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

fractal reinforcement of elastic membranes we describe a homogenization model of an elastic membrane reinforced by the inclusion of a fractal string we follow a variational approach consisting in proving the convergence of certain energy functionals this leads to the spectral convergence of a sequence of weighted second order elliptic partial differential operators to a singular elliptic operator with a fractal term

composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

variational convergence over metric spaces we introduce a natural definition of l p convergence of maps p 1 in the case where the domain is a convergent sequence of measured metric space with respect to the measured gromov hausdorff topology and the target is a gromov hausdorff convergent sequence with the l p convergence we establish a theory of variational convergences we prove that the poincare inequality with some additional condition implies the asymptotic compactness the asymptotic compactness is equivalent to the gromov hausdorff compactness of the energy sublevel sets supposing that the targets are cat 0 spaces we study convergence of resolvents as applications we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of ricci curvature

a harnack inequality for weighted degenerate parabolic equations abstract in this paper we generalize the recent result of dibenedetto gianazza vespri on the harnack inequality for degenerate parabolic equations to the case of a weighted p laplacian type operator in the spatial part the weight is assumed to belong to the suitable muckenhoupt class

homogenization of non uniformly bounded operators critical barrier for nonlocal effects this paper deals with the homogenization of a class of highly oscillating monotonic operators which are uniformly elliptic but non uniformly bounded we determine an asymptotic barrier below which we obtain a classical behaviour and above which nonlocal effects appear we also prove that this condition is optimal in the case of a fibre reinforced medium

γ convergence of equi coercive nonlinear energies defined on vector valued functions with non uniformly bounded coefficients abstract the present paper deals with the asymptotic behavior of equi coercive sequences n of nonlinear functionals defined over vector valued functions in w 0 1 p m where p 1 m 1 and is a bounded open set of r n n 2 the strongly local energy density f n d u of the functional n satisfies a lipschitz condition with respect to the second variable which is controlled by a positive sequence a n which is only bounded in some suitable space l r we prove that the sequence n converges for the strong topology of l p m to a functional which has a strongly local density f d u for sufficiently regular functions u this compactness result extends former results on the topic which are based either on maximum principle arguments in the nonlinear scalar case or adapted div curl lemmas in the linear case here the vectorial character and the nonlinearity of the problem need a new approach based on a careful analysis of the asymptotic minimizers associated with the functional n the relevance of the conditions which are imposed to the energy density f n d u is illustrated by several examples including some classical hyperelastic energies

mosco convergence of dirichlet forms in infinite dimensions with changing reference measures let e be an infinite dimensional locally convex space let n be a weakly convergent sequence of probability measures on e and let en be a sequence of dirichlet forms on e such that en is defined on l2 n general sufficient conditions for mosco convergence of the gradient dirichlet forms are obtained applications to gibbs states on a lattice and to the gaussian case are given weak convergence of the associated processes is discussed

analysis of degenerate elliptic operators of grusin type we analyse degenerate second order elliptic operators h in divergence form on l 2 r n r m we assume the coefficients are real symmetric and a 1 h h a 2 h for some a 1 a 2 0 where h delta nabla x 1 cdot c delta 1 delta 1 x 1 nabla x 1 c delta 2 delta 2 x 1 nabla x 2 2 here x 1 r n x 2 r m and c delta i delta i are positive measurable functions such that c delta i delta i x behaves like x delta i as x 0 and x delta i as x to infty with delta 1 delta 1 in 0 1 rangle and delta 2 delta 2 geq 0 our principal results state that the submarkovian semigroup s t e th is conservative and its kernel k t satisfies bounds 0 leq k t x y leq a b x t 1 2 b y t 1 2 1 2 where b x r denotes the volume of the ball b x r centred at x with radius r measured with respect to the riemannian distance associated with h the proofs depend on detailed subelliptic estimations on h a precise characterization of the riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation we discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive nor continuous

properties of predictors for autoregressive time series abstract the prediction of the n s th observation of the pth order autoregressive process is investigated the mean square of the predictor error through terms of order n 1 conditional on yn y n 1 y n p 1 is obtained for the stationary normal process the mean squared error expression is similar to the usual regression formula for the variance of the predictor error the usual regression formula for the estimated variance of a predictor error and its generalization to s period prediction is shown to provide a consistent estimator of the mean squared error of the least squares predictor for both stationary and non stationary processes

