perturbation of symmetric markov processes we present a path space integral representation of the semigroup associated with the quadratic form obtained by a lower order perturbation of the l 2 infinitesimal generator mathcal l of a general symmetric markov process an illuminating concrete example for mathcal l is delta d delta s d where d is a bounded euclidean domain in mathbb r d s in 0 1 delta d is the laplace operator in d with zero dirichlet boundary condition and delta s d is the fractional laplacian in d with zero exterior condition the strong markov process corresponding to mathcal l is a levy process that is the sum of brownian motion in mathbb r d and an independent symmetric 2s stable process in mathbb r d killed upon exiting the domain d this probabilistic representation is a combination of feynman kac and girsanov formulas crucial to the development is the use of an extension of nakao s stochastic integral for zero energy additive functionals and the associated ito formula both of which were recently developed in chen et al stochastic calculus for dirichlet processes preprint 2006

quasi regular dirichlet forms and markov processes abstract we identify an analytic property of a non symmetric dirichlet form on a general topological state space called quasi regularity which we prove to be equivalent with the existence of an associated standard markov process respectively the existence of an associated pair of right processes in addition we prove a one to one correspondence between all quasi regular dirichlet forms and all equivalence classes of pairs of m sectorial right processes

on the moments of the integrated geometric brownian motion abstract this note demonstrates how the divided differences characterization for the moments of the integrated geometric brownian process arises naturally from the solution to their differential equations the characterization was introduced by baxter and brummelhuis in their paper baxter and brummelhuis 2011 where they demonstrate its consistency with the results found by oshanin et al 1993 here we demonstrate its validity with more general expressions presented in other papers and explore the divided differences characterization of these moments further

accurate computation of divided differences of the exponential function abstract the traditional recurrence for the computation of exponential divided differences along with a new method based on the properties of the exponential function are studied in detail in this paper our results show that it is possible to combine these two methods to compute exponential divided differences accurately a hybrid algorithm is presented for which our error bound grows quite slowly with the order of the divided difference author

a massively parallel exponential integrator for advection diffusion models this work considers the real leja points method relpm m caliari m vianello l bergamaschi interpolating discrete advection diffusion propagators at spectral leja sequences j comput appl math 172 2004 79 99 for the exponential integration of large scale sparse systems of odes generated by finite element or finite difference discretizations of 3 d advection diffusion models we present an efficient parallel implementation of relpm for polynomial interpolation of the matrix exponential propagators exp dta v and f dta v f z exp z 1 z a scalability analysis of the most important computational kernel inside the code the parallel sparse matrix vector product has been performed as well as an experimental study of the communication overhead as a result of this study an optimized parallel sparse matrix vector product routine has been implemented the resulting code shows good scaling behavior even when using more than one thousand processors the numerical results presented on a number of very large test cases gives experimental evidence that relpm is a reliable and efficient tool for the simulation of complex hydrodynamic processes on parallel architectures

lp independence of spectral bounds of feynman kac semigroups by continuous additive functionals abstract we establish conditions for the l p independence of spectral bounds of feynman kac semigroup by continuous additive functionals whose revuz measures are smooth measures of kato class having non negative order green tightness our continuous additive functionals do not necessarily admit bounded variation in general examples of cauchy principal value and hilbert transform of brownian local time and for relativistic symmetric stable processes are presented

potential theoretic nature of hardy s inequality for dirichlet forms we prove that the hardy inequality for dirichlet forms of symmetric markov processes recently proved by p j fitzsimmons can be established within the framework of our potential inequality this clearly emphasizes the role played by the maximum principle in the inequality

riesz decompositions in markov process theory riesz decompositions of excessive measures and excessive functions are obtained by probabilistic methods without regularity assumptions the decomposition of excessive measures is given for borel right processes the results for excessive functions are formulated within the framework of weak duality these results extend and generalize the pioneering work of hunt in this area

asymptotic properties of additive functionals of brownian motion in this paper we study the asymptotic behavior of additive functionals of brownian motion which are not necessarily of bounded variation the result is then applied to the hilbert transform of the brownian local time

conditional gauge with unbounded potential let b be a ball in r d and q be a borel function on it we prove that if mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaacawg4bgaeyici4sabmoqayaaraaabeaa ieaakiaa bcadawdrbqaamaalaaabagaaiifaiaadghacaggoagaam yeaiaacmcacagg8baabagaaiifaiaadihacqghsislcawg5bgaaiif amaacaaaleqabagaamizaiabgkhitiaaikdaaaaaaogaa8hiagqaci aa rgacagf5baaleaacawgcbaabeqdcqghrii8aaaa 4fb6 mathop sup limits x in bar b int limits b frac q y x y d 2 dy is sma11 enough then mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaadawfqaqaaiaadihacqghiiizcawgcbaa meaacawg6bgaeyici4saeyoaiyraamoqaaqabaaaleqaagqaaogaa8 hiagqaciaa veadaqhaawcbagaa4neaaqaaiaa hhaaagcdawadaqa aigacwgacagg4bgaaiicaiaa bcadawdxbqaaiaa fhacawfoagaa4 heamaabaaaleaacagf0baabeaakiaacmcacawgkbgaamidaawcbaga a8hmaaqaaiaadshadawgaaadbagaamoqaaqabaaaniabguiiydaaki aawufacagldbaacawfgagaa8hpaiaa bcacawfragaa8hiaiabg6hi lcaa 5a6b mathop sup limits mathop x in b limits z in partial b e z x left exp int limits 0 t b q x t dt right infty this paper gives a new proof of one of the two main results by aizenman and simon in 1 by a simple and elementary method a basic theorem in chung and rao 2 is extended to the class of q treated in 1

fast and stable contour integration for high order divided differences via elliptic functions in this paper we will present a new method for evaluating high order divided differences for certain classes of analytic possibly operator values functions this is a classical problem in numerical mathematics but also arises in new applications such as e g the use of generalized convolution quadrature to solve retarded potential integral equations the functions which we will consider are allowed to grow exponentially to the left complex half plane and the interpolation points are scattered in a large real interval our approach is based on the representation of divided differences as contour integral and we will employ a subtle parameterization of the contour in combination with a quadrature approximation by the trapezoidal rule

on the numerical stability of newton s formula for lagrange interpolation abstract it was previously noticed in the literature that the computation of newton s formula can be very sensitive to the ordering of the input data and it may be numerically unstable even for interpolation at the chebyshev nodes nevertheless to the best of our knowledge the effects of rounding errors in the computation of newton s formula have not been properly quantized yet in this note we make a full stability analysis of the propagation of rounding errors in the computation of newton s formula for lagrange interpolation we derive sharp upper bounds for the numerical error in the computations of three algorithms for computing polynomial interpolants in newton form that can be used to understand the instabilities mentioned above we throw light on and give a practical meaning to the condition number defined in carnicer et al 2019 by showing that it is the right thermometer for measuring the effects of rounding errors in the computation of newton s formula

absorption semigroups their generators and schrodinger semigroups on presente des resultats de perturbations pour des c 0 semi groupes on considere le probleme aux valeurs initiales pour l equation de la chaleur avec absorption avec une solution dans l p r v avec une mauvaise fonction v r r on considere le meme probleme dans l 2 r pour l equation de schrodinger

strong continuity of generalized feynman kac semigroups necessary and sufficient conditions let e d e be a strongly local quasi regular symmetric dirichlet form on l2 e m and xt t 0 px x e the diffusion process associated with e d e for u d e e u has a quasi continuous version u and u xt has fukushima s decomposition u xt u x0 mtu ntu where mtu is the martingale part and ntu is the zero energy part in this paper we study the strong continuity of the generalized feynman kac semigroup defined by ptuf x ex entuf xt t 0 two necessary and sufficient conditions for ptu t 0 to be strongly continuous are obtained by considering the quadratic form qu d e b where qu f f e f f e u f2 for f d e b and the energy measure u of u respectively an example is also given to show that ptu t 0 is strongly continuous when u is not a measure of the kato class but of the hardy class with the constant u e 12 cf definition 4 5

uniformly integrable operators and large deviations for markov processes abstract in this paper we introduce and investigate the notion of uniformly integrable operators on l p e its relations to classical compactness and hypercontractivity are exhibited several consequences of this notion are established such as perron frobenius type theorems independence on p of the spectral radius in l p p continuity of spectral radius and especially the existence of spectral gap in the irreducible case we also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent for a essentially irreducible markov process we show that the uniform integrability in l p of some type of resolvent associated with the transition semigroup implies the large deviation principle of level 3 with some rate function given by a modified donsker varadhan entropy functional we also prove that the uniform integrability condition becomes even necessary in the symmetric case finally we present several applications of our results to feynman kac semigroups to the thermodynamical limits of grand ensembles and to non symmetric markov processes given by girsanov s formula

