phase diagram for turbulent transport sampling drift eddy diffusivity and variational principles we study the long time large scale transport in a three parameter family of isotropic incompressible velocity fields with power law spectra scaling law for transport is characterized by the scaling exponent q and the hurst exponent h as functions of the parameters the parameter space is divided into regimes of scaling laws of different em functional forms of the scaling exponent and the hurst exponent we present the full three dimensional phase diagram the limiting process is one of three kinds brownian motion h 1 2 persistent fractional brownian motions 1 2

the center of the generic division ring and twisted multiplicative group actions abstract let f be a field and let p be a prime the problem we study is whether the center cp of the division ring of p p generic matrices is stably rational over f given a finite group g and a z g lattice we let f m be the quotient field of the group algebra of the abelian group m procesi and formanek linear multilinear algebra 7 1979 203 212 have shown that for all n there is a z s n lattice gn such that cn is stably isomorphic to the fixed field under the action of sn of f gn let h be a p sylow subgroup of sp let a be the root lattice and let l f z s p h we show that there exists an action of sp on l z s p z h a twisted by an element ext 1 s p z s p z h a l such that l z s p z h a s p is stably isomorphic to cp the extension corresponds to an element of the relative brauer group of l over lh since z s p z h a and z s p h are quasi permutations l z s p z h a s p is stably rational over f however it is not known whether l z s p z h a s p is stably rational over f thus the result represents a reduction on the problem since z s p z h a is quasi permutation however the twist introduces a new level of complexity

super diffusivity in a shear flow model from perpetual homogenization this paper is concerned with the asymptotic behavior solutions of stochastic differential equations dy t d t y t dt y 0 0 and d 2 is a 2 times 2 skew symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales 12 21 h x 1 with h x 1 n 0 n h n x 1 r n where h n are smooth functions of period 1 h n 0 0 n and r n grow exponentially fast with n we can show that y t has an anomalous fast behavior y t 2 t 1 with 0 and obtain quantitative estimates on the anomaly using and developing the tools of homogenization

homogenization of an advection equation with locally stationary random coefficients abstract in the paper we consider the solution of an advection equation with rapidly changing coefficients t u e 1 e v t e 2 x e x u e 0 for t t and u e t x u 0 x x r d here e 0 is some small parameter and the drift term v t x t x r 1 d is assumed to be a d dimensional vector valued random field with incompressible spatial realizations we prove that when the field is gaussian locally stationary quasi periodic in the x variable and strongly mixing in time the solutions u e t x converge in law as e 0 to u 0 x t t x where x s t x s t is a diffusion satisfying x t t x x the averages of u e t x converge then to the solution of the corresponding kolmogorov backward equation

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

diffusion approximation for the advection of particles in a strongly turbulent random environment in this paper we prove several theorems concerning the motion of a particle in a random environment the trajectory of a particle is the solution of the differential equation dx t dt v x t where v x v e 1 f x 0 1 v is a constant vector f is a mean zero fluctuation field and e 1 is a parameter measuring the size of the fluctuations we show that both in case of a motion of a single particle and of a particle system considered in the macroscopic coordinate system moving along with velocity v i e x x vt e t t e 2 the diffusion approximation holds provided that f is divergence free moreover we show how to renormalize trajectories to obtain a similar result for non divergence free fields these results generalize theorems due to khasminskii and to kesten and papanicolaou

retract rational fields abstract let k be an infinite field the notion of retract k rationality was introduced by saltman in the study of noether s problem and other rationality problems we will investigate the retract rationality of a field in this paper theorem 1 let k k l be fields if k is retract k rational and l is retract k rational then l is retract k rational theorem 2 for any finite group g containing an abelian normal subgroup h such that g h is a cyclic group for any complex representation g gl v the fixed field c v g is retract c rational theorem 3 if g is a finite group then all the sylow subgroups of g are cyclic if and only if c m g is retract c rational for all g lattices m for all short exact sequences 0 c m m 0 because the unramified brauer group of a retract c rational field is trivial theorems 2 and 3 generalize previous results of bogomolov and barge respectively see theorems 5 9 and 6 1