semilinear evolution transmission problems across fractal layers abstract a semilinear parabolic transmission problem across either a fractal layer s or the corresponding prefractal layer s h is studied local existence uniqueness and regularity results for the mild solution in both cases are established as well as a sufficient condition on the initial datum for global existence is given the asymptotic behaviour of the solutions of the approximating problems is studied

fractal snowflake domain diffusion with boundary and interior drifts abstract we study a parabolic ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain we prove that under standard conditions a related cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial ventsell problem as a second result we prove that functions that are intrinsically lipschitz on the snowflake boundary admit euclidean lipschitz extensions to the closure of the entire domain our methods combine the fractal membrane analysis the vector analysis for local dirichlet forms and pde on fractals coercive closed forms and the analysis of lipschitz functions

variations on a strange semi continuity result abstract this paper is devoted to the study of some lower semi continuity results whose model example is lim inf n u n v n u v for any sequence u n v n which weakly converges to u v in h 0 1 2 and satisfies the positivity assumptions u n 0 and v n 0 extensions of this semi continuity result are obtained in the following cases two fixed linear operators two monotone operators two varying linear operators varying domains and non uniformly bounded operators in these various cases the operators under consideration are second order elliptic operators in divergence form and truncation arguments play an essential role finally the case of higher order operators is treated by a quite different approach

on the exterior calculus of invariant theory the full extent of the tragedy of hermann grassmann which has been unfolding since his death by a succession of misadventures and misunderstandings of his work unique in the history of modern mathematics is yet to be fully realized it began with a deliberate ambiguity on the part of the author of the ausdehnungslehre whereby one and the same notation was chosen to designate two different algebraic operations originally known as the progressive and the regressive product this choice was not an oversight on grassmann s part as we shall see but was rather dictated by the author s deep understanding of the philosophical reach of his discovery an understanding which lay well beyond the powers of the

sharp regularity for functionals with p q growth we prove regularity theorems for minimizers of integral functionals of the calculus of variations f x du x dx with non standard growth conditions of p q type z p f x z l z q 1 p q in particular we find that a sufficient condition for minimizers to be regular is qp n n where the function f x z is holder continuous with respect to the x variable and x rn this condition is also sharp we include results in the setting of orlicz spaces moreover we treat certain relaxed functionals too finally we address a problem posed by marcellini in 43 showing a minimizer with an isolated singularity

weak convergence of diffusion processes on wiener space let be a gaussian measure on a suslin space x h be the corresponding cameron martin space and e i h be an orthonormal basis of h suppose that n n is a sequence of probability measures which converges weakly to a probability measure consider a sequence of dirichlet forms varepsilon n where varepsilon n f int x nabla h f 2 h rho n rm d gamma and sqrt rho n in w 2 1 gamma we prove some sufficient conditions for mosco convergence varepsilon n to varepsilon where varepsilon f int x nabla h f 2 h rho rm d gamma in particular if x is a hilbert space sup n sqrt rho n w 2 1 gamma infty and partial e i rho n rho n can be uniformly approximated by finite dimensional conditional expectations rm ie mathcal f n mu n big partial e i rho n rho n big for every fixed e i then under broad assumptions varepsilon n to varepsilon mosco and the distributions of the associated stochastic processes converge weakly

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

an elementary proof of the first fundamental theorem of vector invariant theory 1 introduction for a long time the problem of extending to arbitrary characteristic the fundamental theorems of classical invariant theory remained untouched the technique of young bitableaux introduced by doubilet rota and stein succeeded in providing a simple combinatorial proof of the first fundamental theorem valid over every infinite field since then it was generally thought that the use of bitableaux or some equivalent device as the cancellation lemma proved by hodge after a suggestion of d e littlewoodpwould be indispensable in any such proof and that single tableaux would not suffice in this note we provide a simple combinatorial proof of the first fun damental theorem of invariant theory valid in any characteristic which uses only single tableaux or brackets as they were named by cayley we believe the main combinatorial lemma theorem 2 2 used in our proof can be fruitfully put to use in other invariant theoretic settings as we hope to show elsewhere 2 a combinatorial lemma let k be a commutative integral domain with unity given an alphabet of d symbols x1 x2 xd and a positive integer n consider the d n variables xi j i 1 2 d j 1 2 n which are supposed to be algebraically independent let k x ij be the letter place algebra 6 namely the polynomial algebra generated over k by the variables xi 1 j

particle approximation of the wasserstein diffusion abstract we construct a system of interacting two sided bessel processes on the unit interval and show that the associated empirical measure process converges to the wasserstein diffusion von renesse and sturm 2009 25 assuming that markov uniqueness holds for the generating wasserstein dirichlet form the proof is based on the variational convergence of an associated sequence of dirichlet forms in the generalized mosco sense of kuwae and shioya 2003 19

