statistical theory for the stochastic burgers equation in the inviscid limit a statistical theory is developed for the stochastic burgers equation in the inviscid limit master equations for the probability density functions of velocity velocity difference and velocity gradient are derived no closure assumptions are made instead closure is achieved through a dimension reduction process namely the unclosed terms are expressed in terms of statistical quantities for the singular structures of the velocity field here the shocks master equations for the environment of the shocks are further expressed in terms of the statistics of singular structures on the shocks namely the points of shock generation and collisions the scaling laws of the structure functions are derived through the analysis of the master equations rigorous bounds on the decay of the tail probabilities for the velocity gradient are obtained using realizability constraints we also establish that the probability density function q of the velocity gradient decays as jj 7 2 as 1 c 2000 john wiley sons inc

on the rate of convergence to the normal law for solutions of the burgers equation with singular initial data we study the scaling limit of random fields which are the solutions of a nonlinear partial differential equation known as the burgers equation under stochastic initial condition these are assumed to be a gaussian process with long range dependence we present some results on the rate of convergence to the normal law

diffusion approximation for self similarity of stochastic advection in burgers equation self similarity of burgers equation with stochastic advection is studied in self similar variables a stationary solution is constructed which establishes the existence of a stochastically self similar solution for the stochastic burgers equation the analysis assumes that the stochastic coefficient of advection is transformed to a white noise in the self similar variables furthermore by a diffusion approximation the long time convergence to the self similar solution is proved in the sense of distribution

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

random periodic solutions of spdes via integral equations and wiener sobolev compact embedding abstract in this paper we study the existence of random periodic solutions for semilinear spdes on a bounded domain with a smooth boundary we identify them as the solutions of coupled forward backward infinite horizon stochastic integral equations on l 2 d in general cases for this we use mercer s theorem and eigenvalues and eigenfunctions of the second order differential operators in the infinite horizon integral equations we then use the argument of the relative compactness of wiener sobolev spaces in c 0 0 t l 2 d and generalized schauder s fixed point theorem to prove the existence of a solution of the integral equations this is the first paper in literature to study random periodic solutions of spdes our result is also new in finding semi stable stationary solution for non dissipative spdes while in literature the classical method is to use the pull back technique so researchers were only able to find stable stationary solutions for dissipative systems

stratified structure of the universe and burgers equation a probabilistic approach the model of the potential turbulence described by the 3 dimensional burgers equation with random initial data was developped by zeldovich and shandarin in order to explain the existing large scale structure of the universe most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results the gaussian scenario under suitable moment assumptions on the initial velocity field these results and some open questions are discussed in sect 2 where we concentrate on the gaussian model and the shot noise model in sect 3 we construct a probabilistic model of strong initial fluctuations a zero range shot noise field with high amplitudes which reveals an intermittent large time behaviour with the velocity vec v t x determined by the position of the largest initial fluctuation discounted by the heat kernelg t x in a neighborhood ofx the asymptoties of such local maximum ast can be analyzed with the help of the theory of records sect 4 finally in sect 5 we introduce a global definition of a point process oft local maxima and show the weak convergence of the suitably rescaled process to a non trivial limit ast

one dimensional stochastic burgers equation driven by levy processes abstract in this paper we proved the global existence and uniqueness of the strong weak and mild solutions for one dimensional burgers equation perturbed by a poisson form process a poisson form and q wiener process with the dirichlet bounded condition we also proved the existence of the invariant measure of these models

distribution of the first particle in discrete orthogonal polynomial ensembles we show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure if the difference or q log derivative of the weight function is rational in a number of classical special cases the recurrence procedure is equivalent to the difference and q painleve equations of 10 17 our approach is based on the formalism of discrete integrable operators and discrete riemann hilbert problems developed in 3 4

stochastic dissipative pde s and gibbs measures we study a class of dissipative nonlinear pde s forced by a random force omega t x with the space variable x varying in a bounded domain the class contains the 2d navier stokes equations under periodic or dirichlet boundary conditions and the forces we consider are those common in statistical hydrodynamics they are random fields smooth in t and stationary short correlated in time t in this paper we confine ourselves to kick forces of the form where the k s are smooth bounded identically distributed random fields the equation in question defines a markov chain in an appropriately chosen phase space a subset of a function space that contains the zero function and is invariant for the random flow of the equation concerning this markov chain we prove the following main result see theorem 2 2 the markov chain has a unique invariant measure to prove this theorem we present a construction assigning to any invariant measure a gibbs measure for a 1d system with compact phase space and apply a version of ruelle perron frobenius uniqueness theorem to the corresponding gibbs system we also discuss ergodic properties of the invariant measure and corresponding properties of the original randomly forced pde

deformation analysis of matrix models abstract in this paper we show that the non linear equations that play a role in tracy and widom s analysis of scaling limits for hermitian matrix models are special cases of the monodromy preserving deformation equations first discussed by sato miwa and jimbo in the early 1980 s