mathematical models with exact renormalization for turbulent transport ii fractal interfaces non gaussian statistics and the sweeping effect this paper continues the study of a model for turbulent transport with an exact renormalization theory which has recently been proposed and developed by the authors three important topics are analyzed with complete mathematical rigor for this model 1 renormalized higher order statistics of a passively advected scalar such as the pair distance distribution and the fractal dimension of interfaces 2 the effect of non gaussian turbulent velocity statistics on renormalization theory 3 the sweeping effect of additional large scale mean velocities a special emphasis is placed on renormalization theory in the vicinity of the value of the analogue of the kolmogorov spectrum in the model in the authors earlier paper it was established that the kolmogorov value is at a phase transition boundary in the exact renormalization theory it is found here that the qualitative model despite its simplicity contains in the vicinity of the kolmogorov value a remarkable amount of the qualitative behavior of turbulent transport which has been uncovered in recent experiments and proposed in phenomenological theories in particular the richardson 4 3 law for pair dispersion and interfaces with fractal dimension defect of 2 3 occur in the model rigorously as limits when the kolmogorov spectrum is approached as a limit from one side of the phase transition boundary alternative corrections to the richardson law with the same form as those proposed heuristically in the recent literature and interfaces with fractal dimension defect 1 3 occur in the model when the kolmogorov spectrum is approached from the other side of the phase transition it is very interesting that fractal dimension defects of roughly the value either 1 3 or 2 3 for level sets and interfaces of passive scalars have been ubiquitous in recent turbulence experiments as regards non gaussian the asymptotic normality of normalized integrals b 56 corresponding to compactly supported blobs with mean zero the proof of this latter fact is done in the same way as step 2 proposition b 3 using the fact that the corresponding random processes tilde v delta s have finite domain of dependence this concludes the proof of proposition b 4

the rationality problem for finite subgroups of gl4 q abstract let g be a finite subgroup of gl 4 q the group g induces an action on q x 1 x 2 x 3 x 4 the rational function field of four variables over q theorem the fixed subfield q x 1 x 2 x 3 x 4 g f q x 1 x 2 x 3 x 4 f f for any g is rational i e purely transcendental over q except for two groups which are images of faithful representations of c 8 and c 3 c 8 into gl 4 q both fixed fields for these two exceptional cases are not rational over q there are precisely 227 such groups in gl 4 q up to conjugation the answers to the rationality problem for most of them were proved by kitayama and yamasaki 2009 ky except for four cases we solve these four cases left unsettled by kitayama and yamasaki thus the whole problem is solved completely

on the arithmetic of phase locking coupled neurons as a lattice on r2 abstract using methods from the geometry of numbers we derive an explicit global solution for the phase locking behavior of a simple integrate and fire model of coupled neurons the solution gives the ratios of phase locking rotation numbers attained as functions of the parameters of natural frequency and bidirectional coupling the ordering of the ratios is related to farey type series and to simple continued fractions a transition between two ratios say a b to c d is possible if and only if ad bc 1 empirically similar ordering is evident in published data from various neuron analogues we compare and contrast the present results with those from models based on caianiello s equation and on more general mappings on the torus

a finite dimensional approximation of the effective diffusivity for a symmetric random walk in a random environment we consider a nearest neighbor symmetric random walk on a homogeneous ergodic random lattice z d the jump rates of the walk are independent identically bernoulli distributed random variables indexed by the bonds of the lattice a standard result from the homogenization theory see a de masi p a ferrari s goldstein w d wick an invariance principle for reversible markov processes applications to random walks in random environments j statist phys 55 3 4 1989 787 855 asserts that the scaled trajectory of the particle satisfies the functional central limit theorem the covariance matrix of the limiting normal distribution is called the effective diffusivity of the walk we use the duality structure corresponding to the product bernoulli measure to construct a numerical scheme that approximates this parameter when d 3 the estimates of the convergence rates are also provided