weighted sobolev spaces on metric measure spaces abstract we investigate weighted sobolev spaces on metric measure spaces x d m denoting by the weight function we compare the space w x d m which always concides with the closure h x d m of lipschitz functions with the weighted sobolev spaces w 1 p x d m and h 1 p x d m defined as in the euclidean theory of weighted sobolev spaces under mild assumptions on the metric measure structure and on the weight we show that w x d m h x d m we also adapt the results proved by muckenhoupt and the ones proved by zhikov to the metric measure setting considering appropriate conditions on that ensure the equality w 1 p x d m h x d m

nonlocal effects in two dimensional conductivity the paper deals with the asymptotic behaviour as e 0 of a two dimensional conduction problem whose matrix valued conductivity ae is e periodic and not uniformly bounded with respect to e we prove that only under the assumptions of equi coerciveness and l1 boundedness of the sequence ae the limit problem is a conduction problem of same nature this new result points out a fundamental difference between the two dimensional conductivity and the three dimensional one indeed under the same assumptions of periodicity equi coerciveness and l1 boundedness it is known that the high conductivity regions can induce nonlocal effects in three or greater dimensions

on singularity of energy measures on self similar sets we provide general criteria for energy measures of regular dirichlet forms on self similar sets to be singular to bernoulli type measures in particular every energy measure is proved to be singular to the hausdorff measure for canonical dirichlet forms on 2 dimensional sierpinski carpets

insulating layers and robin problems on koch mixtures abstract this paper deals with a reinforcement problem for a plane domain whose boundary is a deterministic or random mixture of self similar koch curves we construct an e thin polygonal 2 dimensional fiber e n n n 0 e 1 around pre fractal approximating domains n and related suitable energy functionals the aim of this paper is to study the asymptotic behavior of the reinforced energy functionals while simultaneously the thickness of the fibers and the conductivity of the functionals on the fibers converges to 0 as n

the fundamental theorems of vector invariants abstract new proofs are given of the first fundamental theorems of vector invariants for the special linear and special orthogonal groups the novelty of the proofs is that they rely on reducing the theorems to the case of 2 2 matrices there is a close connection between the vector invariants of the special linear group or special orthogonal group of n n matrices for all n 2 and those of the analogous group of 2 2 matrices this connection is described using the notion of leading monomials properties of leading monomials are also used to give new proofs of the second fundamental theorems of vector invariants for the general linear and orthogonal groups

homogenization of stiff plates and two dimensional high viscosity stokes equations the paper deals with the homogenization of stiff heterogeneous plates assuming that the coefficients are equi bounded in l 1 we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients this compactness result is based on a div curl lemma for fourth order equations on the other hand using an intermediate stream function we deduce from the plates case a similar result for high viscosity stokes equations in dimension two so that the nature of the stokes equation is preserved in the homogenization process finally we show that the l 1 boundedness assumption cannot be relaxed indeed in the case of the stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem which violates the compactness result obtained under the l 1 boundedness condition

determination of the closure of the set of elasticity functionals we determine the closure for mosco convergence in l2 3 of the set of elasticity functionals we prove that this closure coincides with the set of all non negative lower semicontinuous quadratic functionals which are objective i e which vanish for rigid motions the result is still valid if we consider only the set of isotropic elasticity functionals which have a prescribed poisson coefficient this shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients

poincare inequality for weighted first order sobolev spaces on loop spaces abstract let e be the loop space over a compact connected riemannian manifold with a torsion skew symmetric connection let l d be the ornstein uhlenbeck operator on a nonempty connected component d of the loop space e and let v d r be the restriction on d of the potential in the logarithmic sobolev inequality found by l gross on the loop group by s aida and by f z gong and z m ma on the loop space respectively we prove that the schrodinger operator l v l d v always has a spectral gap at the bottom 0 v of its spectrum and thus has its ground state transformed operator 1 l v 0 v where is the unique ground state of l v in particular our result proves l gross s conjecture about the existence of a spectral gap for the ground state transform of the schrodinger operator studied by him on the loop group in addition in all the above cases we identify the domain of the dirichlet forms associated with the ground state transforms as weighted first order sobolev spaces with weight given by 2 thus establishing a poincare inequality for them all these results are consequences from some new results in this paper on dirichlet forms characterizing certain classes with spectral gaps and from results by s aida and m hino

