global heat kernel estimate for relativistic stable processes in exterior open sets in this paper sharp two sided estimates for the transition densities of relativistic stable processes with mass m 0 1 in c1 1 exterior open sets are established for all time t 0 these transition densities are also the dirichlet heat kernels of m m2 2 with m 0 1 in c1 1 exterior open sets the estimates are uniform in m in the sense that the constants are independent of m 0 1 as a corollary of our main result we establish sharp two sided green function estimates for relativistic stable processes with mass m 0 1 in c1 1 exterior open sets

rank based optimal tests of the adequacy of an elliptic varma model we are deriving optimal rank based tests for the adequacy of a vector autoregressive moving average varma model with elliptically contoured innovation density these tests are based on the ranks of pseudo mahalanobis distances and on narmed residuals computed from tyler s ann statist 15 1987 234 251 scatter matrix they generalize the univariate signed rank procedures proposed by hallin and puri j multivariate anal 39 1991 1 29 two types of optimality properties are considered both in the local and asymptotic sense a la le cam a fixed score procedures local asymptotic minimaxity at selected radial densities and b estimated score procedures local asymptotic minimaxity uniform over a class f of radial densities contrary to their classical counterparts based on cross covariance matrices these tests remain valid under arbitrary elliptically symmetric innovation densities including those with infinite variance and heavy tails we show that the ares of our fixed score procedures with respect to traditional gaussian methods are the same as for the tests of randomness proposed in hallin and paindaveine bernoulli 8 2002b 787 815 the multivariate serial extensions of the classical chernoff savage and hodges lehmann results obtained there thus also hold here in particular the van der waerden versions of our tests are uniformly more powerful than those based on cross covariances as for our estimated score procedures they are fully adaptive hence uniformly optimal over the class of innovation densities satisfying the required technical assomptions

optimal tests for multivariate location based on interdirections and pseudo mahalanobis ranks we propose a family of tests based on randies 1989 concept of inter directions and the ranks of pseudo mahalanobis distances computed with respect to a multivariate m estimator of scatter due to tyler 1987 for the multivariate one sample problem under elliptical symmetry these tests which generalize the univariate signed rank tests are affine invariant depending on the score function considered van der waerden laplace they allow for locally asymptotically maximin tests at selected densities multivariate normal multivariate double exponential local powers and asymptotic relative efficiencies are derived with respect to hotelling s test randies 1989 multivariate sign test peters and randies 1990 wilcoxon type test and with respect to the oja median tests we moreover extend to the multivariate setting two famous univariate results the traditional chemoff savage 1958 property showing that hotelling s traditional procedure is uniformly dominated in the pitman sense by the van der waerden version of our tests and the celebrated hodges lehmann 1956 864 result providing for any fixed space dimension k the lower bound for the asymptotic relative efficiency of wilcoxon type tests with respect to hotelling s these asymptotic results are confirmed by a monte carlo investigation and application to a real data set

eigenvalue gaps for the cauchy process and a poincare inequality abstract a connection between the semigroup of the cauchy process killed upon exiting a domain d and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem was established in r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 211 2004 355 423 from this a variational characterization for the eigenvalues n n 1 of the cauchy process in d was obtained in this paper we obtain a variational characterization of the difference between n and 1 we study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for 1 where is the eigenvalue corresponding to the first antisymmetric eigenfunction for d the proof is based on a variational characterization of 1 and on a weighted poincare type inequality the poincare inequality is valid for all symmetric stable processes 0 2 and any other process obtained from brownian motion by subordination we also prove upper bound estimates for the spectral gap 2 1 in bounded convex domains

nonparametric tests for conditional symmetry in dynamic models this article proposes omnibus tests for conditional symmetry around a parametric function in a dynamic context conditional moments may not exist or may depend on the explanatory variables test statistics are suitable functionals of the empirical process of residuals and explanatory variables whose limiting distribution under the null is nonpivotal the tests are implemented with the assistance of a bootstrap method which is justified assuming very mild regularity conditions on the specification of the center of symmetry and the underlying serial dependence structure finite sample properties are examined by means of a monte carlo experiment

estimation risk for the var of portfolios driven by semi parametric multivariate models abstract joint estimation of market and estimation risks in portfolios is investigated when the individual returns follow a semi parametric multivariate dynamic model and the asset composition is time varying under ellipticity of the conditional distribution asymptotic theory for the estimation of the conditional value at risk var is developed an alternative method the filtered historical simulation which does not rely on ellipticity is also studied asymptotic confidence intervals for the conditional var which allow for simultaneous quantification of the market and estimation risks are derived the particular case of minimum variance portfolios is analyzed in more detail potential usefulness feasibility and drawbacks of the two approaches are illustrated via monte carlo experiments and an empirical study based on stock returns

