stochastic uncertain systems subject to relative entropy constraints induced norms and monotonicity properties of minimax games entropy and relative entropy are fundamental concepts on which information theory is founded on and in general telecommunication systems design on the other hand dissipation inequalities minimax strategies and induced norms are the basic concepts on which robustness of uncertain control and estimation of systems are founded on in this paper the precise relation between these notions is investigated in particular it will be shown that the higher the dissipation the higher the entropy of the system which has implications in computing the induced norm associated with robustness these connections are obtained by considering stochastic optimal uncertain control systems in which uncertainty is described by a relative entropy constraint between the nominal and uncertain measures while the pay off is a linear functional of the uncertain measure this is a minimax game in which the controller measure seeks to minimize the pay off while the disturbance measure aims at maximizing the pay off salient properties of the minimax solution are derived including a characterization of the optimal sensitivity reduction computation of the induced norm monotonicity properties of minimax solution and relations between dissipation and relative entropy of the system the theory is developed in an abstract setting and then applied to nonlinear partially observable continuous time uncertain controlled systems in which the nominal and uncertain systems are described by conditional distributions in addition existence of the optimal control policy among the class of policies known as wide sense control laws is shown and an explicit formulae for the worst case conditional measure is derived the results are applied to linear quadratic gaussian problems

prime ends for domains in metric spaces in this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions we compare our prime ends to those of caratheodory and nakki modulus ends and

the martin boundary in non lipschitz domains the martin boundary with respect to the laplacian and with respect to uniformly elliptic operators in divergence form can be identified with the euclidean boundary in c domains where x bx loglog 1 x logloglog 1 x b small a counterexample shows that this result is very nearly sharp

quantum risk sensitive estimation and robustness this paper studies a quantum risk sensitive estimation problem and investigates robustness properties of the filter this is a direct extension to the quantum case of analogous classical results all investigations are based on a discrete approximation model of the quantum system under consideration this allows us to study the problem in a simple mathematical setting we close the paper with some examples that demonstrate the robustness of the risk sensitive estimator

martin points on open manifolds of non positive curvature the martin boundary of a cartan hadamard manifold describes a fine geometric structure at infinity which is a sub space of positive harmonic functions we describe conditions which ensure that some points of the sphere at infinity belong to the martin boundary as well in the case of the universal cover of a compact manifold with ballmann rank one we show that martin points are generic and of full harmonic measure the result of this paper provides a partial answer to an open problem of s t yau

dirichlet conditions at infinity for parabolic and elliptic equations we investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in rn r under specific assumptions we establish existence of solutions satisfying prescribed conditions at infinity depending on the direction along which infinity is approached moreover the large time behavior of such solutions is addressed we consider also elliptic equations in rn with similar conditions at infinity

risk sensitive filtering and smoothing for hidden markov models abstract in this paper we address the problem of risk sensitive filtering and smoothing for discrete time hidden markov models hmm with finite discrete states the objective of risk sensitive filtering is to minimise the expectation of the exponential of the squared estimation error weighted by a risk sensitive parameter we use the so called reference probability method in solving this problem we achieve finite dimensional linear recursions in the information state and thereby the state estimate that minimises the risk sensitive cost index also fixed interval smoothing results are derived we show that l2 or risk neutral filtering for hmms can be extracted as a limiting case of the risk sensitive filtering problem when the risk sensitive parameter approaches zero

the poisson boundary of the mapping class group a theory of random walks on the mapping class group and its non elementary subgroups is developed we prove convergence of sample paths in the thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the poisson boundary of random walks the methods are based on an analysis of the asymptotic geometry of teichmuller space and of the contraction properties of the action of the mapping class group on the thurston boundary we prove in particular that teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees using our analysis of the mapping class group action on the thurston boundary we prove that no non elementary subgroup of the mapping class group can be a lattice in a higher rank semi simple lie group

boundary value problems with measures for elliptic equations with singular potentials we study the boundary value problem with radon measures for nonnegative solutions of lvu u vu 0 in a bounded smooth domain when v is a locally bounded nonnegative function introducing some specific capacity we give sufficient conditions on a radon measure on so that the problem can be solved we study the reduced measure associated to this equation as well as the boundary trace of positive solutions in appendix a a ancona solves a question raised by m marcus and l veron concerning the vanishing set of the poisson kernel of lv for an important class of potentials v

moveout approximations for p and sv waves in dip constrained transversely isotropic media we generalize the p and sv wave moveout formulas obtained for transversely isotropic media with vertical axes of symmetry vti based on the weak anisotropy approximation we generalize them for 3d dip constrained transversely isotropic dti media a dti medium is a transversely isotropic medium whose axis of symmetry is perpendicular to a dipping reflector the formulas are derived in the plane defined by the source receiver line and the normal to the reflector in this configuration they can be easily obtained from the corresponding vti formulas it is only necessary to replace the expression for the normalized offset by the expression containing the apparent dip angle the final results apply to general 3d situations in which the plane reflector may have arbitrary orientation and the source and the receiver may be situated arbitrarily in the dti medium the accuracy of the proposed formulas is tested on models with varying dip of the reflector and for several orientations of the horizontal source receiver line with respect to the dipping reflector

