generalized random walk in a random environment random walk in a random environment a process where a randomly chosen transition probability is assigned to each point has been exhaustively studied in the one dimensional nearest neighbor case but is much more difficult in any more general case this paper develops crude techniques for handling transience results and zero one laws in a generalized set up

the speed of a biased random walk on a percolation cluster at high density we study the speed of a biased random walk on a percolation cluster on z d in function of the percolation parameter p we obtain a first order expansion of the speed at p 1 which proves that percolating slows down the random walk at least in the case where the drift is along a component of the lattice

number of degrees of freedom of two dimensional turbulence we derive upper bounds for the number of degrees of freedom of two dimensional navier stokes turbulence freely decaying from a smooth initial vorticity field omega x y 0 omega0 this number denoted by n is defined as the minimum dimension such that for n or n arbitrary n dimensional balls in phase space centered on the solution trajectory omega x y t for t 0 contract under the dynamics of the system linearized about omega x y t in other words n is the minimum number of greatest lyapunov exponents whose sum becomes negative it is found that n or c1re when the phase space is endowed with the energy norm and n or c2re 1 ln re 1 3 when the phase space is endowed with the enstrophy norm here c1 and c2 are constant and re is the reynolds number defined in terms of omega0 the system length scale and the viscosity nu the linear or nearly linear dependence of n on re is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number this result is in a sharp contrast to the forced case for which well known estimates for the hausdorff dimension dh of the global attractor scale highly superlinearly with nu 1 we argue that the extra dependence of dh on nu 1 is not an intrinsic property of the turbulent dynamics rather it is a removable artifact brought about by the use of a time independent forcing as a model for energy and enstrophy injection that drives the turbulence

universality of probability distributions among two dimensional turbulent flows we study statistical properties of two dimensional turbulent flows three systems are considered the navier stokes equation surface quasigeostrophic flow and a model equation for thermal convection in the earth s mantle direct numerical simulations are used to determine one point fluctuation properties comparative study shows universality of probability density functions pdfs across different types of flow for instance the pdfs for derivatives of the advected quantity are the same for the three flows once normalized by the average size of fluctuations the single point statistics is surprisingly robust with respect to the nature of the nonlinearity

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

quenched large deviations for simple random walks on percolation clusters including long range correlations we prove a quenched large deviation principle ldp for a simple random walk on a supercritical percolation cluster srwpc on mathbb z d d geq 2 the models under interest include classical bernoulli bond and site percolation as well as models that exhibit long range correlations like the random cluster model the random interlacement and the vacant set of random interlacements for d geq 3 and the level sets of the gaussian free field d geq 3 inspired by the methods developed by kosygina et al commun pure appl math 59 1489 1521 2006 for proving quenched ldp for elliptic diffusions with a random drift and by yilmaz commun pure appl math 62 8 1033 1075 2009 and rosenbluth quenched large deviations for multidimensional random walks in a random environment a variational formula ph d thesis nyu arxiv 0804 1444v1 for similar results regarding elliptic random walks in random environment we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures of the environment markov chain in the non elliptic case of srwpc via a contraction principle this reduces easily to a quenched ldp for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas the main difficulty in our set up lies in the inherent non ellipticity as well as the lack of translation invariance stemming from conditioning on the fact that the origin belongs to the infinite cluster we develop a unifying approach for proving quenched large deviations for srwpc based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster

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

effective degrees of nonlinearity in a family of generalized models of two dimensional turbulence we study the small scale behavior of generalized two dimensional turbulence governed by a family of model equations in which the active scalar theta delta alpha 2 psi is advected by the incompressible flow u psi y psi x here psi is the stream function delta is the laplace operator and alpha is a positive number the dynamics of this family are characterized by the material conservation of theta whose variance theta2 is preferentially transferred to high wave numbers direct transfer as this transfer proceeds to ever smaller scales the gradient nublatheta grows without bound this growth is due to the stretching term nablatheta nabla u whose effective degree of nonlinearity differs from one member of the family to another this degree depends on the relation between the advecting flow u and the active scalar theta i e on alpha and is wide ranging from approximately linear to highly superlinear linear dynamics are realized when nablau is a quantity of no smaller scales than theta so that it is insensitive to the direct transfer of the variance of theta which is nearly passively advected this case corresponds to alpha or 2 for which the growth of nablatheta is approximately exponential in time and nonaccelerated for alpha 2 superlinear dynamics are realized as the direct transfer of theta2 entails a growth in nablau thereby enhancing the production of nablatheta this superlinearity reaches the familiar quadratic nonlinearity of three dimensional turbulence at alpha 1 and surpasses that for alpha 1 the usual vorticity equation alpha 2 is the border line where nablau and theta are of the same scale separating the linear and nonlinear regimes of the small scale dynamics we discuss these regimes in detail with an emphasis on the locality of the direct transfer