gaugeability and conditional gaugeability new kato classes are introduced for general transient borel right processes for which gauge and conditional gauge theorems hold these new classes are the genuine extensions of the green tight measures in the classical brownian motion case however the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability we show that gaugeability conditional gaugeability and the subcriticality for the associated schrodinger operators are equivalent for transient borel right processes with strong duals analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric markov processes these analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases connections with the positivity of the spectral radii of the associated schrodinger operators are also established

probabilistic approach to the dirichlet problem of perturbed stable processes letx t be a symmetric stable process of index 0 2 inr d d 2 in this paper we deal with the perturbation ofx t by a multiplicative functional of the following form m t exp left sum limits s mathop leqslant limits t f x s x s right withf being a function onr d r d satisfying certain conditions first we prove the following gauge theorem ifd is a bounded open domain ofr d then the functiong x e x m d is either identically infinite ond or bounded ond where d is the first exit time fromd then we formulate the dirichlet problem associated with the perturbed symmetric stable process by using dirichlet form theory finally we apply the gauge theorem to prove the existence and uniqueness of solutions to the dirichlet problem mentioned above

sharp essential self adjointness of relativistic schrodinger operators with a singular potential abstract this paper is devoted to the study of essential self adjointness of a relativistic schrodinger operator with a singular homogeneous potential from an explicit condition on the coefficient of the singular term we provide a sufficient and necessary condition for essential self adjointness

relativistic schrodinger operators asymptotic behavior of the eigenfunctions nonrelativistic schrodinger operators are perturbations of the negative laplacian and the connection with stochastic processes and brownian motion in particular is well known and usually goes under the name of feynman and kac we present a similar connection between a class of relativistic schrodinger operators and a class of processes with stationary independent increments in particular we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur

universal constraints on the location of extrema of eigenfunctions of non local schrodinger operators abstract we derive a lower bound on the location of global extrema of eigenfunctions for a large class of non local schrodinger operators in convex domains under dirichlet exterior conditions featuring the symbol of the kinetic term the strength of the potential and the corresponding eigenvalue and involving a new universal constant we show a number of probabilistic and spectral geometric implications and derive a faber krahn type inequality for non local operators our study also extends to potentials with compact support and we establish bounds on the location of extrema relative to the boundary edge of the support or level sets around minima of the potential

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

estimating multivariate volatility models equation by equation summary the paper investigates the estimation of a wide class of multivariate volatility models instead of estimating an m multivariate volatility model a much simpler and numerically efficient method consists in estimating m univariate generalized auto regressive conditional heteroscedasticity type models equation by equation in the first step and a correlation matrix in the second step strong consistency and asymptotic normality of the equation by equation estimator are established in a very general framework including dynamic conditional correlation models the equation by equation estimator can be used to test the restrictions imposed by a particular multivariate generalized auto regressive conditional heteroscedasticity specification for general constant conditional correlation models we obtain the consistency and asymptotic normality of the two step estimator comparisons with the global method in which the model parameters are estimated in one step are provided monte carlo experiments and applications to financial series illustrate the interest of the approach

testing multivariate distributions in garch models in this paper we consider testing distributional assumptions in multivariate garch models based on empirical processes using the fact that joint distribution carries the same amount of information as the marginal together with conditional distributions we first transform the multivariate data into univariate independent data based on the marginal and conditional cumulative distribution functions we then apply the khmaladze s martingale transformation k transformation to the empirical process in the presence of estimated parameters the k transformation eliminates the effect of parameter estimation allowing a distribution free test statistic to be constructed we show that the k transformation takes a very simple form for testing multivariate normal and multivariate t distributions the procedure is applied to a multivariate financial time series data set

cauchy noise and affiliated stochastic processes by departing from the previous attempt phys rev e 51 4114 1995 we give a detailed construction of conditional and perturbed markov processes under the assumption that the cauchy law of probability replaces the gaussian law appropriate for the wiener process as the model of primordial noise all considered processes are regarded as probabilistic solutions of the so called schrodinger interpolation problem whose validity is thus extended to the jump type processes and their step process approximants

limit theorems for branching markov processes abstract we establish almost sure limit theorems for a branching symmetric hunt process in terms of the principal eigenvalue and the ground state of an associated schrodinger operator here the branching rate and the branching mechanism can be state dependent in particular the branching rate can be a measure belonging to a certain kato class and is allowed to be singular with respect to the symmetrizing measure for the underlying hunt process x the almost sure limit theorems are established under the assumption that the associated schrodinger operator of x has a spectral gap such an assumption is satisfied if the underlying process x is a brownian motion a symmetric stable like process on r n or a relativistic symmetric stable process on r n

spectral properties of the massless relativistic harmonic oscillator abstract an explicit solution of the spectral problem of the non local schrodinger operator obtained as the sum of the square root of the laplacian and a quartic potential in one dimension is presented the eigenvalues are obtained as zeroes of special functions related to the fourth order airy function and closed formulae for the fourier transform of the eigenfunctions are derived these representations allow to derive further spectral properties such as estimates of spectral gaps heat trace and the asymptotic distribution of eigenvalues as well as a detailed analysis of the eigenfunctions a subtle spectral effect is observed which manifests in an exponentially tight approximation of the spectrum by the zeroes of the dominating term in the fourier representation of the eigenfunctions and its derivative