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

aubry mather theory and periodic solutions of the forced burgers equation consider a hamiltonian system with hamiltonian of the form h x t p where h is convex in p and periodic in x a ndt and x2 r 1 it is well known that its smooth invariant curves correspond to smoothz 2 periodic solutions of the pde ut h x t u x 0 in this paper we establish a connection between the aubry mather theory of invariant sets of the hamiltonian system andz 2 periodic weak solutions of this pde by realizing the aubry mather sets as closed subsets of the graphs of these weak solutions we show that the complement of the aubry mather set on the graph can be viewed as a subset of the generalized unstable manifold of the aubry mather set defined in 2 24 the graph itself is a backward invariant set of the hamiltonian system the basic idea is to embed the globally minimizing orbits used in the aubry mather theory into the characteristic fields of the above pde this is done by making use of one and two sided minimizers a notion introduced in 12 and inspired by the work of morse on geodesics of type a 26 the asymptotic slope of the minimizers also known as the rotation number is given by the derivative of the homogenized hamiltonian defined in 21 as an application we prove that thez 2 periodic weak solution of the above pde with given irrational asymptotic slope is unique a similar connection also exists in multidimensional problems with the convex hamiltonian except that in higher dimensions two sided minimizers with a specified asymptotic slope may not exist c 1999 john wiley sons inc

the stochastic burgers equation we study burgers equation perturbed by a white noise in space and time we prove the existence of solutions by showing that the cole hopf transformation is meaningful also in the stochastic case the problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise an explicit solution of the latter is constructed through a generalized feynman kac formula typical properties of the trajectories are then discussed a technical result concerning the regularizing effect of the convolution with the heat kernel is proved for stochastic integrals

on the structure of correlation functions in the normal matrix model we study the structure of the normal matrix model nmm we show that all correlation functions of the model with an axially symmetric potential can be expressed in terms of a single holomorphic function of one variable this observation is used to demonstrate the exact solvability of the model the two point correlation function is calculated in the continuum limit the answer is proven to be universal i e potential independent up to a change of the scale in connection with nmm with a general polynomial potential we have developed a two dimensional free fermion formalism and constructed a family of completely integrable hierarchies of non linear differential equations which we call the extended kp n hierarchies the well known kp hierarchy is a lower dimensional reduction of this family the extended kp 1 hierarchy contains the 2 1 dimensional burgers equations the partition function of the n n nmm is the function of the extended kp n hierarchy which is invariant with respect to a subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane

stationary solutions of spdes and infinite horizon bdsdes with non lipschitz coefficients abstract we prove a general theorem that the l 2 r d r 1 l 2 r d r d valued solution of an infinite horizon backward doubly stochastic differential equation if exists gives the stationary solution of the corresponding stochastic partial differential equation we prove the existence and uniqueness of the l 2 r d r 1 l 2 r d r d valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming lipschitz conditions but under the monotonicity condition therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations and the solution of the infinite horizon problem gives the stationary solution of the spdes according to our general result

two results concerning asymptotic behavior of solutions of the burgers equation with force we consider the burgers equation with an external force for the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast if the force is the product of a periodic function ofx and white noise in time we prove the existence of an invariant distribution concentrated on the space of space periodic functions which is the limit of a wide class of distributions ast