random walk on the infinite cluster of the percolation model we consider random walk on the infinite cluster of bond percolation on d we show that in the supercritical regime whend 3 this random walk is a s transient this conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance it is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite

homogenization of richardsʼ equation of van genuchten mualem model abstract this paper is devoted to the homogenization of richards equation of van genuchten mualem model which is a nonlinear degenerate parabolic differential equation it is usually used to model the motion of saturated unsaturated water flow in porous media we firstly apply the kirchhoff transformation to the equation and obtain a simpler equivalent equation with a linear oscillated diffusion term then under the real assumption for van genuchten mualem model we obtain the homogenized equation based on the two scale convergence theory some results on the first order corrector are also presented

on the dual cascade in two dimensional turbulence we study the dual cascade scenario for two dimensional turbulence driven by a spectrally localized forcing applied over a finite wavenumber range kmin kmax with kmin 0 such that the respective energy and enstrophy injection rates and satisfy k 2 k 2 the classical kraichnan leith batchelor paradigm based on the simultaneous conservation of energy and enstrophy and the scaleselectivity of the molecular viscosity requires that the domain be unbounded in both directions for two dimensional turbulence either in a doubly periodic domain or in an unbounded channel with a periodic boundary condition in the acrosschannel direction a direct enstrophy cascade is not possible in the usual case where the forcing wavenumber is no greater than the geometric mean of the integral and dissipation wavenumbers constant spectral slopes must satisfy 5 and 8 where is the asymptotic slope of the range of wavenumbers lower higher than the forcing wavenumber the influence of a large scale dissipation on the realizability of a dual cascade is analyzed we discuss the consequences for numerical simulations attempting to mimic the classical unbounded picture in a bounded domain

on the chemical distance for supercritical bernoulli percolation we prove large deviation estimates at the correct order for the graph distance of two sites lying in the same cluster of an independent percolation process we improve earlier results of gartner and molchanov and grimmett and marstrand and answer affirmatively a conjecture of kozlov

a zero one law for planar random walks in random environment we solve the problem posed by s a kalikow whether the event that the x coordinate of a random walk in a two dimensional random environment approaches has necessarily probability either zero or one the answer is yes if we assume the environment to be i i d and in general no if we allow the environment to be just stationary and ergodic

inverse energy cascade in two dimensional turbulence deviations from gaussian behavior high resolution numerical simulations of stationary inverse energy cascade in two dimensional turbulence are presented deviations from gaussian behavior of velocity differences statistics are quantitatively investigated the level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd order moments which bring the signature of the inverse energy flux no measurable intermittency corrections could be found in their scaling laws the seventh order skewness increases by almost two orders of magnitude with respect to the third thus becoming of order unity

multiscale analysis of diffusion processes in composite media the goal of this paper is to present some homogenization results for a nonlinear problem arising in the modeling of diffusion in a periodic structure formed by two media with different properties separated by an active interface our setting is relevant for modeling heat diffusion in composite materials with imperfect interfaces or electrical conduction in biological tissues the approach we follow is based on the periodic unfolding method which allows us to deal with general media

convergence to fractional kinetics for random walks associated with unbounded conductances we consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail we prove that the scaling limit of this process is a fractional kinetics process that is the time change of a d dimensional brownian motion by the inverse of an independent stable subordinator we further show that the same process appears in the scaling limit of the non symmetric bouchaud s trap model

percolation beyond ℤd the contributions of oded schramm oded schramm 1961 2008 influenced greatly the development of percolation theory beyond the usual d setting in particular the case of nonamenable lattices here we review some of his work in this field

on the regularity of a class of generalized quasi geostrophic equations abstract in this article we consider the following generalized quasi geostrophic equation t u 0 u r x r 2 where 0 0 1 and 0 2 we first show a general conditional criterion yielding the nonlocal maximum principles for the whole space active scalars then mainly by applying the general criterion for the case 0 1 and 1 2 we obtain the global well posedness of the system with smooth initial data and for the case 0 1 and 2 1 we prove the local smoothness and the eventual regularity of the weak solution of the system with appropriate initial data

diminishing inverse transfer and non cascading dynamics in surface quasi geostrophic turbulence abstract the inverse transfer in the forced dissipative surface quasi geostrophic equation is studied when the natural dissipation operator 1 2 is employed the nonlinear transfer of this system conserves the two quadratic quantities 1 1 4 2 2 and 2 1 2 2 2 kinetic energy where is the stream function and denotes a spatial average in the limit of infinite domain the kinetic energy density 2 remains bounded for the natural dissipation operator for the power law inverse transfer region the inverse flux of 1 diminishes as it proceeds toward sufficiently low wavenumbers whenever the kinetic energy 2 remains bounded this implies that no persistent non dissipative inverse cascade of 1 to ever lower wavenumbers is sustainable as long as the dissipation parameter is held fixed this result does not rule out the possibility that for sufficiently small a finite inverse flux would reach a certain low wavenumber numerical results supporting the theoretical predictions are presented