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

conditional gauge theorem for non local feynman kac transforms feynman kac transforms driven by discontinuous additive functionals are studied in this paper for a large class of markov processes general gauge and conditional gauge theorems are established for such transforms furthermore the l 2 infinitesimal generator of the schrodinger semigroup given by a non local feynman kac transform is determined in terms of its associated bilinear form

time reversal and elliptic boundary value problems in this paper we prove that there exists a unique bounded continuous weak solution to the dirichlet boundary value problem for a general class of second order elliptic operators with singular coefficients which does not necessarily have the maximum principle our method is probabilistic the time reversal of symmetric markov processes and the theory of dirichlet forms play a crucial role in our approach

absolute continuity of symmetric diffusions let x and y be symmetric diffusion processes with a common state space and let p m resp q be the law of x resp y with its symmetry measure m resp as initial distribution we study the consequences of the absolute continuity condition q t 1 t the density p also serves to link the dirichlet forms of x and y in a simple way our identification of l relies on notions of even and odd for additive functionals these notions complement fukushima s decomposition and the forward backward martingale decomposition of lyons and zheng

on the lp theory of c0 semigroups associated with second order elliptic operators ii abstract we study positive c 0 semigroups on l p associated with second order uniformly elliptic divergence type operators with singular lower order terms subject to a wide class of boundary conditions we obtain an interval p min p max in the l p scale where these semigroups can be defined including the case 2 p min p max we present an example showing that the result is optimal we also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p for the whole range of p where the semigroups are defined

perturbation of dirichlet forms lower semiboundedness closability and form cores we study perturbations e e qu of dirichlet forms e on some l2 space l2 m given by quadratic forms q g g d with a signed borel measure whose positive and negative parts are smooth measures with respect to the given dirichlet form we give necessary and sufficient conditions for e to be a lower semibounded closed quadratic form with a unique associated continuous semigroup pt on l2 m we also study the associated resolvent semigroup as well as the relation between e and the largest closed quadratic form which is smaller than e in particular we exhibit form cores

kullback leibler information and its applications in multi dimensional adaptive testing this paper first discusses the relationship between kullback leibler information kl and fisher information in the context of multi dimensional item response theory and is further interpreted for the two dimensional case from a geometric perspective this explication should allow for a better understanding of the various item selection methods in multi dimensional adaptive tests mat which are based on these two information measures the kl information index ki method is then discussed and two theorems are derived to quantify the relationship between ki and item parameters due to the fact that most of the existing item selection algorithms for mat bear severe computational complexity which substantially lowers the applicability of mat two versions of simplified kl index ski built from the analytical results are proposed to mimic the behavior of ki while reducing the overall computational intensity

on the connection between schrodinger and dirichlet forms relations between schrodinger forms associated with schrodinger operators in l2 dnx rn open n 1 and the corresponding dirichlet forms are investigated various concrete examples are presented

hardy s inequality for dirichlet forms abstract we prove an abstract form of hardy s l2 inequality in which the dirichlet integral is replaced by the dirichlet form of a general symmetric markov process a number of examples are provided

stochastic calculus on distorted brownian motion relations between malliavin s calculus and the formulation of schrodinger dynamics in terms of local dirichlet forms are investigated it is proved that under mild regularity conditions there exists a directional derivative of the solution of the stochastic equation associated with the corresponding dirichlet form also sufficient conditions are given so that e f xt x is a strong solution of the cauchy problem t h t x q x h t x where xt x t 0 x rd is a distorted brownian motion

non symmetric perturbations of symmetric dirichlet forms abstract we provide a path space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric local dirichlet form the representation is a combination of feynman kac and girsanov formulas and extends previously known results in the framework of symmetric diffusion processes through the use of the hardy class of smooth measures which contains the kato class of smooth measures