negative solutions to three dimensional monomial noether problem three dimensional monomial noether problem can have negative solutions for 8 groups by the suitable choice of the coefficients we find the necessary and sufficient condition for the coefficients to have a negative solution the results are obtained by two criteria of irrationality using galois cohomology

on the superdiffusive behavior of passive tracer with a gaussian drift in the present article we consider a motion of a passive tracer particle whose trajectory satisfies the ito stochastic differential equation d x t v t x t dt sqrt 2 kappa d w t where w is a brownian motion v is a stationary gaussian random field with incompressible realizations independent of w and 0 we prove the superdiffusive character of the motion under certain conditions on the energy spectrum of the velocity field the result is shown both for steady time independent and time dependent and markovian velocity fields in addition we provide explicit upper and lower bounds for the hurst exponent of the trajectory all previous rigorous results concerned explicitely solvable shear flows cases

quenched invariance principles for random walks on percolation clusters we consider a supercritical bernoulli percolation model in d2 and study the simple symmetric random walk on the infinite percolation cluster the aim of this paper is to prove the almost sure q

random walks on supercritical percolation clusters we obtain gaussian upper and lower bounds on the transition density qt x y of the continuous time simple random walk on a supercritical percolation cluster c1 in the euclidean lattice the bounds analogous to aronsen s bounds for uniformly elliptic divergence form diusions hold with constants ci depending only on p the percolation probability and d the irregular nature of the medium means that the bound for qt x only holds for t sx where the constant sx depends on the percolation congura tion

invariant distributions and scaling limits for some diffusions in time varying random environments we consider a family of one dimensional diffusions in dynamical wiener mediums which are random perturbations of the ornstein uhlenbeck diffusion process we prove quenched and annealed convergences in distribution and under weigh ted total variation norms we find two kind of stationary probability measures which are either the standard normal distribution or a quasi invariant measure depending on the environment and which is naturally connected to a random dynamical system we apply these results to the study of a model of time inhomogeneous brox s diffusions which generalizes the diffusion studied by brox ann probab 14 4 1206 1218 1986 and those investigated by gradinaru and offret ann inst henri poincare probab stat 2011 we point out two distinct diffusive behaviours and we give the speed of convergences in the quenched situations

reduction in the rationality problem for multiplicative invariant fields abstract for a faithful zg lattice a and a field k on which the group g acts by field automorphisms let r be the normal subgroup generated by the elements of g which act trivially on k and act as reflections on a we prove that the rationality of the multiplicative invariant field k a g over k ar g is equivalent to the rationality of k a g over k ar g where g is a particular subgroup of g such that g r g we then use this reduction result to prove that k a g is rational over k where g is the automorphism group of a crystallographic root system g acts trivially on k and a is any lattice on the space q

invariance principle for the random conductance model we study a continuous time random walk x in an environment of i i d random conductances mu e in 0 infty in mathbb z d we assume that mathbb p mu e 0 p c so that the bonds with strictly positive conductances percolate but make no other assumptions on the law of the e we prove a quenched invariance principle for x and obtain green s functions bounds and an elliptic harnack inequality

an invariance principle for non symmetric markov processes and reflecting diffusions in random domains we study an invariance principle for additive functionals of nonsymmetric markov processes with singular mean forward velocities we generalize results of kipnis and varadhan kv and de masi et al de in two directions markov processes are non symmetric and mean forward velocities are distributions we study continuous time markov processes we use our result to homogenize non symmetric reflecting diffusions in random domains

superdiffusion in nearly stratified flows in classical work matheron and the marsilly showed that superdiffusive scaling of mean square displacements occurs in transport diffusion for stratified flows with steady simple shear layers and long range spatial correlations more recently the authors have calculated a formula for the non gaussian large scale long time renormalized green function for these problems here the scaling laws and renormalized green functions for diffusion in nearly stratified flows are studied in such flows the simple shear layer with long range correlations is perturbed by incompressible flows with short range correlations here it is established that these flows belong to the same universality class as the simple shear layers with a renormalized green function with a similar structure but reflecting homogenization by the transverse displacements the tools in the analysis involve a modification of homogenization theory and also rigorous diagrammatic perturbation theory

