scalar product of bethe vectors from functional equations in this work the scalar product of bethe vectors for the six vertex model is studied by means of functional equations the scalar products are shown to obey a system of functional equations originated from the yang baxter algebra and its solution is given as a multiple contour integral

on the elliptic 2 solid on solid model functional relations and determinants in this work we study an elliptic solid on solid model with domain wall boundaries having the elliptic quantum group ep gl2 as its underlying symmetry algebra we elaborate on results previously presented in galleas phys rev e 94 010102 2016 and extend our analysis to include continuous families of single determinantal representations for the model s partition function interestingly our families of representations are parameterized by two continuous complex variables which can be arbitrarily chosen without affecting the partition function in this work we study an elliptic solid on solid model with domain wall boundaries having the elliptic quantum group ep gl2 as its underlying symmetry algebra we elaborate on results previously presented in galleas phys rev e 94 010102 2016 and extend our analysis to include continuous families of single determinantal representations for the model s partition function interestingly our families of representations are parameterized by two continuous complex variables which can be arbitrarily chosen without affecting the partition function

large deviations for stochastic pde with levy noise abstract we prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small levy noise we use general large deviations theorems of varadhan and bryc coupled with the techniques of feng and kurtz 2006 15 viscosity solutions of integro partial differential equations in hilbert spaces and deterministic optimal control methods the laplace limit is identified as a viscosity solution of a hamilton jacobi bellman equation of an associated control problem we also establish exponential moment estimates for solutions of stochastic evolution equations driven by levy noise general results are applied to stochastic hyperbolic equations perturbed by subordinated wiener process

on differential games for infinite dimensional systems with nonlinear unbounded operators abstract we consider a two player zero sum differential game governed by an abstract nonlinear differential equation of accretive type in an infinite dimensional space we prove that the value function of the game is the unique viscosity solution of the corresponding hamilton jacobi isaacs equation in the sense of 12 we also discuss some properties of this notion of solution

a comparison principle for hamilton jacobi equations related to controlled gradient flows in infinite dimensions we develop new comparison principles for viscosity solutions of hamilton jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures our examples are optimal control of ginzburg landau and fokker planck equations they arise in limit considerations of externally forced non equilibrium statistical mechanics models or through the large deviation principle for interacting particle systems our approach is based on two key ingredients an appropriate choice of geometric structure defining the gradient flow and a free energy inequality resulting from such gradient flow structure the approach allows us to handle hamiltonians with singular state dependency in the nonlinear term as well as hamiltonians with a state space which does not satisfy the radon nikodym property in the case where the state space is a hilbert space the method simplifies existing theories by avoiding the perturbed optimization principle

perron s method for hamilton jacobi equations on considere l existence des solutions d equations aux derivees partielles non lineaires scalaires d ordre 1 f x u du 0 dans ou est un sous ensemble ouvert de r n f r r n r est continue u r est l inconnue

what portion of the sample makes a partial sum asymptotically stable or normal let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0 2 be given we show that if at each stage n a number k n depending on n of the lower and upper order statistics are removed from the n th partial sum of the given random variables then under appropriate conditions on k n the remaining sum can be normalized to converge in distribution to a standard normal random variable a further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour our main tool is a new brownian bridge approximation to the uniform empirical process in weighted supremum norms

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

holder gradient estimates for fully nonlinear elliptic equations in this paper we prove interior and global holder estimates for lipschitz viscosity solutions of second order nonlinear uniformly elliptic equations the smoothness hypotheses on the operators are more general than previously considered for classical solutions so that our estimates are also new in this case and readily extend to embrace obstacle problems in particular isaac s equations of stochastic differential game theory constitute a special case of our results and moreover our techniques in combination with recent existence theorems of ishii lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic isaac s equation

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

a dynamic programming approach to nonlinear boundary control problems of parabolic type abstract in this paper we study a hamilton jacobi equation related to the boundary control of a parabolic equation with neumann boundary conditions the state space of this problem is a hilbert space and the equation is defined classically only on a dense subset of the state space moreover the hamiltonian appearing in the equation contains fractional powers of an unbounded operator these facts render the problem difficult in this paper we give a revised definition of a viscosity solution to accommodate the unboundness of the hamiltonian we then obtain existence and uniqueness results for viscosity solutions in particular we show that under suitable assumptions the value function of the boundary control problem is the unique viscosity solution of the related hamilton jacobi equation

microscopic derivations of several hamilton jacobi equations in infinite dimensions and large deviation of stochastic systems abstract we consider hamilton jacobi equations which characterize optimal controlled partial differential equations of the following types the allen cahn equation the cahn hilliard equation a nonlinear fokker planck equation and a vlasov fokker planck equation in each of the examples the optimal control problem and its associated cost functional can be derived as limit from a microscopically defined stochastic system using the probabilistic theory of large deviation the physical context here makes it natural to derive a free energy inequality which is very useful in proving the well posedness of the hamilton jacobi equation the article is written using informal arguments rigorous results will appear elsewhere

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

partial differential equations from integrable vertex models in this work we propose a mechanism for converting the spectral problem of vertex models transfer matrices into the solution of certain linear partial differential equations this mechanism is illustrated for the uq sl 2 invariant six vertex model and the resulting partial differential equation is studied for particular values of the lattice length

viscosity solutions of hamilton jacobi equations in infinite dimensions iv hamiltonians with unbounded linear terms in this work we show how to interpret s and e in the viscosity sense when the solutions may be expected to be weakly sequentially continuous in particular we introduce notions which allow us to handle the terms and for which we can obtain existence and uniqueness results in a variety of circumstances

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions iii uniqueness of viscosity solutions for general second order equations we prove here the uniqueness and existence of viscosity solutions for a general class of fully nonlinear second order equations in an infinite dimensional hilbert space the uniqueness is proved using the so called inf and sup convolutions which allow some reduction to finite dimensional problems

on a class of first order hamilton jacobi equations in metric spaces abstract we establish well posedness of a class of first order hamilton jacobi equation in geodesic metric spaces the result is then applied to solve a hamilton jacobi equation in the wasserstein space of probability measures which arises from the variational formulation of a compressible euler equation

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

representation of solutions of hamilton jacobi equations on etudie le probleme de cauchy pour l equation de hamilton jacobi u t h t x du 0 dans o t r n avec la condition initiale u o x u o x pour x r n on a t 0 hi o t r n r n r et u o r n r sont donnes on represente la solution viscosite comme la fonction valeur d un jeu differentiel approprie

continuous representations of scalar products of bethe vectors we present families of single determinantal representations of on shell scalar products of bethe vectors our families of representations are parameterized by a continuous complex variable which can be fixed at one s convenience here we consider bethe vectors in two versions of the six vertex model the case with boundary twists and the case with open boundaries

hamilton jacobi equations in infinite dimensions iii on etudie des equations de la forme f x u du 0 dans ou est un sous ensemble ouvert d un espace de banach v la fonction inconnue u r est continue du x est la derivee de frechet de u en x et f est une fonction non lineaire qui est une application continue de r v dans r on introduit la notion de solution viscosite stricte afin d eliminer l hypothese que v a la propriete de radon nikodym

the problem of quantum integrability abstract the notion of integrability in quantum mechanics is investigated in order to prepare rigorous grounds for the study of regular and irregular behaviour of quantum systems its common sense definition turns out to have deficiencies which are illustrated by various explicit examples part of the ambiguities are shown to have their origin in the difficulty of transferring the concept of independent constants of motion into quantum mechanics due to a fundamental theorem on sets of commuting operators by von neumann taking into account the classical limit with coherent states does not resolve the problems as a result it is pointed out why the appealing phenomenological distinction between regular and chaotic quantum systems cannot be traced back to the present notion of quantum integrability in a mathematically rigorous way

six vertex model and non linear differential equations i spectral problem in this work we relate the spectral problem of the toroidal six vertex model s transfer matrix with the theory of integrable non linear differential equations more precisely we establish an analogy between the classical inverse scattering method and previously proposed functional equations originating from the yang baxter algebra the latter equations are then regarded as an auxiliary linear problem allowing us to show that the six vertex model s spectrum solves riccati type non linear differential equations generating functions of conserved quantities are expressed in terms of determinants and we also discuss a relation between our riccati equations and a stationary schrodinger equation

a viscosity solution approach to the infinite dimensional hjb equation related to a boundary control problem in a transport equation the paper concerns the infinite dimensional hamilton jacobi bellman equation related to an optimal control problem regulated by a linear transport equation with boundary control a suitable viscosity solution approach is needed in view of the presence of the unbounded control related term in the state equation in the hilbert setting an existence and uniqueness result is obtained

a viscosity approach to infinite dimensional hamilton jacobi equations arising in optimal control with state constraints we consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous we prove that the value function is a viscosity solution of the bellman equation and is in fact the minimal nonnegative supersolution

hamilton jacobi equations in infinite dimensions ii existence of viscosity solutions on demontre des resultats d existence generaux pour 2 problemes 1 le probleme de cauchy u t h x t u du 0 dans v 0 t u x 0 x dans v et 2 le probleme stationnaire u h x u du 0 dans v ou h et sont donnees

on the rate of convergence of sums of extremes to a stable law letx1 x2 be a sequence of independent and identically distributed random variables in the domain of normal attraction of a nonnormal stabler law it is known that only the sum of thek n largest andk n smallest extreme values in thenth partial sum withk n andk n n 0 are responsible for the asymptotic stable distribution of the whole sum we investigate the rate at which such sums of extreme values converge to a stable law in conjunction with the rate at which the sums of the middle terms become asymptotically negligible in terms of rates of convergence our results provide in many cases a quantitative measure of exactly what portion of the sample is asymptotically stable

semigroups of relations and subdifferentials of the minimal time function and of the geodesic distance given a closed subset s of a banach space x we study the minimal time function to reach a point x of x starting from s we relate this problem to the study of the geodesic distance from x to s associated with a riemannian metric or a finsler metric it appears that both cases can be treated simultaneously and yield solutions to a hamilton jacobi equation we detect simple links between the general framework we propose and multivalued semigroups new concepts of infinitesimal generators of such semigroups are considered

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

viscosity solutions of hamilton jacobi equations in infinite dimensions v unbounded linear terms and b continuous solutions we continue the systematic study of first order hamilton jacobi equations in infinite dimensions begun in parts i iv of this series 121 our approach relies on as before an appropriate interpretation of the notion of viscosity solutions as introduced in 11 lo in the interim the strong development of viscosity solution theory in many directions has continued unabated in addition to the many references given in earlier papers in this series which we will not repeat here see 12 we would like to point out that a complete theory of second order fully nonlinear and possibly degenerate elliptic equations in finite dimensions has been made possible by the use of viscosity solutions s 9 18 20 22 19 32 331 for example and there is now a substantial body of work concerning second order equations in infinite dimensions 25 271 however in this work we come back to the particular situation studied in 12 part iv namely we come back to hamilton jacobi equations in

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

the necessary conditions for optimal control in hilbert spaces obtention du principe maximum de pontryagin pour une classe generale de problemes de commande de type lagragien afin qu associe avec un ensemble de conditions necessaires une commande soit satisfaite le principe est demontre en utilisant la programmation dynamique et l equation resultante d hamilton jacobi bellman de dimension finie

delay optimal control and viscosity solutions to associated hamilton jacobi bellman equations abstractin this article optimal control problems of differential equations with delays are investigated for which the associated hamilton jacobi bellman hjb equations are nonlinear partial diffe

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

on the rate of convergence of the sum of the sample extremes let x1 x2 x n be independent random variables having a common distribution in the domain of normal attraction of a completely asymmetric stable law with characteristic exponent 0 1 and support bounded below let x n n x n n 1 x n 1 denote the ordered sample we obtain the rate of convergence of n 1 x n n x n n k n 1 to the stable limit law as both n and k n as a consequence we obtain a representation of the sum x n n x n n k n 1

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

the baxter revolution i review the revolutionary impact rodney baxter has had on statistical mechanics beginning with his solution of the 8 vertex model in 1971 and the invention of corner transfer matrices in 1976 to the creation of the rsos models in 1984 and his continuing current work on the chiral potts model