energy spectra of steady two dimensional turbulent flows more than 30 years ago the energy spectrum of twodimensional turbulence was theoretically predicted yet the evidence for this spectrum has remained weak in some cases classical theory 1 3 predicts the energy spectrum based on a cascade of either energy or enstrophy once the relevant physical quantities have been identified simple dimensional analysis leads to the power spectrum as a function of wave number k however the spectra measured in many numerical simulations decay substantially faster with k than predicted 4 the range of wave numbers larger than the inverse system size and up to the dissipative damping is called inertial range in this paper we assume power laws for the inertial range as a working hypothesis since possible deviations from a power behavior cannot be determined from numerical data with confidence a general review of two dimensional turbulence and the slope of the energy spectrum can be found in refs 5 6 following we summarize recent results on the inertial range spectrum

random dirichlet environment viewed from the particle in dimension d ge 3 we consider random walks in random dirichlet environment rwde which is a special type of random walks in random environment where the exit probabilities at each site are i i d dirichlet random variables on zd rwde are parameterized by a 2d tuple of positive reals called weights in this paper we characterize for d 3 the weights for which there exists an absolutely continuous invariant probability distribution for the process viewed from the particle we can deduce from this result and from ann inst henri poincare probab stat 47 2011 1 8 a complete description of the ballistic regime for d 3

the statistical mechanics of random copolymers random copolymers are polymers with two or more types of monomer where the monomer sequence is determined by some random process once determined the sequence is fixed so random copolymers are an example of a system with quenched randomness we review the statistical mechanics of random copolymers focusing on self avoiding walk models where there are two types of monomers a and b which are randomly distributed along the polymer chain theoretical approximate and numerical results are reviewed for models of the random copolymer adsorption localization and collapse phase transitions we concentrate on what is known about the existence of phase transitions the morita approximation and results about self averaging we also discuss in less detail the replica trick and numerical methods including monte carlo methods exact enumeration and transfer matrix methods important open problems are identified throughout and highlighted in the conclusions

numerical homogenization of the richards equation for unsaturated water flow through heterogeneous soils homogenized equations and the corresponding effective constitutive relations are generally necessary for numerically modeling large scale unsaturated flow processes in soils recently based on the kirchhoff transformation and the two scale convergence theory a homogenization method for the richards equation with the mualem van genuchten model has been proposed with a constant model parameter relating to the inverse of the air entry pressure and the soil pore size distribution the homogenized model is computationally efficient and convenient to use because of its explicit expression in this study we generalize this method allowing to be a spatially distributed random field and proposing a homogenized richards equation in the mixed form h under the condition that the effective hydraulic conductivity tensor is diagonal this generalization eliminates the limitation of a constant in practical applications the proposed homogenized model is meaningful in most situations because the flow problems are influenced mainly by the diagonal terms of conductivity and the off diagonal terms are often neglected two dimensional numerical tests are conducted in soil profiles with different degrees of spatial heterogeneity structure to illustrate that the homogenized model can capture the fine scale flow behaviors on coarse grids effectively homogenization for the richards equation with other two commonly used constitutive relations the brooks corey model and the gardner russo model is also illustrated in this study

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models

linear stability analysis of parallel shear flows for an inviscid generalized two dimensional fluid system the linear stability of parallel shear flows for an inviscid generalized two dimensional 2d fluid system the so called turbulence system is studied this system is characterized by the relation q 2 between the advected scalar q and the stream function here is a real number not exceeding 3 and q is referred to as the generalized vorticity in this study a sufficient condition for linear stability of parallel shear flows is derived using the conservation of wave activity a stability analysis is then performed for a sheet vortex that violates the stability condition the instability of a sheet vortex in the 2d euler system 2 is referred to as a kelvin helmholtz kh instability such an instability for the generalized 2d fluid system is investigated for 0 3 the sheet vortex is unstable in the sense that a sinusoidal perturbation applied to it grows exponentially with time the growth rate is finite and depends on the wavenumber of the perturbation as k3 for 1 3 where k is the wavenumber of the perturbation in contrast for 0 1 the growth rate is infinite in other words a transition of the growth rate of the perturbation occurs at 1 a physical model for kh instability in the generalized 2d fluid system which can explain the transition of the growth rate of the perturbation at 1 is proposed

invariance principle for the random conductance model with unbounded conductances we study a continuous time random walk x in an environment of i i d random conductances e 1 we obtain heat kernel bounds and prove a quenched invariance principle for x this holds even when e e