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

estimates for derivatives of the poisson kernels on homogeneous manifolds of negative curvature we obtain estimates for derivatives of the poisson kernels for the second order differential operators on homogeneous manifolds of negative curvature both in the coercive and noncoercive case

moveout approximations for p and sv waves in vti media abstractwe tested alternative expressions for the p and sv wave moveout formulas in vti media based on the weak anisotropy wa approximation our moveout formulas represent expansions with respect to small parameters which are related to deviations of anisotropy from isotropy first order p wave formulas depend on four parameters two way zero offset traveltime t0 related to the vertical velocity 0 the depth h of the single horizontal reflector and two wa parameters w and w the first order sv wave formulas depend on three parameters again on t0 now related to the sv wave vertical velocity 0 depth h and the wa version of parameter the second order formulas are slightly more complicated the p and sv wave formulas depend on an additional parameter r the ratio of the sv and p wave vertical velocities the sv wave formula depends in addition on the wa parameter w because the dependence of the moveout formulas on r is very weak r can be specified as a typical sv to p wave velocity rat

weak anisotropy moveout approximations for p waves in homogeneous layers of monoclinic or higher anisotropy symmetries we have used so called weak anisotropy wa parameterization as an alternative to the parameterization of generally anisotropic media by a stiffness tensor wa parameters consist of linear combinations of normalized stiffness tensor elements controlling various seismic signatures hence they are theoretically extractable from seismic data they are dimensionless and can be designed to have the same order of magnitude wa parameters similarly to thomsen type parameters have a clear physical interpretation they are however applicable to anisotropy of any symmetry strength and orientation they are defined in coordinate systems independent of the symmetry elements of the studied media expressions using wa parameters naturally simplify as the anisotropy becomes weaker or as the anisotropy symmetry increases we expect that due to these useful properties wa parameterization can potentially provide a framework for seismic data processing in generally anisotropic media using the wa parameterization we have derived and tested approximate p wave moveout formulas for a homogeneous layer of up to monoclinic symmetry underlain by a horizontal reflector coinciding with a symmetry plane the derived traveltime formulas represent an expansion of the traveltime with respect to small wa parameters for the comparison with standard moveout formulas we expressed ours in the common form of nonhyperbolic moveout containing normal moveout velocity and a quartic coefficient as functions of the wa parameters the accuracy of our formulas depends strongly on the deviation of ray and phase velocity directions controlled by the deviation of the ray and phase velocities the errors do not generally increase with increasing offset nor do they increase with decreasing anisotropy symmetry the accuracy of our formulas is comparable with or better than the accuracy of commonly used formulas

the dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality we study the spatial behaviour of random walks on infinite graphs which are not necessarily invariant under some transitive group action and whose transition probabilities may have infinite range we assume that the underlying graphg satisfies a strong isoperimetric inequality and that the transition operatorp is strongly reversible uniformly irreducible and satisfies a uniform first moment condition we prove that under these hypotheses the random walk converges almost surely to a random end ofg and that the dirichlet problem forp harmonic functions is solvable with respect to the end compactification if in addition the graph as a metric space is hyperbolic in the sense of gromov then the same conclusions also hold for the hyperbolic compactification in the place of the end compactification the main tool is the exponential decay of the transition probabilities implied by the strong isoperimetric inequality finally it is shown how the same technique can be applied to brownian motion to obtain analogous results for riemannian manifolds satisfying cheeger s isoperimetric inequality in particular in this general context new and simpler proofs of well known results on the dirichlet problem for negatively curved manifolds are obtained

local limit theorem for symmetric random walks in gromov hyperbolic groups completing a strategy of gouezel and lalley we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non elementary gromov hyperbolic group denoting by r the inverse of the spectral radius of the random walk the probability to return to the identity at time n behaves like c r n n 3 2 an important step in the proof is to extend ancona s results on the martin boundary up to the spectral radius we show that the martin boundary for r harmonic functions coincides with the geometric boundary of the group in an appendix we explain how the symmetry assumption of the measure can be dispensed with for surface groups

