tunnelling in multidimensional systems tunnelling is examined from the point of view of analytic continuation of lagrangian manifolds this enables an intrinsically multidimensional approach to many problems we illustrate the ideas by computing energy splittings between congruent tori in a number of integrable systems the examples include an integrable but non separable system which cannot be solved by reduction to one dimensional systems the calculations can all be viewed in a geometrical way that does not make explicit use of separation of coordinates

the distance to a polyhedron abstract we give some exact formulas and some estimates for the distance to a polyhedron in a normed linear space e we show that in the case when e r n endowed with the l norm these estimates are in general better than a recent estimate by cook gerards schrijver and tardos 2

vicious walks with a wall noncolliding meanders and chiral and bogoliubov de gennes random matrices spatially and temporally inhomogeneous evolution of one dimensional vicious walkers with wall restriction is studied we show that its continuum version is equivalent with a noncolliding system of stochastic processes called brownian meanders here the brownian meander is a temporally inhomogeneous process introduced by yor as a transform of the bessel process that is the motion of radial coordinate of the three dimensional brownian motion represented in spherical coordinates it is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of gaussian ensembles of bogoliubov de gennes hamiltonians of the mean field theory of superconductivity which have a particle hole symmetry we report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type c to type ci in the nonstandard classes of random matrix theory of altland and zirnbauer the relation between the noncolliding systems of the generalized meanders of yor which are associated with the even dimensional bessel processes and the chiral random matrix theory is also clarified

polyhedral sweeping processes with unbounded nonconvex valued perturbation abstract a polyhedral sweeping process with a multivalued perturbation whose values are nonconvex unbounded sets is studied in a separable hilbert space polyhedral sweeping processes do not satisfy the traditional assumptions used to prove existence theorems for convex sweeping processes we consider the polyhedral sweeping process as an evolution inclusion with subdifferential operators depending on time the widely used assumption of lipschitz continuity for the multivalued perturbation term is replaced by a weaker notion of h lipschitzness the existence of solutions is proved for this sweeping process

linear differential equations and multiple zeta values iii zeta 3 we consider the hypergeometric equation 1 t t t g x3g 0 whose unique analytic solution 1 t x 1 o t near t 0 for t 1 becomes a generating function for multiple zeta values 1 1 x f3 x 1 3 x3 3 3 x6 we apply the so called wkb method to study solutions of the hypergeometric equation for large x and we calculate corresponding stokes matrices we prove that the function f3 x near x is also expressed via wkb type functions which subject to some stokes phenomenon this implies that f3 x satisfies a sixth order linear differential equation with irregular singularity at infinity

sweeping process with unbounded nonconvex perturbation abstract a convex sweeping process with unbounded nonconvex perturbation term of a rather general form is investigated in a separable hilbert space we consider our sweeping process as an evolution inclusion with subdifferential operators this allows to prove the theorem on existence of solutions under conditions generalizing the usual assumptions for convex sweeping processes in particular we show that multivalued mappings whose values are hyperplanes half spaces etc may be chosen as multivalued mappings generating the sweeping process such mappings do not satisfy the conditions under which convex sweeping processes are usually studied

step fluctuations for a faceted crystal a statistical mechanics model for a faceted crystal is the 3d ising model at zero temperature it is assumed that in one octant all sites are occupied by atoms the remaining ones being empty allowed atom configurations are such that they can be obtained from the filled octant through successive removals of atoms with breaking of precisely three bonds if v denotes the number of atoms removed then the grand canonical boltzmann weight is q v 0 q 1 as shown by cerf and kenyon in the limit q 1 a deterministic shape is attained which has the three facets 100 010 001 and a rounded piece interpolating between them we analyse the step statistics as q 1 in the rounded piece it is given by a determinantal process based on the discrete sine kernel exactly at the facet edge the steps have more space to meander their statistics is again determinantal but this time based on the airy kernel in particular the border step is well approximated by the airy process which has been obtained previously in the context of growth models our results are based on the asymptotic analysis for space time inhomogeneous transfer matrices

interacting brownian motions in infinite dimensions with logarithmic interaction potentials we investigate the construction of diffusions consisting of infinitely numerous brownian particles moving in mathbb r d and interacting via logarithmic functions two dimensional coulomb potentials these potentials are very strong and act over a long range in nature the associated equilibrium states are no longer gibbs measures we present general results for the construction of such diffusions and as applications thereof construct two typical interacting brownian motions with logarithmic interaction potentials namely the dyson model in infinite dimensions and ginibre interacting brownian motions the former is a particle system in mathbb r while the latter is in mathbb r 2 both models are translation and rotation invariant in space and as such are prototypes of dimensions d 1 2 respectively the equilibrium states of the former diffusion model are determinantal or pfaffian random point fields with sine kernels they appear in the thermodynamical limits of the spectrum of the ensembles of gaussian random matrices such as goe gue and gse the equilibrium states of the latter diffusion model are the thermodynamical limits of the spectrum of the ensemble of complex non hermitian gaussian random matrices known as the ginibre ensemble