interpretation of the lavrentiev phenomenon by relaxation we consider functionals of the calculus of variations of the form f u 01 f x u u dx defined for u w1 0 1 and we show that the relaxed functional f with respect to weak w1 1 0 1 convergence can be written as f u 01 f x u u dx l u where the additional term l u called the lavrentiev term is explicitly identified in terms of f

homogenization on nested fractals we study the homogenization problem on nested fractals letxt be the continuous time markov chain on the pre nested fractal given by puttingi i d random resistors on each cell it is proved that under some conditions alpha n x t e n t converges in law to a constant time change of the brownian motion on the fractal asn where is the contraction rate andte is a time scale constant as the brownian motion on fractals is not a semi martingale we need a different approach from the well developed martingale method

nonlocal limits in the study of linear elliptic systems arising in periodic homogenization in the present paper we obtain the two scale limit system of a sequence of linear elliptic periodic problems with varying coefficients we show that this system has not the same structure than the classical one obtained when the coefficients are fixed this is due to the apparition of nonlocal effects our results give an example showing that the homogenization of elliptic problems with varying coefficients depending on one parameter gives in general a nonlocal limit problem

semilinear fractal problems approximation and regularity results abstract we prove existence and uniqueness of the weak solution for a second order semilinear transmission problem in a bounded domain in r 3 in the case of a koch type fractal layer s or the corresponding prefractal layer approximating s we study the asymptotic behavior of the solutions of the corresponding prefractal problem and we prove regularity results for the weak solution via a generalized green formula for domains with fractal boundaries

mean estimation bias in least squares estimation of autoregressive processes abstract estimation of the parameters of an autoregressive process with a mean that is a function of time is considered approximate expressions for the bias of the least squares estimator of the autoregressive parameters that is due to estimating the unknown mean function are derived for the case of a mean function that is a polynomial in time a reparameterization that isolates the bias is given using the approximate expressions a method of modifying the least squares estimator is proposed a monte carlo study of the second order autoregressive process is presented the monte carlo results agree well with the approximate theory and generally speaking the modified least squares estimators performed better than the least squares estimator for the second order process we also considered the empirical properties of the estimated generalized least squares estimator of the mean function and the error made in predicting the process one two and three periods in the future

continuity of eigenvalues of subordinate processes in domains let x x t t 0 be a symmetric markov process in a state space e and d an open set of e let s n s n t t 0 be a subordinator with laplace exponent n and s s t t 0 a subordinator with laplace exponent suppose that x is independent of s and s n in this paper we consider the subordinate processes open image in new window and open image in new window and their subprocesses open image in new window and x d killed upon leaving d suppose that the spectra of the semigroups of open image in new window and x d are all discrete with open image in new window being the eigenvalues of the generator of open image in new window and open image in new window being the eigenvalues of the generator of x d we show that if lim n n for every 0 then

on the construction and convergence of traces of forms abstract we elaborate a new method for constructing traces of quadratic forms in the framework of hilbert and dirichlet spaces our method relies on monotone convergence of quadratic forms and the canonical decomposition into regular and singular part we give various situations where the trace can be described more explicitly and compute it for some illustrating examples we then show that mosco convergence of dirichlet forms implies mosco convergence of a subsequence of their approximating traces

small sample properties of predictions from the regression model with autoregressive errors abstract monte carlo methods are used to examine the small sample properties of various estimators of asymptotic prediction variance apv from the regression model with autoregressive errors two practical estimators of apv one of which includes terms reflecting parameter estimation and one of which excludes these terms are compared to the mean squared error of prediction for regression models with different autocorrelation of the errors different sample sizes and different period ahead predictions the inclusion of terms reflecting the estimation of parameters is found to be worthwhile particularly in small samples

bias of autoregressive spectral estimators abstract bias of the least squares estimator of the log of the spectral density of an autoregression attenuates the peaks of the estimator under the assumption of an autoregressive generating process of known finite order we obtain an expression for the order 1 t bias where t is the sample length of the observed series this approximation is a sum of several simple functions of the unknown coefficients when the spectral density has sharp peaks one of these functions dominates the bias the attenuation from this dominant component can be substantial when the spectral peak is well defined and several examples illustrate this effect since the integral of the order 1 t bias components that are frequency dependent is 0 unbiased estimation of entropy to this order is possible for autoregressive processes these bias expressions extend to autoregressive models in which the mean is a polynomial function of time similar results obtain for the log of the yule walker spectral estimator for which the order