integrable fredholm operators and dual isomonodromic deformations the fredholm determinants of a special class of integrable integral operators k supported on the union of m curve segments in the complex plane are shown to be the functions of an isomonodromic family of meromorphic covariant derivative operators having regular singular points at the 2m endpoints of the curve segments and a singular point of poincare index 1 at infinity the rank r of the corresponding vector bundle over the riemann sphere equals the number of distinct terms in the exponential sum defining the numerator of the integral kernel the matrix riemann hilbert problem method is used to deduce an identification of the fredholm determinant as a function in the sense of segal wilson and sato i e in terms of abelian group actions on the determinant line bundle over a loop space grassmannian an associated dual isomonodromic family of covariant derivative operators having rank n 2m and r finite regular singular points located at the values of the exponents defining the kernel of k is derived the deformation equations for this family are shown to follow from an associated dual set of riemann hilbert data in which the roles of the r exponential factors in the kernel and the 2m endpoints of its support are interchanged the operators are analogously associated to an integral operator whose fredholm determinant is equal to that of k

time asymptotics for solutions of the burgers equation with a periodic force we consider the burgers equation with a periodic force frac partial partial t u u cdot nabla u frac12 delta u nabla v x which presents a simplified model for turbulence we are interested in the asymptotic behaviour of solutions for t to infty this problem has been studied by sinai who uses a probabilistic and very technical approach using methods from spectral theory we get similar results this functional analytic approach gives an easier proof for certain initial data periodic or some random perturbations of those we show time convergence towards a deterministic periodic limit solution related to the ground state of a certain schrodinger operator

a dynamical polynomial chaos approach for long time evolution of spdes abstract we propose a dynamical generalized polynomial chaos dgpc method to solve time dependent stochastic partial differential equations spdes with white noise forcing the long time simulation of spde solutions by polynomial chaos pc methods is notoriously difficult as the dimension of the stochastic variables increases linearly with time exploiting the markovian property of white noise dgpc 1 implements a restart procedure that allows us to expand solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future white noise the dimension of the representation is kept minimal by application of a karhunen loeve kl expansion using frequent restarts and low degree polynomials on sparse multi index sets the method allows us to perform long time simulations including the calculation of invariant measures for systems which possess one we apply the method to the numerical simulation of stochastic burgers and navier stokes equations with white noise forcing our method also allows us to incorporate time independent random coefficients such as a random viscosity we propose several numerical simulations and show that the algorithm compares favorably with standard monte carlo methods

isomonodromy transformations of linear systems of difference equations we introduce and study isomonodromy transformations of the matrix linear difference equation y z 1 a z y z with polynomial a z our main result is construction of an isomonodromy action of zm n 1 1 on the space of coefficients a z here m is the size of matrices and n is the degree of a z the birational action of certain rank n subgroups can be described by difference analogs of the classical schlesinger equations and we prove that for generic initial conditions these difference schlesinger equations have a unique solution we also show that both the classical schlesinger equations and the schlesinger transformations known in isomonodromy theory can be obtained as limits of our action in two different limit regimes similarly to the continuous case for m n 2 the difference schlesinger equations and their q analogs yield discrete painlev e equations examples include dpii dpiv dpv and q pvi

pathwise random periodic solutions of stochastic differential equations abstract in this paper we study the existence of random periodic solutions for semilinear stochastic differential equations we identify these as the solutions of coupled forward backward infinite horizon stochastic integral equations in general cases we then use the argument of the relative compactness of wiener sobolev spaces in c 0 0 t l 2 and generalized schauder s fixed point theorem to prove the existence of a solution of the coupled stochastic forward backward infinite horizon integral equations the condition on f is then further weakened by applying the coupling method of forward and backward gronwall inequalities the results are also valid for stationary solutions as a special case when the period can be an arbitrary number

ergodic attractors and almost everywhere asymptotics of scalar semilinear parabolic differential equations abstract we consider dynamics of scalar semilinear parabolic equations on bounded intervals with periodic boundary conditions and on the entire real line with a general nonlinearity g t x u u x either not depending on t or periodic in t while the topological and geometric structure of their attractors has been investigated in depth we focus here on ergodic theoretical properties the main result is that the union of supports of all the invariant measures projects one to one to r 2 we rely on a novel application of the zero number techniques with respect to evolution of measures on the phase space and on properties of the flux of zeroes and the dissipation of zeroes as an example of an application we prove uniqueness of an invariant measure for a large family of considered equations which conserve a certain quantity mass thus generalizing the results by sinai for the scalar viscous burgers equation

almost sure asymptotic stability of stochastic partial differential equations with jumps in this paper we investigate stochastic partial differential equations with jumps in infinite dimensions the key motivation of this paper is under a local lipschitz condition but without a linear growth condition to give an existence and uniqueness theorem khasminskii type theorem where the classical existence and uniqueness result can be regarded as a special case and then to discuss the almost sure asymptotic stability of the solutions moreover as a by product we also derive that the solutions are weakly attracted finally several examples are constructed to demonstrate our results