distribution free specification tests of conditional models this article proposes a class of asymptotically distribution free specification tests for parametric conditional distributions these tests are based on a martingale transform of a proper sequential empirical process of conditionally transformed data standard continuous functionals of this martingale provide omnibus tests while linear combinations of the orthogonal components in its spectral representation form a basis for directional tests finally neyman type smooth tests a compromise between directional and omnibus tests are discussed as a special example we study in detail the construction of directional tests for the null hypothesis of conditional normality versus heteroskedastic contiguous alternatives a small monte carlo study shows that our tests attain the nominal level already for small sample sizes

spines skeletons and the strong law of large numbers for superdiffusions consider a supercritical superdiffusion x t t 0 on a domain d r d with branching mechanism x z x z x z 2 0 e zy 1 zy x dy the skeleton decomposition provides a pathwise description of the process in terms of immigration along a branching particle diffusion we use this decomposition to derive the strong law of large numbers slln for a wide class of superdiffusions from the corresponding result for branching particle diffusions that is we show that for suitable test functions f and starting measures f x t p f x t w p almost surely as t where w is a finite non deterministic random variable characterized as a martingale limit our method is based on skeleton and spine techniques and offers structural insights into the driving force behind the slln for superdiffusions the result covers many of the key examples of interest and in particular proves a conjecture by fleischmann and swart stochastic process appl 106 2003 141 165 for the super wright fisher diffusion

periodic solutions for a pseudo relativistic schrodinger equation abstract we study the existence and the regularity of non trivial t periodic solutions to the following nonlinear pseudo relativistic schrodinger equation 0 1 x m 2 m u x f x u x in 0 t n where t 0 m is a non negative real number f is a regular function satisfying the ambrosetti rabinowitz condition and a polynomial growth at rate p for some 1 p 2 1 we investigate such problem using critical point theory after transforming it to an elliptic equation in the infinite half cylinder 0 t n 0 with a nonlinear neumann boundary condition by passing to the limit as m 0 in 0 1 we also prove the existence of a non trivial t periodic weak solution to 0 1 with m 0

absolute continuity of symmetric markov processes we study girsanov s theorem in the context of symmetric markov processes extending earlier work of fukushima takeda and fitzsimmons on girsanov transformations of gradient type we investigate the most general girsanov transformation leading to another symmetric markov process this investigation requires an extension of the forward backward martingale method of lyons zheng to cover the case of processes with jumps

stochastic models for relativistic diffusion the diffusion equation is related to the schroedinger equation by analytic continuation the formula e sup 2 p sup 2 c sup 2 m sup 2 c sup 4 leads to a relativistic schroedinger equation and analytic continuation yields a relativistic diffusion equation that involves fractional calculus this paper develops stochastic models for relativistic diffusion and equivalent differential equations with no fractional derivatives connections to anomalous diffusion are also discussed along with alternative models

tests for conditional ellipticity in multivariate garch models tests are proposed for the assumption that the conditional distribution of a multivariate garch process is elliptic these tests are of kolmogorov smirnov and cramer von mises type and make use of the common geometry underlying the characteristic function of any spherically symmetric distribution the asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions it is shown that both the finite sample and the asymptotic null distribution depend on the unknown distribution of the euclidean norm of the innovations therefore a conditional monte carlo procedure is used to actually carry out the tests the validity of this resampling scheme is formally justified results on the behavior of the new tests in finite samples are included along with comparisons with other tests

functional integral approach to semi relativistic pauli fierz models abstract by means of functional integrations spectral properties of semi relativistic pauli fierz hamiltonians h p a 2 m 2 m v h rad in quantum electrodynamics are considered here p is the momentum operator a a quantized radiation field on which an ultraviolet cutoff is imposed v an external potential h rad the free field hamiltonian and m 0 describes the mass of electron two self adjoint extensions of a semi relativistic pauli fierz hamiltonian are defined the feynman kac type formula of e t h is given a self adjointness a spatial decay of bound states a gaussian domination of the ground state and the existence of a measure associated with the ground state are shown all the results are independent of values of coupling constant and it is emphasized that m 0 is included

