convergent finite difference solvers for viscosity solutions of the elliptic monge ampere equation in dimensions two and higher the elliptic monge ampere equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail novel solution methods are required for stability and convergence to the weak viscosity solution in this article we build a wide stencil finite difference discretization for the monge ampere equation the scheme is monotone so the barles souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation solutions of the scheme are found using a damped newton s method we prove convergence of newton s method and provide a systematic method to determine a starting point for the newton iteration computational results are presented in two and three dimensions which demonstrates the speed and accuracy of the method on a number of exact solutions which range in regularity from smooth to nondifferentiable

an analytic formula for u 3 boson matrix elements the u 3 boson lie algebra is the liquid limit of the symplectic algebra sp 3 r an analytic formula is given for the u 3 boson matrix elements in irreducible unitary representations corresponding to the sp 3 r discrete series the formula also applies to the generators of the mathematically isomorphic but physically different interacting boson algebra

regularity of laws and ergodicity of hypoelliptic sdes driven by rough paths we consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior in particular we focus on the case when the driving noise is a rough path valued fractional brownian motion with hurst parameter h 2 1 1 our contribution in this work is twofold first when the driving vector fields satisfy hormander s c elebrated lie bracket condition we derive explicit quantitative bounds on the i nverse of the malliavin ma trix en route to this we provide a novel deterministic version of norris s lemma for differential equations driven by rough paths this result with the added assumption that the linearized equation has moments will then yield that the transition laws have a smooth density with respect to lebesgue measure our second main result states that under hormander s condi tion the solutions to rough differential equations driven by fractional brownian motion with h 2 1 1 enjoy a suitable version of the strong feller property under a standard controllability condition this implies that they admit a unique stationary solution that is physical in the sense that it does not look into the future

an approximation algorithm for nonholonomic systems in siam j control optim 37 1997 to appear limiting process of control affine systems with holder continuous inputs submitted we have studied the limiting behavior of trajectories of control affine systems sigma dot x sum k 1 m u k f k x generated by a sequence u j subseteq l 1 0 t rr m where the f k are smooth vector fields on a smooth manifold m we have shown that under very general conditions the trajectories of sigma generated by the u j converge to trajectories of an extended system of sigma of the form sigma ext dot x sum k 1 r v kf k x where f k k 1 ldots m are the same as in sigma and f m 1 ldots f r are lie brackets of f 1 ldots f m in this paper we will apply these convergence results to solve the inverse problem i e given any trajectory gamma of an extended system sigma ext find trajectories of sigma that converge to gamma uniformly this is done by means of a universal construction that only involves the knowledge of the v k k 1 ldots r and the structure of the lie brackets in sigma ext but does not depend on the manifold m and the vector fields f 1 ldots f m these results can be applied to approximately track an arbitrary smooth path in m for controllable systems sigma which in particular gives an alternative approach to the motion planning problem for nonholonomic systems

rough path limits of the wong zakai type with a modified drift term abstract the wong zakai theorem asserts that odes driven by reasonable e g piecewise linear approximations of brownian motion converge to the corresponding stratonovich stochastic differential equation with the aid of rough path analysis we study non reasonable approximations and go beyond a well known criterion of ikeda watanabe north holland 1989 in the sense that our result applies to perturbations on all levels exhibiting additional drift terms involving any iterated lie brackets of the driving vector fields in particular this applies to the approximations by mcshane 72 and sussmann 91 our approach is not restricted to brownian driving signals at last these ideas can be used to prove optimality of certain rough path estimates

pointwise rates of convergence for the oliker prussner method for the monge ampere equation we study the oliker prussner method exploiting its geometric nature we derive discrete stability and continuous dependence estimates in the max norm by using a discrete alexandroff estimate and the brunn minkowski inequality we show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain nodes still conform to the domain boundary this gives a suitable notion of operator consistency which combined with stability leads to pointwise rates of convergence for classical and non classical solutions of the monge ampere equation

convergent difference schemes for degenerate elliptic and parabolic equations hamilton jacobi equations and free boundary problems convergent numerical schemes for degenerate elliptic partial differential equations are constructed and implemented simple conditions are identified which ensure that nonlinear finite difference schemes are monotone and nonexpansive in the maximum norm explicit schemes endowed with an explicit cfl condition are built for time dependent equations and are used to solve stationary equations iteratively explicit and implicit formulations of monotonicity for first and second order equations are unified bounds on orders of accuracy are established an example of a scheme which is stable but nonmonotone and nonconvergent is presented schemes for hamilton jacobi equations obstacle problems one phase free boundary problems and stochastic games are built and computational results are presented