white noise perturbation of the viscous shock fronts of the burgers equation we study the front dynamics of solutions of the initial value problem of the burgers equation with initial data being the viscous shock front plus the white noise perturbation in the sense of distribution the solutions propagate with the same speed as the unperturbed front however the front location is random and satisfies a central limit theorem with the variance proportional to the timet ast goes to infinity with probability arbitrarily close to one the front width iso 1 for large time

stochastic burgers equations and their semi classical expansions in this paper we use the hopf cole logarithmic transformation and the stochastic hamilton jacobi theory to study stochastic heat equations and burgers equations before the caustics the stochastic inviscid burgers equation gives the first term of our semi classical expansion i e the inviscid limit of the viscous stochastic burgers equation in order to push our results beyond the inviscid limit we construct solutions for iterated stochastic hamilton jacobi continuity equations then we give the semi classical asymptotic expansions for stochastic heat equations and burgers equations by using nelson s stochastic mechanical processes with drifts given by the solution of the iterated stochastic hamilton jacobi continuity equations the explicit formula for the remainder term is given by a path integral

harmonic analysis on the infinite dimensional unitary group and determinantal point processes the infinite dimensional unitary group u is the inductive limit of growing compact unitary groups u n in this paper we solve a problem of harmonic analysis on u stated in ol3 the problem consists in computing spectral decomposition for a remarkable 4 parameter family of characters of u these characters generate representations which should be viewed as analogs of nonexisting regular representation of u the spectral decomposition of a character of u is described by the spectral measure which lives on an infinite dimensional space of indecomposable characters the key idea which allows us to solve the problem is to embed into the space of point configurations on the real line without two points this turns the spectral measure into a stochastic point process on the real line the main result of the paper is a complete description of the processes corresponding to our concrete family of characters we prove that each of the processes is a determinantal point process that is its correlation functions have determinantal form with a certain kernel our kernels have a special integrable form and are expressed through the gauss hypergeometric function from the analytic point of view the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by richard askey and peter lesky one difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0 we do this by expressing the polynomials in terms of a solution of a discrete riemann hilbert problem and computing the nonoscillatory asymptotics of this solution from the point of view of statistical physics we study thermodynamic limit of a discrete log gas system an interesting feature of this log gas is that its density function is asymptotically equal to the characteristic function of an interval our point processes describe how different the random particle configuration is from the typical densely packed configuration in simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers bo1 bo2 bo4

fredholm determinants jimbo miwa ueno τ functions and representation theory the authors show that a wide class of fredholm determinants arising in the representation theory of big groups such as the infinite dimensional unitary group solve painleve equations their methods are based on the theory of integrable operators and the theory of riemann hilbert problems 2002 wiley periodicals inc

hyperbolic asymptotics in burgers turbulence and extremal processes large time asymptotics of statistical solutionu t x 1 2 of the burgers equation 1 1 is considered where x l x is a stationary zero mean gaussian process depending on a large parameterl 0 so that xi l x sim sigma l eta x l l to infty where sigma l l 2 2 log l 1 2 and x is a given standardized stationary gaussian process we prove that asl the hyperbolicly scaled random fieldsu l2t l2x converge in distribution to a random field with saw tooth trajectories defined by means of a poisson process on the plane related to high fluctuations of x which corresponds to the zero viscosity solutions at the physical level of rigor such asymptotics was considered before by gurbatov malakhov and saichev 1991

the burgers equation with a random force and a general model for directed polymers in random environments the study of the burgers equation with a random force leads via a hopf cole type transformation to a stochastic heat equation having a white noise with spatial parameters type potential the latter can be studied by means of a general model of directed polymers in random environments with two point random potentials these models exhibit a gaussian behavior at large times and have certain stationary distributions which yield the corresponding results for the above stochastic heat and burgers equations

burgers turbulence and random lagrangian systems we consider a spatially periodic inviscid random forced burgers equation in arbitrary dimension and the random time dependent lagrangian system related to it we construct a unique stationary distribution for viscosity solutions of the burgers equation we also show that with probability 1 there exists a unique minimizing trajectory for the random lagrangian system which generates a non trivial ergodic invariant measure for the non random skew product extension of the lagrangian system