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

exponential decay of eigenfunctions of brown ravenhall operators we prove the exponential decay of eigenfunctions of reductions of brown ravenhall operators to arbitrary irreducible representations of rotation reflection and permutation symmetry groups under the assumption that the corresponding eigenvalues are below the essential spectrum

pointwise eigenfunction estimates and intrinsic ultracontractivity type properties of feynman kac semigroups for a class of levy processes we introduce a class of l evy processes subject to specific regularity conditions and consider their feynman kac semigroups given under a kato class potential using new techniques first we analyze the rate of decay of eigenfunctions at infinity we prove bounds on subaveraging functions from which we derive two sided sharp pointwise estimates on the ground state and obtain upper bounds on all other eigenfunctions next by using these results we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions we establish the relationships of these properties derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the l evy density and the potential at infinity define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations these results are amply illustrated by key examples

central limit theorems for supercritical branching markov processes abstract in this paper we establish spatial central limit theorems for a large class of supercritical branching markov processes with general spatial dependent branching mechanisms these are generalizations of the spatial central limit theorems proved in 1 for branching ou processes with binary branching mechanisms compared with the results of 1 our central limit theorems are more satisfactory in the sense that the normal random variables in our theorems are non degenerate

dirichlet forms and semilinear elliptic equations with measure data we propose a probabilistic definition of solutions of semilinear elliptic equations with possibly nonlocal operators associated with regular dirichlet forms and with measure data using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right hand side of the equation is monotone and satisfies mild integrability assumption and the measure is smooth we also study regularity of solutions under the assumption that the measure is smooth and has finite total variation some applications of our general results are given

ground states solutions for a non linear equation involving a pseudo relativistic schrodinger operator in this paper we prove the existence regularity and symmetry of a ground state for a nonlinear equation in the whole space involving a pseudo relativistic schrodinger operator

properties and estimation of asymmetric exponential power distribution the new distribution class asymmetric exponential power distribution aepd proposed in this paper generalizes the class of skewed exponential power distributions sepd in a way that in addition to skewness introduces different decay rates of density in the left and right tails our parametrization provides an interpretable role for each parameter we derive moments and moment based measures skewness kurtosis expected shortfall it is demonstrated that a maximum entropy property holds for the aepd distributions we establish consistency asymptotic normality and efficiency of the maximum likelihood estimators over a large part of the parameter space by dealing with the problems created by non smooth likelihood function and derive explicit analytical expressions of the asymptotic covariance matrix where the results apply to the sepd class they enlarge on the current literature also we give a convenient stochastic representation of the distribution our monte carlo study illustrates the theoretical results we also provide some empirical evidence for the usefulness of employing aepd errors in garch type models for predicting downside market risk of financial assets

on enhanced binding and related effectsin the non and semi relativistic pauli fierz models we prove enhanced binding and increase of binding energies in the non and semi relativistic pauli fierz models for arbitrary values of the fine structure constant and the ultra violet cut off and discuss the resulting improvement of exponential localization of ground state eigenvectors for the semi relativistic model we also discuss the increase of the renormalized electron mass and determine the linear leading order term in the asymptotics of the self energy as the ultra violet cut off goes to infinity

asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes strong consistency and asymptotic normality of the quasi maximum likelihood estimator qmle are given for a general class of multidimensional causal processes for particular cases already studied in the literature for instance univariate or multivariate garch arch arma garch processes the assumptions required for establishing these results are often weaker than existing conditions the qmle asymptotic behavior is also given for numerous new examples of univariate or multivariate processes for instance tarch or nlarch processes

a probabilistic principle and generalized schrodinger perturbation abstract we present a probabilistic principle for an additive functional of certain markov processes its smallness with time is equivalent to its smalness with space and apply it to a perturbation problem for relativistic schrodinger operators

borel weil theorem for configuration varieties and schur modules we present a generalization of the classical schur modules ofgl n exhibiting the same interplay among algebra geometry and combinatorics a generalized young diagramdis an arbitrary finite subset ofn n for eachd we define the schur modulesdofgl n we introduce a projective variety fdand a line bundle ld and describe the schur module in terms of sections of ld for diagrams with the northeast property i1 j1 i2 j2 d min i1 i2 max j1 j2 d which includes the skew diagrams we resolve the singularities of fdand show analogues of bott s and kempf s vanishing theorems finally we apply the atiyah bott fixed point theorem to establish a weyl type character formula of the form charsd x txwt t i j 1 xix 1j dij t wheretruns over certain standard tableaux ofd our results are valid over fields of arbitrary characteristic

global heat kernel estimates for symmetric jump processes in this paper we study sharp heat kernel estimates for a large class of symmetric jump type processes in r d for all t 0 a prototype of the processes under consideration are symmetric jump processes on r d with jumping intensity

nonparametric model checks for time series this paper studies a class of tests useful for testing the goodness of fit of an autoregressive model these tests are based on a class of empirical processes marked by certain residuals the paper first gives their large sample behavior under null hypotheses then a martingale transformation of the underlying process is given that makes tests based on it asymptotically distribution free consistency of these tests is also discussed briefly