generalized moveout approximation for qp and qsv waves in a homogeneous transversely isotropic medium the moveout approximations can be used in kinematic modeling velocity analysis and time migration the generalized moveout approximation involves five approximation parameters and has several known approximations as special cases a method is demonstrated for determining parameters of the generalized nonhyperbolic moveout approximation for qp and qsv waves in a homogeneous transversely isotropic medium with vertical symmetry axis vti medium the additional parameters for the generalized approximation are computed from the hyperbolic asymptote at infinite offset comparison with a few well known moveout approximations for higher order terms in the taylor series and asymptotic behavior shows that the generalized moveout approximation is superior to other nonhyperbolic approximations a few numerical examples for qp and qsv waves in a vti medium also indicate that the generalized approximation performs the best

martin compactification of a complete surface with negative curvature in this paper we consider the martin compactification associated with the operator mathcal l delta 1 of a complete non compact surface sigma 2 with negative curvature in particular we investigate positive eigenfunctions with eigenvalue one of the laplace operator delta of sigma 2 and prove a uniqueness result such eigenfunctions are unique up to a positive multiplicative constant if they vanish on the part of the geometric boundary s infty sigma 2 of sigma 2 where the curvature is bounded above by a negative constant and satisfy some growth estimate on the other part of s infty sigma 2 where the curvature approaches zero this uniqueness result plays an essential role in our recent paper cao and he infinitesimal rigidity of steady gradient ricci soliton in dimension three arxiv 1412 2714 2014 preprint in which we prove an infinitesimal rigidity theorem for deformations of certain three dimensional collapsed gradient steady ricci soliton with a non trivial killing vector field

risk sensitive control and dynamic games for partially observed discrete time nonlinear systems solves a finite horizon partially observed risk sensitive stochastic optimal control problem for discrete time nonlinear systems and obtains small noise and small risk limits the small noise limit is interpreted as a deterministic partially observed dynamic game and new insights into the optimal solution of such game problems are obtained both the risk sensitive stochastic control problem and the deterministic dynamic game problem are solved using information states dynamic programming and associated separated policies a certainty equivalence principle is also discussed the authors results have implications for the nonlinear robust stabilization problem the small risk limit is a standard partially observed risk neutral stochastic optimal control problem

estimation of hidden markov models risk sensitive filter banks and qualitative analysis of their sample paths a sequential filtering scheme for the risk sensitive state estimation of partially observed markov chains is presented the previously introduced risk sensitive filters are unified in the context of risk sensitive maximum a posterior probability estima tion structural results for the filter banks are given the influence of the availability of information and the transition probabilities on the decision regions and the behavior of risk sensitive estima tors are studied index terms hidden markov models hmms risk sensitive estimation

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

the boundary of a square tiling of a graph coincides with the poisson boundary answering a question of benjamini and schramm ann probab 24 3 1219 1238 1996 we show that the poisson boundary of any planar uniquely absorbing e g one ended and transient graph with bounded degrees can be realised geometrically as a circle which arises from a discrete version of riemann s mapping theorem this implies a conjecture of northshield potential anal 2 4 299 314 1993 some of our technique apply to the non planar case and might have further applications when the graph is also hyperbolic then under mild conditions we prove the equivalence of several boundary constructions

nonlinear estimation for a class of systems this paper considers nonlinear estimation problems for classes of models and employs relative entropy to describe the uncertainty classes two optimization problems are formulated on general banach spaces and their solutions are sought 1 when the transition probability between the signal to be estimated x and the measurement y or stochastic kernel is unknown and 2 when the joint probability induced by the random variables rvs x y is unknown for both problems the uncertainty is described by a relative entropy constraint between the unknown distribution and a fixed nominal distribution the results include existence of the optimal measures using weak convergence techniques and properties associated with the estimate of the true distribution classical examples are chosen to illustrate the applicability of the results

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

brief paper risk sensitive filtering for jump markov linear systems in this paper a risk sensitive multiple model filtering algorithm is derived using the reference probability methods first the approximation of the interacting multiple model imm algorithm is identified in the reference probability domain then the same type of approximation is used to derive the finite dimensional risk sensitive filtering algorithm the derived algorithm reduces to the imm filter when the risk sensitive parameter goes to zero and reduces to the risk sensitive filter for linear gauss markov systems when the number of models is unity the algorithm performs better in a simulated uncertain parameter scenario than the imm filter

robustness analysis of a maximum correntropy framework for linear regression abstract in this paper we formulate a solution of the robust linear regression problem in a general framework of correntropy maximization our formulation yields a unified class of estimators which includes the gaussian and laplacian kernel based correntropy estimators as special cases an analysis of the robustness properties is then provided the analysis includes a quantitative characterization of the informativity degree of the regression which is appropriate for studying the stability of the estimator using this tool a sufficient condition is expressed under which the parametric estimation error is shown to be bounded explicit expression of the bound is given and discussion on its numerical computation is supplied for illustration purpose two special cases are numerically studied

