a note on estimation algebras on nonlinear filtering theory abstract the idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by brockett and mitter independently it turns out that the concept of estimation algebra plays a crucial role in the investigation of finite dimensional nonlinear filters in his talk at the international congress of mathematicians in 1983 brockett proposed to classify all finite dimensional estimation algebras in this paper we consider some filtering systems in a special filtering system 1 we have some structure results 2 for any arbitrary finite dimensional state space under the condition that the drift term is a linear vector field plus a gradient vector field we classify all finite dimensional estimation algebras with maximal rank 3 we classify all finite dimensional estimation algebras with maximal rank if the dimension of the state space is less than or equal to three a more general filtering system is considered the above three results can be used locally therefore from the algebraic point of view we have now understood generically some finite dimensional filters

self adjoint elliptic operators with boundary conditions on not closed hypersurfaces abstract the theory of self adjoint extensions of symmetric operators is used to construct self adjoint realizations of a second order elliptic differential operator on r n with linear boundary conditions on a relatively open part of a compact hypersurface our approach allows to obtain kre n like resolvent formulae where the reference operator coincides with the free operator with domain h 2 r n this provides an useful tool for the scattering problem from a hypersurface concrete examples of this construction are developed in connection with the standard boundary conditions dirichlet neumann robin and type assigned either on a n 1 dimensional compact boundary or on a relatively open part schatten von neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven these give existence and completeness of the wave operators of the associated scattering systems

on a necessary and sufficient condition for finite dimensionality of estimation ever since the technique of the kalman bucy filter was popularized there has been an intense interest in finding new classes of finite dimensional recursive filters in the late seventies the concept of the estimation algebra of a filtering system was introduced it has proven to be an invaluable tool in the study of nonlinear filtering problems in this paper a simple algebraic necessary and sufficient condition is established for an estimation algebra of a special class of filtering systems to be finite dimensional also presented is a rigorous proof of the wei norman program which allows one to construct finite dimensional recursive filters from finite dimensional estimation algebras

mitter conjecture for low dimensional estimation algebras in non linear filtering it is well known that a systematical way to construct finite dimensional filter is to classify all finite dimensional estimation algebras mitter conjecture which states that all functions in any finite dimensional estimation algebra are necessarily degree one polynomial plays a crucial role in classifying all finite dimensional estimation algebras the purpose of this paper is to prove the mitter conjecture for estimation algebra of dimension at most 5 with arbitrary state space dimension dedicated to professor han fu chen on the occasion of his 70th birthday

finite dimensional filters with nonlinear drift ii brockett s problem on classification of finite dimensional estimation algebras the idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by brockett and mitter independently it turns out that the concept of estimation algebra plays a crucial role in the investigation of finite dimensional nonlinear filters in his talk at the international congress of mathematics in 1983 brockett proposed classifying all finite dimensional estimation algebras in this paper all finite dimensional algebras with maximal rank are classified if the dimension of the state space is less than or equal to two therefore from the lie algebraic point of view all finite dimensional filters are understood generically in the case where the dimension of state space is less than three

mitter conjecture and structure theorem for six dimensional estimation algebras the problem of classification of finite dimensional estimation algebras was formally proposed by brockett in his lecture at international congress of mathematicians in 1983 due to the difficulty of the problem in the early 1990s brockett suggested that one should understand the low dimensional estimation algebras first in this article we extend yau and his coauthors work of the mitter conjecture for low dimensional estimation algebras in nonlinear filtering problem and we apply the results to give classification of estimation algebras of dimension six

an extension of the benes filter and some identification problems solved by nonlinear filtering methods abstract an extension of bennes filtering theorem to time dependent parameter dependent and unknown initial conditions filtering problems is proved a new approach for the identification of linear diffusions with unknown drift coefficients follows new finite dimensional filters which may also be regarded as identification filters are derived and related to w s wong s earlier work which used algebraic methods

on the littlewood paley stein g function we consider semigroups tt which are contractive on lp m for all p e q q and q e 1 2 we give an example on symmetric spaces of the noncompact type which shows that the littlewood paley stein g function associated to the infinitesimal generator of tt may be unbounded on lq m and on l m we prove that variants of the g function are bounded on these lebesgue spaces in this paper we present some new results concerning the lp boundedness of the littlewood paley stein g function suppose that tt is a co semigroup of operators on lp m where m is a a finite measure space and let a e n the littlewood paley stein function ga associated to the given semigroup is defined by the rule ga f 1 ta ttfi2 dt for all f for which the right hand side makes sense in classical cases where tt is the poisson semigroup acting on lp r or lp t the g function is an important tool in fourier analysis for instance it is frequently used to prove multiplier theorems and pointwise convergence results the reader is referred to 1 1 for a survey of its role in a general context the g function was considered by e m stein he proved that if tt is a symmetric diffusion semigroup then ga satisfies the estimate 1 aplifiip liga f iip bp fllp vf e lp m whenever p e 1 oo a proof of this result can be found in 10 a simpler and more general approach via transference techniques is presented in 2 a third proof is in 4 and 5 it is of an entirely different nature in that it is based on the fact that generators of symmetric diffusion semigroups and even more general semigroups 3 have an hx functional calculus one of the interesting applications of g functions is to functional calculus for infinitesimal generators of co semigroups stein proved a marcinkiewicz type result for those as a corollary of the theory he developed for the g function the relationship between g functions and functional calculus for the infinitesimal generator of tt was investigated in 4 and 5 our treatment here owes received by the editors march 7 1994 originally communiated to the proceedings of the ams by j marshall ash 1991 mathematics subject classification primary 42b25 47a60 secondary 43a85

lie algebraic methods in optimal control of stochastic systems with exponential of integral cost abstract the purpose of this paper is to formulate and study the optimal control of partially observed stochastic systems with exponential of integral sample cost known as risk sensitive problems using lie algebraic tools this leads to the introduction of the sufficient statistic algebra l s through which one can determine a priori the maximum order of the controller when dim l s the construction of the control laws is addressed through extensions of the wei norman method as in nonlinear filtering problems aside from specific known finite dimensional examples which are studied in order to delineate the application of the lie algebraic tools new classes of finite dimensional controllers are identified as well in addition relations with minimax dynamic games are explored to best assess the importance and generality of the finite dimensional control systems

global c regularity for second order quasilinear elliptic euqations in divergence form on considere des equations elliptiques quasilineaires d ordre 2 sous forme divergence sous des conditions aux limites de dirihlet

explicit solution of dmz equation in nonlinear filtering via solution of odes in this note we develop a real time and accurate solution for nonlinear filtering problems based on the gaussian distribution specifically we present an explicit solution of the duncan mortensen zakai dmz equation of the yau filtering system which includes the linear filtering system and exact filtering system the solution is given in terms of a solution of a system of ordinary differential equations in particular our method can be implemented in hardware the complexity of our algorithms is the same as those of kalman bucy filters in the case of linear filtering systems

what is not in the domain of the laplacian on sierpinski gasket type fractals abstract we consider the analog of the laplacian on the sierpinski gasket and related fractals constructed by kigami a function f is said to belong to the domain of if f is continuous and f is defined as a continuous function we show that if f is a nonconstant function in the domain of then f2 is not in the domain of we give two proofs of this fact the first is based on the analog of the pointwise identity f2 2f f f 2 where we show that f 2 does not exist as a continuous function in fact the correct interpretation of f2 is as a singular measure a result due to kusuoka we give a new proof of this fact the second is based on a dichotomy for the local behavior of a function in the domain of at a junction point x0 of the fractal in the typical case nonvanishing of the normal derivative we have upper and lower bounds for f x f x0 in terms of d x x0 for a certain value and in the nontypical case vanishing normal derivative we have an upper bound with an exponent greater than 2 this method allows us to show that general nonlinear functions do not operate on the domain of

structure and classification theorems of finite dimensional exact estimation algebras estimation algebra turns out to be a crucial concept in the investigation of finite dimensional nonlinear filters in an earlier paper by the authors a necessary and sufficient algebraic condition was derived for an exact estimation algebra to be finite dimensional in this paper the investigation of the properties of finite dimensional exact estimation algebras is continued and some structure and partial classification theorems for such algebras are proved

hardy littlewood theory for semigroups soit t t t 0 un semi groupe symetrique sousmarkovien et soit n 2 alors on a les proprietes equivalentes a dim t t n b il existe c 2 0 s t t t f c 2 t n 2 f 1 f l 1 x t 0 en fait a b meme dans le cas n 2

finite dimensional filters with nonlinear drift viii classification of finite dimensional estimation algebras of maximal rank with state space dimension 4 the idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by brockett and mitter independently it turns out that the concept of estimation algebra plays a crucial role in the investigation of finite dimensional nonlinear filters in his talk at the international congress of mathematics in 1983 brockett proposed a classification of all finite dimensional estimation algebras chiou and yau classify all finite dimensional estimation algebras of maximal rank with dimension of the state space less than or equal to two in this paper we succeed in classifying all finite dimensional estimation algebras of maximal rank with state space dimension equal to three thus from the lie algebraic point of view we have now understood generically all finite dimensional filters with state space dimension less than four

on the existence of weak solutions for a nonlinear time dependent dirac equation in this paper we prove the existence of a weak solution of the cauchy problem for the nonlinear dirac equation in where x r is the characteristic function of a compact interval of 0

a gaussian estimate for the heat kernel on differential forms and application to the riesz transform let m g be a complete riemannian manifold which satisfies a sobolev inequality of dimension n and on which the volume growth is comparable to the one of mathbb r n for big balls if there is no non zero l 2 harmonic 1 form and the ricci tensor is in l frac n 2 varepsilon cap l infty for an varepsilon 0 then we prove a gaussian estimate on the heat kernel of the hodge laplacian acting on 1 forms this allows us to prove that under the same hypotheses the riesz transform d vardelta 1 2 is bounded on l p for all 1 3

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

classification of finite dimensional estimation algebras of maximal rank with arbitrary state space dimension and mitter conjecture in the late seventies the concept of the estimation algebra of a filtering system was introduced it was proven to be an invaluable tool in the study of non linear filtering problems in the early eighties brockett proposed to classify finite dimensional estimation algebras and mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one despite the massive effort in understanding the finite dimensional estimation algebras the 20 year old problem of brockett and mitter conjecture remains open in this paper we give a classification of finite dimensional estimation algebras of maximal rank and solve the mitter conjecture affirmatively for finite dimensional estimation algebras of maximal rank in particular for an estimation algebra e of maximal rank we give a necessary and sufficient conditions for e to be finite dimensional in terms of the drift fi x and observation hj x as an important corollary we show that the numb

finite dimensional filters with non linear drift ix construction of finite dimensional estimation algebras of non maximal rank the idea of using estimation algebra to construct finite dimensional nonlinear filters was first proposed by brockett and mitter independently it has proven to be an invaluable tool in the study of nonlinear filtering problem in 1983 brockett proposed to classify all finite dimensional estimation algebras in this paper we give the construction of finite dimensional estimation algebras of non maximal rank these non maximal rank finite dimensional estimation algebras play an important role in brockett s classification problem

regularity of a minimal surface at its free boundary in this paper we shall consider a stationary minimal surface x whose free or at least partially free boundary lies on a given supporting surface s it is well known that if the supporting surface s has no boundary then x possesses the same smoothness as s does if on the other hand the boundary of s is nonvoid one can at most expect c t 1 2 regularity of x at its free boundary c f 1 5 6 this c 1 1 2 boundary regularity was proved in the paper 6 by s hildebrandt and j c c nitsche and in the paper 5 by m griiter s hildebrandt and j c c nitsche under the condition that the supporting surface is of class c 3 for some ae 0 1 now combining the ideas of d kinderlehrer 7 and m giaquinta e giusti 4 and modifying the argument in 6 concerning the conformality relation we are able to show the c 1 1 2 regularity under a c2 assumption which in turn seems to be the weakest even if the minimal surface under consideration are minima of the area we also treat the regularity problem of general stationary surfaces under the c 1 1 condition and of minima of the dirichlet integral under a c 1 condition thus without conformality assumption

martingale solutions for the stochastic nonlinear schrodinger equation in the energy space we consider a stochastic nonlinear schrodinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic nlse in h 1 on compact manifolds and bounded domains we construct a martingale solution using a modified faedo galerkin method based on the littlewood paley decomposition for the 2d manifolds with bounded geometry we use the strichartz estimates to show the pathwise uniqueness of solutions

a bound approach to asymptotic optimality in non linear filtering of diffusion processes abstract the asymptotic behavior as a small parameter e 0 is investigated for one dimensional non linear filtering problems both weakly non linear systems wnl and systems measured through a low noise channel are considered upper and lower bounds on the optimal mean square error combined with perturbation methods are used to show that in the case of wnl the kalman filter formally designed for the underlying linear systems is asymptotically optimal in some sense in the case of systems with low measurement noise three asymptotically optimal filters are provided one of which is linear examples with simulation results are provided

endpoint strichartz estimates and global solutions for the nonlinear dirac equation abstract we prove endpoint strichartz estimates for the klein gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions as an application global wellposedness of the nonlinear dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable

the boson star equation with initial data of low regularity abstract the cauchy problem for the l 2 critical boson star equation with initial data of low regularity in spatial dimension d 3 is studied local well posedness in h s for s 1 4 is proved moreover for radial initial data local well posedness is established in h s for s 0 both results are shown to be almost optimal by providing complementary ill posedness results

solution of the zakai equation by separation of variables an exact expression for the unnormalized conditional density is derived for certain nonlinear estimation problems the basic method is to solve the zakai equation using separation of variables the new filter is a generalization of the kalman bucy filter the benes filter and other nonlinear filters within a unified framework

on a new class of finite dimensional estimation algebras abstract we introduce a new class of solvable finite dimensional estimation algebras that contains both the linear and the benes cases the condition for finite dimensionality is expressed in terms of two matrices they play an important role in understanding the structure of estimation algebras

new classes of finite dimensional nonlinear filters abstract we derive sufficient conditions for the estimation algebras of two special classes of nonlinear estimation systems to be finite dimensional by carrying out the brockett wei norman program new classes of finite dimensional filters are obtained these systems can be regarded respectively as extensions of the kalman bucy filters and the benes filters

small data scattering of semirelativistic hartree equation abstract in this paper we study the small data scattering of hartree type semirelativistic equation in space dimension 3 the hartree type nonlinearity is v u 2 u and the potential v which generalizes the yukawa has some growth condition we show that the solution scatters to linear solution if an initial data given in h s 1 is sufficiently small and s 1 4 here h s 1 is sobolev type space taking in angular regularity with norm defined by h s 1 h s s h s to establish the results we employ the recently developed strichartz estimate which is l 2 averaged on the unit sphere s 2 and construct the resolution space based on u p v p space

strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds abstract we prove the global in time strichartz estimates for wave equations on the nontrapping asymptotically conic manifolds we obtain estimates for the full set of wave admissible indices including the endpoint the key points are the properties of the microlocalized spectral measure of laplacian on this setting showed in 18 and a littlewood paley squarefunction estimate as applications we prove the global existence and scattering for a family of nonlinear wave equations on this setting

the novel classes of finite dimensional filters with non maximal rank estimation algebra on state dimension four and rank of one ever since the technique of kalman bucy filter was popularised finding new classes of finite dimensional recursive filters has drawn much concern the idea of using estimation algebra to construct

some remarks on estimation algebras abstract a necessary condition for the finite dimensionality of the estimation algebra of a single input single output nonlinear system is derived and then extended to cover the multi input case a result of hijab concerning the finite computability of the output function is also generalised

ground state solutions for fractional choquard equations involving upper critical exponent abstract in this article we study the following fractional choquard equation involving upper critical exponent s u v x u f x u x u 2 s u 2 s 2 u x r n where 0 0 s 1 s denotes the fractional laplacian of order s n 2 s 0 2 s and 2 s 2 n n 2 s when v and f are asymptotically periodic in x we prove that the equation has a ground state solution for large by nehari method