stationary solutions of spdes and infinite horizon bdsdes in this paper we study the existence of stationary solutions for stochastic partial differential equations we establish a new connection between l 2 rd r1 l 2 rd rd valued solutions of backward doubly stochastic differential equations bdsdes on infinite horizon and the stationary solutions of the spdes moreover we prove the existence and uniqueness of the solutions of bdsdes on both finite and infinite horizons so obtain the solutions of initial value problems and the stationary solutions independent of any initial value of spdes the connection of the weak solutions of spdes and bdsdes has independent interests in the areas of both spdes and bsdes

nonlinear fluctuations of weakly asymmetric interacting particle systems we introduce what we call the second order boltzmann gibbs principle which allows one to replace local functionals of a conservative one dimensional stochastic process by a possibly nonlinear function of the conserved quantity this replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states as an application of this second order boltzmann gibbs principle we introduce the notion of energy solutions of the kpz and stochastic burgers equations under minimal assumptions we prove that the density fluctuations of one dimensional stationary weakly asymmetric conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic burgers equation we also show that the fluctuations of the height function associated to these models are given by energy solutions of the kpz equation in this sense unfortunately we lack a uniqueness result for these energy solutions we conjecture these solutions to be unique and we show some regularity results for energy solutions of the kpz burgers equation supporting this conjecture

solutions of a system of forced burgers equation in this article we obtain explicit solutions of a system of forced burgers equation subject to some classes of bounded and compactly supported initial data and also subject to certain unbounded initial data in a series of papers rao and yadav 2010 1 3 obtained explicit solutions of a nonhomogeneous burgers equation in one dimension subject to certain classes of bounded and unbounded initial data earlier kloosterziel 1990 4 represented the solution of an initial value problem for the heat equation with initial data in l 2 r n e x 2 2 as a series of self similar solutions of the heat equation in r n here we express the solutions of certain classes of cauchy problems for a system of forced burgers equation in terms of self similar solutions of some linear partial differential equations

difference system for selberg correlation integrals the selberg correlation integrals are averages of the products with respect to the selberg density our interest is in the case m 1 1 when this corresponds to the th moment of the corresponding characteristic polynomial we give the explicit form of an n 1 n 1 matrix linear difference system in the variable which determines the average and we give the gauss decomposition of the corresponding n 1 n 1 matrix for a positive integer the difference system can be used to efficiently compute the power series defined by this average

front speed in the burgers equation with a random flux we study the large time asymptotic shock front speed in an inviscid burgers equation with a spatially random flux function this equation is a prototype for a class of scalar conservation laws with spatial random coefficients such as the well known buckley leverett equation for two phase flows and the contaminant transport equation in groundwater flows the initial condition is a shock located at the origin the indicator function of the negative real line we first regularize the equation by a special random viscous term so that the resulting equation can be solved explicitly by a cole hopf formula using the invariance principle of the underlying random processes and the laplace method we prove that for large times the solutions behave like fronts moving at averaged constant speeds in the sense of distribution however the front locations are random and we show explicitly the probability of observing the head or tail of the fronts finally we pass to the inviscid limit and establish the same results for the inviscid shock fronts

scaling laws for the multidimensional burgers equation with quadratic external potential the reordering of the multidimensional exponential quadratic operator in coordinate momentum space see x wang c h oh and l c kwek 1998 j phys a math gen 31 4329 4336 is applied to derive an explicit formulation of the solution to the multidimensional heat equation with quadratic external potential and random initial conditions the solution to the multidimensional burgers equation with quadratic external potential under gaussian strongly dependent scenarios is also obtained via the hopf cole transformation the limiting distributions of scaling solutions to the multidimensional heat and burgers equations with quadratic external potential are then obtained under such scenarios

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

computing the tutte polynomial of archimedean tilings abstract we describe an algorithm to compute the tutte polynomial of large fragments of archimedean tilings by squares triangles hexagons and combinations thereof our algorithm improves a well known method for computing the tutte polynomial of square lattices we also address the problem of obtaining tutte polynomial evaluations from the symbolic expressions generated by our algorithm improving the best known lower bound for the asymptotics of the number of spanning forests and the lower and upper bounds for the asymptotics of the number of acyclic orientations of the square lattice

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure