large deviation principles for sequences of logarithmically weighted means abstract in this paper we consider several examples of sequences of partial sums of triangular arrays of random variables x n n 1 in each case x n converges weakly to an infinitely divisible distribution a poisson distribution or a centered normal distribution for each sequence we prove large deviation results for the logarithmically weighted means 1 log n k 1 n 1 k x k n 1 with speed function v n log n we also prove a sample path large deviation principle for x n n 1 defined by x n i 1 n u i 2 n where 2 0 and u n n 1 is a sequence of independent standard brownian motions

maximal cohen macaulay modules over the cone of an elliptic curve let r k y1 y2 y3 f f y13 y23 y33 where k is an algebraically closed field with chark 3 using atiyah bundle classification over elliptic curves we describe the matrix factorizations of the graded indecomposable reflexive r modules equivalently we describe explicitly the indecomposable bundles over the projective curve v f pk2 using the fact that over the completion r of r every reflexive module is gradable we obtain a description of the maximal cohen macaulay modules over r k y1 y2 y3 f

convex invariant means and a pathwise central limit theorem on etudie le probleme de choisir un point aleatoirement a partir de la ligne des reels on definit une notion de distribution d espace uniforme on considere trois sources supplementaires d uniformite l invariance d echelle convexe la consistance avec des methodes de moyennes et la linearite mesure de mokobodski

fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

factoring the adjoint and maximal cohen macaulay modules over the generic determinant a question of bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices we show that such factorizations indeed exist over any coefficient ring when the matrix has even size establishing a correspondence between such factorizations and extensions of maximal cohen macaulay modules over the generic determinant we exhibit all factorizations where one of the factors has determinant equal to the generic determinant the classification shows not only that the cohen macaulay representation theory of the generic determinant is wild in the tame wild dichotomy but that it is quite wild even in rank two the isomorphism classes cannot be parametrized by a finite dimensional variety over the coefficients we further relate the factorization problem to the multiplicative structure of the mathop rm ext nolimits algebra of the two nontrivial rank one maximal cohen macaulay modules and determine it completely

the well posedness of the surface quasi geostrophic equations in the besov morrey spaces abstract in this paper we develop the boundedness of riesz transform in morrey spaces which generalize the corresponding result in the usual l p spaces as an application of this boundedness we establish the well posedness and blow up criterion for the surface quasi geostrophic equations in the framework of besov morrey spaces

function theory random paths and covering spaces pour des varietes de riemann connexes m on etudie la relation entre la theorie de fonction harmonique sur m les proprietes statistiques des chemins aleatoires sur m et la structure geometrique globale de m

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

the radiative transfer equation in the forward peaked regime in this work we study the radiative transfer equation in the forward peaked regime in free space specifically it is shown that the equation is well posed by proving instantaneous regularization of weak solutions for arbitrary initial datum in l1 classical techniques for hypo elliptic operators such as averaging lemma are used in the argument among the interesting aspects of the proof are the use of the stereographic projection and the presentation of a rigorous expression for the scattering operator given in terms of a fractional laplace beltrami operator on the sphere or equivalently a weighted fractional laplacian analog in the projected plane such representations may be used for accurate numerical simulations of the model as a bonus given by the methodology we show convergence of henyey greenstein scattering models and vanishing of the solution at time algebraic rate due to scattering diffusion

rank 2 arithmetically cohen macaulay bundles on a nonsingular cubic surface abstract rank 2 indecomposable arithmetically cohen macaulay bundles e on a nonsingular cubic surface x in p 3 are classified by means of the possible forms taken by the minimal graded free resolution of e over p 3 the admissible values of the chern classes of e are listed and the vanishing locus of a general section of e is studied properties of e such as slope semi stability and simplicity are investigated the number of relevant families is computed together with their dimension

