jacobian whose desingularizations are explicit k3 surfaces the main difference between the methods used in this paper and those of logan and van luijk is that the obstructions are obtained here from explicitly constructed quaternion algebras rather than elliptic fibrations

nontrivial elements of sha explained through k3 surfaces we present a new method to show that a principal homogeneous space of the jacobian of a curve of genus two is nontrivial the idea is to exhibit a brauer manin obstruction to the existence of rational points on a quotient of this principal homogeneous space in an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose jacobians have nontrivial tate shafarevich group

perturbation theory for a stochastic process with ornstein uhlenbeck noise the ornstein uhlenbeck process may be used to generate a noise signal with a finite correlation time if a one dimensional stochastic process is driven by such a noise source it may be analysed by solving a fokker planck equation in two dimensions in the case of motion in the vicinity of an attractive fixed point it is shown how the solution of this equation can be developed as a power series the coefficients are determined exactly by using algebraic properties of a system of annihilation and creation operators

spatial entropy of central potentials and strong asymptotics of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy sn of the classical orthogonal polynomials here an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained it is shown that this entropy is intimately related to the relative entropy i 0 of the equilibrium measure 0 x and the weight function x of the polynomials to do so the theory of strong asymptotics of orthogonal polynomials on compact sets is used

strong asymptotics of laguerre polynomials and information entropies of two dimensional harmonic oscillator and one dimensional coulomb potentials the information entropies of the two dimensional harmonic oscillator v x y 1 2 x2 y2 and the one dimensional hydrogen atom v x 1 x can be expressed by means of some entropy integrals of laguerre polynomials whose values have not yet been analytically determined here we first study the asymptotical behavior of these integrals in detail by extensive use of strong asymptotics of laguerre polynomials then this result which is also important by itself in a context of both approximation theory and potential theory is employed to analyze the information entropies of the aforementioned quantum mechanical potentials for the very excited states in both position and momentum spaces it is observed in particular that the sum of position and momentum entropies has a logarithmic growth with respect to the main quantum number which characterizes the corresponding physical state finally the rate of convergence of the entropies is numerically examined

calculating effective diffusivities in the limit of vanishing molecular diffusion in this paper we study the problem of the numerical calculation by monte carlo methods of the effective diffusivity for a particle moving in a periodic divergent free velocity field in the limit of vanishing molecular diffusion in this limit traditional numerical methods typically fail since they do not represent accurately the geometry of the underlying deterministic dynamics we propose a stochastic splitting method that takes into account the volume preserving property of the equations of motion in the absence of noise and when inertial effects can be neglected an extension of the method is then proposed for the cases where the noise has a non trivial time correlation structure and when inertial effects cannot be neglected the method of modified equations is used to explain failings of euler based methods the new stochastic geometric integrators are shown to outperform standard euler based integrators various asymptotic limits of physical interest are investigated by means of numerical experiments using the new integrators

asymptotic upper bounds for the entropy of orthogonal polynomials in the szegő class we give an asymptotic upper bound as n for the entropy integral en w pn2 x log pn2 x w x dx where pn is the nth degree orthonormal polynomial with respect to a weight w x on 1 1 which belongs to the szeg class we also study two functionals closely related to the entropy integral first their asymptotic behavior is completely described for weights w in the bernstein class then as for the entropy we obtain asymptotic upper bounds for these two functionals when w x belongs to the szeg class in each case we give conditions for these upper bounds to be attained

aggregation of inertial particles in random flows we consider the trajectories of particles suspended in a randomly moving fluid if the lyapunov exponent of these trajectories is negative the paths of these particles coalesce so that particles aggregate here we give a detailed account of a method b mehlig and m wilkinson phys rev lett 92 250602 2004 for calculating this exponent it is expressed as the expectation value of a random variable evolving under a stochastic differential equation we analyze this equation in detail in the limit where the correlation time of the velocity field of the fluid is very short such that the stochastic differential equation is a langevin equation we derive an asymptotic perturbation expansion of the lyapunov exponent for particles suspended in three dimensional flows in terms of a dimensionless measure of the inertia of the particles epsilon and a measure of the relative intensities of potential and solenoidal components of the velocity field gamma we determine the phase diagram in the epsilon gamma plane

wiener chaos solutions of linear stochastic evolution equations a new method is described for constructing a generalized solution of a stochastic evolution equation existence uniqueness regularity and a probabilistic representation of this wiener chaos solution are established for a large class of equations as an application of the general theory new results are obtained for several types of the passive scalar equation

asymptotics for entropy integrals associated with exponential weights we establish a first order asymptotic for the entropy integrals 1pn2 log pn2 w2 and 1pn2 log pnw 2 w2 where pn n 0 are the orthonormal polynomials associated with the exponential weight w2

bounded variation singular stochastic control and dynkin game we consider a bounded variation singular stochastic control problem with value v in a general situation with control of a diffusion and nonlinear cost functional defined as solution to a backward stochastic differential equation bsde associated with this is a dynkin game with value u we establish the well known relation frac partial partial x v u for this general situation a saddle point for the dynkin game is given by the pair of first action times of an optimal control the methods are from stochastic analysis and include a priori estimates pathwise construction and comparison theorems for forward stochastic differential equations fsde and bsde

functionals of gegenbauer polynomials and d dimensional hydrogenic momentum expectation values the system of gegenbauer or ultraspherical polynomials cn x n 0 1 is a classical family of polynomials orthogonal with respect to the weight function x 1 x2 1 2 on the support interval 1 1 integral functionals of gegenbauer polynomials with integrand f x cn x 2 x where f x is an arbitrary function which does not depend on n or are considered in this paper first a general recursion formula for these functionals is obtained then the explicit expression for some specific functionals of this type is found in a closed and compact form namely for the functionals with f x equal to 1 x 1 x log 1 x2 and 1 x log 1 x which appear in numerous physico mathematical problems finally these functionals are used in the explicit evaluation of the momentum expectation values p and log p of the d dimensional hydrogenic atom with nuclear charge z 1 the power expectation values p are given by means of a terminating 5f4 hypergeometric function with unit argument which is

stochastic suspensions of heavy particles turbulent suspensions of heavy particles in incompressible flows have gained much attention in recent years a large amount of work focused on the impact that the inertia and the dissipative dynamics of the particles have on their dynamic and statistical properties substantial progress followed from the study of suspensions in model flows which although much simpler reproduce most of the important mechanisms observed in real turbulence this paper presents recent developments made on the relative motion of a pair of particles suspended in time uncorrelated and spatially self similar gaussian flows this review is complemented by new results by introducing a time dependent stokes number it is demonstrated that inertial particle relative dispersion recovers asymptotically richardson s diffusion associated to simple tracers a perturbative homogeneization technique is used in the small stokes number asymptotics and leads to interpreting first order corrections to tracer dynamics in terms of an effective drift this expansion implies that the correlation dimension deficit behaves linearly as a function of the stokes number the validity and the accuracy of this prediction is confirmed by numerical simulations

convergence of passive scalar fields in ornstein uhlenbeck flows to kraichnan s model we prove that the passive scalar field in the ornstein uhlenbeck velocity field with wave number dependent correlation times converges in the white noise limit to that of kraichnan s model with higher spatial regularity

bounds for orthogonal polynomials for exponential weights abstract orthogonal polynomials pn w2 x for exponential weights w2 e 2q on a finite or infinite interval i have been intensively studied in recent years we discuss efforts of the authors to extend and unify some of the theory our deepest result is the bound p n w 2 x w x x a n x a n 1 4 c x i with c independent of n and x here a n are the mhaskar rahmanov saff numbers for q and q must satisfy some smoothness conditions on i