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

structure theorem for five dimensional estimation algebras abstract the problem of classification of finite dimensional estimation algebras was formally proposed by brockett in his lecture at international congress of mathematicians in 1983 due to the difficulty of the problem in the early 1990s brockett suggested that one should understand the low dimensional estimation algebras first in this paper we give classification of estimation algebras of dimension at most five although the classification of finite dimensional estimation algebra of maximal rank was completed by yau and his coworkers chen chiou hu wong and wu the problem of classification of non maximal rank finite dimensional estimation algebra is still wide open except for the case of state space dimension 2 hopefully the result of this paper will shed some light on the non maximal rank estimation algebras

global well posedness of the energy critical nonlinear schrodinger equation with small initial data in h1 t3 a refined trilinear strichartz estimate for solutions to the schrodinger equation on the flat rational torus t3 is derived by a suitable modification of critical function space theory this is applied to prove a small data global well posedness result for the quintic nonlinear schrodinger equation in hs t3 for all s 1 this is the first energy critical global well posedness result in the setting of compact manifolds

heat semigroup and singular pdes we provide in this work a semigroup approach to the study of singular pdes in the line of the paracontrolled approach developed recently by gubinelli imkeller and perkowski starting from a heat semigroup we develop a functional calculus and introduce a paraproduct based on the semigroup for which commutator estimates and schauder estimates are proved together with their paracontrolled extensions this machinery allows us to investigate singular pdes in potentially unbounded riemannian manifolds under mild geometric conditions as an illustration we study the generalized parabolic anderson model equation and prove under mild geometric conditions its well posed character in small time on a potentially unbounded 2 dimensional riemannian manifold for an equation driven by a coloured noise and for all times for the linear parabolic anderson model equation in 2 dimensional unbounded manifolds this machinery can be extended to an even more singular setting and deal with sobolev spaces rather than holder spaces

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

riesz transforms and lie groups of polynomial growth abstract let g be a lie group of polynomial growth we prove that the second order riesz transforms on l 2 g dg are bounded if and only if the group is a direct product of a compact group and a nilpotent group in which case the transforms of all orders are bounded

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

sobolev algebras on lie groups and riemannian manifolds we prove that on any connected unimodular lie group g the space l p g l g where l p g is the sobolev space of order 0 associated with a sublaplacian is an algebra under pointwise product this generalizes results due to strichartz in the euclidean case to bohnke in the case of stratified groups and others a global version of this fact holds for groups with polynomial growth we give similar results for riemannian manifolds with ricci curvature bounded from below respectively nonnegative

finite dimensional filters with nonlinear drift xi explicit solution of the generalized kolmogorov equation in brockett mitter program abstract ever since the technique of the kalman bucy filter was popularized there has been an intense interest in finding new classes of finite dimensional recursive filters in the late 1970s the concept of the estimation algebra of a filtering system was introduced brockett clark and mitter proposed to use the wei norman approach to solve the nonlinear filtering problem in 1990 tam wong and yau presented a rigorous proof of the brocket mitter program which allows one to construct finite dimensional recursive filters from finite dimensional estimation algebras later yau wrote down explicitly a system of ordinary differential equations and generalized kolmogorov equation to which the robust duncan mortenser zakai equation can be reduced thus there remains three fundamental problems in brockett mitter program the first is the problem of finding explicit solution to the generalized kolmogorov equation the second is the problem of finding real time solution of a system of odes the third is the brockett s problem of classification of finite dimensional estimation algebras in this paper we solve the first problem

complete classification of finite dimensional estimation algebras of maximal rank the idea of using estimation algebras to construct finite dimensional non linear filters was first proposed by brockett and clark and mitter independently in his famous talk at the international congress of mathematics in 1983 brockett proposed to classify all finite dimensional estimation algebras in this paper we explain why the theory of estimation algebras plays an important role in non linear filtering we show how to use the wei norman approach to construct finite dimensional filters from finite dimensional estimation algebras we survey some results in estimation algebras after 1984 we give a self contained proof of complete classification of finite dimensional estimation algebras of maximal rank in one place the proof given here is simpler than those proofs scattered in several papers this provides the readers with a complete coherent view of the important topic of the classification of finite dimensional estimation algebras

besov and triebel lizorkin spaces on lie groups in this paper we develop a theory of besov and triebel lizorkin spaces on general noncompact connected lie groups endowed with a sub riemannian structure such spaces are defined by means of hypoelliptic sub laplacians with drift and endowed with a measure whose density with respect to a right haar measure is a continuous positive character of the group we prove several equivalent characterizations of their norms we establish comparison results also involving sobolev spaces of recent introduction and investigate their complex interpolation and algebra properties

theorems on the structure of finite dimensional estimation algebras abstract in this paper we proved several theorems concerning the structure of finite dimensional estimation algebras in particular under proper technical assumptions we proved the following 1 the observation of a filtering system must be linear if the estimation algebra is finite dimensional 2 all elements of a finite dimensional estimation algebra belong to a special class of polynomial differential operators 3 all finite dimensional estimation algebras are solvable

solution of filtering problem with nonlinear observations for all known finite dimensional filters one always needs the conditon that the observation terms are degree one polynomial on the other hand in many practical examples e g tracking problem the observation terms may be nonlinear our new method in this paper can treat filtering problems with nonlinear observation terms in the first time which includes kalman bucy filter as a special case

nonlinear system immersion observers and finite dimensional filters abstract nonlinear deterministic and stochastic systems which can be immersed into linear observable ones are identified by necessary and sufficient conditions those systems are shown to admit observers in the deterministic case and exact finite dimensional filters in the stochastic case the sufficient conditions obtained generalize earlier results in both cases

sobolev spaces on lie groups embedding theorems and algebra properties abstract let g be a noncompact connected lie group denote with a right haar measure and choose a family of linearly independent left invariant vector fields x on g satisfying hormander s condition let be a positive character of g and consider the measure whose density with respect to is in this paper we introduce sobolev spaces l p adapted to x and 1 p 0 and study embedding theorems and algebra properties of these spaces as an application we prove local well posedness and regularity results of solutions of some nonlinear heat and schrodinger equations on the group

finite dimensional estimation algebras with state dimension 3 and rank 2 mitter conjecture in this paper we study the structure of finite dimensional estimation algebras with state dimension 3 and rank 2 arising from a nonlinear filtering system by using the theories of the euler operat

formes de dirichlet generales et densite des variables aleatoires reelles sur l espace de wiener abstract it is shown that the main properties of dirichlet spaces can be proved without the classical local compacity hypothesis to this purely measurable framework we extend the functional calculus and the energy image density property these results are then applied to the ornstein uhlenbeck semi group on wiener space where it is obtained that the real random variables which are solutions of lipschitzian stochastic differential equations always have probability densities after perhaps an initial deterministic phase

optimal convergence rate and regularity of nonrelativistic limit for the nonlinear pseudo relativistic equations abstract in this paper we are concerned with the nonrelativistic limit of the following pseudo relativistic equation with hartree nonlinearity or power type nonlinearity 2 c 2 m 2 c 4 m c 2 u u n u where c denotes the speed of light we prove that as c the ground states of this equation converges to the ground state of its nonrelativistic counterpart 2 2 m u u n u with convergence rate 1 c 2 in every h s norms moreover we show that this rate is optimal

small data scattering for semi relativistic equations with hartree type nonlinearity abstract we prove that the initial value problem for the equation i t u m 2 u e 0 x x u 2 u in r 1 3 m 0 0 0 is globally well posed and the solution scatters to free waves asymptotically as t if we start with initial data which is small in h s r 3 for s 1 2 and if m 0 moreover if the initial data is radially symmetric we can improve the above result to m 0 and s 0 which is almost optimal in the sense that l 2 r 3 is the critical space for the equation the main ingredients in the proof are certain endpoint strichartz estimates l 2 r 1 3 bilinear estimates for free waves and an application of the u p and v p function spaces

gromov s compactness theorem for pseudo holomorphic curves we give a complete proof for gromov s compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary

real time solution of the nonlinear filtering problem without memory ii it is well known that the nonlinear filtering problem has important applications in both military and commercial industries the central problem of nonlinear filtering is to solve the duncan mortensen zakai dmz equation in real time and in a memoryless manner the purpose of this paper is to show that under very mild conditions which essentially say that the growth of the observation h is greater than the growth of the drift f the dmz equation admits a unique nonnegative weak solution u which can be approximated by a solution u r of the dmz equation on the ball b r with u r pat b r 0 the error of this approximation is bounded by a function of r which tends to zero as r goes to infinity the solution u r can in turn be approximated efficiently by an algorithm depending only on solving the observation independent kolmogorov equation on b r in theory our algorithm can solve basically all engineering problems in real time specifically we show that the solution obtained from ou