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

simple random walk on long range percolation clusters ii scaling limits supported in part at the technion by a landau fellowship supported in part by an alfred sloan fellowship in mathematics

exponential decay for a nonlinear model for electrical conduction in biological tissues abstract we study the asymptotic convergence to a periodic steady state of the solution of a nonlinear system of equations with periodic boundary data modeling electrical conduction in biological tissues both in the microscopic and in the homogenized version such model keeps into account the resistive behavior of the intracellular and extracellular domains and also the capacitive resistive behavior of the lipidic cellular membrane the rate of convergence is analyzed and the systems of equations satisfied by the asymptotic limits are exhibited when the resistive behavior of the membrane is described by a strictly monotone and coercive nonlinear function the special case of homogeneous boundary conditions is also investigated where the coercivity assumption can be relaxed

α regularization of the pod galerkin dynamical systems of the kuramoto sivashinsky equation abstract in order to improve the dynamics and stability of the pod galerkin models of strongly stiff systems an like regularization is suggested and assessed in the present article in this method the pod eigenmodes of the non linear terms are replaced by their helmholtz filtered counterparts while the other terms are remained unchanged as an example the method is applied to the pod galerkin models of the one dimensional kuramoto sivashinsky ks equation in a full chaotic regime and the fidelity of the original and regularized models to the direct numerical simulations dns are investigated moreover the effects of regularization on the dynamics of various terms and whole of the systems are analyzed via eigenvalue analysis of each term separately and the total dynamical system as a whole the numerical experiments show definite effectiveness of the method and excellent improvements in the predicted dynamics and stability by minimum number of free parameters

random walks in random environments random walks in random environments and their diffusion analogues have been a source of surprising phenomena and challenging problems especially in the non reversible situation since they began to be studied in the 1970s we review the model available results and techniques and point out several gaps in the understanding of these processes

comparison of quenched and annealed invariance principles for random conductance model we show that there exists an ergodic conductance environment such that the weak annealed invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle does not hold

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

large deviations for random walks in a random environment we prove two large deviation principles for random walks in a random environment one for quenched walks in general ergodic environments and the other for averaged walks in a product environment 2003 wiley periodicals inc

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

front formation in an active scalar equation we study the formation of thermal fronts in an active scalar equation that is similar to the euler equation for a particular initial condition an earlier candidate for finite time blowup the front forms in a generalized self similar way with constant hyperbolicity at the center the behavior belongs to a class of scenarios for which finite time blowup is impossible a systematic exploration of many different initial conditions reveals no evidence of singular solutions copyright ital 1999 ital the american physical society

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

nearest neighbor walks with low predictability profile and percolation in 2 epsilon dimensions a few years ago grimmett kesten and zhang proved that for supercritical bond percolation on z 3 simple random walk on the infinite cluster is a s transient we generalize this result to a class of wedges in z 3 including for any e 0 1 the wedge w e x y z z 3 x 0 z x e which can be thought of as representing a 2 e dimensional lattice our proof builds on recent work of benjamini pemantle and peres and involves the construction of finite energy flows using nearest neighbor walks on z with low predictability profile along the way we obtain some new results on attainable decay rates for predictability profiles of nearest neighbor walks

two scale homogenization of non linear degenerate evolution equations using the notion of two scale convergence developed by allaire the homogenization of a degenerate non linear evolution equation with periodically oscillating coefficients is presented a two scale homogenized system is obtained as the limit of the periodic problem monotone operator methods and two scale convergence are employed to show that the solutions of the periodic problem converge to the unique solution of the homogenized system homogenized initial conditions are also obtained and the sense in which they hold for the homogenized initial value problem is made specific

micro structure models of diffusion in fissured media abstract two diffusion models are developed which recognize the local geometry of the individual cells or storage sites and the exchange of flux on the micro scale of these cells the cauchy problems for these model systems are shown to be resolved by holomorphic semigroups and various classical models are obtained as limits of these disributed micro structure models