constrained rough paths we introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural on the way we develop a notion of rough integration and an ecient and intrinsic theory of rough dierential equations rdes on manifolds the theory of rdes is then used to construct parallel translation along manifold valued rough paths finally this framework is used to show there is a one to one correspondence between rough paths on a d dimensional manifold and rough paths on d dimensional euclidean space this last result is a rough path analogue of cartan s development map and its stochastic version which was developed by eells and elworthy and malliavin

on uniformly subelliptic operators and stochastic area let x a be a markov process with generator i j i aij j where a is a uniformly elliptic symmetric matrix thanks to the fundamental works of t lyons stochastic differential equations driven by x a can be solved in the rough path sense that is pathwise by using a suitable stochastic area process our construction of the area which generalizes previous works of lyons stoica and then lejay is based on dirichlet forms associated to subellitpic operators this enables us in particular to discuss large deviations and support descriptions in suitable rough path topologies as typical rough path corollary freidlin wentzell theory and the stroock varadhan support theorem remain valid for stochastic differential equations driven by x a

smoothness of densities for area like processes of fractional brownian motion we consider a process given by a n dimensional fractional brownian motion with hurst parameter frac 1 4 h frac 1 2 along with an associated levy area like process and prove the smoothness of the density for this process with respect to lebesgue measure

representations of the groups sp n r and sp n in a u n basis the explicit expressions for the infinitesimal operators and the finite matrix elements with respect to a u n basis are obtained for the representations of the most degenerate series of sp n r and for the irreducible unitary representations of sp n with the highest weights m 0 0

a solution to the su n external state labelling problem based upon a u n 1 n 1 group i general theory the state labelling problem arising in the reduction of the direct product of a p positive row u n irreducible representation h1 hp0 with a q negative row one 0 h q h 1 into a sum of mixed u n irreducible representations k1 kp0 k q k 1 is solved by using the complementarity between u n and u p q within some positive discrete series irreducible representations of u pn qn this complementarity allows analysis of the problem in terms of the group chain u p q contains implies u p u q instead of u n u n contains implies u n for the most general su n irreducible representations corresponding to p q n 1 the relevant group chain is therefore u n 1 n 1 contains implies u n 1 u n 1 in such a case the additional labels include those of an intermediate u n 1 irreducible representation h1s hn 1s as well as the additional labels solving the state labelling problems for the products k1 kn 1 h1s hn 1s and k 1 k n 1 h1s hn 1s of u n 1 irreducible representations hence the proposed solution reflects in a direct way the operation of king s branching rule for the chain u n u n contains implies u n supplemented whenever necessary with king s modification rule

structure and matrix elements of the degenerate series representations of u p q and u p q in a u p u q basis the representations of the most degenerate series of the group u p q which are induced by the representations of the maximal parabolic subgroup are considered in this article by making use of the infinitesimal operators of these representations in the u p u q basis the conditions are derived which are necessary and sufficient for irreducibility for the reducible representations we describe their structure composition series we select from among the irreducible representations which are obtained in this article all representations of u p q which admit unitarization as a result we obtain the principal degenerate series the supplimentary degenerate series the discrete degenerate series and the exceptional degenerate series of unitary representations of u p q the u p u q spectrum of the representations of u p q with highest weights 1 0 0 2 is defined we obtain the integral representation for the matrix elements of the degenerate representations of u p q in the u p u q basis t

analytic expressions for the matrix elements of generators of sp 6 in an sp 6 u 3 basis work done by many authors indicates that important tools for calculations in microscopic collective models are the matrix elements of the generators of the symplectic group sp 6 in an sp 6 u 3 basis rosensteel has derived recursion relations for these matrix elements while filippov has determined them using generating function technique but it would also be convenient to have explicit and analytic formulas for them this is what we do in this paper for the case of closed shells i e when the irreducible representation irrep of sp 6 is characterized by equal values for the three weight generators in the lowest weight state we also indicate how our results can be extended to the case of arbitrary irreps of sp 6 i e when we have open shells

euler estimates for rough differential equations abstract we consider controlled ordinary differential equations and give new estimates for higher order euler schemes our proofs are inspired by recent work of a m davie who considers first and second order schemes in order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group such euler estimates have powerful applications by a simple limit argument they apply to rough path differential equations rdes in the sense of t lyons and hence also to stochastic differential equations driven by brownian motion or other random rough paths with sufficient integrability in the context of the latter we obtain strong remainder estimates in stochastic taylor expansions a la azencott ben arous castell and platen although our findings appear novel even in the case of driving brownian motion our main insight is the genuine rough path nature of quantitative remainder estimates in stochastic taylor expansions there are several other applications of which we discuss in detail l q convergence in lyons universal limit theorem and moment control of rde solutions

