uniform boundedness of conditional gauge and schrodinger equations we prove that for a bounded domaind rn withc2 boundary and q in k n loc n geqq 3 if e x exp int limits 0 tau d q x t dt mathop ddag limits infty ind then mathop sup limits mathop x in d limits z in partial d e z x exp int limits 0 tau d q x t dt infty xt brownian motion

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

general gauge and conditional gauge theorems general gauge and conditional gauge theorems are established for a large class of not necessarily symmetric strong markov processes including brownian motions with singular drifts and symmetric stable processes furthermore new classes of functions are introduced under which the general gauge and conditional gauge theorems hold these classes are larger than the classical kato class when the process is brownian motion in a bounded c 1 1 domain

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

isolated singularities of the schrodinger equation with a good potential we study the behaviour near an isolated singularity say 0 of non negative solutions of the schrodinger equation au vu 0 defined in a punctured ball 0 3 though the results have natural counterparts in two dimensions in the simplest case where v and vanish identically this behaviour is well known in fact given any nonnegative harmonic function u u x defined in br there exists the limit

robust wiener filtering for multiple inputs with channel distortion corresp robust wiener filtering has previously been considered for the single input scalar case where there is no channel distortion and where the signal to be estimated is the source signal itself here these results are extended to the multiple input vector case where linear channel distortion is allowed and the signal to be estimated is a linear filtered version of the source signal the results are obtained from those for the single input ease by modifying the constraints on signal and noise characteristics such a modification is motivated by examining the expression of the mean squared error for the optimum filter

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

extinction of superdiffusions and semilinear partial differential equations abstract a superdiffusion is a measure valued branching process associated with a nonlinear operator lu u where l is a second order elliptic differential operator and is a function from r d r to r in the case l 1 0 the so called subcritical case the expectation of the total mass does not increase and the mass vanishes at a finite time with probability 1 most results on connections between the superdiffusion and differential equations involving lu u were obtained for the subcritical case in the present paper we assume only that l 1 is a bounded function in this more general setting the probability of extinction can be smaller than 1 and we show that his happens if and only if there exists a strictly positive solution of the equation lu u 0 in r d we also establish a relation between the probability of extinction in a domain d and strictly positive solutions of equation lu u 0 in d that are equal to 0 on the boundary of d we call such solutions special for the equation c u u where is the beltrami laplace operator on a complete riemannian manifold e strictly positive solutions in e were studied in connection with a geometrical problem which two functions represent scalar curvatures of two riemannian metrics related by a conformal mapping see 1 and references there

on minimax robustness a general approach and applications the minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary the existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications in particular it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust the efficacy of the methods proposed for a general game is tested in the problems of matched filtering wiener filtering quadratic detection and output energy filtering in which uncertainties in their respective signal and noise models are assumed to exist these problems are analyzed in a common hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

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

the martin kernel and infima of positive harmonic functions let d be a bounded lipschitz domain in bbr n and let k x z x in d z in partial d be the martin kernel based at x 0 in d for x y in d let k x y inf h x h mbox positive harmonic in d h y 1 we show that the function k completely determines the family of positive harmonic functions on d precisely for every z in partial d lim y rightarrow z k x y k x 0 y k x z the same result is true for second order uniformly elliptic operators and schr o dinger operators

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

subcriticality and gaugeability of the schrodinger operator we investigate a schrodinger operator 2 v in r d d 3 with a potential v in the class k d satisfying a similar kato condition at infinity and prove an equivalence theorem connecting various conditions on subcriticality strong positivity and gaugeability of the operator

positivity for equations involving polyharmonic operators with dirichlet boundary conditions cranston fabes and zhao 26 5 established the uniform bound sup x y 2 x 6 y r g1 n x z g1 n z y dz g1 n x y m 1 1 where g1 n x y is the green function for the laplacian with dirichlet boundary conditions on a lipschitz domain rn with n 3 see 27 for n 2 this estimate was used in 23 and 18 to obtain positivity uniformly with respect to f 0 for noncooperative elliptic systems as