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

risk sensitive filtering and smoothing via reference probability methods we address the risk sensitive filtering problem which is minimizing the expectation of the exponential of the squared estimation error multiplied by a risk sensitive parameter such filtering can be more robust to plant and noise uncertainty than minimum error variance filtering although optimizing a differently formulated performance index to that of the so called h sub spl infin filtering risk sensitive filtering leads to a worst case deterministic noise estimation problem given from the differential game associated with h sub spl infin filtering we consider a class of discrete time stochastic nonlinear state space models we present linear recursions in the information state and the result for the filtered estimate that minimizes the risk sensitive cost index we also present fixed interval smoothing results for each of these signal models in addition a brief discussion is included on relations of the risk sensitive estimation problem to minimum variance estimation and a worst case estimation problem in a deterministic noise scenario related to minimax dynamic games the technique used in this paper is the so called reference probability method which defines a new probability measure where the observations are independent and translates the problem to the new measure the optimization problem is solved using simple estimation theory in the new measure and the results are interpreted as solutions in the original measure

boundary singularities of solutions of semilinear elliptic equations with critical hardy potentials we study the boundary behaviour of positive functions u satisfying e gd u frac xk d 2 x u g u 0 in a bounded domain gw of bbr n where 0 xk leq frac 1 4 g is a continuous nonndecreasing function and d is the distance function to prt gw we first construct the martin kernel associated to the the linear operator cl xk gd frac xk d 2 x and give a general condition for solving equation e with any radon measure gm for boundary data when g u u q 1 u we show the existence of a critical exponent q c q c n xk 1 whith the following properties when 0 q q c any measure is eligible for solving e with gm for boundary data if q geq q c a necessary and sufficient condition is expressed in terms of the absolute continuity of gm with respect to some besov capacity the same capacity characterizes the removable compact boundary sets at end any positive solution f gd u frac xk d 2 x u u q 1 u 0 with q 1 admits a boundary trace which is a positive outer regular borel measure when 1 q q c we prove that to any positive outer regular borel measure we can associate a positive solutions of f with this boundary trace

finite dimensional optimal controllers for nonlinear plants abstract optimal risk sensitive feedback controllers are now available for very general stochastic nonlinear plants and performance indices they consist of nonlinear static feedback of so called information states from an information state filter in general these filters are linear but infinite dimensional and the information state feedback gains are derived from doubly infinite dimensional dynamic programming the challenge is to achieve optimal finite dimensional controllers using finite dimensional calculations for practical implementation this paper derives risk sensitive optimality results for finite dimensional controllers the controllers can be conveniently derived for linearized approximate models applied to nonlinear stochastic systems performance indices for which the controllers are optimal for the nonlinear plants are revealed that is inverse risk sensitive optimal control results for nonlinear stochastic systems with finite dimensional linear controllers are generated it is instructive to see from these results that as the nonlinear plants approach linearity the risk sensitive finite dimensional controllers designed using linearized plant models and risk sensitive indices with quadratic cost kernels are optimal for a risk sensitive cost index which approaches one with a quadratic cost kernel also even far from plant linearity as the linearized model noise variance becomes suitably large the index optimized is dominated by terms which can have an interesting and practical interpretation limiting versions of the results as the noise variances approach zero apply in a purely deterministic nonlinear h setting risk neutral and continuous time results are summarized more general indices than risk sensitive indices are introduced with the view to giving useful inverse optimal control results in non gaussian noise environments

reflection moveout approximations for p waves in a moderately anisotropic homogeneous tilted transverse isotropy layer abstractwe have developed approximate nonhyperbolic p wave moveout formulas applicable to weakly or moderately anisotropic media of arbitrary anisotropy symmetry and orientation instead of the commonly used taylor expansion of the square of the reflection traveltime in terms of the square of the offset we expand the square of the reflection traveltime in terms of weak anisotropy wa parameters no acoustic approximation is used we specify the formulas designed for anisotropy of arbitrary symmetry for the transversely isotropic ti media with the axis of symmetry oriented arbitrarily in the 3d space resulting formulas depend on three p wave wa parameters specifying the ti symmetry and two angles specifying the orientation of the axis of symmetry tests of the accuracy of the more accurate of the approximate formulas indicate that maximum relative errors do not exceed 0 3 or 2 5 for weak or moderate p wave anisotropy respectively