unique continuation for elliptic operators a geometric variational approach dans un sous ensemble connexe ouvert de r n n 3 on considere l equation d ordre 2 lu div a x u b x u v x u 0 ou a x a id j x est une matrice symetrique reelle on etudie les proprietes d unicite de l operateur

uniqueness of schrodinger operators restricted in a domain abstract let s 2 v be the schrodinger s operator defined on c 0 d where d is a open domain in r d by means of the asymptotic behavior of v near the boundary d we give the necessary and sufficient conditions to the essential markovian self adjointness of s for the nonnegative potential v and to the uniqueness of s in l 1 d for general v

a note on an inequality involving the normal distribution the following inequality is useful in studying a variation of the classical isoperimetric problem let x be normally distributed with mean 0 and variance 1 if g is absolutely continuous and g x has finite variance then e lbrack g x rbrack 2 geq operatorname var lbrack g x rbrack with equality if and only if g x is linear in x the proof involves expanding g x in hermite polynomials

estimating difficulty from polytomous categorical data a comprehensive analysis of difficulty for multiple choice items requires information at different levels the test the items and the alternatives this paper introduces a new parameterization of the nominal categories model ncm for analyzing difficulty at these three levels the new parameterization is referred to as the ne ncm and is statistically equivalent to the ncm the ne ncm is applied to a sample of responses from a logical analysis test the results suggest that the individuals execute a self terminated response process that is mostly determined by working memory load

multidimensional adaptive testing with optimal design criteria for item selection several criteria from the optimal design literature are examined for use with item selection in multidimensional adaptive testing in particular it is examined what criteria are appropriate for adaptive testing in which all abilities are intentional some should be considered as a nuisance or the interest is in the testing of a composite of the abilities both the theoretical analyses and the studies of simulated data in this paper suggest that the criteria of a optimality and d optimality lead to the most accurate estimates when all abilities are intentional with the former slightly outperforming the latter the criterion of e optimality showed occasional erratic behavior for this case of adaptive testing and its use is not recommended if some of the abilities are nuisances application of the criterion of as optimality or ds optimality which focuses on the subset of intentional abilities is recommended for the measurement of a linear combination of abilities the criterion of c optimality yielded the best results the preferences of each of these criteria for items with specific patterns of parameter values was also assessed it was found that the criteria differed mainly in their preferences of items with different patterns of values for their discrimination parameters

on the potential theory of symmetric markov processes on reprend des resultats de dynkin et on considere un processus de markov fort symetrique continu a droite on etudie la relation existant entre le cone de mesures positives d energie finie et la capacite

on the fly backward error estimate for matrix exponential approximation by taylor algorithm abstract in this paper we show that it is possible to estimate the backward error for the approximation of the matrix exponential on the fly without the need to precompute in high precision quantities related to specific accuracies in this way the scaling parameter s and the degree m of the truncated taylor series the underlying method are adapted to the input matrix and not to a class of matrices sharing some spectral properties such as with the classical backward error analysis the result is a very flexible method which can approximate the matrix exponential at any desired accuracy moreover the risk of overscaling that is the possibility to select a value s larger than necessary is mitigated by a new choice as the sum of two powers of two finally several numerical experiments in matlab with data in double and variable precision and with different tolerances confirm that the method is accurate and often faster than available good alternatives

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

stochastic calculus for symmetric markov processes using time reversal we introduce a stochastic integral for zero energy additive functionals of symmetric markov processes extending earlier work of s nakao various properties of such stochastic integrals are discussed and an ito formula for dirichlet processes is obtained

the feller property for absorption semigroups letu u t t 0 be a substochastic strongly continuous semigroup onl1 x m wherexis locally compact andma borel measure onx we give conditions on absorption ratesvimplying that the strong feller property carries over fromu tou v these conditions are essentially in terms of the kato class associated withu preparing these results we discuss the perturbation theory of strongly continuous semigroups and properties of one parameter semigroups onl m in the symmetric case of dirichlet forms we generalize the results to measure perturbations for the case of the heat equation on rdwe show that the results are close to optimal

reporting of subscores using multidimensional item response theory recently there has been increasing interest in reporting subscores this paper examines reporting of subscores using multidimensional item response theory mirt models e g reckase in appl psychol meas 21 25 36 1997 c r rao and s sinharay eds handbook of statistics vol 26 pp 607 642 north holland amsterdam 2007 beguin glas in psychometrika 66 471 488 2001 a mirt model is fitted using a stabilized newton raphson algorithm haberman in the analysis of frequency data university of chicago press chicago 1974 sociol methodol 18 193 211 1988 with adaptive gauss hermite quadrature haberman von davier lee in ets research rep no rr 08 45 ets princeton 2008 a new statistical approach is proposed to assess when subscores using the mirt model have any added value over i the total score or ii subscores based on classical test theory haberman in j educ behav stat 33 204 229 2008 haberman sinharay puhan in br j math stat psychol 62 79 95 2008 the mirt based methods are applied to several operational data sets the results show that the subscores based on mirt are slightly more accurate than subscore estimates derived by classical test theory

nonautonomous kato classes of measures and feynman kac propagators the behavior of the feynman kac propagator corresponding to a time dependent measure on r n is studied we prove the boundedness of the propagator in various function spaces on r n and obtain a uniqueness theorem for an exponentially bounded distributional solution to a nonautonomous heat equation

multidimensional cat item selection methods for domain scores and composite scores theory and applications multidimensional computer adaptive testing mcat can provide higher precision and reliability or reduce test length when compared with unidimensional cat or with the paper and pencil test this study compared five item selection procedures in the mcat framework for both domain scores and overall scores through simulation by varying the structure of item pools the population distribution of the simulees the number of items selected and the content area the existing procedures such as volume segall in psychometrika 61 331 354 1996 kullback leibler information veldkamp van der linden in psychometrika 67 575 588 2002 minimize the error variance of the linear combination van der linden in j educ behav stat 24 398 412 1999 and minimum angle reckase in multidimensional item response theory springer new york 2009 are compared to a new procedure minimize the error variance of the composite score with the optimized weight proposed for the first time in this study the intent is to find an item selection procedure that yields higher precisions for both the domain and composite abilities and a higher percentage of selected items from the item pool the comparison is performed by examining the absolute bias correlation test reliability time used and item usage three sets of item pools are used with the item parameters estimated from real live cat data results show that volume and minimum angle performed similarly balancing information for all content areas while the other three procedures performed similarly with a high precision for both domain and overall scores when selecting items with the required number of items for each domain the new item selection procedure has the highest percentage of item usage moreover for the overall score it produces similar or even better results compared to those from the method that selects items favoring the general dimension using the general model segall in psychometrika 66 79 97 2001 the general dimension method has low precision for the domain scores in addition to the simulation study the mathematical theories for certain procedures are derived the theories are confirmed by the simulation applications

orthogonal polynomial expansions for the matrix exponential many different algorithms have been suggested for computing the matrix exponential in this paper we put forward the idea of expanding in either chebyshev legendre or laguerre orthogonal polynomials in order for these expansions to converge quickly we cluster the eigenvalues into diagonal blocks and accelerate using shifting and scaling

integrability of functionals of dirichlet processes probabilistic representations of semigroups and estimates of heat kernels abstract this paper consists of three parts in part i we obtain results on the integrability of functional of exponential type of dirichlet processes in part ii we give a striking probabilistic representation of semigroup probably non markovian associated with a non divergence operator part iii is devoted to perturbation bounds on the operator norm of semigroups and a new short proof of the off diagonal estimates of the heat kernel associated with a divergence operator the theory of dirichlet forms and forward backward martingales decompositions play a central role in the whole paper

quadratic forms corresponding to the generalized schrodinger semigroups suppose that xt is the standard brownian motion in rd d 3 that h1 rd is a bounded continuous function such that 2 belongs to the kato class and is a measure belonging to the kato class let a t be defined as a t xt x0 t0 xs dxs and let a t be the continuous additive functional with as its revuz measure define at as the sum of the two additive functionals above then the semigroup defined as ttf x ex eatf xt is called a generalized schrodinger semigroup in this paper we identify the quadratic form corresponding to tt as e h1 rd with e u v 12 u x v x dx rd uv x x dx rdu x v x dx