diffusion in turbulence we prove long time diffusive behavior homogenization for convection diffusion in a turbulent flow that it incompressible and has a stationary and square integrable stream matrix simple shear flow examples show that this result is sharp for flows that have stationary stream matrices

on noether s problem for central extensions of symmetric and alternating groups for a field k and a finite group g acting regularly on a set of indeterminates x xg g g let k g denote the invariant field k x g we first prove for the alternating group an that if n is odd then q an is rational over q an 1 we then obtain an analogous result where an is replaced by an arbitrary finite central extension of either an or sn valid over q n for suitable n concrete applications of our results yield 1 a new proof of maeda s result on the rationality of q x1 x5 a5 q 2 an affirmative answer to noether s problem over q for both a5 and s5 3 an affirmative answer to noether s problem over c for every finite central extension group of either an or sn with n 5

on superdiffusive behavior of a passive tracer in a random flow abstract in this note we consider a passive tracer model describing particle dispersion in a turbulent flow the trajectory of the particle is given by the solution of an ordinary differential equation x t f x t x 0 x 0 where f x is a divergence free random vector field that is spatially homogeneous and isotropic we show that trajectories of the tracer display superdiffusive behavior when the random velocity f x decorrelates at large distances but does it not rapidly but rather at some moderate rate the main tools used in the proofs are variational principles and tauberian type theorems

generic extensions and generic polynomials for linear algebraic groups abstract we construct generic extensions and polynomials for certain linear algebraic groups over finite fields in particular we show the existence of generic polynomials for the cyclic group c 2 m in characteristic p for all odd primes p

quenched invariance principles for walks on clusters of percolation or among random conductances in this work we principally study random walk on the supercritical infinite cluster for bond percolation on d we prove a quenched functional central limit theorem for the walk when d 4 we also prove a similar result for random walk among i i d random conductances along nearest neighbor edges of d when d 1

fractional gaussian noise enhanced information capacity of a nonlinear neuron model with binary signal input this paper reveals the effect of fractional gaussian noise with hurst exponent h 1 2 1 on the information capacity of a general nonlinear neuron model with binary signal input the fgn and its corresponding fractional brownian motion exhibit long range strong dependent increments it extends standard brownian motion to many types of fractional processes found in nature such as the synaptic noise in the paper for the subthreshold binary signal sufficient conditions are given based on the forbidden interval theorem to guarantee the occurrence of stochastic resonance while for the suprathreshold binary signal the simulated results show that additive fgn with hurst exponent h 1 2 1 could increase the mutual information or bits count the investigation indicated that the synaptic noise with the characters of long range dependence and self similarity might be the driving factor for the efficient encoding and decoding of the nervous system

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

central limit theorem for random walks in doubly stochastic random environment mathscr h _ 1 suffices we prove a central limit theorem under diffusive scaling for the displacement of a random walk on zdzd in stationary and ergodic doubly stochastic random environment under the h 1h 1 condition imposed on the drift field the condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable this improves the best existing result fluctuations in markov processes time symmetry and martingale approximation 2012 springer where it is assumed that the stream tensor is in lmax 2 d lmax 2 d with 0 0 our proof relies on an extension of the relaxed sector condition of bull inst math acad sin n s 7 2012 463 476 and is technically rather simpler than existing earlier proofs of similar results by oelschlager ann probab 16 1988 1084 1126 and komorowski landim and olla fluctuations in markov processes time symmetry and martingale approximation 2012 springer