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

sharp estimates for the green function in lipschitz domains we give sharp global estimates for the green function and martin kernel in lipschitz domains we apply the estimates to obtain short proofs of some classical results

a conformal inequality related to the conditional gauge theorem we prove the inequality h x 1 g x y h y cg x y c where g is the green function of a plane domain d h is positive and harmonic on d and c is a constant whose value depends on the topological nature of the domain in particular for the class of proper simply connected domains c may be taken to be an absolute constant as an application we prove the conditional gauge theorem for plane domains of finite area for which the constant c in the above inequality is finite

a probabilistic approach to multivariable robust filtering and open loop control a new approach to robust filtering prediction and smoothing of discrete time signal vectors is presented linear time invariant filters are designed to be insensitive to spectral uncertainty in signal models the goal is to obtain a simple design method leading to filters which are not overly conservative modeling errors are described by sets of models parameterized by random variables with known covariances these covariances could either be estimated from data or be used as robustness tuning knobs a robust design is obtained by minimizing the spl hscrsub 2 norm or equivalently the mean square estimation error averaged with respect to the assumed model errors a polynomial solution based on an averaged spectral factorization and a unilateral diophantine equation is derived the robust estimator is referred to as a cautious wiener filter it turns out to be only slightly more complicated to design than an ordinary wiener filter the methodology can be applied to any open loop filtering or control problem in particular we illustrate this for the design of robust multivariable feedforward regulators decoupling and model matching filters

robust least squares estimation with a relative entropy constraint given a nominal statistical model we consider the minimax estimation problem consisting of finding the best least squares estimator for the least favorable statistical model within a neighborhood of the nominal model the neighborhood is formed by placing a bound on the kullback leibler kl divergence between the actual and nominal models for a gaussian nominal model and a finite observations interval or for a stationary gaussian process over an infinite interval the usual noncausal wiener filter remains optimal however the worst case performance of the filter is affected by the size of the neighborhood representing the model uncertainty on the other hand standard causal least squares estimators are not optimal and a characterization is provided for the causal estimator and the corresponding least favorable model the causal estimator takes the form of a risk sensitive estimator with an appropriately selected risk sensitivity coefficient

holder continuity of normalized solutions of the schrodinger equation definition 1 assume that v l a function u is called local weak solution of the schr6dinger equation 89 v u 0 on an open subset d c r d if u l 162 v u l and 1 f u r f v u r 0 v e mp o 2d o

gauge theorems for resolvents with application to markov processes letv v 0 be a not necessarily sub markovian resolvent such that the kernelv for some 0 is compact and irreducible we prove the following general gauge theorem if there exists at least onev excessive function which is notv inviriant thenv0 is bounded

robust wiener filtering with non parametric spectral uncertainty in this article robust least squares filtering problems are considered for non parametric multivariate spectral uncertainty defined by the so called spectral band and generalised moment constraints its major aim is to provide a basis for computing approximate solutions to worst case wiener filtering optimisation problems involving causal filters and multivariate signals it hinges upon associating upper and lower bounds on the minimum worst case performance achievable with causal filters with linear cost linear matrix inequality lc lmi constraint optimisation problems on the basis of a lagrangean duality formulation for the worst case least squares performance of a given filter upper bounds on it are obtained as the optimal values of lc lmi problems then for linearly parameterised classes of filter transfer functions a causal filter which optimises such an upper bound on worst case performance can also be obtained from an lc lmi optimisation problem to estimate the amount of conservatism incur

robust wiener kolmogorov theory a minimax formulation is considered for the problem of designing robust linear causal estimators of linear functions of discrete time wide sense stationary signals when knowledge of the signal and or noise spectra is inexact the solution is given under mild regularity conditions in terms of a least favorable pair of spectra thus reducing the minimax problem to a direct maximization problem which in many cases can be solved easily it is noted that this design method leads in particular to robust n step predictors robust causal filters and robust n lag smoothers the method of design is illustrated by a thorough development of the special case of one step noiseless prediction further solutions are given explicitly for the problem of robust causal filtering of an uncertain signal in white noise and numerical examples are given for this case which illustrate the effectiveness of this design

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

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

robust reduced order nonstrictly proper state estimation via the optimal projection equations with petersen hollot bounds a state estimation design problem involving parametric plant uncertainties is considered an error bound suggested by recent work of petersen and hollot is utilized for guaranteeing robust estimation necessary conditions which generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator which minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators an additional feature is the presence of a static estimation gain in conjunction with the dynamic kalman estimator i e a nonstrictly proper estimator

h 2 robust filter design with performance certificate via convex programming in this paper a new approach to h 2 robust filter design is proposed both continuous and discrete time invariant systems subject to polytopic parameter uncertainty are considered after a brief discussion on some of the most expressive methods available for h 2 robust filter design a new one based on a performance certificate calculation is presented the performance certificate is given in terms of the gap produced by the robust filter between lower and upper bounds of a minimax programming problem where the h 2 norm of the estimation error is maximized with respect to the feasible uncertainties and minimized with respect to all linear rational and causal filters the calculations are performed through convex programming methods developed to deal with linear matrix inequality lmi many examples borrowed from the literature to date are solved and it is shown that the proposed method outperforms all other designs

robust kalman filtering for uncertain discrete time systems this paper is concerned with the problem of a kalman filter design for uncertain discrete time systems the system under consideration is subjected to time varying norm bounded parameter uncertainty in both the state and output matrices the problem addressed is the design of a linear filter such that the variance of the filtering error is guaranteed to be within a certain bound for all admissible uncertainties furthermore the guaranteed cost can be optimized by appropriately searching a scaling design parameter