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

wigner eckart theorem and infinitesimal operators of group representations the new notion of a tensor operator which transforms under a representation of the compact lie group is introduced this tensor operator is a linear space v of operators in order to obtain the traditional tensor operator it is sufficient to choose a basis in the space v the wigner eckart theorem is valid for the tensor operators the notion of irreducibility and the equivalence relation for tensor operators are formulated necessary and sufficient conditions of irreducibility and of equivalence are proved using the wigner eckart theorem the author gives the method of evaluation of infinitesimal operators of the representations of compact and noncompact lie groups the new invariants of irreducible representations of compact groups are found their quantity is equal to that of independent casimir operators

optimal mass transport for registration and warping image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times in this paper we present a method for computing elastic registration and warping maps based on the monge kantorovich theory of optimal mass transport this mass transport method has a number of important characteristics first it is parameter free moreover it utilizes all of the grayscale data in both images places the two images on equal footing and is symmetrical the optimal mapping from image a to image b being the inverse of the optimal mapping from b to a the method does not require that landmarks be specified and the minimizer of the distance functional involved is uniques there are no other local minimizers finally optimal transport naturally takes into account changes in density that result from changes in area or volume although the optimal transport method is certainly not appropriate for all registration and warping problems this mass preservation property makes the monge kantorovich approach quite useful for an interesting class of warping problems as we show in this paper our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the l2 kantorovich wasserstein or earth mover s distance under a mass preservation constraint we show how this approach leads to practical algorithms and demonstrate our method with a number of examples including those from the medical field we also extend this method to take into account changes in intensity and show that it is well suited for applications such as image morphing

representations of the groups gl n r and su n in an so n basis the explicit form for the infinitesimal operators and the finite matrix elements with respect to an so n basis is obtained for the representations of the most degenerate series of the group sl n r and for the irreducible unitary representations of the group su n with highest weight m 0 0

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

a limit theorem for financial markets with inert investors we study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time it turns out that semi markov processes are tailor made for modelling inert investors with a suitable scaling we show that when the price is driven by the market imbalance the log price process is approximated by a process with long range dependence and non gaussian returns distributions driven by a fractional brownian motion consequently investor inertia may lead to arbitrage opportunities for sophisticated market participants the mathematical contributions are a functional central limit theorem for stationary semi markov processes and approximation results for stochastic integrals of continuous semimartingales with respect to fractional brownian motion

infinitesimal operators of group representations in noncanonical bases explicit expressions are obtained for the action of the infinitesimal operators of some classes of representations of the groups u p q so p q su n sp n sl n r and gl n c in the bases that differ from the gel fand zetlin ones

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

smoothness of the density for solutions to gaussian rough differential equations we consider stochastic differential equations driven by a multi dimensional gaussian process under the assumption that the vector fields satisfy hormander s bracket condition we demonstrate that the solution admits a smooth density for any strictly positive time t provided the driving noise satisfies certain non degeneracy assumptions our analysis relies on an interplay of rough path theory malliavin calculus and the theory of gaussian processes our result applies to a broad range of examples including fractional brownian motion with hurst parameter greater than 1 4 the ornstein uhlenbeck process and the brownian bridge returning after time t

on the numerical solution of the equation frac partial 2 z partial x 2 frac partial 2 z partial y 2 left frac partial 2 z partial x partial y right 2 f and its discretizations i the equation indicated in the title is the simplest representative of the class of nonlinear equations of monge ampere type equations with such nonlinearities arise in dynamic meteorology geometric optics elasticity and differential geometry in some special cases heuristic procedures for numerical solution are available but in order for them to be successful a good initial guess is required for a bounded convex domain nonnegativef and dirichlet data we consider a special discretization of the equation based on its geometric interpretation for the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence no special information about an initial guess is required and to initiate the iterates a routine step is made the method is self correcting and is structurally suitable for a parallel computer the computer program modules and several examples are presented in two appendices

complementary group with respect to so n we look for a complementary group with respect to so n within either irreducible representation 1 2 dn or 1 2 dn 1 3 2 of the group sp 2dn r of linear canonical transformations in a 2dn dimensional phase space we prove that i such a group is sp 2d r when n 2q 1 or n 2q 2d ii it is su d d when n 2 and d 1 iii it does not exist when n 2 d 1 or 2 n 2q 2d