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

self consistent four point closure for transport in steady random flows ensemble averaging of advection dispersion equations describing transport of a passive scalar in incompressible random velocity fields requires a closure approximation commonly used approaches such as the direct interaction approximation and large eddy simulations as well as equivalent renormalization schemes employ so called two point or one loop closures these approaches have proven to be adequate for transport in zero mean unbiased time dependent random velocity fields with increasing accuracy for decreasing temporal coherence in the opposite limit of steady velocity fields with finite bias however these schemes fail to predict effective transport properties both quantitatively and qualitatively leading to an obvious inconsistency for transverse dispersion in two spatial dimensions for this case two point closures predict that macroscopic transverse dispersion increases as the square root of the disorder variance while it has been proven rigorously that there is no disorder induced contribution to macroscopic transverse dispersion for purely advective transport furthermore two point closures significantly underestimate the disorder induced contribution to longitudinal dispersion we derive a four point closure for stochastically averaged transport equations that goes beyond classical one loop schemes and demonstrate that it is exact for transverse dispersion and correctly predicts an increase of the longitudinal disorder induced dispersion coefficient with the square of the variance of the strong disorder the predicted values of asymptotic longitudinal dispersion coefficients are consistent with those obtained via monte carlo random walk simulations

diffusive and nondiffusive limits of transport in nonmixing flows we study the passive scalar transport in a class of nonmixing markovian flows with power law spectra and correlation times we establish a new diffusion regime under an optimal condition convergent kubo formula on the spatial temporal structure of this family of flows under such a condition the peclet number of the problem may be infinite we propose a general criterion for the diffusion regime that takes into account of both the effect of molecular diffusion and the spatial temporal structure of the velocity field we conjecture the criterion to be applicable in general for temporally ergodic reversible markovian flows we show heuristically that the violation of this criterion may lead to a nondiffusive scaling limit

quenched invariance principle for random walks with time dependent ergodic degenerate weights we study a continuous time random walk xx on zdzd in an environment of dynamic random conductances taking values in 0 0 we assume that the law of the conductances is ergodic with respect to space time shifts we prove a quenched invariance principle for the markov process xx under some moment conditions on the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

discrete velocity fields with explicitly computable lagrangian law we introduce a class of random velocity fields on the periodic lattice and in discrete time having a certain hidden markov structure the generalized lagrangian velocity the velocity field as viewed from the location of a single moving particle has similar hidden markov structure and its law is found explicitly its rate of convergence to equilibrium is studied in small numerical examples and in rigorous results giving absolute and relative bounds on the size of the second largest eigenvalue modulus the effect of molecular diffusion on the rate of convergence is also investigated in some cases it slows convergence to equilibrium after repeating the velocity field periodically throughout the integer lattice it is shown that with the usual diffusive rescaling the single particle motion converges to brownian motion in both compressible and incompressible cases an exact formula for the effective diffusivity is given and numerical examples are shown

approximation of the effective conductivity of ergodic media by periodization this paper is concerned with the approximation of the effective conductivity a associated to an elliptic operator xa x x where for x d d 1 a x is a bounded elliptic random symmetric d d matrix and takes value in an ergodic probability space x writing an x the periodization of a x on the torus tdn of dimension d and side n we prove that for almost all we extend this result to non symmetric operators x a e x x corresponding to diffusions in ergodic divergence free flows a is d d elliptic symmetric matrix and e x an ergodic skew symmetric matrix and to discrete operators corresponding to random walks on d with ergodic jump rates

the sector constants of continuous state branching processes with immigration abstract continuous state branching processes with immigration are studied we are particularly concerned with the associated non symmetric dirichlet form after observing that gamma distributions are only reversible distributions for this class of models we prove that every generalized gamma convolution is a stationary distribution of the process with suitably chosen branching mechanism and with continuous immigration for such non reversible processes the strong sector condition is discussed in terms of a characteristic called the thorin measure in addition some connections with notion from non commutative probability theory will be pointed out through calculations involving the stieltjes transform

division algebras over surfaces abstract we consider here a number of topics concerning the theory of division algebras over the function field of a surface one result relates the obstruction for ramification data to be from a division algebra and third etale cohomology another result shows this obstruction is always zero when the surface is spec of a regular local ring with some mild assumptions at the same time we study the brauer group of this function field as it relates to the brauer group of the function field of the henselization finally we prove a result which says that brauer group elements which look like they are of prime index q unequal to any characteristic must have all their ramification split by a cyclic galois extension of the same degree this last result requires a primitive q root of one

quasi monomial actions and some 4 dimensional rationality problems abstract let g be a finite group acting on k x 1 x n the rational function field of n variables over a field k the action is called a purely monomial action if x j 1 i n x i a i j for all g for 1 j n where a i j 1 i j n gl n z the main question is that under what situations the fixed field k x 1 x n g is rational purely transcendental over k this rationality problem has been studied by hajja kang hoshi rikuna when n 3 in this paper we will prove that k x 1 x 2 x 3 x 4 g is rational over k provided that the purely monomial action is decomposable to prove this result we introduce a new notion the quasi monomial action which is a generalization of previous notions of multiplicative group actions moreover we determine the rationality problem of purely quasi monomial actions of k x y g over k where k k g

limit cycles of a bineuronal network model abstract a biological system of two synaptically interacting pacemaker neurons is studied mathematically modelling each neuron with a general relaxation oscillator and the synaptic action by jumps in the postsynaptic neuron potential this results in a nonlinear dynamical system with some defined discontinuities on a two dimensional torus we prove that generic noninhibitory systems must exhibit a periodic behavior a finite number of limit cycles and two at most binary codes defined by the periodic neuron discharges we also study the structural stability persistence of these features and conclude that the bineuronal system acts as a stable memory when both neurons are inhibitory two mutually exclusive facts are deduced either the system has a periodic asymptotic behavior or it has a non trivial compact set attracting all the orbits

diffusive limits for true or myopic self avoiding random walks and self repellent brownian polymers in d 3 the problems considered in the present paper have their roots in two different cultures the true or myopic self avoiding walk model tsaw was introduced in the physics literature by amit et al phys rev b 27 1635 1645 1983 this is a nearest neighbor non markovian random walk in mathbb z d which prefers to jump to those neighbors which were less visited in the past the self repelling brownian polymer model srbp initiated in the probabilistic literature by durrett and rogers probab theory relat fields 92 337 349 1992 independently of the physics community is the continuous space time counterpart a diffusion in mathbb r d pushed by the negative gradient of the mollified occupation time measure of the process in both cases similar long memory effects are caused by a path wise self repellency of the trajectories due to a push by the negative gradient of softened local time we investigate the asymptotic behaviour of tsaw and srbp in the non recurrent dimensions first we identify a natural stationary in time and ergodic distribution of the environment the local time profile as seen from the moving particle the main results are diffusive limits in the case of tsaw for a wide class of self interaction functions we establish diffusive lower and upper bounds for the displacement and for a particular more restricted class of interactions we prove full clt for the finite dimensional distributions of the displacement in the case of srbp we prove full clt without restrictions on the interaction functions these results settle part of the conjectures based on non rigorous renormalization group arguments equally valid for the tsaw and srbp cases in amit et al 1983 the proof of the clt follows the non reversible version of kipnis varadhan theory on the way to the proof we slightly weaken the so called graded sector condition

homogenization of a diffusion process in a divergence free random field on etudie le probleme d homogeneisation pour un processus de diffusion dans un champ de derive aleatoire independant de la divergence dans le cas particulier d un petit champ gaussien on derive un developpement asymptotique de la matrice de diffusion efficace en termes de sa densite spectrale

invariance principle for the random conductance model in a degenerate ergodic environment we study a continuous time random walk x on zd in an environment of random conductances taking values in 0 we assume that the law of the conductances is ergodic with respect to space shifts we prove a quenched invariance principle for x under some moment conditions of the environment the key result on the sublinearity of the corrector is obtained by moser s iteration scheme

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

convection enhanced diffusion for random flows we analyze the effective diffusivity of a passive scalar in a two dimensional steady incompressible random flow that has mean zero and a stationary stream function we show that in the limit of small diffusivity or large peclet number with convection dominating there is substantial enhancement of the effective diffusivity our analysis is based on some new variational principles for convection diffusion problems and on some facts from continuum percolation theory some of which are widely believed to be correct but have not been proved yet we show in detail how the variational principles convert information about the geometry of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result of isichenko and kalda that the effective diffusivity behaves like 3 13 when the molecular diffusivity is small assuming some percolation theoretic facts we also analyze the effective diffusivity for a special class of convective flows random cellular flows where the facts from percolation theory are well established and their use in the variational principles is more direct than for general random flows