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

robust kalman filtering for uncertain systems abstract this paper studies the problem of kalman filter design for uncertain systems the system under consideration is subjected to time varying norm bounded parameter uncertainties in both the state and measurement matrices the problem we address is the design of a state estimator such that the covariance of the estimation error is guaranteed to be within a certain bound for all admissible uncertainties a riccati equation approach is proposed to solve the above problem furthermore a suboptimal covariance upper bound can be computed by a convex optimization

on upper bounds on worst case h 2 performance obtained with dynamic multipliers in this note an iterative procedure is presented for obtaining a decreasing sequence of upper bounds on the worst case h 2 performance of a given stabilizing controller in the presence of normalized coprime factor perturbations to obtain such bounds a descent procedure is introduced for a dual lagrangean functional which gives upper bounds on the worst case h 2 performance index and is defined on the set of real rational and non negative functions dynamic multipliers specific ways are presented for selecting feasible and descent directions in this set and lower bounds are derived for the corresponding decreases on the dual functional at any step the dynamics of the current multiplier gives rise to a linear class over which the optimization of the dual functional is shown to be equivalent to linear optimization subject to linear matrix inequalities lmi this allows for a combination of function space and lmi techniques in the process of obtaining increasingly tighter upper bounds on worst case h

boundary harnack principle for symmetric stable processes in this paper we study potential theoretic properties of the symmetric stable processes 0 2 establishing the boundary harnack principle for ratios of harmonic functions on any open sets identifying the martin boundary with the euclidean boundary for open sets with a certain interior fatness property and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets

robust discrete time minimum variance filtering the bounded variance filtered estimation of the state of an uncertain linear discrete time system with an unknown norm bounded parameter matrix is considered an upper bound on the variance of the estimation error is found for all admissible systems and estimators are derived that minimize the latter bound we treat the finite horizon time varying case and the infinite time case where the nominal system model is time invariant in the special stationary case where it is known that the uncertain system is time invariant we provide a robust filter for all uncertainties that still keep the system asymptotically stable

feynman kac functionals and positive solutions of 1 2δu qu 0 let d be a bounded c 2 domain in d and let q be a bounded borel function in d for x d and z d suppose x t under the law p x z is brownian motion in d starting at x and conditioned to converge to z let be the lifetime of x t we show that if the quantity e x z left exp left int limits 0 t q x s ds right right is finite for one x d and one z d then this quantity remains bounded as x varies over d and z varies over d this may be considered one quantitative expression of the qualitative statement that no matter where brownian motion in d eventually hits d it goes all over d before it gets there we apply this result to show that if the equation 1 2 u qu 0 admits a non negative solution in d which is strictly positive on a subset of d of positive harmonic measure then for any non negative bounded borel function f on d it admits a unique bounded solution u satisfying u f on d and this solution u is non negative

positive solutions with a time independent boundary singularity of semilinear heat equations in bounded lipschitz domains abstract we study time global positive solutions of semilinear heat equations of the form u t u f x u in a bounded lipschitz domain in r n in particular we show the existence of a positive solution with a time independent singularity at a boundary point of which converges to a positive solution with the behavior like the martin kernel at of the corresponding elliptic equation at time infinity a nonlinear term f is conditioned in terms of a certain lipschitz continuity with respect to the second variable and a generalized kato class associated with the martin kernel at and admits not only usual one v x u p log 1 u q but also one with variable exponents

a martin boundary in the plane let e be an open connected subset of euclidean space with a green function and let a be harmonic measure on the martin boundary i of e we will show that except for a a x a null set of x y g i 2 x is an entrance point for brownian motion conditioned to leave e at y r s martin gave examples in dimension 3 or higher for which there exist minimal accessible martin boundary points x 7 y for which this condition fails we will give a similar example in dimension 2