on large deviations for some sequences of weighted means of gaussian processes in this paper we study some sequences of weighted means of continuous real valued gaussian processes more precisely we consider suitable generalizations of both arithmetic and logarithmic means of a gaussian process with covariance function which satisfies either an exponential decay condition or a power decay condition our aim is to provide limits of variances of functionals of such weighted means which allow the application of some large deviation results in the literature

on the differentiability of the solution to the hamilton jacobi equation with critical fractional diffusion abstract we prove that the hamilton jacobi equation for an arbitrary hamiltonian h locally lipschitz but not necessarily convex and fractional diffusion of order one critical has classical c 1 solutions the proof is achieved using a new holder estimate for solutions of advection diffusion equations of order one with bounded vector fields that are not necessarily divergence free

global well posedness for the 2d inviscid benard system with fractional diffusivity and yudovichʼs type data abstract in this paper we consider the cauchy problem of the two dimensional inviscid benard system with fractional diffusivity we show that there is a global unique solution to this system with yudovich s type data

global well posedness in the super critical dissipative quasi geostrophic equations we consider the quasi geostrophic equation with the dissipation term in the case constantin cordoba wu 6 proved the global existence of strong solution in h 1 and h 2 under the assumption of small l norm of initial data in this paper we prove the global existence in the scale invariant besov space b 2 2 2 1 for initial data small in the b 2 2 2 1 norm we also prove a global stability result in b 1 2 1

tangent measure distributions of fractal measures tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in euclidean spaces the idea is loosely speaking to attach to every point of the set a family of random measures called the alpha dimensional tangent measure distributions at the point which describe asymptotically the alpha dimensional scenery seen by an observer zooming down towards this point this tool has been used by bandt ba and graf g to study the regularity of the local geometry of self similar sets but in this paper we show that its scope goes much beyond this situation and in fact it may be used to describe a strong regularity property possessed by every measure we show that for every measure mu on a euclidean space and any dimension alpha at mu almost every point all alpha dimensional tangent measure distributions are palm measures this means that the local geometry of every dimension of general measures can be described like the local geometry of self similar sets by means of a family of statistically self similar random measures we believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities

a pde approach to fractional diffusion a space fractional wave equation we study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers of order s in 0 1 of symmetric coercive linear elliptic second order operators in bounded domains varomega we realize fractional diffusion as the dirichlet to neumann map for a nonuniformly elliptic problem posed on the semi infinite cylinder mathcal c varomega times 0 infty we thus rewrite our evolution problem as a quasi stationary elliptic problem with a dynamic boundary condition and derive space time and space time regularity estimates for its solution the latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation we propose and analyze two fully discrete schemes the discretization in time is based on finite difference discretization techniques the trapezoidal and leapfrog schemes the discretization in space relies on the tensorization of a first degree fem in varomega with a suitable hp fem in the extended variable for both schemes we derive stability and error estimates we consider a first degree fem in varomega with mesh refinement near corners and the aforementioned hp fem in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open bounded polytopal but not necessarily convex domains varomega subset mathbb r 2 we discuss implementation details and report several numerical examples

regularity for a local nonlocal transmission problem we formulate and study an elliptic transmission like problem combining local and nonlocal elements let mathbb r n be separated into two components by a smooth hypersurface on one side of a function satisfies a local second order elliptic equation on the other it satisfies a nonlocal one of lower order in addition a nonlocal transmission condition is imposed across the interface which has a natural variational interpretation we deduce the existence of solutions to the corresponding dirichlet problem and show that under mild assumptions they are holder continuous following the method of de giorgi the principal difficulty stems from the lack of scale invariance near which we circumvent by deducing a special energy estimate which is uniform in the scaling parameter we then turn to the question of optimal regularity and qualitative properties of solutions and show that in the case of constant coefficients and flat they satisfy a kind of transmission condition where the ratio of fractional conormal derivatives on the two sides of the interface is fixed a perturbative argument is then given to show how to obtain regularity for solutions to an equation with variable coefficients and smooth interface throughout we pay special attention to a nonlinear version of the problem with a drift given by a singular integral operator of the solution which has an interpretation in the context of quasigeostrophic dynamics

a generalization of the strassen converse this note is devoted to a generalization of the strassen converse let gn r 0 n 1 be a sequence of measurable functions such that for every n 1 gn x y2 c gn x gn y and gn x y2 c gn x gn y for all x y r where 0 c is a constant which is independent of n let x xn n 1 be a sequence of i i d random variables assume that there exist r 1 and a function 0 0 with limt t depending only on the sequence gn n 1 such that lim supn gn x1 x2 e x r a s whenever e x r and ex 0 we prove the converse result namely that lim supn gn x1 x2 a s implies e x r and ex 0 if in addition lim supn gn c c for all c 0 some applications are provided to illustrate this result

almost sure kallianpur robbins laws for brownian motion in the plane the kallianpur robbins law describes the long term asymptotic behaviour of integrable additive functionals of brownian motion in the plane in this paper we prove an almost sure version of this result it turns out that differently from many known results this requires an iterated logarithmic average a similar result is obtained for the small scales asymptotic by means of an ergodic theorem of chacon ornstein type which allows an exceptional set of scales

pointwise estimates for solutions of fractal burgers equation abstract in this paper we provide two sided estimates and uniform asymptotics for the solution of d dimensional critical fractal burgers equation u t 2 u b u u q 0 1 2 b r d for q 1 d and u 0 l 1 r d we consider also q 1 d under additional condition u 0 l r d in both cases we assume u 0 0 which implies that the solution is non negative the estimates are given in the terms of the function p t u 0 where p t denotes the semigroup for the operator t 2

radial symmetry results for fractional laplacian systems abstract in this paper we generalize the direct method of moving planes for the fractional laplacian to the system case considering a coupled nonlinear system with fractional laplacian we first establish a decay at infinity principle and a narrow region principle using these principles we then obtain two radial symmetry results for the decaying solutions of the fractional laplacian systems finally we apply our method to fractional schrodinger systems and fractional henon systems

global well posedness for the euler boussinesq system with axisymmetric data abstract in this paper we prove the global well posedness for the three dimensional euler boussinesq system with axisymmetric initial data without swirl this system couples the euler equation with a transport diffusion equation governing the temperature

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

foliations the ergodic theorem and brownian motion regarding the leaves of a compact foliated riemannian manifold as the orbits of a dynamical system we will prove a birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves there are several unexpected developments in the foundations first there always are measures on any compact foliated riemannian manifold m f with respect to which an adequate brownian motion along the leaves ergodic theorem holds second relative to these measures which are called harmonic there is a liouville type theorem for any bounded bore1 function on the manifold which is harmonic on each leaf namely for almost all leaves relative to any one of these harmonic measures this leaf harmonic function is constant on each leaf third the harmonic measures have a very reasonable local characterization in any foliation coordinate system they are transversal sums of h l dx l where h l is a positive harmonic function on the leaf l and dx l is the riemannian measure on l these statements comprise theorem 1 which is stated more precisely later in the introduction the first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measures of plante 141 and ruelle sullivan 151 and some foliations do not admit these the second fact is unexpected because there are foliations where all the leaves are hyperbolic planes and each hyperbolic plane being conformal to the disk admits 285 0022 1236183 3 00

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

nonlinear fractional laplacian problems with nonlocal gradient terms let n 2 be a smooth bounded domain for s 1 2 1 we consider a problem of the form where 0 is a real parameter f belongs to a suitable lebesgue space and is a nonlocal gradient square term given by depending on the real parameter 0 we derive existence and non existence results the proof of our existence result relies on sharp calderon zygmund type regularity results for the fractional poisson equation with low integrability data we also obtain existence results for related problems involving different nonlocal diffusion terms

commutator estimate and its application to regularity criteria of the dissipative quasi geostrophic equation abstract a new commutator estimate with respect to a nonlinear convection upper bounded by a single partial derivative component in hilbert spaces is obtained as an application regularity criteria on the supercritical quasi geostrophic equation are obtained provided that solution growth conditions are assumed to involve a single partial derivative component

long time dynamics of 2d quasi geostrophic equations with damping in lp abstract the long time behavior of solutions of 2d quasi geostrophic equations with damping on the whole space is studied the existence of the global attractor in l p is proved new difficulties come from nonlocal terms and the banach space l p we overcome them by using commutator estimates and positivity inequalities associated with fractional laplacian

note on the well posedness of a slightly supercritical surface quasi geostrophic equation abstract for the following slightly supercritical surface quasi geostrophic equation t u d 0 u d 2 m d 0 1 where m c r 2 0 is a radial non decreasing positive function which roughly has a logarithmic growth near infinity we apply the method of nonlocal maximum principle to show the global well posedness of smooth solutions

on the cauchy problem for integro differential operators in sobolev classes and the martingale problem abstract the existence and uniqueness in sobolev spaces of solutions of the cauchy problem to parabolic integro differential equation with variable coefficients of the order 0 2 is investigated the principal part of the operator has kernel m t x y y d with a bounded nondegenerate m holder in x and measurable in y the lower order part has bounded and measurable coefficients the result is applied to prove the existence and uniqueness of the corresponding martingale problem

a frequency localized maximum principle applied to the 2d quasi geostrophic equation in this paper we prove a maximum principle for a frequency localized transport diffusion equation as an application we prove the local well posedness of the supercritical quasi geostrophic equation in the critical besov spaces mathring b 1 alpha infty q and global well posedness of the critical quasi geostrophic equation in mathring b 0 infty q for all 1 q here mathring b s infty q is the closure of the schwartz functions in the norm of b s infty q

remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations abstract in this paper we are concerned with the partial regularity for suitable weak solutions of the tri dimensional magnetohydrodynamic equations with the help of the de giorgi iteration method we obtain the results proved by he and xin c he z xin partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations j funct anal 227 2005 113 152 namely the one dimensional parabolic hausdorff measure of the possible singular points of the velocity field and the magnetic field is zero

rank 2 cohen macaulay modules over singularities of type x13 x23 x33 x43 abstract we describe by matrix factorizations all graded rank 2 maximal cohen macaulay modules over singularities of type x 1 3 x 2 3 x 3 3 x 4 3

temporal distributional limit theorems for dynamical systems suppose t t is a borel flow on a complete separable metric space x f x rightarrow mathbb r is borel and x in x a temporal distributional limit theorem is a scaling limit for the distributions of the random variables x t int 0 t f t s x ds where t is chosen randomly uniformly from 0 t x is fixed and t rightarrow infty we discuss such laws for irrational rotations anosov flows and horocycle flows

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

partial regularity of suitable weak solutions to the fractional navier stokes equations in this paper we study the partial regularity of fractional navier stokes equations in mathbb r 3 times 0 infty with 3 4 s 1 we show that the suitable weak solution is regular away from a relatively closed singular set whose 5 4s dimentional hausdorff measure is zero the result is a generalization of the partial regularity for the classical navier stokes system in caffarelli et al commun pure appl math 35 771 831 1982

asymptotic expansion of solutions to the drift diffusion equation with fractional dissipation abstract the initial value problem for the drift diffusion equation arising from the model of semiconductor device simulations is studied the dissipation on this equation is given by the fractional laplacian 2 large time behavior of solutions to the drift diffusion equation with 0 1 is discussed when 1 large time behavior of solutions is known however when 0 1 the perturbation methods used in the preceding works would not work in this paper the asymptotic expansion of solutions with high order is derived

on almost sure local and global central limit theorems lets1 s2 be a sequence of sums of i i d random variables the properties of the logarithmic average frac 1 log n sum limits k 1 n frac i a k leqq s k b k kp a k leqq s k b k will be studied under some conditions