frozen path approximation for turbulent diffusion and fractional brownian motion in random flows we establish the conditions for the frozen path approximation for turbulent transport in a class of nonmixing gaussian flows with long range correlation we identify the regimes of fractional brownian motion limit as well as the brownian motion limit

on the poisson equation and diffusion approximation 3 three different results are established which turn out to be closely connected so that the first one implies the second one which in turn implies the third one the first one states the smoothness of an invariant diffusion density with respect to a parameter the second establishes a similar smoothness of the solution of the poisson equation in rd the third one states a diffusion approximation result or in other words an averaging of singularly perturbed diffusion for fully coupled sde systems or sde systems with complete dependence

computation of the effective diffusivity tensor for transport of a passive scalar in a turbulent incompressible flow we consider the passive scalar transport in an incompressible random flow our basic result is a proof of the convergence of a certain numerical scheme for the computation of the eddy diffusivity tensor the scheme leads to the formula for the diffusivity expressed in terms of an infinite series we give a rigorous proof of the geometric bounds on the magnitude of the terms of the series provided that the eulerian field is markovian and gaussian and its temporal dynamics has a sufficiently large spectral gap the principal tools used in the proofs are the decomposition of the space of square integrable fields formed over the possible realizations of the eulerian velocity field in the gaussian chaos and the hypercontractivity properties of gaussian measures

stationarity of lagrangian velocity in compressible environments we study the transport of a passive tracer particle by a random d dimensional gaussian compressible velocity field it is well known since the work of lumley see 13 and port and stone see 20 that the observations of the velocity field from the moving particle the so called lagrangian velocity process are statistically stationary when the field itself is incompressible in this paper we study the question of stationarity of lagrangian observations in compressible environments we show that given sufficient temporal decorrelation of the velocity statistics there exists a transformation of the original probability measure under which the lagrangian velocity process is time stationary the transformed probability is equivalent to the original measure as an application of this result we prove the law of large numbers for the particle trajectory

homogenization of divergence form operators with lower order terms in random media the probabilistic machinery central limit theorem feynman kac formula and girsanov theorem is used to study the homogenization property for pde with second order partial differential operator in divergence form whose coefficients are stationary ergodic random fields furthermore we use the theory of dirichlet forms so that the only conditions required on the coefficients are non degeneracy and boundedness

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

superdiffusive bounds on self repellent brownian polymers and diffusion in the curl of the gaussian free field in d 2 we consider two models of random diffusion in random environment in two dimensions the first example is the self repelling brownian polymer this describes a diffusion pushed by the negative gradient of its own occupation time measure local time the second example is a diffusion in a fixed random environment given by the curl of massless gaussian free field

an invariance principle for diffusion in turbulence we prove an almost sure invariance principle for diffusion driven by velocities with unbounded stationary vector potentials the result generalizes to multiple particles motion driven by a common velocity field and independent molecular brownian motions

twisted multiplicative field invariants noether s problem and galois extensions noether s problem concerns fields f xg 1 g e g g f g where f is a field g is a finite group g acts by permuting the indeterminants x in the obvious way and f g refers to the field of invariant elements in showing the f g s were nonrational for certain g the author was led to consider multiplicative invariant fields ss which are defined as follows let g be a finite group and q a g lattice that is q is a finitely generated z g module which is free as an abelian group form the group algebra f g and its field of fractions f q g acts naturally on f q and the multiplicative invariant field is f q in ss it was shown that for certain q f q g was essentially equivalent to actually stably isomorphic to f g for g a split extension of g with abelian kernel this leads one to ask whether f g can be expressed as a kind of invariant field for g when g is a nonsplit extension of g the answer is yes and is the core of this paper in fact f g will be shown to be a so called a twisted multiplicative invariant field of g the goal of this paper is to present this fact and then draw a series of conclusions using it such a twisted invariant fields have appeared explicitly in previous works in ss these fields arise as another way of describing the invariant fields of reductive algebraic groups note that in s6 the finite group is the weyl group and the lattice is derived from the root lattice in s7 the unramified brauer groups of a twisted invariant fields are calculated

an integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection diffusion by a steady incompressible velocity field are developed here under these conditions a rigorous stieltjes integral representation for effective diffusivity in turbulent transport is derived this representation is valid for all peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the peclet number one consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all peclet numbers a nonlocal variational principle for effective diffusivity is developed along with applications to advection diffusion by random arrays of vortices a new class of rigorous examples is introduced these examples have an explicit stieltjes measure for the effective diffusivity furthermore the effective diffusivity behaves likek0 pe 1 2 in the limit of large peclet numbers wherek0 is the molecular diffusivity formal analogies with the theory of composite materials are exploited systematically

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

frobenius groups and retract rationality abstract let k be any field g be a finite group acting on the rational function field k x g g g by h x g x h g for any h g g define k g k x g g g g noether s problem asks whether k g is rational purely transcendental over k a weaker notion retract rationality introduced by saltman is also very useful for the study of noether s problem we prove that if g is a frobenius group with abelian frobenius kernel then k g is retract k rational for any field k satisfying some mild conditions as an application we show that for any algebraic number field k for any frobenius group g with frobenius complement isomorphic to s l 2 f 5 there is a galois extension field k over k whose galois group is isomorphic to g i e the inverse galois problem is valid for the pair g k the same result is true for any non solvable frobenius group if k 8 is a cyclic extension of k

on discrete wirtinger northcott problems abstract some discrete wirtinger northcott inequalities are established in this paper let dx and x be the periodic difference and non periodic difference of a vector x r n respectively for 1 p we give the maximum and the minimum of d x x d x and x in the l p ball b p 0 n in particular we prove that for p the result min b 0 n d x 4 n n is even 4 n n 2 1 n is odd is equivalent to the well known discrete first northcott inequality established by fan taussky and todd

diffusion approximation for the convection diffusion equation with random drift we consider the asymptotic behavior of the solutions ofscaled convection diffusion equations t u t x x t x 1 v t 2 x x u t x with the initial condition u 0 x u 0 x as the parameter 0 under the assumptions that 0 and v t x t x r d is a d dimensional stationary zero mean incompressible gaussian random field markovian and mixing in t we show that the laws of u t t 0 in an appropriate functional space converge weakly as 0 to a type measureconcentrated on a solution of a certain constant coefficient heat equation

anchored nash inequalities and heat kernel bounds for static and dynamic degenerate environments abstract we introduce anchored versions of the nash inequality they allow to control the l 2 norm of a function by dirichlet forms that are not uniformly elliptic we then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments as an example we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium

mathematical models with exact renormalization for turbulent transport the advection diffusion of a passive scalar by incompressible velocity fields which admit a statistical description and involve a continuous range of excited spatial and or temporal scales is very important in applications ranging from fully developed turbulence to the diffusion of tracers in heterogeneous porous media a variety of renormalization theories which typically utilize partial resummation of divergent perturbation series according to various recipes have been applied to this problem in various contexts in this paper a simple model problem for the advection diffusion of a passive scalar is introduced and the complete renormalization theory is developed with full mathematical rigor explicit formulas for the anomalous time scaling in various regimes as well as the green s function for the large scale long time ensemble average are developed here formulas for the renormalized higher order statistics are also developed the simple form of the model problem is deceptive the renormalization theory for this problem exhibits a remarkable range of different renormalization phenomena as parameters in the velocity statistics are varied these phenomena include the existence of several distinct anomalous scaling regimes as the spectral parameter tilde varepsilon is varied as well as explicit regimes in tilde varepsilon where the effective equation for the ensemble average is not a simple diffusion equation but instead involves an explicit random nonlocal eddy diffusivity we use fourier analysis and the feynman kac formula as main tools in the explicit exact renormalization theory developed here

a martingale approach to homogenization of unbounded random flows we study the asymptotic behavior of brownian motion in steady unbounded incompressible random flows we prove an invariance principle for almost all realizations of random flows the key compactness result is obtained by moser s iterative scheme in pde theory

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

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