canonical solution of the state labelling problem for su n so n and littlewood s branching rule i general formulation the internal state labelling problem for the d row irreducible representations of su n where 2d or n when reduced with respect to so n is shown to amount to the external state labelling problem for u d the canonical solution of the latter due to biedenharn et al 1967 provides a canonical solution of the former which reflects the operation of littlewood s branching rule for u n contains implies o n in a very simple way

gaussian type lower bounds for the density of solutions of sdes driven by fractional brownian motions in this paper we obtain gaussian type lower bounds for the density of solutions to stochastic differential equations sdes driven by a fractional brownian motion with hurst parameter h in the one dimensional case with additive noise our study encompasses all parameters h 0 1 while the multidimensional case is restricted to the case h 1 2 we rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools

complementarity relation between the u p q and u n lie groups and some applications to atomic physics the complementarity relation between the unitary groups u d and u n within the symmetrical irreducible representations of the larger unitary group u dn is extended to non compact groups it is proved that the pseudo unitary group u p q is complementary with respect to u n within some positive discrete series irreducible representations of the larger pseudo unitary group u pn qn the latter arise when reducing the metaplectic irreducible representations 1 2 dn and 1 2 dn 13 2 of the real symplectic group sp 2dn r where d p q and they are characterised by a single label the eigenvalue of the first order casimir operator some applications of the u p q u n complementarity to atomic physics are outlined for such purposes the isomorphism between the lie algebras of su 2 2 and so 4 2 is used extensively the mathematical framework underlying the kibler negadi approach 1983 4 of the hydrogen atom dynamical group is extended to the independent electron dynamical group of intrashell many electron states as well as to the correlated electron dynamical group of intrashell doubly excited states

representations of the quantum algebras uq ur s and uq ur s related to the quantum hyperboloid and sphere representations of the principal degenerate series of the quantum algebra uq ur s are introduced structure of these representations is studied several classes of unitary irreducible representations of the algebra uq ur s are separated in the set of irreducible representations and irreducible constituents of reducible representations the formulas of action of operators of irreducible representations of the quantum algebra uq ur s in the basis corresponding to restriction of representations onto the subalgebra uq ur us are given

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

flow of diffeomorphisms induced by a geometric multiplicative functional in this paper it is shown that the unique multiplicative functional solution to a differential equation driven by a geometric multiplicative functional consitutes a flow of local diffeomorphisms in the case where the driving geometric multiplicative functional is generated by a brownian motion the result in particular presents an answer to an open problem proposed in ikeda and watanabe 4

the dirichlet problem for the convex envelope the convex envelope of a given function was recently characterized as the solution of a fully nonlinear partial differential equation pde in this article we study a modified problem the dirichlet problem for the underlying pde the main result is an optimal regularity result differentiability c1 regularity of the boundary data implies the corresponding result for the solution in the interior despite the fact that the solution need not be continuous up to the boundary secondary results are the characterization of the convex envelope as i the value function of a stochastic control problem and ii the optimal underestimator for a class of nonlinear elliptic pdes

canonical solution of the state labelling problem for su n so n and littlewood s branching rule iii su 3 so 3 case for pt ii see ibid vol 17 no 4 p 777 89 1984 parts i and ii discussed in general terms a new solution to the state labelling problem for the d row irreducible representations irreps of su n when reduced with respect to so n this solution was termed canonical because it reflects the operation of littlewood s branching rule for u n o n in a straightforward way the su 3 so 3 case is now worked out in detail explicit expressions of the canonical basis states of so 3 irreps l belonging to an su 3 irrep h1h2 are obtained the matrix of the transformation from the bargmann moshinsky basis to the canonical one is also calculated it is shown that in both cases the extra label necessary to completely specify the states can be chosen as the label js characterising an intermediate su 2 irrep in the reduction of the produce representation jl js into su 2 irreps j where j 1 2 h1 h2 and jl 1 2l or 1 2 l 1 whenever h1 h2 l is even or odd

averaging theorems for highly oscillatory differential equations and iterated lie brackets using averaging techniques and developing proper algebraic formalisms we study the limiting process of ordinary differential equations with highly oscillatory right hand sides we give sufficient conditions generalizing earlier work by kurzweil and jarnik for a sequence u j u 1 j ldots u m j subseteq l 1 0 t bbb r m to be such that for every choice of smooth vector fields f k k 1 ldots m on a smooth manifold the trajectories of dot x sum k 1 mu k j t f k x converge to the trajectories of an extended system dot x sum k 1 r v k t f k x where the new directions f m 1 ldots f r are lie brackets of f 1 ldots f m

determination of the sp 2d r generator matrix elements through a boson mapping for the discrete series irreducible representations lambda n 2 d of sp 2d r the determination of the sp 2d r generator matrix elements in an sp 2d r contains implies u d basis is reduced to the much simpler calculation of boson operator matrix elements between u nu contains implies u d boson states where nu d d 1 2 the key of this reduction is the previously derived holstein primakoff boson representation of the sp 2d r generators as an illustration the case of sp 6 r is worked out in detail