exponential asymptotics and boundary value problems keeping both sides happy at all orders we introduce templates for exponential asymptotic expansions that in contrast to matched asymptotic approaches enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter graphic 1 and have a single boundary layer at one end of the interval for linear equations the template is a transseries that takes the form of a sliding ladder of exponential scales for nonlinear equations the transseries template is a two dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed an exponential asymptotic approach also reveals how boundary value problems force the surprising presence of transseries in the linear case and negative powers of terms in the series beyond all orders in the nonlinear case we also demonstrate how these transseries can be resummed to generate multiple scales type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter finally we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion 1 embed inline graphic 1 gif

third order equation for harmonic generation complex canonical transformation and jwkb solution the quantum description of third harmonic generation can be formulated as an eigenvalue problem for a third order linear differential equation we perform a semiclassical study of this third order equation generalizing the familiar jwkb theory for the second order schrodinger equation and deriving explicit albeit approximate formulas for the eigenvalues within this semiclassical context a central role in this analysis is played by a nonlinear complex canonical transformation which permits a complete description of the classical motion generated by a complex polynomial hamiltonian function in the complexified position and momentum planes

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

error bounds for the method of steepest descents new error bounds are found for the asymptotic expansions which arise in the method of steepest descents these bounds are found by exploiting the reformulation of the method of steepest descents given in recent work by berry howls a major difference between this and previous work is that the global properties of the integrand and not just those on the path of steepest descent play an essential role the bounds are significantly easier to compute than those in the current literature applications of the error bounds are given for asymptotic expansions arising from the pearcey integral the airy function the modified bessel function and the integral of the airy function

resurgent transseries and the holomorphic anomaly nonperturbative closed strings in local mathbb c mathbb p 2 the holomorphic anomaly equations describe b model closed topological strings in calabi yau geometries having been used to construct perturbative expansions it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries these yield formal nonperturbative solutions showing integrability of the holomorphic anomaly equations at the nonperturbative level this paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local mathbb c mathbb p 2 toric calabi yau background and by addressing the associated resurgent large order analysis of both perturbative and multi instanton sectors in particular analyzing the asymptotic growth of the perturbative free energies one finds contributions from three different instanton actions related by mathbb z 3 symmetry alongside another action related to the kahler parameter resurgent transseries methods then compute from the extended holomorphic anomaly equations higher instanton sectors and it is shown that these precisely control the asymptotic behavior of the perturbative free energies as dictated by resurgence the asymptotic large order growth of the one instanton sector unveils the presence of resonance i e each instanton action is necessarily joined by its symmetric contribution the structure of different resurgence relations is extensively checked at the numerical level both in the holomorphic limit and in the general nonholomorphic case always showing excellent agreement with transseries data computed out of the nonperturbative holomorphic anomaly equations the resurgence relations further imply that the string free energy displays an intricate multi branched borel structure and that resonance must be properly taken into account in order to describe the full transseries solution

overlapping stokes smoothings survival of the error function and canonical catastrophe integrals we derive doubly uniform approximations for the remainder in the optimally truncated saddle point expansion for an integral containing a large parameter double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate as described by catastrophe theory and while the subdominant exponentials they contribute switch on and off as described by the error function smoothing of the stokes phenomenon two sorts of asymptotic singularity are thereby united in a common framework the formula for the remainder incorporates both the stokes error function and the canonical catastrophe integrals a numerical illustration is given in which the distant cluster contains two saddles the asymptotic theory gives an accurate description of the details of the fractional remainder even when this is of order exp 36

polynuclear growth on a flat substrate and edge scaling of goe eigenvalues we consider the polynuclear growth png model in 1 1 dimension with flat initial condition and no extra constraints through the robinson schensted knuth rsk construction one obtains the multilayer png model which consists of a stack of non intersecting lines the top one being the png height the statistics of the lines is translation invariant and at a fixed position the lines define a point process we prove that for large times the edge of this point process suitably scaled has a limit this limit is a pfaffian point process and identical to the one obtained from the edge scaling of the gaussian orthogonal ensemble goe of random matrices our results give further insight to the universality structure within the kpz class of 1 1 dimensional growth models

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

harmonic maps to buildings and singular perturbation theory the notion of a uni versal building associated with a point in the hitchin base is introduced the universal building is a building equipped with a harmonic map from the universal cover of the given riemann surface that is initial among harmonic maps which induce the given cameral cover of the riemann surface the notion of a versal building is obtained by relaxing the uniqueness condition in the definition of an initial object in the rank one case the universal building is the leaf space of the quadratic differential defining the point in the hitchin base the main conjectures of this paper are 1 a versal building always exists 2 the asymptotics of the riemann hilbert correspondence and the non abelian hodge correspondence are controlled by the harmonic map associated to any versal building 3 spectral networks arise as inverse images of singularities of the versal building and 4 versal buildings encode the data of a 3d calabi yau category whose space of stability conditions has a connected component that contains the hitchin base the main theorem establishes the existence of the universal building conjecture 3 as well as the riemann hilbert part of conjecture 2 in the case of the rank two example introduced in the seminal work of berk nevins roberts on higher order stokes phenomena it is also shown that the asymptotics of the riemann hilbert correspondence are always controlled by a harmonic map to a certain building which is constructed as the asymptotic cone of a symmetric space

existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation abstract an evolution inclusion is considered in a separable hilbert space the right hand side of the inclusion contains the subdifferential of a time dependent proper convex lower semicontinuous function and a multivalued perturbation with nonempty closed not necessarily bounded values along with this inclusion we consider the inclusion with the perturbation term being convexified we prove the existence of solutions and a density of the solution set of the original inclusion in a closure of the solution set of the inclusion with the convexified perturbation in contrast to the known results of this kind we do not suppose that the convex function has the compactness property and that the values of the perturbation are bounded sets an example of a perturbation with closed unbounded values satisfying the conditions of the main theorems is given

janossy densities ii pfaffian ensembles we extend the main result of the companion paper j stat phys 113 595 610 to the case of the pfaffian ensembles

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

smoothing of the stokes phenomenon for high order differential equations we extend the class of functions for which the smooth transition of a stokes multiplier across a stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n the equation chosen admits solutions of hypergeometric function type which in the case n 2 are related to the parabolic cylinder functions in general the solutions of this equation involve compound asymptotic expansions valid in certain sectors of the complex z plane with more than one dominant and subdominant series the functional form of the stokes multipliers expressed in terms of an appropriately scaled variable describing transition across a stokes line is found to obey the error function smoothing law derived by berry

noncolliding squared bessel processes we consider a particle system of the squared bessel processes with index 1 conditioned never to collide with each other in which if 1 0 the origin is assumed to be reflecting when the number of particles is finite we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel when the number of particles is infinite we give sufficient conditions for initial configurations so that the system is well defined there the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the weierstrass canonical product whose zeros on the positive part of the real axis are given by the particle positions in the initial configuration from the class of infinite particle initial configurations satisfying our conditions we report one example in detail which is a fixed configuration such that every point of the square of positive zero of the bessel function j is occupied by one particle the process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process which is determinantal with the extended bessel kernel in the long term limit

a bifurcation phenomenon of stokes curves around a double turning point and influence of virtual turning points upon the transition probabilities for three level systems abstract we study the non adiabatic transition problem of three level systems from the viewpoint of the exact wkb analysis we show the importance of virtual turning points and new stokes curves in this problem by presenting some concrete examples where the contribution of virtual turning points to transition probabilities becomes greater than that of ordinary turning points in the course of our discussion we encounter a bifurcation phenomenon of stokes curves around a double turning point and we investigate this issue as well

janossy densities i determinantal ensembles we derive an elementary formula for janossy densities for determinantal point processes with a finite rank projection type kernel in particular for 2 polynomial ensembles of random matrices we show that the janossy densities on an interval i can be expressed in terms of the christoffel darboux kernel for the orthogonal polynomials on the complement of i

janossy densities of coupled random matrices we explicitly calculate the janossy densities for a special class of finite determinantal random point processes with several types of particles introduced by prahofer and spohn and in the full generality by johansson in connection with the analysis of polynuclear growth models the results of this paper generalize the theorem we proved earlier with borodin about the janossy densities in biorthogonal ensembles in particular our results can be applied to ensembles of random matrices coupled in a chain which provide a very important example of determinantal ensembles we study

maximum distributions of bridges of noncolliding brownian paths one dimensional brownian motion starting from the origin at time t 0 conditioned to return to the origin at time t 1 and to stay positive during time interval 0 t 1 is called the bessel bridge with duration 1 we consider an n particle system of such bessel bridges conditioned never to collide with each other in 0 t 1 which is the continuum limit of the vicious walk model in watermelon configuration with a wall distributions of maximum values of paths attained in the time interval t epsilon 0 1 are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane for the outermost path the distribution function of maximum value is exactly determined for general n we show that the present n path system of noncolliding bessel bridges is realized as the positive eigenvalue process of the 2nx2n matrix valued brownian bridge in the symmetry class c using this fact computer simulations are performed and numerical results on the n dependence of the maximum value distributions of the inner paths are reported the present work demonstrates that the extreme value problems of noncolliding paths are related to random matrix theory the representation theory of symmetry and number theory

exponential asymptotics and stokes lines in nonlinear ordinary differential equations a technique for calculating exponentially small terms beyond all orders in singularly perturbed ordinary differential equations is presented the approach is based on the application of a wkbj type ansatz to the late terms in the naive asymptotic expansion and the identification of stokes lines and is closely related to the well known stokes line smoothing phenomenon in linear ordinary differential equations the method is illustrated by application to examples

infinite systems of noncolliding generalized meanders and riemann liouville differintegrals yor s generalized meander is a temporally inhomogeneous modification of the 2 1 dimensional bessel process with 1 in which the inhomogeneity is indexed by kappa in 0 2 nu 1 we introduce the noncolliding particle systems of the generalized meanders and prove that they are pfaffian processes in the sense that any multitime correlation function is given by a pfaffian in the infinite particle limit we show that the elements of matrix kernels of the obtained infinite pfaffian processes are generally expressed by the riemann liouville differintegrals of functions comprising the bessel functions j used in the fractional calculus where orders of differintegration are determined by as special cases of the two parameters the present infinite systems include the quaternion determinantal processes studied by forrester nagao and honner and by nagao which exhibit the temporal transitions between the universality classes of random matrix theory

discrete approximations relaxation and optimization of one sided lipschitzian differential inclusions in hilbert spaces we study discrete approximations of nonconvex differential inclusions in hilbert spaces and dynamic optimization optimal control problems involving such differential inclusions and their discrete approximations the underlying feature of the problems under consideration is a modified one sided lipschitz condition imposed on the right hand side i e on the velocity sets of the differential inclusion which is a significant improvement of the conventional lipschitz continuity our main attention is paid to establishing efficient conditions that ensure the strong approximation in the w1 p norm as p 1 of feasible trajectories for the one sided lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations to proceed with the latter issue we derive a new extension of the bogolyubov type relaxation density theorem to the case of differential inclusions satisfying the modified one sided lipschitzian condition all the results obtained are new not only in the infinite dimensional hilbert space framework but also in finite dimensional spaces

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

dispersive hyperasymptotics and the anharmonic oscillator hyperasymptotic summation of steepest descent asymptotic expansions of integrals is extended to functions that satisfy a dispersion relation we apply the method to energy eigenvalues of the anharmonic oscillator for which there is no known integral representation but for which there is a dispersion relation hyperasymptotic summation exploits the rich analytic structure underlying the asymptotics and is a practical alternative to borel summation of the rayleigh schrodinger perturbation series

hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one in this paper we discuss the special properties of hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one we show that the re expansions are independent of the inhomogeneity we illustrate how this leads to a symmetry breaking in the stokes constants within a pair of formal solutions of a differential equation a consequence is that stokes constants may exactly vanish in higher order equations leading to dramatic simplifications in the hyperasymptotic structures two examples are included

fluctuations of the one dimensional polynuclear growth model in half space we consider the multi point equal time height fluctuations of the one dimensional polynuclear growth model in half space for special values of the nucleation rate at the origin the multi layer version of the model is reduced to a process with a determinantal weight for which the asymptotics can be analyzed in the scaling limit the fluctuations near the origin are shown to be equivalent to those of the largest eigenvalue of the orthogonal symplectic to unitary transition ensemble at soft edge in random matrix theory

exact wkb analysis of non adiabatic transition probabilities for three levels the schrodinger equation id dt h with a large parameter and an appropriate 3 3 matrix h is studied by the exact wkb method i e wkb analysis based on the borel resummation transition probabilities of solutions of the equation are explicitly calculated with the help of the connection formula for wkb solutions since a 3 3 system is considered addition of new stokes curves is necessary to obtain the correct connection formula berk nevins roberts observation and we present a practically useful procedure for finding required new stokes curves

hyperasymptotics and hyperterminants exceptional cases a new method is introduced for the computation of hyperterminants it is based on recurrence relations and can also be used to compute the parameter derivatives of the hyperterminants these parameter derivatives are needed in hyperasymptotic expansions in exceptional cases numerical illustrations and an application are included

eynard mehta theorem schur process and their pfaffian analogs we give simple linear algebraic proofs of the eynard mehta theorem the okounkov reshetikhin formula for the correlation kernel of the schur process and pfaffian analogs of these results we also discuss certain general properties of the spaces of all determinantal and pfaffian processes on a given finite set

exact wkb and abelianization for the t_3 equation we describe the exact wkb method from the point of view of abelianization both for schrodinger operators and for their higher order analogues opers the main new example which we consider is the t 3 equation an order 3 equation on the thrice punctured sphere with regular singularities at the punctures in this case the exact wkb analysis leads to consideration of a new sort of darboux coordinate system on a moduli space of flat mathrm sl 3 connections we give the simplest example of such a coordinate system and verify numerically that in these coordinates the monodromy of the t 3 equation has the expected asymptotic properties we also briefly revisit the schrodinger equation with cubic potential and the mathieu equation from the point of view of abelianization

on the use of hadamard expansions in hyperasymptotic evaluation of laplace type integrals iii clusters of saddle points it is shown how the recently developed hadamard expansion procedure can be applied to the hyperasymptotic evaluation of laplace type integrals containing a large variable when the phase function has a cluster of close lying saddle points the modification to this procedure that is required when the saddles in the cluster coalesce to form a single higher order saddle is discussed an example is also considered in which there is both a coalescence of saddles and a stokes phenomenon as the phase of the large variable is allowed to vary numerical examples are given to illustrate the accuracy that can be obtained with this new procedure

discrete polynuclear growth and determinantal processes we consider a discrete polynuclear growth png process and prove a functional limit theorem for its convergence to the airy process this generalizes previous results by prahofer and spohn the result enables us to express the f1 goe tracy widom distribution in terms of the airy process we also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions

symmetry of matrix valued stochastic processes and noncolliding diffusion particle systems as an extension of the theory of dyson s brownian motion models for the standard gaussian random matrix ensembles we report a systematic study of hermitian matrix valued processes and their eigenvalue processes associated with the chiral and nonstandard random matrix ensembles in addition to the noncolliding brownian motions we introduce a one parameter family of temporally homogeneous noncolliding systems of the bessel processes and a two parameter family of temporally inhomogeneous noncolliding systems of yor s generalized meanders and show that all of the ten classes of eigenvalue statistics in the altland zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems as a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process a stochastic calculus proof of a version of the harish chandra itzykson zuber formula of integral over unitary group is established

two bessel bridges conditioned never to collide double dirichlet series and jacobi theta function it is known that the moments of the maximum value of a one dimensional conditional brownian motion the three dimensional bessel bridge with duration 1 started from the origin are expressed using the riemann zeta function we consider a system of two bessel bridges in which noncolliding condition is imposed we show that the moments of the maximum value is then expressed using the double dirichlet series or using the integrals of products of the jacobi theta functions and its derivatives since the present system will be provided as a diffusion scaling limit of a version of vicious walker model the ensemble of 2 watermelons with a wall the dominant terms in long time asymptotics of moments of height of 2 watermelons are completely determined for the height of 2 watermelons with a wall the average value was recently studied by fulmek by a method of enumerative combinatorics

stokes multiplier expansion in an inhomogeneous differential equation with a small parameter accurate approximations to the solutions of a second order inhomogeneous equation with a small parameter are derived using exponential asymptotics the subdominant homogeneous solutions that are sw

optimal control of the sweeping process over polyhedral controlled sets abstract the paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given bolza type functional which depends on control and state variables as well as their velocities besides the highly non lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types all of this creates serious challenges for deriving necessary optimality conditions we develop here the method of discrete approximations and combine it with advanced tools of first order and second order variational analysis and generalized differentiation this approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions as a by product of the developed approach we prove the strong w 1 2 convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous time system and derive necessary optimality conditions for the discrete counterparts the established necessary optimality conditions for the sweeping process are illustrated by several examples

stokes geometry of higher order linear ordinary differential equations and middle convolution abstract the middle convolution introduced by katz and developed by dettweiler reiter oshima and others defines an operation of reduction of linear ordinary differential equations with polynomial coefficients in this paper employing an idea of the exact steepest descent method proposed by aoki kawai takei we study how to determine the complete stokes geometry and how to obtain the borel summability of wkb solutions of a higher order linear ordinary differential equation with a large parameter when it is reduced to a second order equation via middle convolution to show the practical usefulness of the method we also investigate some concrete examples numerically

on higher order stokes phenomena of an inhomogeneous linear ordinary differential equation in this paper we discuss the higher order stokes phenomenon of the particular integral of an inhomogeneous linear ordinary differential equation with a large parameter we determine the complete stokes geometry that are all the active and inactive stokes curves and the higher order stokes curve the main tool that we use is the growth of the coefficients in the dominating divergent asymptotic expansion of the particular integral we also discuss a strategy that might work in more complicated problems

exponentially improved asymptotic solutions of ordinary differential equations ii irregular singularities of rank one re expansions are found for the optimal remainder terms in the well known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one the re expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions as well being considerably more accurate and providing a smooth interpretation of the stokes phenomenon they are also of strikingly simple form in addition explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained

on the higher order stokes phenomenon during the course of a stokes phenomenon an asymptotic expansion can change its form as a further series prefactored by an exponentially small term and a stokes multiplier appears in the representation the initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters in this paper we introduce the concept of a higher order stokes phenomeno at which a stokes multiplier itself can change value we show that the higher order stokes phenomenon can be used to explain the apparent sudden birth of stokes lines at regular points and how it is indispensable to the proper derivation of expansions that involve three or more possible asymptotic contributions we provide an example of how the higher order stokes phenomenon can have important effects on the large time behaviour of partial differential equations

exponential asymptotics and stokes lines in a partial differential equation a singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered a steepest descent analysis of the fourier transform solution identifies asymptotic contributions from saddle points end points and poles and the stokes lines across which these may be switched on and off these results are then derived directly from the equation by optimally truncating the naive perturbation expansion and smoothing the stokes discontinuities the analysis reveals two new types of stokes switching a higher order stokes line which is a stokes line in the approximation of the late terms of the asymptotic series and which switches on or off stokes lines themselves and a second generation stokes line in which a subdominant exponential switched on at a primary stokes line is itself responsible for switching on another smaller exponential the new stokes lines discussed by berk et al berk et al 1982 j math phys 23 988 1002 are second generation stokes lines while the vanishing stokes lines discussed by aoki et al aoki et al 1998 in microlocal analysis and complex fourier analysis ed k f t kawai pp 165 176 are switched off by a higher order stokes line

unfolding the high orders of asymptotic expansions with coalescing saddles singularity theory crossover and duality we study the leading behaviour of the late coefficients high orders r of asymptotic expansions in a large parameter k for contour integrals involving a cluster of coalescing saddles and thereby establish the form of the divergence of the expansions the two principal cases are saddle to cluster where the integral is through a simple saddle and its expansion diverges because of a distant cluster and cluster to saddle where the integral is through a cluster and its expansion diverges because of a distant simple saddle in both the large r coefficients are dominated by the factorial divided by power familiar in asymptotics but this changes its form as the saddles in the cluster are made to coalesce and separate by varying parameters a a 1 a 2 in the integrand the crossover between different forms is described by a series of canonical integrals built from the cuspoid catastrophe polynomials of singularity theory that describe the geometry of the coalescence the arguments of these integrals involve not only the a but also fractional powers of r which by a curious duality replace the powers of the original large parameter k which occur in uniform approximations involving these integrals a by product of the cluster to saddle analysis is a new exact formula for the coefficients of uniform asymptotic expansions

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

pfaffian expressions for random matrix correlation functions it is well known that pfaffian formulas for eigenvalue correlations are useful in the analysis of real and quaternion random matrices moreover the parametric correlations in the crossover to complex random matrices are evaluated in the forms of pfaffians in this article we review the formulations and applications of pfaffian formulas for that purpose we first present the general pfaffian expressions in terms of the corresponding skew orthogonal polynomials then we clarify the relation to eynard and mehta s determinant formula for hermitian matrix models and explain how the evaluation is simplified in the cases related to the classical orthogonal polynomials applications of pfaffian formulas to random matrix theory and other fields are also mentioned

virtual turning points and bifurcation of stokes curves for higher order ordinary differential equations for a higher order linear ordinary differential operator p its stokes curve bifurcates in general when it hits another turning point of p this phenomenon is most neatly understandable by taking into account stokes curves emanating from virtual turning points together with those from ordinary turning points this understanding of the bifurcation of a stokes curve plays an important role in resolving a paradox recently found in a system of linear differential equations associated with the fourth painleve equation

hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one a sequence of re expansions is developed for the remainder terms in the well known poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one these re expansions are a series whose terms are a product of stokes multipliers coefficients of the original poincare series expansions and certain multiple integrals the so called hyperterminants each step of the process reduces the estimate of the error term by an exponentially small factor the method of this paper is based on the borel laplace transform which makes it applicable to other problems at the end of the paper the method is applied to integrals with saddles also a powerful new method is presented to compute the stokes multipliers a numerical example is included

noncolliding brownian motion and determinantal processes a system of one dimensional brownian motions bms conditioned never to collide with each other is realized as i dyson s bm model which is a process of eigenvalues of hermitian matrix valued diffusion process in the gaussian unitary ensemble gue and as ii the h transform of absorbing bm in a weyl chamber where the harmonic function h is the product of differences of variables the vandermonde determinant the karlin mcgregor formula gives determinantal expression to the transition probability density of absorbing bm we show from the karlin mcgregor formula if the initial state is in the eigenvalue distribution of gue the noncolliding bm is a determinantal process in the sense that any multitime correlation function is given by a determinant specified by a matrix kernel by taking appropriate scaling limits spatially homogeneous and inhomogeneous infinite determinantal processes are derived we note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix kernels are expressed using spectral projections of appropriate effective hamiltonians on the common structure of matrix kernels continuity of processes in time is proved and general property of the determinantal processes is discussed

hyperasymptotics for integrals with finite endpoints berry howls 1991 hereinafter called bh refined the method of steepest descent to study exponentially accurate asymptotics of a general class of integrals involving exp kf z along doubly infinite contours in the complex plane passing over saddlepoints of f z here we derive analogous results for integrals with integrands of a similar form but whose local expansions in powers of 1 k are made about the finite endpoints of semi infinite contours of integration we treat endpoints where f z behaves locally linearly or quadratically generically local endpoint expansions made by the method of steepest descent diverge because of the presence of saddles of f z we derive resurgence relations which express the original integral exactly as a truncated endpoint expansion plus a remainder involving the global saddle structure of f z via integrals through certain adjacent saddles the saddles adjacent to the endpoint are determined by a topological rule if the least term of the endpoint expansion is the n 0 k th summing to here calculates the endpoint integral up to an error of approximately exp n 0 k we develop a scheme involving iteration of the new resurgence relations with a similar one derived in bh which can reduce this error down to exp 2 386n 0 k this hyperasymptotic formalism parallels that of bh and incorporates automatically any change in the complete asymptotic expansion as the endpoint moves in the complex plane provided that it does not coincide with other saddles we illustrate the analytical and numerical use of endpoint hyperasymptotics by application to the complementary error function erfc x and a constructed incomplete airy function

new stokes line in wkb theory the wkb theory for differential equations of arbitrary order or integral equations in one dimension is investigated the rules previously stated for the construction of stokes lines for nth order differential equations n 3 or integral equations are found to be incomplete because these rules lead to asymptotic forms of the solutions that depend on path this paradox is resolved by the demonstration that new stokes lines can arise when previously defined stokes lines cross a new formulation of the wkb problem is given to justify the new stokes lines with the new stokes lines the asymptotic forms can be shown to be independent of path in addition the wkb eigenvalue problem is formulated and the global dispersion relation is shown to be a functional of loop integrals of the action

interacting brownian particles and the wigner law in this paper we study interacting diffusing particles governed by the stochastic differential equationsdx j t n db j t d jon x 1 x n dt j 1 2 n here theb jare independent brownian motions in d and o n x 1 x n n i j v x i x j n i u x 1 the potentialv has a singularity at 0 strong enough to keep the particles apart and the potentialu serves to keep the particles from escaping to infinity our interest is in the behaviour as the number of particles increases without limit which we study through the empirical measure process we prove tightness of these processes in the case ofd 1 v x log x u x x 2 2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure valued process exists this process is deterministic and converges to the wigner law ast some information on the rates of convergence is derived and the case of a cauchy initial distribution is analysed completely

determinantal process starting from an orthogonal symmetry is a pfaffian process when the number of particles n is finite the noncolliding brownian motion bm and the noncolliding squared bessel process with index 1 besq are determinantal processes for arbitrary fixed initial configurations in the present paper we prove that if initial configurations are distributed with orthogonal symmetry they are pfaffian processes in the sense that any multitime correlation functions are expressed by pfaffians the 2 2 skew symmetric matrix valued correlation kernels of the pfaffians processes are explicitly obtained by the equivalence between the noncolliding bm and an appropriate dilatation of a time reversal of the temporally inhomogeneous version of noncolliding bm with finite duration in which all particles start from the origin n 0 and by the equivalence between the noncolliding besq and that of the noncolliding squared generalized meander starting from n 0

on the asymptotic and numerical solution of linear ordinary differential equations an investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank we introduce a classification of the solutions into two types explicit and implicit for the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached no such sector exists for implicit solutions in consequence the two types of solution have different uniqueness properties another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions we also investigate the computation of the solutions by numerical integration of the differential equation it is shown that it is the exception rather than the rule for the integration process to be stable particularly so for implicit solutions to overcome these instabilities we develop boundary value methods complete with error analysis numerical examples illustrate the computation of both explicit and implicit solutions and also the associated stokes multipliers

on the exact steepest descent method a new method for the description of stokes curves in order to determine the region where the borel sum of a wentzel kramers brillouin wkb solution of an mth order m 3 ordinary differential equation p 0 with polynomial coefficients is well defined we apply the steepest descent method to the laplace integral of a wkb solution of p 0 with p denoting the laplace transform of p as a counterpart of the connection formula for we introduce a new notion of the exact steepest descent path which is the union of bifurcated steepest descent paths both theoretical and computer assisted experimental studies are given to show the importance of this notion

differential equations for dyson processes we call a dyson process any process on ensembles of matrices in which the entries undergo diffusion we are interested in the distribution of the eigenvalues or singular values of such matrices in the original dyson process it was the ensemble of n n hermitian matrices and the eigenvalues describe n curves given sets x1 xm the probability that for each k no curve passes through xk at time k is given by the fredholm determinant of a certain matrix kernel the extended hermite kernel for this reason we call this dyson process the hermite process similarly when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the laguerre process for which there is a corresponding extended laguerre kernel scaling the hermite process at the edge leads to the airy process which was introduced by prahofer and spohn as the limiting stationary process for a polynuclear growth model and in the bulk to the sine process scaling the laguerre process at the edge leads to the bessel process

third harmonic generation as a third order quasi exactly solvable system the two degrees of freedom in the usual third harmonic generation effective hamiltonian can be separated both classically and quantum mechanically into a harmonic oscillator and a nonlinear oscillator in turn the quantum hamiltonian of the nonlinear oscillator can be written as a cubic polynomial in the generators of a quasi exactly solvable lie algebra and the physically relevant eigenstates are precisely those that can be determined exactly although in general not explicitly since the standard jeffreys wentzel kramers brillouin methods are not easily applicable to general third order differential equations we use a bohr sommerfeld quantization of the classical orbits to obtain approximate explicit formulas for the corresponding quantum eigenvalues

global asymptotics for multiple integrals with boundaries under convenient geometric assumptions the saddle point method for multidimensional laplace integrals is extended to the case where the contours of integration have boundaries the asymptotics are studied in the case of nondegenerate and of degenerate isolated critical points the incidence of the stokes phenomenon is related to the monodromy of the homology via generalized picard lefschetz formulae and is quantified in terms of geometric indices of intersection exact remainder terms and the hyperasymptotics are then derived a direct consequence is a numerical algorithm to determine the stokes constants and indices of intersections examples are provided

scaling limit of vicious walks and two matrix model we consider the diffusion scaling limit of the one dimensional vicious walker model of fisher and derive a system of nonintersecting brownian motions the spatial distribution of n particles is studied and it is described by use of the probability density function of eigenvalues of nxn gaussian random matrices the particle distribution depends on the ratio of the observation time t and the time interval t in which the nonintersecting condition is imposed as t t is going on from 0 to 1 there occurs a transition of distribution which is identified with the transition observed in the two matrix model of pandey and mehta despite of the absence of matrix structure in the original vicious walker model in the diffusion scaling limit accumulation of contact repulsive interactions realizes the correlated distribution of eigenvalues in the multimatrix model as the particle distribution

polynuclear growth model with external source and random matrix model with deterministic source we present a random matrix interpretation of the distribution functions which have appeared in the study of the one dimensional polynuclear growth png model with external sources it is shown that the distribution goe2 which is defined as the square of the gaussian orthogonal ensemble goe tracy widom distribution can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source which have been studied in a different context previously compared to the original interpretation of the goe2 as the square of goe ours has an advantage in that it can also describe the transition from the gaussian unitary ensemble gue tracy widom distribution to the goe2 we further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain noncolliding brownian motion model and the multilayer png model with an external source this provides us with a multimatrix model interpretation of the multipoint height distributions of the png model with an external source

quantitative stability of variational systems i the epigraphical distance this paper proposes a global measure for the distance between the elements of a variational system parametrized families of optimization problems

on the exact wkb analysis of operators admitting infinitely many phases abstract to analyze differential operators whose wkb solutions admit infinitely many phases we introduce a class of differential operators of wkb type and analyze their exact wkb theoretic structure near their turning points our analysis makes full use of techniques and ideas in microlocal analysis we use a quantized contact transformation to construct a wkb solution of a differential equation of wkb type and we use a spath type division theorem for a differential operator of wkb type to study its structure near turning points as an application we show a connection formula for wkb solutions near a simple turning point

hyperasymptotics for multidimensional integrals exact remainder terms and the global connection problem the method of steepest descents for single dimensional laplace type integrals involving an asymptotic parameter k was extended by berry howls in 1991 to provide exact remainder terms for truncated asymptotic expansions in terms of contributions from certain non local saddlepoints this led to an improved asymptotic expansion hyperasymptotics which gave exponentially accurate numerical and analytic results based on the topography of the saddle distribution in the single complex plane of the integrand in this paper we generalize these results to similar well behaved multidimensional integrands with quadratic critical points integrated over infinite complex domains as previously pointed out the extra complex dimensions give rise to interesting problems and phenomena first the conventionally defined surfaces of steepest descent are no longer unique second the stokes s phenomenon whereby contributions from subdominant saddles enter the asymptotic representation is of codimension one third we can collapse the representation of the integral onto a single complex plane with branch cuts at the images of critical points the new results here demonstrate that dimensionality only trivially affects the form of the exact multidimensional remainder thus the growth of the late terms in the expansion can be identified and a hyperasymptotic scheme implemented we show by a purely algebraic method how to determine which critical points contribute to the remainder and hence resolve the global connection problem riemann sheet structure and homology associated with the multidimensional topography of the integrand