positive solutions of schrodinger equations and fine regularity of boundary points given a lipschitz domain in mathbb r n and a nonnegative potential v in such that v x d x 2 is bounded we study the fine regularity of boundary points with respect to the schrodinger operator l v v in using potential theoretic methods several conditions are shown to be equivalent to the fine regularity of z in partial omega the main result is a simple explicit if is smooth necessary and sufficient condition involving the size of v for z in partial omega to be finely regular an intermediate result consists in a majorization of int a vert frac u d partial omega vert 2 dx for u positive harmonic in and a subset omega conditions for almost everywhere regularity in a subset a of are also given as well as an extension of the main results to a notion of fine mathcal l 1 vert mathcal l 0 regularity if mathcal l j mathcal l v j v 0 v 1 being two potentials with v 0 v 1 and mathcal l a second order elliptic operator

optimal control of partially observable stochastic systems with an exponential of integral performance index the stochastic control problem with linear stochastic differential equations driven by brownian motion processes and as cost functional the exponential of a quadratic form is considered the soluti

on families of optimal hardy weights for linear second order elliptic operators abstract we construct families of optimal hardy weights for a subcritical linear second order elliptic operator using a one dimensional reduction more precisely we first characterize all optimal hardy weights with respect to one dimensional subcritical sturm liouville operators on a b a b and then apply this result to obtain families of optimal hardy inequalities for general linear second order elliptic operators in higher dimensions as an application we prove a new rellich inequality

dirichlet problem at infinity on gromov hyperbolic metric measure spaces we study the asymptotic dirichlet problem for p harmonic functions in a very general setting of gromov hyperbolic metric measure spaces

existence of non trivial harmonic functions on cartan hadamard manifolds of unbounded curvature the liouville property of a complete riemannian manifold m i e the question whether there exist non trivial bounded harmonic functions on m attracted a lot of attention for cartan hadamard manifolds the role of lower curvature bounds is still an open problem we discuss examples of cartan hadamard manifolds of unbounded curvature where the limiting angle of brownian motion degenerates to a single point on the sphere at infinity but where nevertheless the space of bounded harmonic functions is as rich as in the non degenerate case to see the full boundary the point at infinity has to be blown up in a non trivial way such examples indicate that the situation concerning the famous conjecture of greene and wu about existence of non trivial bounded harmonic functions on cartan hadamard manifolds is much more complicated than one might have expected

finite dimensional risk sensitive filters and smoothers for discrete time nonlinear systems finite dimensional optimal risk sensitive filters and smoothers are obtained for discrete time nonlinear systems by adjusting the standard exponential of a quadratic risk sensitive cost index to one involving the plant nonlinearity it is seen that these filters and smoothers are the same as those for a fictitious linear plant with the exponential of squared estimation error as the corresponding risk sensitive cost index such finite dimensional filters do not exist for nonlinear systems in the case of minimum variance filtering and control

on the existence of weak solutions of semilinear elliptic equations and systems with hardy potentials abstract let r n n 3 be a bounded c 2 domain and x dist x put l 2 with 0 in this paper we provide various necessary and sufficient conditions for the existence of weak solutions to l u u p in u on where 0 p 0 and are measures on and respectively we then establish existence results for the system l u v p in l v u p in u v on where 1 p 0 p 0 and are measures on and are measures on we also deal with elliptic systems where the nonlinearities are more general

risk sensitive particle filters for mitigating sample impoverishment risk sensitive filters rsf put a penalty to higher order moments of the estimation error compared to conventional filters as the kalman filter minimizing the mean square error mse the result is a more cautious filter which can be interpreted as an implicit and automatic way to increase the state noise covariance on the other hand the process of jittering or roughening is well known in particle filters to mitigate sample impoverishment the purpose of this contribution is to introduce risk sensitive particle filters rspf as an alternative approach to mitigate sample impoverishment based on constructing explicit risk functions from a general class of factorizable functions it is first shown that rsf can be done in nonlinear systems using a recursion of an infinite dimensional information state which involves general risk functions then this information state calculation is carried out using particle approximations some alternative approaches generalizations specific cases comparison to existing methods of sample impoverishment mitigation and issues related to the selection of risk functions and parameters are examined performance of the resulting filter using various risk functions is illustrated on a simulated scenario and compared with the roughening method

on positive solutions of second order elliptic equations stability results and classification soit p un operateur uniformement elliptique d ordre 2 defini dans un domaine cr n soit p le cone convexe de toutes les solutions positives de l equation pu 0 dans on etudie les proprietes generales de p pour p et generaux

on the estimation of parameters of takagi sugeno fuzzy filters this study derives a class of filtering algorithms for takagi sugeno fuzzy models via solving a nonlinear parameters estimation problem the considered estimation problem is related to the problem of minimizing the expected value of the exponential of filtering errors energy under some stochastic assumptions the filtering criteria which involve an expectation operator are replaced by the deterministic quadratic optimization problems whose solutions provide a class of fuzzy filtering algorithms from a viewpoint of errors in the estimation of linear parameters of the fuzzy filter the derived filtering algorithms were analyzed with emphasis on stability robustness and steady state error issues the stability and robustness analyses have been made deterministically without making any assumption