conditional transformation of drift formula and potential theory for 1 2 b using conditional brownian motion and the transformation of drift formula of cameron martin girsarov maruyama we give integral conditions on a vector fieldb which imply the harmonic measures and green functions for 1 2 and 1 2 b on a bounded lipschitz domaind are equivalent by equivalent we mean there exist two sided inequalities with constants depending only onb andd this enables one to conclude the potential theory for 1 2 b ond and 1 2 ond are the same

the h path distribution of the lifetime of conditioned brownian motion for non smooth domains consider the h paths of doob and let d be the lifetime of brownian motion killed at d where d is a bounded lipschitz or nta domain in n then log p x h tau d t sim lambda d t as t to infty where d is the first positive eigenvalue of 1 2 on d

conditioned brownian motion in planar domains we give an upper bound for the green functions of conditioned brownian motion in planar domains a corollary is the conditional gauge theorem in bounded planar domains

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

on energy inequalities we consider the schrodinger operator h 2 q in a bounded domain d in rd d 3 with q in the kato class kd d and finite gauge g x ex exp 0q xs ds where xt is a brownian motion and is the first exit time of the brownian motion xt from the domain d let k and g denote the green operators in d for h and 2 respectively we prove that there is a positive constant such that1 gf f kf f gf f where f is a real measurable function for which g f f is finite as a direct consequence of this double inequality we have that the potential gf is in the sobolev space h10 d if and only if kf h10 d

optimal linear filtering under parameter uncertainty this paper addresses the problem of designing a guaranteed minimum error variance robust filter for convex bounded parameter uncertainty in the state output and input matrices the design procedure is valid for linear filters that are obtained from the minimization of an upper bound of the error variance holding for all admissible parameter uncertainty the results provided generalize the ones available in the literature to date in several directions first all system matrices can be corrupted by parameter uncertainty and the admissible uncertainty may be structured assuming the order of the uncertain system is known the optimal robust linear filter is proved to be of the same order as the order of the system in the present case of convex bounded parameter uncertainty the basic numerical design tools are linear matrix inequality lmi solvers instead of the riccati equation solvers used for the design of robust filters available in the literature to date the paper that contains the most important and very recent results on robust filtering is used for comparison purposes in particular it is shown that under the same assumptions our results are generally better as far as the minimization of a guaranteed error variance is considered some numerical examples illustrate the theoretical results

robust h 2 control based on dynamic multipliers in this article upper bounds on the worst case h 2 performance index relative to structured feedback perturbations are considered which are based on the minimisation of dual lagrangean functionals over linearly parametrised finite dimensional classes of dynamic multipliers it is shown that the minimisation problems in question can be recast as optimisation problems with linear cost functional and linear matrix inequality lmi constraints an iterative scheme is suggested to generate linearly parametrised classes of multipliers of increasing dynamic order so that progressively tighter upper bounds can be obtained as illustrated by two simple numerical examples finally with a view to synthesis procedures based on d k iterations relative to multipliers and controllers it is shown that the minimisation of the upper bounds corresponding to given multipliers with respect to linearly parametrised classes of youla parameters can also be cast as linear cost lmi constraint problems

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

the d module structure of r f modules let r be a regular ring essentially of finite type over a perfect field k an r module m is called a unit r f module if it comes equipped with an isomorphism f e m m where f denotes the frobenius map on spec r and f e is the associated pullback functor it is well known that m then carries a natural d r module structure in this paper we investigate the relation between the unit r f structure and the induced d r structure on m in particular it is shown that if k is algebraically closed and m is a simple finitely generated unit r f module then it is also simple as a d r module an example showing the necessity of k being algebraically closed is also given

noncausal minimax linear state estimation for systems with uncertain second order statistics this note considers the problem of minimax state estimation of the states of a linear time invariant system which is driven by and observed in the presence of noise processes with uncertain second order statistics when the process noise and observations are scalars the problem is shown to be equivalent to a scalar minimax estimation problem the existence of a minimax solution is thereby established and the minimax filter is shown to be a linear transformation of the minimax filter for the scalar problem

the intersection homology d module in finite characteristic for y a closed normal subvariety of codimension c of a smooth variety x brylinski and kashiwara showed that the local cohomology module c y x x contains a unique simple x submodule denoted by y x in this paper the analogous result is shown for x and y defined over a perfect field of finite characteristic moreover a local construction of y x is given relating it to the theory of tight closure from the construction one obtains a criterion for the x simplicity of c y x x

on the lifetime of a conditioned brownian motion in the ball consider the brownian motion conditioned to start in x to converge to y with x y and to be killed at the boundary here is a bounded domain in rn for which x and y is the lifetime of this brownian motion maximal one would guess for x and y being opposite boundary points and we will show that this holds true for balls in rn as a consequence we find the best constant for the positivity preserving property of some elliptic systems and an identity between this constant and a sum of inverse dirichlet eigenvalues