random walks on infinite percolation clusters in models with long range correlations for a general class of percolation models with long range correlations on zdzd d 2d 2 introduced in j math phys 55 2014 083307 we establish regularity conditions of barlow ann probab 32 2004 3024 3084 that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak poincare inequality as immediate corollaries we deduce quenched heat kernel bounds parabolic harnack inequality and finiteness of the dimension of harmonic functions with at most polynomial growth heat kernel bounds and the quenched invariance principle of probab theory related fields 166 2016 619 657 allow to extend various other known results about bernoulli percolation by mimicking their proofs for instance the local central limit theorem of electron j probab 14 209 1 27 or the result of ann probab 43 2015 2332 2373 that the dimension of at most linear harmonic functions on the infinite cluster is d 1d 1 in terms of specific models all these results are new for random interlacements at every level in any dimension d 3d 3 as well as for the vacant set of random interlacements ann of math 2 171 2010 2039 2087 comm pure appl math 62 2009 831 858 and the level sets of the gaussian free field comm math phys 320 2013 571 601 in the regime of the so called local uniqueness which is believed to coincide with the whole supercritical regime for these models

a law of large numbers for random walks in random mixing environments we prove a law of large numbers for a class of ballistic multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions which hold when the environment is a weak mixing field in the sense of dobrushin and shlosman our result holds if the mixing rate balances moments of some random times depending on the path it applies in the nonnestling case but we also provide examples of nestling walks that satisfy our assumptions the derivation is based on an adaptation using coupling of the regeneration argument of sznitman and zerner

spectral measure and approximation of homogenized coefficients this article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations the originality of this work is the use of a well known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients in particular we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error and conversely combined with results by otto and the first author in low dimension and results by the second author in high dimension this allows us to prove that for any dimension d 2 there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem

quenched invariance principle for random walks in balanced random environment we consider random walks in a balanced random environment in mathbb z d d 2 we first prove an invariance principle for d 2 and the transience of the random walks when d 3 recurrence when d 2 in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition then using percolation arguments we show that under mere ellipticity the above results hold for random walks in i i d balanced environments

effective polynomial ballisticity conditions for random walk in random environment the conditions t 0 1 which were introduced by sznitman in 2002 have had a significant impact on research in random walk in a random environment among others these conditions entail a ballistic behavior as well as an invariance principle they require the stretched exponential decay of certain slab exit probabilities for the random walk under the averaged measure and are asymptotic in nature the main goal of this paper is to show that in all relevant dimensions i e d 2 in order to establish the conditions t it is actually enough to check a corresponding condition of polynomial type in addition to only requiring an a priori weaker decay of the corresponding slab exit probabilities than t another advantage of the condition is that it is effective in the sense that it can be checked on finite boxes in particular this extends the conjectured equivalence of the conditions t 0 1 to all relevant dimensions 2014 wiley periodicals inc

hydrodynamic nambu brackets derived by geometric constraints a geometric approach to derive the nambu brackets for ideal two dimensional 2d hydrodynamics is suggested the derivation is based on two forms with vanishing integrals in a periodic domain and with resulting dynamics constrained by an orthogonality condition as a result 2d hydrodynamics with vorticity as dynamic variable emerges as a generic model with conservation laws which can be interpreted as enstrophy and energy functionals generalized forms like surface quasi geostrophy and fractional poisson equations for the stream function are also included as results from the derivation the formalism is extended to a hydrodynamic system coupled to a second degree of freedom with the rayleigh benard convection as an example this system is reformulated in terms of constitutive conservation laws with two additive brackets which represent individual processes a first representing inviscid 2d hydrodynamics and a second representing the coupling between hydrodynamics and thermodynamics the results can be used for the formulation of conservative numerical algorithms that can be employed for example for the study of fronts and singularities

random walk on the incipient infinite cluster for oriented percolation in high dimensions we consider simple random walk on the incipient infinite cluster for the spread out model of oriented percolation on mathbb z d times mathbb z in dimensions d 6 we obtain bounds on exit times transition probabilities and the range of the random walk which establish that the spectral dimension of the incipient infinite cluster is frac 4 3 and thereby prove a version of the alexander orbach conjecture in this setting the proof divides into two parts one part establishes general estimates for simple random walk on an arbitrary infinite random graph given suitable bounds on volume and effective resistance for the random graph a second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d 6 by extending results about critical oriented percolation obtained previously via the lace expansion

an almost sure invariance principle for random walks in a space time random environment we consider a discrete time random walk in a space time i i d random environment we use a martingale approach to show that the walk is diffusive in almost every fixed environment we improve on existing results by proving an invariance principle and considering environments with an l2 averaged drift we also state an a s invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean under an ergodic invariant measure for the environment chain

a law of large numbers for random walks in random environment we derive a law of large numbers for a class of multidimensional random walks in random environment satisfying a condition which first appeared in the work of kalikow the approach is based on the existence of a renewal structure under an assumption of transience in the direction l this extends to a multidimensional context previous work of kesten our results also enable proving the convergence of the law of the environment viewed from the particle toward a limiting distribution