random walks and induced dirichlet forms on self similar sets abstract let k be a self similar set satisfying the open set condition following kaimanovich s elegant idea 25 it has been proved that on the symbolic space x of k a natural augmented tree structure e exists it is hyperbolic and the hyperbolic boundary h x with the gromov metric is holder equivalent to k in this paper we consider certain reversible random walks with return ratio 0 1 on x e we show that the martin boundary m can be identified with h x and k with this setup and a device of silverstein 41 we obtain precise estimates of the martin kernel and the naim kernel in terms of the gromov product moreover the naim kernel turns out to be a jump kernel satisfying the estimate where is the hausdorff dimension of k and depends on for suitable the kernel defines a regular non local dirichlet form on k this extends the results of kigami 27 concerning random walks on certain trees with cantor type sets as boundaries see also 5

brief risk sensitive filtering prediction and smoothing for discrete time singular systems this paper is concerned with steady state risk sensitive filtering prediction and smoothing problems for discrete time singular systems it is shown that a risk sensitive estimator can be obtained by ensuring the minimum of an indefinite quadratic form to be maximum minimum when the risk sensitivity parameter q is negative positive an auxiliary state space signal model and an innovation sequence in krein space are introduced to simplify the derivation of the estimator the estimator is calculated based on one j spectral factorization for risk seeking q 0 a numerical example is given to demonstrate the applicability of the result

a new approach to the martin boundary via diffusions conditioned to hit a compact set let l generate a transient diffusion x t on rd and let d be an exterior domain let h be the smallest positive solution of lh 0 in d and h 1 on dd define xh t to be the process x t conditioned to hit dd by doob s h transform theory xh t is also a markov diffusion and its generator lh is defined by lhf l h l hf letting rd be the hitting time of dd define the harmonic measure for xh t on dd starting from x e d by i4x dy p r xh trd e dy let xn n 1 c d be a sequence satisfying limp xn i for which i4h converges weakly call two such sequences xn 1 and x 1 equivalent if lim k limn o x we call the set of equivalence classes thus generated the harmonic measure boundary at infinity for lh this boundary is independent of the particular exterior domain d we prove that the harmonic measure boundary at infinity for lh coincides with the martin boundary for l on rd the formal adjoint of the operator l on rd in the case that l generates a reversible diffusion the martin boundaries of l and l coincide and hence the harmonic measure boundary of lh coincides with the martin boundary for l on rd a similar probabilistic description of the martin boundary for l on rd can be given in the nonreversible case these results are then used to give explicit representations of the martin boundaries of l and l for several classes of diffusion processes

boundary behavior of harmonic maps on non smooth domains and complete negatively curved manifolds abstract in this paper we develop a theory for harmonic maps which is analogous to the classical theory for bounded harmonic functions in harmonic analysis we solve the dirichlet problem in bounded lipschitz domains and manifolds with negative sectional curvatures generalize the wienner criterion and prove the fatou theorem for harmonic maps into convex balls

on the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and dirichlet boundary conditions we give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains d subset bbb r n n ge 3 with dirichlet boundary conditions we impose mild conditions on the domains and lower order nonlinear coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by kato class functions our approach uses an implicit probabilistic representation schauder s fixed point theorem and new a priori estimates for solutions of the corresponding linear elliptic equations in the course of deriving these a priori estimates we show that the green functions for operators of the form 1 over 2 delta b cdot nabla on d are comparable when one modifies the drift term b on a compact subset of d this generalizes a previous result of ancona 2 obtained under an l p condition on b to a kato condition on b 2

the oshima sekiguchi and liouville theorems on heintze groups abstract let l be an elliptic operator on a riemannian manifold m a function f annihilated by l is said to be l harmonic f is said to have moderate growth if and only if f grows at most exponentially in the riemannian distance if m is a rank one symmetric space and l is the laplace beltrami operator for m the oshima sekiguchi theorem t oshima j sekiguchi eigenspaces of invariant differential operators on an affine symmetric space invent math 57 1980 1 81 states that a l harmonic function f has moderate growth if and only if f is the poisson integral of a distribution on the furstenberg boundary in this work we prove that this result generalizes to a very large class of homogeneous riemannian manifolds of negative curvature we also i prove a liouville type theorem that characterizes the polynomial like harmonic functions which vanish on the boundary in terms of their growth properties ii describe all polynomial like harmonic functions and iii give asymptotic expansions for the poisson kernel one consequence of this work is that every schwartz distribution on the boundary is the boundary value for a l harmonic function f which is uniquely determined modulo polynomial like harmonic functions