extensions of levy khintchine formula and beurling deny formula in semi dirichlet forms setting abstract the levy khintchine formula or more generally courrege s theorem characterizes the infinitesimal generator of a levy process or a feller process on r d for more general markov processes the formula that comes closest to such a characterization is the beurling deny formula for symmetric dirichlet forms in this paper we extend these celebrated structure results to include a general right process on a metrizable lusin space which is supposed to be associated with a semi dirichlet form we start with decomposing a regular semi dirichlet form into the diffusion jumping and killing parts then we develop a local compactification and an integral representation for quasi regular semi dirichlet forms finally we extend the formulae of levy khintchine and beurling deny in semi dirichlet forms setting through introducing a quasi compatible metric

on the lavrentiev phenomenon for multiple integral scalar variational problems abstract we prove the non occurrence of lavrentiev gaps between lipschitz and sobolev functions for functionals of the form i u f u u u when r n r is lipschitz and belongs to a wide class of open bounded sets in r n containing lipschitz domains the lagrangian f is assumed to be either convex in both variables or a sum of functions f s a s g b s with g convex and s a s g 0 b s satisfying a non oscillatory condition at infinity we thus derive the non occurrence of the lavrentiev phenomenon for unnecessarily convex functionals of the gradient no growth conditions are assumed

mosco convergence and wiener measures for conductive thin boundaries abstract the mosco convergence of energy functionals and the convergence of associated wiener measures are proved for a domain with highly conductive thin boundary we obtain those results for matrix valued conductivities and a family of speed measures measures of the underlying domain in particular this family includes the lebesgue measure and the one which makes the energy functional superposition the expectation of the displacement of the associated processes close to the boundary goes to due to the explosion of the conductivity at the limit

quenched invariance principles for random walks and elliptic diffusions in random media with boundary via a dirichlet form extension theorem and making full use of two sided heat kernel estimates we establish quenched invariance principles for random walks in random environments with a boundary in particular we prove that the random walk on a supercritical percolation cluster or amongst random conductances bounded uniformly from below in a half space quarter space etc converges when rescaled diffusively to a reflecting brownian motion which has been one of the important open problems in this area we establish a similar result for the random conductance model in a box which allows us to improve existing asymptotic estimates for the relevant mixing time furthermore in the uniformly elliptic case we present quenched invariance principles for domains with more general boundaries

some diffusion processes associated with two parameter poisson dirichlet distribution and dirichlet process the two parameter poisson dirichlet distribution pd is the distribution of an infinite dimensional random discrete probability it is a generalization of kingman s poisson dirichlet distribution the two parameter dirichlet process pi alpha theta nu 0 is the law of a pure atomic random measure with masses following the two parameter poisson dirichlet distribution in this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures pd and pi alpha theta nu 0 the methods used come from the theory of dirichlet forms

non lipschitz uniform domain shape optimization in linear acoustics we introduce new parametrized classes of shape admissible domains in r n n 2 and prove that they are compact with respect to the convergence in the sense of characteristic functions the hausdorff sense the sense of compacts and the weak convergence of their boundary volumes the domains in these classes are bounded e domains with possibly fractal boundaries that can have parts of any non uniform hausdorff dimension greater or equal to n 1 and less than n we prove the existence of optimal shapes in such classes for maximum energy dissipation in the framework of linear acous tics a by product of our proof is the result that the class of bounded e domains with fixed e is stable under hausdorff convergence an additional and related result is the mosco convergence of robin type energy functionals on converging domains

discrete approximation of symmetric jump processes on metric measure spaces in this paper we give general criteria on tightness and weak convergence of discrete markov chains to symmetric jump processes on metric measure spaces under mild conditions as an application we investigate discrete approximation for a large class of symmetric jump processes we also discuss some application of our results to the scaling limit of random walk in random conductance

homogenization for conductive thin layers of pre fractal type the aim of this work is to obtain convergence results for the solutions of transmission problems across highly conductive layers of pre fractal type from the point of view of homogenization we prove the m convergence of the energy functionals to an energy functional which incorporates a singular term supported within the layer from the convergence of the energy functionals we deduce the convergence of the approximating solutions to the limit solution in a suitable sense

bias of some commonly used time series estimates summary we study the bias of yule walker and least squares estimates for univariate and multivariate autoregressive processes we obtain explicit formulae for the large sample bias of yule walker estimates in the scalar first and second order cases and for least squares estimates in the general case both simulations and theory indicate that yulewalker estimates are inferior to least squares estimates for strongly autocorrelated processes yule walker estimates can be severely biased even for comparatively largesample sizes