on the conditioned exit measures of super brownian motion in this paper we present a martingale related to the exit measures of super brownian motion by changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure this measure is shown to be identical to a measure generated by a non homogeneous branching particle system with immigration of mass an application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain the results are similar in flavor to the immortal particle picture of conditioned super brownian motion but more general as the change of measure is given by a martingale which need not arise from a single harmonic function

on a parabolic equation with a singular lower order term abstract we obtain the existence of the weak green s functions of parabolic equations with lower order coefficients in the so called parabolic kato class which is being proposed as a natural generalization of the kato class in the study of elliptic equations as a consequence we are able to prove the existence of solutions of some initial boundary value problems moreover based on a lower and an upper bound of the green s function we prove a harnack inequality for the non negative weak solutions

robust minimum variance filtering this paper deals with the robust minimum variance filtering problem for linear systems subject to norm bounded parameter uncertainty in both the state and the output matrices of the state space model the problem addressed is the design of linear filters having an error variance with a guaranteed upper bound for any allowed uncertainty two methods for designing robust filters are investigated the first one deals with constant parameter uncertainty and focuses on the design of steady state filters that yield an upper bound to the worst case asymptotic error variance this bound depends on an upper bound for the power spectrum density of a signal at a specific point in the system and it can be made tighter if a tight bound on the latter power spectrum can be obtained the second method allows for time varying parameter uncertainty and for general time varying systems and is more systematic we develop filters with an optimized upper bound for the error variance for both finite and infinite horizon filtering problems

lifetime of conditioned brownian motion in lipschitz domains if d d d 3 is bounded and has lipschitz boundary then the expected lifetime of any brownian h path process in d is finite

on ratios of harmonic functions abstract let u and v be harmonic functions in r n with the same zero set z we show that the ratio f of such functions is always well defined and is real analytic moreover it satisfies the maximum and minimum principles for n 3 we also prove the harnack inequality and the gradient estimate for the ratios of harmonic functions namely sup k f c inf k f sup k f c inf k f for any compact subset k of where the constant c depends on k z only in dimension two the first inequality follows from the boundary harnack principle and the second from the gradient estimate recently obtained by mangoubi it is an open question whether these inequalities remain true in higher dimensions n 4

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

the boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations we investigate the boundary growth of positive superharmonic functions u on a bounded domain in mathbb r n n 3 satisfying the nonlinear elliptic inequality 0 le delta u le c delta omega x alpha u p quad rm in omega where c 0 0 and p 0 are constants and delta omega x is the distance from x to the boundary of the result is applied to show a harnack inequality for such superharmonic functions also we study the existence of positive solutions with singularity on the boundary of the nonlinear elliptic equation delta u vu f x u quad rm in omega where v and f are borel measurable functions conditioned by the generalized kato class

the gauge theorem for a class of additive functionals of zero energy in earlier works the gauge theorem was proved for additive functionals of brownian motion of the form 0 q b s ds whereq is a function in the kato class subsequently the theorem was extended to additive functionals with revuz measures in the kato class we prove that the gauge theorem holds for a large class of additive functionals of zero energy which are in general of unbounded variation these additive functionals may not be semi martingales but correspond to a collection of distributions that belong to the kato class in a suitable sense our gauge theorem generalizes the earlier versions of the gauge theorem

stability of martin boundary under non local feynman kac perturbations recently the authors showed that the martin boundary and the minimal martin boundary for a censored or resurrected stable process y in a bounded c 1 1 open set d with 1 2 can all be identified with the euclidean boundary d of d under the gaugeability assumption we show that the martin boundary and the minimal martin boundary for the schrodinger operator obtained from y through a non local feynman kac transform can all be identified with d in other words the martin boundary and the minimal martin boundary are stable under non local feynman kac perturbations moreover an integral representation of nonnegative excessive functions for the schrodinger operator is explicitly given these results in fact hold for a large class of strong markov processes as are illustrated in the last section of this paper as an application the martin boundary for censored relativistic stable processes in bounded c 1 1 smooth open sets is studied in detail

on critical and subcritical sturm liouville operators we study critical and subcritical sturm liouville operators k 1 ddx p ddx v in l2 a b k dx a b r defined as friedrichs extensions under most general conditions on k p and v in particular we prove the equivalence of the notions of criticality as introduced by simon and murata for an important class of potentials v