a consistent test for conditional symmetry in time series models the assumption of conditional symmetry is often invoked to validate adaptive estimation and consistent estimation of arch garch models by quasi maximum likelihood imposing conditional symmetry can increase the efficiency of bootstraps if the symmetry assumption is valid this paper proposes a procedure for testing conditional symmetry the proposed test does not require the data to be stationary or i i d and the dimension of the conditional variables can be infinite the proposed test is consistent and is asymptotically distribution free in addition the test is shown to have nontrivial power against root t local alternatives the size and power of the test are satisfactory even for small samples applying the test to various time series we reject conditional symmetry in inflation exchange rate and stock returns among the nonfinancial time series considered we find that investment the consumption of durables and manufacturing employment also reject conditional symmetry

generic singularities of certain schubert varieties let g be a connected semisimple algebraic group b a borel subgroup t a maximal torus in b with weyl group w and q a subgroup containing b for w in w let x wq denote the schubert variety overline bwq q for y in w such that x yq subseteq x wq one knows that byq q admits a t stable transversal in x wq which we denote by cal n yq wq we prove that under certain hypotheses cal n yq wq is isomorphic to the orbit closure of a highest weight vector in a certain weyl module we also obtain a generalisation of this result under slightly weaker hypotheses further we prove that our hypotheses are satisfied when q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight and x yq is an irreducible component of the boundary of x wq that is the complement of the open orbit of the stabiliser in g of x wq as a consequence we describe the singularity of x wq along byq q and obtain that the boundary of x wq equals its singular locus

the gorenstein locus of minuscule schubert varieties in this article we explicitly describe the gorenstein locus of all minuscule schubert varieties this proves a special case of the conjecture of a woo and a yong alexander woo alexander yong governing singularities of schubert varieties j algebra 320 2 2008 495 520 regarding the gorenstein locus of schubert varieties

on the range of validity of the autoregressive sieve bootstrap we explore the limits of the autoregressive ar sieve bootstrap and show that its applicability extends well beyond the realm of linear time series as has been previously thought in particular for appropriate statistics the ar sieve bootstrap is valid for stationary processes possessing a general wold type autoregressive representation with respect to a white noise in essence this includes all stationary purely nondeterministic processes whose spectral density is everywhere positive our main theorem provides a simple and effective tool in assessing whether the ar sieve bootstrap is asymptotically valid in any given situation in effect the large sample distribution of the statistic in question must only depend on the first and second order moments of the process prominent examples include the sample mean and the spectral density as a counterexample we show how the ar sieve bootstrap is not always valid for the sample autocovariance even when the underlying process is linear

an uncertainty principle for fermions with generalized kinetic energy we derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one particle hamiltoniansh f i v x acting onl2 n these bounds are then used to derive a lower bound on the kinetic energy sum limits j 1 n left langle psi f i nabla j psi right rangle for ann fermion wavefunction we discuss two examples in more detail f p p andf p p2 m2 1 2 m both in three dimensions

lp independence of spectral bounds of schrodinger type semigroups let l be an m symmetric markov generator and a signed measure in the kato class we consider a schrodinger type operator h l on lp m we prove that under certain conditions for the markov semigroup generated by l and the potential the lp spectral bound of h is independent of p if and only if the l2 spectral bound is non positive

sharp heat kernel estimates for relativistic stable processes in open sets in this paper we establish sharp two sided estimates for the transition densities of relativistic stable processes i e for the heat kernels of the operators m m2 2 in c1 1 open sets here m 0 and 0 2 the estimates are uniform in m 0 m for each fixed m 0 letting m 0 we recover the dirichlet heat kernel estimates for 2 2 in c1 1 open sets obtained in 14 sharp two sided estimates are also obtained for green functions of relativistic stable processes in bounded c1 1 open sets

generalized 3g theorem and application to relativistic stable process on non smooth open sets let g x y and gd x y be the green functions of rotationally invariant symmetric stable process in rd and in an open set d respectively where 0 2 the inequality gd x y gd y z gd x z c g x y g y z is a very useful tool in studying local schrodinger operators when the above inequality is true with c c d 0 then we say that the 3g theorem holds in d in this paper we establish a generalized version of 3g theorem when d is a bounded fat open set which includes a bounded john domain the 3g we consider is of the form gd x y gd z w gd x w where y may be different from z when y z we recover the usual 3g the 3g form gd x y gd z w gd x w appears in non local schrodinger operator theory using our generalized 3g theorem we give a concrete class of functions belonging to the non local kato class introduced by chen and song on fat open sets as an application we discuss relativistic stable processes relativistic hamiltonian when 1 in fat open sets we identify the martin boundary and the minimal martin boundary with the euclidean boundary for relativistic stable processes in fat open sets furthermore we show that relative fatou type theorem is true for relativistic stable processes in fat open sets the main results of this paper hold for a large class of symmetric markov processes as are illustrated in the last section of this paper we also discuss the generalized 3g theorem for a large class of symmetric stable levy processes

symmetrization of levy processes and applications abstract it is shown that many of the classical generalized isoperimetric inequalities for the laplacian when viewed in terms of brownian motion extend to a wide class of levy processes the results are derived from the multiple integral inequalities of brascamp lieb and luttinger but the probabilistic structure of the processes plays a crucial role in the proofs