existence of the harmonic measure for random walks on graphs and in random environments we give a sufficient condition for the existence of the harmonic measure from infinity of transient random walks on weighted graphs in particular this condition is verified by the random conductance model on d d 3 when the conductances are i i d and the bonds with positive conductance percolate the harmonic measure from infinity also exists for random walks on supercritical clusters of 2 this is proved using results of barlow ann probab 32 3024 3084 2004 and barlow and hambly electron j probab 14 1 1 27 2009

the speed of biased random walk on percolation clusters we consider biased random walk on supercritical percolation clusters in 2 we show that the random walk is transient and that there are two speed regimes if the bias is large enough the random walk has speed zero while if the bias is small enough the speed of the random walk is positive

green s function for a generalized two dimensional fluid a green s function for a generalized two dimensional 2d fluid in an unbounded domain the so called turbulence system is discussed the generalized 2d fluid is characterized by a relationship between an advected quantity q and the stream function namely q 2 here is a real number and q is referred to as the vorticity in this study the green s function refers to the stream function produced by a delta functional distribution of q i e a point vortex with unit strength the green s function has the form g r r 2 except when is an even number where r is the distance from the point vortex this functional form is known as the riesz potential when is a positive even number the logarithmic correction to the riesz potential has the form g r r 2 in r in contrast when is a negative even number g is given by the higher order laplacian of the delta function the transition of the small scale behavior of q at 2 a well known property of forced and dissipative turbulence is explained in terms of the green s function moreover the azimuthal velocity around the point vortex is derived from the green s function the functional form of the azimuthal velocity indicates that physically realizable systems for the generalized 2d fluid exist only when 3 the green s function and physically realizable systems for an anisotropic generalized 2d fluid are presented as an application of the present study

isoperimetry and heat kernel decay on percolation clusters we prove that the heat kernel on the infinite bernoulli percolation cluster in zd almost surely decays faster than t d 2 we also derive estimates on the mixing time for the random walk confined to a finite box our approach is based on local isoperimetric inequalities some of the results of this paper were previously announced in the note of mathieu and remy c r acad sci paris ser i math 332 2001 927 931

evolving sets mixing and heat kernel bounds we show that a new probabilistic technique recently introduced by the first author yields the sharpest bounds obtained to date on mixing times of markov chains in terms of isoperimetric properties of the state space also known as conductance bounds or cheeger inequalities we prove that the bounds for mixing time in total variation obtained by lovasz and kannan can be refined to apply to the maximum relative deviation p n x y y 1 of the distribution at time n from the stationary distribution we then extend our results to markov chains on infinite state spaces and to continuous time chains our approach yields a direct link between isoperimetric inequalities and heat kernel bounds previously this link rested on analytic estimates known as nash inequalities

scaling limit for a class of gradient fields with nonconvex potentials we consider gradient fields x x d whose law takes the gibbs boltzmann form z 1 exp x y v v x where the sum runs over nearest neighbors we assume that the potential v admits the representation v log d exp 1 2 2 where is a positive measure with compact support in 0 hence the potential v is symmetric but nonconvex in general while for strictly convex v s the translation invariant ergodic gradient gibbs measures are completely characterized by their tilt a nonconvex potential as above may lead to several ergodic gradient gibbs measures with zero tilt still every ergodic zero tilt gradient gibbs measure for the potential v above scales to a gaussian free field

stochastic homogenization of nonconvex hamilton jacobi equations a counterexample this paper provides a counterexample to hamilton jacobi homogenization in the nonconvex case for general stationary ergodic environments

a zero one law for recurrence and transience of frog processes we provide sufficient conditions for the validity of a dichotomy i e zero one law between recurrence and transience of general frog models in particular the results cover frog models with i i d numbers of frogs per site where the frog dynamics are given by quasi transitive markov chains or by random walks in a common random environment including super critical percolation clusters on mathbb z d we also give a sufficient and almost sharp condition for recurrence of uniformly elliptic frog processes on mathbb z d its proof uses the general zero one law

diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice we construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal a dirichlet form can be constructed on the limit set and we consider the properties of the associated laplace operator and diffusion process in particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on diagonal heat kernel for the percolation clusters and for the underlying lattice in this setting a number of features of the lattice are inherited by the critical cluster

a central limit theorem for biased random walks on galton watson trees let mathcal t be a rooted galton watson tree with offspring distribution p k that has p 0 0 mean m kp k 1 and exponential tails consider the biased random walk x n n 0 on mathcal t this is the nearest neighbor random walk which when at a vertex v with d v offspring moves closer to the root with probability d v and moves to each of the offspring with probability 1 d v it is known that this walk has an a s constant speed tt v lim n x n n where x n is the distance of x n from the root with tt v 0 for 0 m and tt v 0 for m for all m we prove a quenched clt for x n n tt v for m the walk is positive recurrent and there is no clt the most interesting case by far is m where the clt has the following form for almost every mathcal t the ratio x nt sqrt n converges in law as n to a deterministic multiple of the absolute value of a brownian motion our approach to this case is based on an explicit description of an invariant measure for the walk from the point of view of the particle previously such a measure was explicitly known only for 1 and the construction of appropriate harmonic coordinates