classification of four dimensional estimation algebras of central importance in nonlinear filtering theory is the classification of finite dimensional estimation algebras through which we can construct recursive filters we classify all t dimensional estimation algebras t spl les 4 for arbitrary state space dimension

brockett s problem of classification of finite dimensional estimation algebras for nonlinear filtering systems in his talk at the international congress of mathematicians in 1982 brockett proposed to classify all finite dimensional estimation algebras arising in nonlinear filters recently chen yau and leung siam j control optim 35 1997 pp 1132 1141 claimed that they had classified all finite dimensional estimation algebras of maximal rank when the dimension of the state space is less than or equal to four in this paper we introduce a series of completely new computations about the estimation algebra and we find two sets of new equations about the omega matrix as a consequence we can prove without any dimension assumption on the state space that the omega matrix is a constant matrix and thus we succeed in classifying all finite dimensional estimation algebras of maximal rank with the state space dimension being arbitrary

partially constrained boundary conditions with energy minimizing mappings on considere la regularite a la frontiere pour une application minimisant l energie u m n pour deux types de conditions aux limites frontiere libre et condition limite partiellement contrainte

information states in stochastic control and filtering a lie algebraic theoretic approach the purpose of the paper is two fold i to introduce the sufficient statistic algebra which is responsible for propagating the sufficient statistics or information state in the optimal control of stochastic systems and ii to apply certain lie algebraic methods and gauge transformations widely used in nonlinear control theory and quantum physics to derive new results concerning finite dimensional controllers this enhances our understanding of the role played by the sufficient statistics the sufficient statistic algebra enables us to determine a priori whether there exist finite dimensional controllers it also enables us to classify all finite dimensional controllers relations to minimax dynamic games are delineated

exact smoothing properties of schrodinger semigroups we study schrodinger semigroups in the scale of sobolev spaces and show that for kato class potentials the range of such semigroups in lp has exactly two more derivatives than the potential this proves a conjecture of b simon we show that eigenfunctions of schrodinger operators are generically smoother by exactly two derivatives in given sobolev spaces than their potentials we give applications to the relation between the potential s smoothness and particle kinetic energy in the context of quantum mechanics and characterize kinetic energies in coulomb systems the techniques of proof involve leibniz and chain rules for fractional derivatives which are of independent interest as well as a new characterization of the kato class 1 introduction in this paper we attempt a precise study of the action of schrodinger semigroups in the scale of sobolev spaces work in this area was begun by b simon si4 in 1985 he proved a number of positive and negative results on such smoothing and also presented an open question in the form of a conjecture regarding necessary and sufficient conditions on the potential v for smoothing of order s by the semigroup of functions in l2 rd study of schrodinger semigroups is of interest for a number of reasons see

conditionally minimax algorithm for nonlinear system state estimation in this note a method of conditionally minimax nonlinear filtering cmnf of processes in nonlinear stochastic discrete time controlled systems is proposed the cmnf is derived by means of local nonparametric optimization of the filtering process given the class of admissible filters sufficient conditions for the existence of the cmnf are considered and the properties of cmnf estimates are investigated results of the cmnf application to control and identification problems are presented

a counterexample of comments on stability margin evaluation for uncertain linear systems in the original paper c gong and s thompson ibid vol 39 p 548 50 1994 there is given what is claimed to be the tightest bound for all unstructured pertubations such that a specified system keeps its asymptotic stability j h su ibid vol 39 p 2523 4 1994 showed that it is not the tightest bound and gives another bound that bound is shown by a counterexample to be incorrect

energy concentration of the focusing energy critical fnls abstract we consider the fractional nonlinear schrodinger equation fnls with non local dispersion and focusing energy critical hartree type nonlinearity x 2 u 2 u we consider the energy concentration phenomena of radial blowup solutions near the maximal existence time we use the concentration compactness approach of 19 for confined case and kinetic energy trapping approach of 21 for unconfined case

algebres de sobolev sur certains groupes nilpotents soit n un groupe de lie nilpotent stratifie on demontre que les espaces de sobolev non isotropes s p sont des algebres de banach pour le produit ordinaire des fonctions lorsque p 1 p ou designe la dimension homogene de n on donne du point de vue de l analyse harmonique dans ces algebres un resultat de synthese spectrale