the ergodic theory of discrete isometry groups on manifolds of variable negative curvature this paper studies the ergodic theory at infinity of an arbitrary discrete isometry group acting on any hadamard manifold h of pinched variable negative curvature most of the results obtained by sullivan in the constant curvature case are generalized to the case of variable curvature we describe connections between measures supported on the limit set of dynamics of the geodesic flow and the geometry of m h we explore the relationship between the growth exponent of the group the hausdorff dimension of the limit set and the topological entropy of the geodesic flow the equivalence of various descriptions of an analogue of the hopf dichotomy is proved as applications we settle a question of j feldman and m ratner about the horocycle flow on a finite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points at the end of this paper we apply our results to the study of some rigidity problems more applications to mostow rigidity for discrete subgroups of rank 1 noncompact semisimple lie groups with infinite covolume will be published in subsequent papers by the author notations h is a hadamard manifold with pinched sectional curvature k 2 k k 1 k2 k1 0 is a non elementary discrete isometry group acting on h freely and properly discontinuously which is called fuchsian h is the ideal boundary of h m h is a complete riemannian manifold sh or sm is the unit tangent bundle g is the geodesic flow on sh or sm v t is the unique geodesic in sh resp sm with the initial velocity v 0 v sh resp sm v resp v is the asymptotic class of the geodesic v t resp v t l is the limit set of on h l is the radical limit set of in l is the nonwandering set of the geodesic flow on sm received by the editors may 7 1995 1991 mathematics subject classification primary 58f17 secondary 58f11 58f15 20h10 research at msri supported by nsf grant dms 8505550 also partially supported by nsf grant dms 9403870 and sfb 170 c 1996 american mathematical society

on asymptotic stability of continuous time risk sensitive filters with respect to initial conditions abstract in this paper we consider the problem of risk sensitive filtering for continuous time stochastic linear gaussian time invariant systems in particular we address the problem of forgetting of initial conditions our results show that suboptimal risk sensitive filters initialized with arbitrary gaussian initial conditions asymptotically approach the optimal risk sensitive filter for a linear gaussian system with gaussian but unknown initial conditions in the mean square sense at an exponential rate provided the arbitrary initial covariance matrix results in a stabilizing solution of the h like riccati equation associated with the risk sensitive problem more importantly in the case of non gaussian initial conditions a suboptimal risk sensitive filter asymptotically approaches the optimal risk sensitive filter in the mean square sense under a boundedness condition satisfied by the fourth order absolute moment of the initial non gaussian density and a slow growth condition satisfied by a certain radon nikodym derivative

entropy and drift in word hyperbolic groups the fundamental inequality of guivarc h relates the entropy and the drift of random walks on groups it is strict if and only if the random walk does not behave like the uniform measure on balls we prove that in any nonelementary hyperbolic group which is not virtually free endowed with a word distance the fundamental inequality is strict for symmetric measures with finite support uniformly for measures with a given support this answers a conjecture of s lalley for admissible measures this is proved using previous results of ancona and blachere haissinsky mathieu for non admissible measures this follows from a counting result interesting in its own right we show that in any infinite index subgroup the number of non distorted points is exponentially small the uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup

brownian motion and dirichlet problems at infinity we discuss angular convergence of riemannian brownian motion on a cartan hadamard manifold and show that the dirichlet problem at infinity for such a manifold is uniquely solvable under the curvature conditions ce 2 ar x km x a2 vspace 0 5pt 0 and cr x 2 km x 1 r x 2 2 2 respectively

robustness and risk sensitive filtering this paper gives a precise meaning to the robustness of risk sensitive filters for problems in which one is uncertain as to the exact value of the probability model it is shown that risk sensitive estimators including filters enjoy an error bound which is the sum of two terms the first of which coincides with an upper bound on the error that one would obtain if one knew exactly the underlying probability model while the second term is a measure of the distance between the true and design probability models the paper includes a discussion of several approaches to estimation including h sub spl infin filtering

integral representations of nonnegative solutions for parabolic equations and elliptic martin boundaries abstract we consider nonnegative solutions of a parabolic equation in a cylinder d 0 t where d is a noncompact domain of a riemannian manifold under the assumption iu i e the associated heat kernel is intrinsically ultracontractive we establish an integral representation theorem any nonnegative solution is represented uniquely by an integral on d 0 m d 0 t where m d is the martin boundary of d for the associated elliptic operator we apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions we also show that iu implies the condition sp i e the constant function 1 is a small perturbation of the elliptic operator on d