robustness of the inverse cascade in two dimensional turbulence we study quasisteady inverse cascades in unbounded and bounded two dimensional turbulence driven by time independent injection and dissipated by molecular viscosity it is shown that an inverse cascade that carries only a fraction r of the energy input to the largest scales requires the enstrophy range energy spectrum to be steeper than k ensuremath 5 ruling out a direct cascade unless 1 ensuremath r ensuremath ll 1 a direct cascade requires the presence of an inverse cascade that carries virtually all energy to the largest scales 1 ensuremath r ensuremath ll 1 these facts underlie the robustness of the kolmogorov kraichnan k ensuremath 5 3 inverse cascade which is readily observable in numerical simulations without an accompanying direct enstrophy cascade we numerically demonstrate an instance where the k ensuremath 5 3 inverse cascading range is realizable with 79 of the energy injection dissipated within the energy range and virtually all of the enstrophy dissipated in the vicinity of the forcing region as equilibrium is approached the respective logarithmic slopes ensuremath alpha and ensuremath beta of the ranges of wave numbers lower and higher than the forcing wave number satisfy ensuremath alpha ensuremath beta ensuremath approx 8 these results are consistent with recent theoretical predictions

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media

symmetries of the turbulent state the emphasis of this review is on fundamental properties degree of universality and symmetries of the turbulent state the central questions are which symmetries remain broken even when the symmetry breaking factor reaches zero and which symmetries in contrast emerge in the state of developed turbulence we shall see that time reversibility is broken in all cases since turbulence is a far from equilibrium state accompanied by dissipation as far as scale invariance is concerned we argue that it is always broken in direct cascades toward small scales no matter how far one goes away from the pumping scale in contrast inverse cascades become scale invariant as they go toward large scales moreover some properties of the inverse cascades seem to be conformal invariant and related to schramm loewner evolution a class of random curves that can be mapped to a 1d brownian walk

the point of view of the particle on the law of large numbers for random walks in a mixing random environment the point of view of the particle is an approach that has proven very powerful in the study of many models of random motions in random media we provide a new use of this approach to prove the law of large numbers in the case of one or higher dimensional finite range transient random walks in mixing random environments one of the advantages of this method over what has been used so far is that it is not restricted to i i d environments

quenched invariance principles for the random conductance model on a random graph with degenerate ergodic weights we consider a stationary and ergodic random field omega e e in e d that is parameterized by the edge set of the euclidean lattice mathbb z d d ge 2 the random variable omega e taking values in 0 infty and satisfying certain moment bounds is thought of as the conductance of the edge e assuming that the set of edges with positive conductances give rise to a unique infinite cluster mathcal c infty omega we prove a quenched invariance principle for the continuous time random walk among random conductances under certain moment conditions an essential ingredient of our proof is a new anchored relative isoperimetric inequality

shortest spanning trees and a counterexample for random walks in random environments we construct forests that span d d 2 that are stationary and directed and whose trees are infinite but for which the subtrees attached to each vertex are as short as possible for d 3 two independent copies of such forests pointing in opposite directions can be pruned so as to become disjoint from this we construct in d 3 a stationary polynomially mixing and uniformly elliptic environment of nearest neighbor transition probabilities on d for which the corresponding random walk disobeys a certain zero one law for directional transience

nonlinear transfer and spectral distribution of energy in α turbulence abstract two dimensional turbulence governed by the so called turbulence equations which include the surface quasi geostrophic equation 1 the navier stokes system 2 and the governing equation for a shallow flow on a rotating domain driven by a uniform internal heating 3 is studied here in both the unbounded and doubly periodic domains this family of equations conserves two inviscid invariants energy and enstrophy in the navier stokes case the dynamics of which are believed to undergo a dual cascade it is shown that an inverse cascade can exist in the absence of a direct cascade and that the latter is possible only when the inverse transfer rate of the inverse cascading quantity approaches its own injection rate constraints on the spectral exponents in the wavenumber ranges lower and higher than the injection range are derived for navier stokes turbulence with moderate reynolds numbers the realization of an inverse energy cascade in the complete absence of a direct enstrophy cascade is confirmed by numerical simulations

limit theory for random walks in degenerate time dependent random environments abstract we study continuous time variable speed random walks in random environments on z d d 2 where at time t the walk at x jumps across edge x y at time dependent rate a t x y the rates which we assume stationary and ergodic with respect to space time shifts are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time we formulate conditions on the environment under which the law of diffusively scaled random walk path tends to brownian motion for almost every sample of the rates the proofs invoke moser iteration to prove sublinearity of the corrector in pointwise sense a key additional input is a conversion of certain weighted energy norms to ordinary ones our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the helffer sjostrand representation of gradient models with certain non strictly convex potentials