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

densities for rough differential equations under hormander s condition we consider stochastic differential equations dy v y dx driven by a multidimensional gaussian process x in the rough path sense t lyons rev mat iberoamericana 14 1998 215 310 using malliavin calculus we show that y t admits a density for t 0 t provided i the vector fields v v 1 v d satisfy hormander s condition and ii the gaussian driving signal x satisfies certain conditions examples of driving signals include fractional brownian motion with hurst parameter h 1 4 the brownian bridge returning to zero after time t and the ornstein uhlenbeck process

a solution to the su n external state labelling problem based upon a u n 1 n 1 group ii recursion relations for the su 2 and su 3 wigner coefficients for pt i see ibid vol 20 p 769 84 1987 the new solution to the su n external state labelling problem proposed in the first paper of this series is analysed in detail in the su 3 case in terms of the two complementary chains u 3 u 3 contains implies u 3 and u 2 2 contains implies u 2 u 2 the classification of su 3 coupled states is completed by the labels of an intermediate u 2 irreducible representation hs hs1hs2 directly related to king s branching rule for u 3 u 3 u 3 for pedagogical purposes the su 2 coupled state construction based upon the complementary chains u 2 u 2 contains implies u 2 and u 1 1 contains implies u 1 u 1 is considered first for both su 2 and su 3 it is proved that in addition to the standard recursion relations the corresponding wigner coefficients satisfy recursion relations of a different type arising from the action of the u 1 1 or u 2 2 complementary group generators a detailed example shows that such complementary recursion relations are quite useful for numerical purposes

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

infinitesimal operators and structure of the representations of the groups so 2n and so 2n in a u n basis and of the groups su 2n and su 2n in an sp n basis the infinitesimal operators of the most degenerate representations of the groups so 2n and su 2n are found in a discrete basis the structure composition series of these representations is studied the classification of unitary irreducible representations of these groups which belong to most degenerate series is given the infinitesimal operators of irredicuble unitary representations of so 2n in a u n basis and of su 2n in an sp n basis are found for the cases of highest weights m m 0 0 m 0

stochastic analysis rough path analysis and fractional brownian motions in this paper we show by using dyadic approximations the existence of a geometric rough path associated with a fractional brownian motion with hurst parameter greater than 1 4 using the integral representation of fractional brownian motions we furthermore obtain a skohorod integral representation of the geometric rough path we constructed by the results in ly1 a stochastic integration theory may be established for fractional brownian motions and strong solutions and a wong zakai type limit theorem for stochastic differential equations driven by fractional brownian motions can be deduced accordingly the method can actually be applied to a larger class of gaussian processes with covariance functions satisfying a simple decay condition

the monge ampere trajectory correction for semi lagrangian schemes requiring that numerical estimates of the flow trajectories comply with the fundamental euler expansion formula that governs the evolution of a volume of fluid leads to a second order nonlinear monge ampere partial differential equation mae in cossette and smolarkiewicz 2011 15 a numerical algorithm based on solving the mae with an inexact newton krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi lagrangian scheme in the context of an incompressible fluid here we extend the theoretical analysis of the elemental rotational and deformational motions presented in cossette and smolarkiewicz 2011 15 in particular closed form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness scalar advection shows that the ma correction improves mass conservation substantially by suppressing anomalous fluid contraction the impact of the ma correction on complex flows is studied in the framework of an ideal magneto fluid in which the formation of current sheets leads to topological changes and reconnection of field lines the use of the ma correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy 2d and 3d examples are presented and the computational performance of the solver is documented

ergodicity of stochastic differential equations driven by fractional brownian motion we study the ergodic properties of finite dimensional syste ms of sdes driven by non degenerate additive fractional brownian motion with arbitrary hurst parameter h 2 0 1 a general framework is constructed to make precise the notions of invariant measure and stationary state for such a system we then prove under rather weak dissipativity conditions that such an sde possesses a unique stationary solution and that the convergence rate of an arbitrary solut ion towards the stationary one is at least algebraic a lower bound on the exponent is also given

evolving communities with individual preferences the goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals we model the population by a measure space f where determines the abundance of individual preferences the preferences of an individual are described by a measurable choice x of a rough path we aim to identify for each individual a choice for the forward evolution yt for an individual in the community these choices yt m ust be consistent so thatyt correctly accounts for the individual s preference and correctly models their interaction with the aggregate behaviour of the community in general solutions are continuum of interacting threads analogous to the huge number of individual atomic trajectories that together make up the motion of a fluid the evolution of the population need not be governed by any over arching partial differential equation pde although one can match the standard non linear parabolic pdes of mckean vlasov type with specific examples of communities in this case the bulk behaviour of the evolving population provides a solution to the pde we focus on the case of weakly interacting systems where we are able to exhibit the existence and uniqueness of consistent solutions an important technical result is continuity of the behaviour of the system with respect to changes in the measure assigning weight to individuals replacing the deterministic with the empirical distribution of an independent and identically distributed sample from leads to many standard models and applying the continuity result allows easy proofs for propagation of chaos the rigorous underpinning presented here leads to uncomplicated models which have wide applicability in both the physical and social sciences we make no presumption that the macroscopic dynamics are modelled by a pde this work builds on the fine probability literature considering the limit behaviour for systems where a large number of particles are interacting with independent preferences there is also work on continuum models with preferences described by a semi martingale measure we mention some of the key papers

reduction of the unitary group to its orthogonal or symplectic subgroup a unified approach based upon complementary groups a unified analysis of the state labelling problems for the d row irreducible representations of u n when reduced with respect to either its orthogonal subgroup o n or its sympletic subgroup usp n the latter in the even n case is carried out by using appropriate metrics and the complementarity relationship between the group o n and sp 2d r or usp n and so 2d in this way a recently proposed canonical solution to the u n contains implies o n state labelling problem is extended to the u n contains implies usp n chain this shows the equivalence between both these internal state labelling problems and the external state labelling problem for u d as expressed in littlewood s branching rules for u n contains implies o n and u n contains implies usp n

the monge ampere equation various forms and numerical solution we present three novel forms of the monge ampere equation which is used e g in image processing and in reconstruction of mass transportation in the primordial universe the central role in this paper is played by our fourier integral form for which we establish positivity and sharp bound properties of the kernels this is the basis for the development of a new method for solving numerically the space periodic monge ampere problem in an odd dimensional space convergence is illustrated for a test problem of cosmological type in which a gaussian distribution of matter is assumed in each localised object and the right hand side of the monge ampere equation is a sum of such distributions

on p q rough paths we extend the work of t lyons 1998 and t lyons and z qian 2002 to define integrals and solutions of differential equations along product of p and q rough paths with 1 p 1 q 1 we use this to write an ito formula at the level of rough paths and to see that any rough path can always be interpreted as a product of a p geometric rough path and a p 2 geometric rough path

partial differential equations driven by rough paths abstract we study a class of linear first and second order partial differential equations driven by weak geometric p rough paths and prove the existence of a unique solution for these equations this solution depends continuously on the driving rough path this allows a robust approach to stochastic partial differential equations in particular we may replace brownian motion by more general gaussian and markovian noise support theorems and large deviation statements all become easy corollaries of the corresponding statements of the driving process in the case of first order equations with gaussian noise we discuss the existence of a density with respect to the lebesgue measure for the solution

boson representations of the real symplectic group and their applications to the nuclear collective model both non hermitian dyson and hermitian holstein primakoff representations of the sp 2d r algebra are obtained when the latter is restricted to a positive discrete series irreducible representation d n 2 1 n 2 for such purposes some results for boson representations recently deduced from a study of the sp 2d r partially coherent states are combined with some standard techniques of boson expansion theories the introduction of usui operators enables the establishment of useful relations between the various boson representations two dyson representations of the sp 2d r algebra are obtained in compact form in terms of d d 1 2 pairs of boson creation and annihilation operators and of an extra u d spin characterized by the irreducible representation 1 d in contrast to what happens when 1 d it is shown that the holstein primakoff representation of the sp 2d r algebra cannot be written in such a compact form for a generic irreducible representation explicit expansions are

complementarity relation between the usp 2ν and so 2d lie groups the object of this paper consists of finding a complementary group with respect to usp n within either irreducible representation 1 2 dn or 1 2 dn 13 2 of the real symplectic group sp 2dn r for such purpose the author deals with the full orthogonal group o n n 2 nu or 2 nu 1 and the unitary symplectic group usp n n 2 nu in a unified way by introducing an appropriate metric a similar treatment applied to their complementary group determination leads to the well known sp 2d r group in the o n case and to the so 2d group in the usp n case

a least squares method for the inverse reflector problem in arbitrary orthogonal coordinates abstract in this article we solve the inverse reflector problem for a light source emitting a parallel light bundle and a target in the far field of the reflector by use of a least squares method we derive the monge ampere equation expressing conservation of energy while assuming an arbitrary coordinate system we generalize a cartesian coordinate least squares method presented earlier by prins et al 13 to arbitrary orthogonal coordinate systems this generalized least squares method provides us the freedom to choose a coordinate system suitable for the shape of the light source this results in significantly increased numerical accuracy decrease of errors by factors up to 104 is reported we present the generalized least squares method and compare its numerical results with the cartesian version for a disk shaped light source

a central limit theorem for the kpz equation we consider the kpz equation in one space dimension driven by a stationary centred space time random field which is sufficiently integrable and mixing but not necessarily gaussian we show that in the weakly asymmetric regime the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the hopf cole solution to the kpz equation driven by space time gaussian white noise while the limiting process depends only on the integrated variance of the driving field the diverging constants appearing in the definition of the reference frame also depend on higher order moments

a version of hormander s theorem for the fractional brownian motion it is shown that the law of an sde driven by fractional brownian motion with hurst parameter greater than 1 2 has a smooth density with respect to lebesgue measure provided that the driving vector fields satisfy hormander s condition the main new ingredient of the proof is an extension of norris lemma to this situation

a numerical method for the elliptic monge ampere equation with transport boundary conditions the problem of optimal mass transport arises in numerous applications including image registration mesh generation reflector design and astrophysics one approach to solving this problem is via the monge ampere equation while recent years have seen much work in the development of numerical methods for solving this equation very little has been done on the implementation of the transport boundary condition in this paper we propose a method for solving the transport problem by iteratively solving a monge ampere equation with neumann boundary conditions to enable mappings between variable densities we extend an earlier discretization of the equation to allow for right hand sides that depend on gradients of the solution b d froese and a m oberman siam j numer anal 49 2011 pp 1692 1714 this discretization provably converges to the viscosity solution the resulting system is solved efficiently with newton s method we provide several challenging computational examples that demonstra

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

good rough path sequences and applications to anticipating stochastic calculus we consider anticipative stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed under a simple condition on the stochastic process we show that the unique solution of the above sde understood in the rough path sense is actually a stratonovich solution we then show that this condition is satisfied by the brownian motion as application we obtain rather flexible results such as support theorems large deviation principles and wong zakai approximations for sdes driven by brownian motion along anticipating vectorfields in particular this unifies many results on anticipative sdes

data driven optimal transport the problem of optimal transport between two distributions x and y is extended to situations where the distributions are only known through a finite number of samples xi and yj a weak formulation is proposed based on the dual of the kantorovich formulation with two main modifications replacing the expected values in the objective function by their empirical means over the xi and yj and restricting the dual variables u x and v y to a suitable set of test functions adapted to the local availability of sample points a procedure is proposed and tested for the numerical solution of this problem based on a fluidlike flow in phase space where the sample points play the role of active lagrangian markers 2016 wiley periodicals inc

malliavin differentiability of solutions of rough differential equations abstract in this paper we study rough differential equations driven by gaussian rough paths from the viewpoint of malliavin calculus under mild assumptions on coefficient vector fields and underlying gaussian processes we prove that solutions at a fixed time are smooth in the sense of malliavin calculus examples of gaussian processes include fractional brownian motion with hurst parameter larger than 1 4

malliavin calculus for regularity structures the case of gpam malliavin calculus is implemented in the context of hairer 2014 16 this involves some constructions of independent interest notably an extension of the structure which accommodates a robust and purely deterministic translation operator in l2l2 directions between models in the concrete context of the generalized parabolic anderson model in 2d one of the singular spdes discussed in the afore mentioned article we establish existence of a density at positive times

numerical solution of the optimal transportation problem using the monge ampere equation a numerical method for the solution of the elliptic monge ampere partial differential equation with boundary conditions corresponding to the optimal transportation ot problem is presented a local representation of the ot boundary conditions is combined with a finite difference scheme for the monge ampere equation email protected s method is implemented leading to a fast solver comparable to solving the laplace equation on the same grid several times theoretical justification for the method is given by a convergence proof in the companion paper 4 solutions are computed with densities supported on non convex and disconnected domains computational examples demonstrate robust performance on singular solutions and fast computational times

fast finite difference solvers for singular solutions of the elliptic monge ampere equation the elliptic monge ampere equation is a fully nonlinear partial differential equation which originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail in this article we build a finite difference solver for the monge ampere equation which converges even for singular solutions regularity results are used to select a priori between a stable provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain this allows singular solutions to be computed using a stable method and regular solutions to be computed more accurately the resulting nonlinear equations are then solved by newton s method computational results in two and three dimensions validate the claims of accuracy and solution speed a computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities

explanation of variability and removal of confounding factors from data through optimal transport a methodology based on the theory of optimal transport is developed to attribute variability in data sets to known and unknown factors and to remove such attributable components of the variability from the data denoting by x the quantities of interest and by z the explanatory factors the procedure transforms x into filtered variables y through a z dependent map so that the conditional probability distributions x z are pushed forward into a target distribution y independent of z among all maps and target distributions that achieve this goal the procedure selects the one that minimally distorts the original data the barycenter of the x z connections are found to unsupervised learning and to fundamental problems in statistics such as conditional density estimation and sampling particularly simple instances of the methodology are shown to be equivalent to k means and principal component analysis an application is shown to a time series of ground temperature hourly data across the united states 2017 wiley periodicals inc

partially coherent states of the real symplectic group in the present paper we introduce partially coherent states for the positive discrete series irreducible representations d n 2 1 n 2 of sp 2d r encountered in physical applications these states are characterized by both continuous and discrete labels the latter specify the row of the irreducible representation 1 2 d of the maximal compact subgroup u d while the former parametrize an element of the factor space sp 2d r h where h is the sp 2d r subgroup leaving the 1 2 d representation space invariant we consider three classes of partially coherent states respectively generalizing the perelomov and barut girardello coherent states as well as some recently introduced intermediate coherent states we prove that each family of partially coherent states forms an overcomplete set in the representation space of d n 2 1 n 2 and study its generating function properties we show that it leads to a representation of the sp 2d r generators in the form of differential operato

non degeneracy of wiener functionals arising from rough differential equations malliavin calculus is about sobolev type regularity of functionals on wiener space the main example being the ito map obtained by solving stochastic differential equations rough path analysis is about strong regularity of the solution to possibly stochastic differential equations we combine arguments of both theories and discuss the existence of a density for solutions to stochastic differential equations driven by a general class of non degenerate gaussian processes including processes with sample path regularity worse than brownian motion

the jain monrad criterion for rough paths and applications to random fourier series and non markovian hormander theory we discuss stochastic calculus for large classes of gaussian processes based on rough path analysis our key condition is a covariance measure structure combined with a classical criterion due to jain and monrad ann probab 11 1983 46 57 this condition is verified in many examples even in absence of explicit expressions for the covariance or volterra kernels of special interest are random fourier series with covariance given as fourier series itself and we formulate conditions directly in terms of the fourier coefficients we also establish convergence and rates of convergence in rough path metrics of approximations to such random fourier series an application to spde is given our criterion also leads to an embedding result for cameron martin paths and complementary young regularity cyr of the cameron martin space and gaussian sample paths cyr is known to imply malliavin regularity and also ito like probabilistic estimates for stochastic integrals resp stochastic differential equations despite their rough pathwise construction at last we give an application in the context of non markovian hormander theory

sensitivity of rough differential equations an approach through the omega lemma the ito map assigns the solution of a rough differential equation a generalization of an ordinary differential equation driven by an irregular path when existence and uniqueness hold by studying how a path is transformed through the vector field which is integrated we prove that the ito map is holder or lipschitz continuous with respect to all its parameters this result unifies and weaken the hypotheses of the regularity results already established in the literature

a note on the notion of geometric rough paths we use simple sub riemannian techniques to prove that every weak geometric p rough path a geometric p rough path in the sense of 20 is the limit in sup norm of a sequence of canonically lifted smooth paths uniformly bounded in p variation thus clarifying the two different definitions of a geometric p rough path our proofs are sufficiently general to include the case of holder and modulus type regularity this allows us to extend a few classical results on holder spaces and p variation spaces to the non commutative setting necessary for the theory of rough paths as an application we give a precise description of the support of enhanced fractional brownian motion and prove a conjecture by ledoux et al

bosonic realisation of sp 4 r and the spectrum of two particle quantum systems a new bosonic holstein primakoff type realisation of sp 4 r is applied in the development of 1 n expansions of non relativistic spin independent two particle hamiltonians as a non trivial example the spectrum of helium like ions is considered and the energies of the low lying levels with given values of total angular momentum are computed the calculation is explicitly performed in an approximation up to the second degree in the small expansion parameter