the discrete spectrum of the spinless one dimensional salpeter hamiltonian perturbed by δ interactions we rigorously define the self adjoint one dimensional salpeter hamiltonian perturbed by an attractive of strength centred at the origin by explicitly providing its resolvent our approach is based on a coupling constant renormalization a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the self adjoint operator representing the negative laplacian perturbed by the in two and three dimensions we show that the spectrum of the self adjoint operator consists of the absolutely continuous spectrum of the free salpeter hamiltonian and an eigenvalue given by a smooth function of the parameter the method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self adjoint hamiltonian consists of two eigenvalues namely the ground state energy and that of the excited antisymmetric state we investigate in detail the dependence of these two eigenvalues on the two parameters of the model that is to say both the aforementioned strength and the separation distance with regard to the latter a remarkable phenomenon is observed differently from the well behaved schrodinger case the 1d salpeter hamiltonian with two identical dirac distributions symmetrically situated with respect to the origin does not converge as the separation distance shrinks to zero to the one with a single centred at the origin having twice the strength however the expected behaviour in the limit in the norm resolvent sense can be achieved by making the coupling of the twin deltas suitably dependent on the separation distance itself

singularities of schubert varieties tangent cones and bruhat graphs let g be a semi simple algebraic group without g 2 factors over an algebraically closed field k of characteristic p 2 3 and suppose b is a borel subgroup t b is a maximal torus and p is a parabolic in g containing b in an earlier paper the authors classified the singular t fixed points x of an arbitrary irreducible t stable subvariety x in g p in all characteristics the key to this being the notion of a peterson translate in particular we showed that if x is cohen macaulay then x is smooth at x if and only if there exists a t invariant curve in x through x which contains a smooth point of x and dim x x dim x where x x is the linear span of the reduced tangent cone to x at x the purpose of this paper is to describe x x when x is a schubert variety in g p and x is a maximal singular t fixed point of x in fact we give two characterizations we first show that in all characteristics x x is the sum of all the peterson translates at x the second characterization involves further study of the peterson translates along the good t invariant curves at x for which the assumption char k 2 3 is needed this leads to the following consequence if x is a maximal singularity of x which is rationally smooth then either the span of the tangent lines to the t stable curves is not a module for the isotropy subgroup of b at x or there exist a pair of orthogonal t invariant curves at x which determine what we call a b 2 pair this characterization gives a nonrecursive algorithm for finding the singular locus of an arbitrary schubert variety in g p in terms of its bruhat graph

perturbation theory for convolution semigroups abstract we deal with convolution semigroups not necessarily symmetric in l p r n and provide a general perturbation theory of their generators by indefinite singular potentials such semigroups arise in the theory of levy processes and cover many examples such as gaussian semigroups stable semigroups relativistic schrodinger semigroups etc we give new generation theorems and feynman kac formulas in particular by using weak compactness methods in l 1 we enlarge the extended kato class potentials used in the theory of markov processes in l 2 setting dirichlet form perturbation theory is finely related to l 1 theory and the extended kato class measures is also enlarged finally various perturbation problems for subordinate semigroups are considered

an almost sure scaling limit theorem for dawson watanabe superprocesses abstract we establish a scaling limit theorem for a large class of dawson watanabe superprocesses whose underlying spatial motions are symmetric hunt processes where the convergence is in the sense of convergence in probability when the underling process is a symmetric diffusion with c b 1 coefficients or a symmetric levy process on r d whose levy exponent is bounded from below by c for some c 0 and 0 2 when is large a stronger almost sure limit theorem is established for the superprocess our approach uses the principal eigenvalue and the ground state for some associated schrodinger operator the limit theorems are established under the assumption that an associated schrodinger operator has a spectral gap

central limit theorem for branching brownian motions in random environment we introduce a model of branching brownian motions in time space random environment associated with the poisson random measure we prove that if the randomness of the environment is moderated by that of the brownian motion the population density satisfies a central limit theorem and the growth rate of the population size is the same as its expectation with strictly positive probability we also characterize the diffusive behavior of our model in terms of the decay rate of the replica overlap on the other hand we show that if the randomness of the environment is strong enough the growth rate of the population size is strictly less than its expectation almost surely to do this we use a connection between our model and the model of brownian directed polymers in random environment introduced by comets and yoshida

a test for symmetries of multivariate probability distributions a monte carlo test for multivariate symmetries is proposed the monte carlo simulations are performed conditionally on a minimal sufficient statistic for the class of distributions with symmetric density additionally a general purpose test statistic based on a distance measure between the probability density function and its symmetrised version is introduced the monte carlo tests for spherical symmetry and multivariate reflection symmetry are studied numerically for this statistic and the results indicate that the tests perform well compared to other tests the method is illustrated with an analysis of a real dataset

conditional gaugeability and subcriticality of generalized schrodinger operators abstract we obtain a necessary and sufficient condition for conditional gaugeability and show the equivalence between conditional gaugeability and subcriticality of generalized schrodinger type operators we apply the condition to concrete examples

nonparametric monte carlo tests for multivariate distributions summary in this paper nonparametric monte carlo test procedures are proposed for composite hypotheses in which the multivariate distribution belongs to a nonparametric family when a random variable is in distribution equal to a product of two independent variables monte carlo simulations can be performed conditionally on one of the variables they can be applied to testing several commonly used distributions including elliptically symmetric reflectively symmetric liouville dirichlet and symmetric scale mixture distributions the tests are exactly valid when the hypothetical distributions are without nuisance location parameters finite sample simulations show that the monte carlo tests outperform bootstrap tests the method is illustrated with analysis of a real dataset

differentiability of spectral functions for symmetric alpha stable processes let be a signed radon measure in the kato class and define a schrodinger type operator h 1 2 2 on r d we show that its spectral bound c inf h is differentiable if d 2 and is green tight

testing for spherical symmetry of a multivariate distribution rotationally invariant tests based on test statistics of the von mises type are proposed under the hypothesis of spherical symmetry of a multivariate distribution the tests are distribution free when the hypothesis of spherical symmetry is true the asymptotic distribution of the test statistics are derived under the null hypothesis and under any fixed alternative a simple criterion for consistency is given the results are illustrated by numerous examples of test statistics which give rise to tests being consistent against all alternatives

when is a schubert variety gorenstein abstract a normal variety is gorenstein if it is cohen macaulay and its canonical sheaf is a line bundle this property is a weakening of smoothness which implies that a variety behaves like a smooth one for various algebro geometric purposes we give a combinatorial characterization of which schubert varieties are gorenstein using in part a new generalization of pattern avoidance we also give an explicit description as a line bundle of the canonical sheaf of a gorenstein schubert variety

on energy critical half wave maps into mathbb s 2 s 2 we consider the energy critical half wave maps equation begin aligned partial t mathbf u mathbf u wedge nabla mathbf u 0 end aligned for mathbf u 0 t times mathbb r rightarrow mathbb s 2 we give a complete classification of all traveling solitary waves with finite energy the proof is based on a geometric characterization of these solutions as minimal surfaces with not necessarily free boundary on mathbb s 2 in particular we discover an explicit lorentz boost symmetry which is implemented by the conformal mobius group on the target mathbb s 2 applied to half harmonic maps from mathbb r to mathbb s 2 complementing our classification result we carry out a detailed analysis of the linearized operator l around half harmonic maps mathbf q with arbitrary degree m geqslant 1 and consisting of m identical blaschke factors here we explicitly determine the nullspace including the zero energy resonances in particular we prove the nondegeneracy of mathbf q moreover we give a full description of the spectrum of l by finding all its l 2 eigenvalues and proving their simplicity furthermore we prove a coercivity estimate for l and we rule out embedded eigenvalues inside the essential spectrum our spectral analysis is based on a reformulation in terms of certain jacobi operators tridiagonal infinite matrices obtained from a conformal transformation of the spectral problem posed on mathbb r to the unit circle mathbb s finally we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half harmonic maps our spectral results also have potential applications to the half harmonic map heat flow which is the parabolic counterpart of the half wave maps equation

drift transforms and green function estimates for discontinuous processes in this paper we consider girsanov transforms of pure jump type for discontinuous markov processes we show that under some quite natural conditions the green functions of the girsanov transformed process are comparable to those of the original process as an application of the general results the drift transform of symmetric stable processes is studied in detail in particular we show that the relativistic a stable process in a bounded c 1 1 smooth open set d can be obtained from symmetric a stable process in d through a combination of a pure jump girsanov transform and a feynman kac transform from this we deduce that the green functions for these two processes in d are comparable

intrinsic ultracontractivity of the feynman kac semigroup for relativistic stable processes let x t be the relativistic stable process in r d a 0 2 d with infinitesimal generator h 0 a m 2 2 m we study intrinsic ultracontractivity iu for the feynman kac semigroup t t for this process with generator h 0 v v 0 v locally bounded we prove that if lim x v x then for every t 0 the operator t t is compact we consider the class v of potentials v such that v 0 lim x v x and v is comparable to the function which is radial radially nondecreasing and comparable on unit balls for v in the class v we show that the semigroup t t is iu if and only if lim x v x x if this condition is satisfied we also obtain sharp estimates of the first eigenfunction 1 for t t in particular when v x x 0 then the semigroup t t is iu if and only if 1 for 1 the first eigenfunction 1 x is comparable to

intrinsic ultracontractivity and conditional gauge for symmetric stable processes abstract it is shown in this paper that the conditional gauge theorem holds for symmetric stable processes on bounded c 1 1 domains in r n where 0 n 2 two of the major tools used to prove this conditional gauge theorem are logarithmic sobolev inequality and intrinsic ultracontractivity

one electron molecules with relativistic kinetic energy properties of the discrete spectrum we discuss the discrete spectrum of the operator h k c left hbar 2 c 2 delta m 2 c 4 right 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits k 1 k z k e 2 left x r k right 1 more specifically we study 1 the behaviour of the eigenvalues when the internuclear distances contract 2 the existence of ac independent lower bound forhk c mc2 3 the nonrelativistic limit of the eigenvalues ofhk c mc2

testing the existence of moments for garch processes abstract it is generally admitted that many financial time series have heavy tailed marginal distributions when time series models are fitted on such data the non existence of appropriate moments may invalidate standard statistical tools used for inference moreover the existence of moments can be crucial for risk management for instance when risk is measured through the expected shortfall this paper considers testing the existence of moments in the framework of garch processes while the second order stationarity condition does not depend on the distribution of the innovation higher order moment conditions involve moments of the independent innovation process we propose tests for the existence of high moments of the returns process which are based on the joint asymptotic distribution of the quasi maximum likelihood qml estimator of the volatility parameters and empirical moments of the residuals a bootstrap procedure is proposed to improve the finite sample performance of our test to achieve efficiency gains we consider non gaussian qml estimators founded on reparameterizations of the garch model and we discuss optimality issues monte carlo experiments and an empirical study illustrate the asymptotic results

smooth points of t stable varieties in g b and the peterson map let g be a connected semi simple algebraic group defined over an algebraically closed field k and let t b p be respectively a maximal torus a borel subgroup and a parabolic subgroup of g inspired by a beautiful result of dale peterson describing the singular locus of a schubert variety in g b we characterize the t fixed points in the singular locus of an arbitrary irreducible t stable subvariety of g p a t variety for short peterson s result cf the deformation theorem 1 says that if k then a schubert variety x g b is smooth at a t fixed point x if and only if it is smooth at every t fixed point y x in the bruhat chevalley order on the fixed point set x t and all the limits c x x lim z x t z x z c c t of the zariski tangent spaces t z x of x coincide as c varies over the set of all t stable curves in x with c t x y where y x using this peterson showed that if g is simply laced and defined over then every rationally smooth point of a schubert variety in g b is smooth more generally the deformation c x x is defined for any k variety x with a t action provided c is what we call good i e c is a curve of the form c overline tz where z is a smooth point of x x t and x c t our first main result theorem 1 4 says that if x x is an attractive fixed point then x is smooth at x if and only if there exist at least two good c containing x such that c x x te x x where te x x denotes the span of the tangent lines of the t stable curves in x containing x in addition if x is cohen macaulay at x and c x x te x x for even one good c then x is smooth at x our second main result theorem 1 6 says that if x is a t variety in g p where g is simply laced then c x x te x x for each good c this is not true for general g but when g has no g2 factors then c x x is always contained in the linear span of the reduced tangent cone to x at x these results lead to several descriptions of the smooth fixed points of a t variety in g p and in particular they give simple proofs of peterson s results valid for any algebraically closed field we also show cf example 7 1 that there can exist t stable subvarieties in g b where g is simply laced which have rationally smooth t fixed points in their singular loci

strict stationarity of generalized autoregressive processes in this paper we consider the multivariate equation x n 1 a n 1 x n b n 1 with i i d coefficients which have only a logarithmic moment we give a necessary and sufficient condition for existence of a strictly stationary solution independent of the future as an application we characterize the multivariate arma equations with general noise which have such a solution

the nash blow up of a cominuscule schubert variety abstract we compute the nash blow up of a cominuscule schubert variety in particular we show that the nash blow up is algebraically isomorphic to another schubert variety of the same lie type as a consequence we give a new characterization of the smooth locus and for grassmannian schubert varieties determine when the nash blow up is a resolution of singularities we also study the induced torus action on the nash blow up and give a bijection between its torus fixed points and peterson translates on the schubert variety

levy targeting and the principle of detailed balance we investigate confining mechanisms for levy flights under premises of the principle of detailed balance in this case the master equation of the jump type process admits a transformation to the levy schrodinger semigroup dynamics akin to a mapping of the fokker planck equation into the generalized diffusion equation this sets a correspondence between above two stochastic dynamical systems within which we address a stochastic targeting problem for an arbitrary stability index e 0 2 of symmetric levy drivers namely given a probability density function specify the semigroup potential and thence the jump type dynamics for which this pdf is actually a long time asymptotic target solution of the master equation here an asymptotic behavior of different motion scenarios ceases to depend on that is exemplified by considering gaussian and cauchy family target pdfs a complementary problem of the reverse engineering is analyzed given a priori a semigroup potential quantify how sensitive upon the choice of the driver is an asymptotic behavior of solutions of the associated master equation and thus an invariant pdf itself this task is accomplished for so called family of levy oscillators