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

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

transience recurrence and critical behavior for long range percolation we study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions d 1 2 where x and y are connected with probability we show that if d s 2d then there is no infinite cluster at criticality this result is extended to the free random cluster model a second corollary is that when d 2 and d s 2d we can erase all long enough bonds and still have an infinite cluster the proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks in particular we show that i i d conductances on a recurrent graph of bounded degree yield a recurrent electrical network

on the scaling of the chemical distance in long range percolation models we consider the unoriented long range percolation on z d in dimensions d 1 where distinct sites x y z d get connected with probability p xy 0 1 assuming p xy x y s o 1 as x y where s 0 and is a norm distance on z d and supposing that the resulting random graph contains an infinite connected component c we let d x y be the graph distance between x and y measured on c our main result is that for s d 2d d x y log x y o 1 x y c x y where 1 is the binary logarithm of 2d s and o 1 is a quantity tending to zero in probability as x y besides its interest for general percolation theory this result sheds some light on a question that has recently surfaced in the context of small world phenomena as part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box

on the anisotropic walk on the supercritical percolation cluster we investigate in this work the asymptotic behavior of an anisotropic random walk on the supercritical cluster for bond percolation on d d 2 in particular we show that for small anisotropy the walk behaves in a ballistic fashion whereas for strong anisotropy the walk is sub diffusive for arbitrary anisotropy we also prove the directional transience of the walk and construct a renewal structure

harmonic functions h transform and large deviations for random walks in random environments in dimensions four and higher we consider large deviations for nearest neighbor random walk in a uniformly elliptic i i d environment on d there exist variational formulae for the quenched and averaged rate functions i q and i a obtained by rosenbluth and varadhan respectively i q and i a are not identically equal however when d 4 and the walk satisfies the so called t condition of sznitman they have been previously shown to be equal on an open set a eq for every a eq we prove the existence of a positive solution to a laplace like equation involving and the original transition kernel of the walk we then use this solution to define a new transition kernel via the h transform technique of doob this new kernel corresponds to the unique minimizer of varadhan s variational formula at it also corresponds to the unique minimizer of rosenbluth s variational formula provided that the latter is slightly modified

correlation structure of the corrector in stochastic homogenization recently the quantification of errors in the stochastic homogenization of divergence form operators has witnessed important progress our aim now is to go beyond error bounds and give precise descriptions of the effect of the randomness in the large scale limit this paper is a first step in this direction our main result is to identify the correlation structure of the corrector in dimension 33 and higher this correlation structure is similar to but different from that of a gaussian free field

analysis of the simulation of single phase flow through a naturally fractured reservoir a general form of the double porosity model for single phase flow through a naturally fractured reservoir is derived by explicitly considering fluid flow in individual matrix blocks the warren and root model is shown to be a crude approximation to this model the general model consists of a parabolic equation coupled to a series of parabolic equations it is shown that the coupling term can be viewed as a positive semidefinite perturbation of the time derivative and hence it is verified that the model is well posed a finite element method is presented to approximate the solution and optimal order l 2 error estimates are derived

growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids the growth rates of scalar gradients are studied numerically and analytically in a family of two dimensional 2d incompressible fluid equations related to the surface quasigeostrophic sqg equation the active scalar is related to the stream function by 2 0 2 a notable difference is observed in a comparison of the instantaneous growth rates in l p and in l norms depending on the stage of the time evolution the crux is the phase shift effect of singular integral operators which displaces the peak location of the scalar gradient from that of the strain rate on this basis a method of detecting such a dislocation is proposed in view of the importance of their coalescence needed for a possible blow up moreover it is found in the long time evolution that a solution of the sqg equation whose regularity is not known is less singular than that of the 2d euler equations known to be regular on the time interval covered by this computation this consistently expands an earlier observation by majda and tabak physica d 98 515 1996 in some detail a 1d model problem is discussed to illustrate the present method and extensions to the 3d case are also are briefly discussed

geometric statistics in turbulence the author presents results regarding certain average properties of incompressible fluids derived from the equations of motion the author estimates the average dissipation rate the average dimension of level sets the role played by the field of direction of vorticity in the three dimensional euler and navier stokes equations is discussed and a class of two dimensional equations that are useful models of incompressible dynamics is described the author presents results concerning scaling exponents in turbulence