schrodinger equations with singular potentials linear and nonlinear boundary value problems let omega subset mathbb r n n ge 3 be a c 2 bounded domain and f subset partial omega be a c 2 submanifold with dimension 0 le k le n 2 denote delta f mathrm dist cdot f v delta f 2 and c h v the hardy constant relative to v in omega we study positive solutions of equations le l gamma v u 0 and ne l gamma v u f u 0 in omega where l gamma v delta gamma v gamma c h v and f in c mathbb r is an odd monotone increasing function we extend the notion of normalized boundary trace introduced in marcus and nguyen ann inst h poincare c non linear anal 34 69 88 2015 and employ it to investigate the linear equation le using these results we obtain properties of moderate solutions of ne finally we determine a criterion for subcriticality of points on partial omega relative to f and study b v p for ne in particular we establish existence and stability results when the data is concentrated on the set of subcritical points

robust event triggered state estimation a risk sensitive approach in this work we investigate a robust event triggered remote state estimation problem for linear gaussian systems with a stochastic event triggering condition the reference measure approach is used to obtain a robust event triggered estimate that minimizes the so called risk sensitive criterion which refers to the expectation of the exponential of the sum of the squared estimation error we introduce the reference measure under which the measurements are identically independently distributed i i d and independent of the states and propose a map to link the real world measure to the reference measure so that the recursions of the information states under the reference measure can be obtained based on these results the risk sensitive criteria are reformulated under the reference measure and closed form expressions of the risk sensitive event triggered posterior and prior estimates are presented which are shown to evolve in simple recursive kalman like structures moreover two sufficient stability conditions for the proposed estimators are given where the first requires the solution of a time varying riccati equation to be positive definite and satisfy a specific inequality which can be further extended to the scenario when the weighting matrix in the risk sensitive criterion is time variant the second gives the range of values of the risk sensitive parameter and covariance of the initial state for which the proposed estimators are stable comparative simulation results demonstrate that the proposed risk sensitive event triggered estimator is more robust to model uncertainties compared with a typical minimum mean squared error mmse estimator with stochastic event triggered sensor scheduling

semilinear elliptic equations with subcritical absorption in lipschitz domains abstract let be a bounded lipschitz domain in r n consider the equation u g x u 0 g c r and g x positive and increasing on r x we say that g is subcritical at y if has a solution u k y with boundary trace k y k 0 for a large family of functions g we establish existence and stability results for boundary value problems for with data given by measures concentrated on the set of subcritical points in addition we describe the precise asymptotic behavior of u k y at y some related results have been obtained in marcus and veron 2011 when g t t q in the case that g satisfies the keller osserman condition we prove if u is a positive solution with strong singularity at y i e u is not dominated by a harmonic function then u lim k u k y finally we extend estimates related to the keller osserman condition that are well known in the case of c 2 domains when g is independent of the space variable to lipschitz domains and a large class of functions g including cases where g x t 0 or g x t as x

the poisson equation on complete manifolds with positive spectrum and applications abstract in this paper we investigate the existence of a solution to the poisson equation on complete manifolds with positive spectrum and ricci curvature bounded from below we show that if a function f has decay f o r 1 e for some e 0 where r is the distance function to a fixed point then the poisson equation u f has a solution u with at most exponential growth we apply this result on the poisson equation to study the existence of harmonic maps between complete manifolds and also existence of hermitian einstein metrics on holomorphic vector bundles over complete manifolds thus extending some results of li tam and ni assuming moreover that the manifold is simply connected and of ricci curvature between two negative constants we can prove that in fact the poisson equation has a bounded solution and we apply this result to the ricci flow on complete surfaces

semilinear elliptic equations admitting similarity transformations in this paper we study the equation u 2 h u 0 in a smooth bounded domain where x dist x 0 and h is a nondecreasing function which satisfies keller osserman condition we introduce a condition on h which implies that the equation is subcritical i e the corresponding boundary value problem is well posed with respect to data given by finite measures under additional assumptions on h we show that this condition is necessary as well as sufficient we also discuss b v problems with data given by positive unbounded measures our results extend results of 13 treating equations of the form u uq 0 with q 1 2

risk sensitive probability for markov chains abstract the probability distribution of a markov chain is viewed as the information state of an additive optimization problem this optimization problem is then generalized to a product form whose information state gives rise to a generalized notion of probability distribution for markov chains the evolution and the asymptotic behavior of this generalized or risk sensitive probability distribution is studied in this paper and a conjecture is proposed regarding the asymptotic periodicity of risk sensitive probability and proved in the two dimensional case the relation between a set of simultaneous non linear and the set of periodic attractors is analyzed

the dirichlet problem at infinity for manifolds of negative curvature on considere l existence de fonctions harmoniques bornees sur des varietes simplement connexes n n de courbure negative