the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

quasipotential and exit time for 2d stochastic navier stokes equations driven by space time white noise we are dealing with the navier stokes equation in a bounded regular domain mathcal o of mathbb r 2 perturbed by an additive gaussian noise partial w q delta partial t which is white in time and colored in space we assume that the correlation radius of the noise gets smaller and smaller as delta searrow 0 so that the noise converges to the white noise in space and time for every delta 0 we introduce the large deviation action functional s delta t and the corresponding quasi potential u delta and by using arguments from relaxation and gamma convergence we show that u delta converges to u u 0 in spite of the fact that the navier stokes equation has no meaning in the space of square integrable functions when perturbed by space time white noise moreover in the case of periodic boundary conditions the limiting functional u is explicitly computed finally we apply these results to estimate of the asymptotics of the expected exit time of the solution of the stochastic navier stokes equation from a basin of attraction of an asymptotically stable point for the unperturbed system

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space abstract in this paper sufficient conditions are given for the controllability of a class of partial stochastic functional differential inclusions with infinite delay in an abstract space with the help of the leray schauder nonlinear alternative an example is provided to illustrate the theory

on stochastic evolution equations for nonlinear bipolar fluids well posedness and some properties of the solution abstract we investigate the stochastic evolution equations describing the motion of a non newtonian fluids excited by multiplicative noise of levy type we show that the system we consider has a unique global strong solution we also give some results concerning the properties of the solution we mainly prove that the unique solution satisfies the markov feller property this enables us to prove by means of some results from ergodic theory that the semigroup associated to the unique solution admits at least an invariant measure which is ergodic and tight on a subspace of the lebesgue space l 2

the maximum corank of graphs with a 2 separation abstract for a graph g v e with vertex set v 1 2 n which is allowed to have parallel edges and for a field f let s g f be the set of all f valued symmetric n n matrices a which represent g the maximum corank of a graph g is the maximum possible corank over all a s g f if g 1 g 2 is a 2 separation we give a formula which relates the maximum corank of g to the maximum corank of some small variations of g 1 and g 2

the strong spectral property for graphs abstract we introduce the set g ssp of all simple graphs g with the property that each symmetric matrix corresponding to a graph g g ssp has the strong spectral property we find several families of graphs in g ssp and in particular characterise the trees in g ssp

large deviations for stationary measures of stochastic nonlinear wave equation with smooth white noise we prove the freidlin wentzell type large deviations principle for the family of stationary measures of stochastic nonlinear wave nlw equation with white noise we do not assume that the limiting equation possesses a unique equilibrium and do not impose roughness on the noise this allows to provide the first such result in the pde setting

zero forcing for sign patterns we introduce a new variant of zero forcing signed zero forcing the classical zero forcing number provides an upper bound on the maximum nullity of a matrix with a given graph i e zero nonzero pattern our new variant provides an analogous bound for the maximum nullity of a matrix with a given sign pattern this allows us to compute for instance the maximum nullity of a z matrix whose graph is l kn the line graph of a clique

conditioning and dichotomy in differential algebraic equations the relationship between the dichotomy of the solution space for a differential algebraic system and the well conditioning of associated boundary value problems is investigated this work extends results that are known for regular explicit ordinary differential equations and it follows the presentation by de hoog and mattheij siam j numer anal 24 1987 pp 89 105

ergodic and mixing properties of the boussinesq equations with a degenerate random forcing we establish the existence uniqueness and attraction properties of an ergodic invariant measure for the boussinesq equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales the central challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system towards this aim we encounter a lie bracket structure in the associated vector fields with a complicated dependence on solutions this leads us to develop a novel hormander type condition for infinite dimensional systems demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the malliavin covariance matrix

large deviations for a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the non gaussian stochastic reaction diffusion equation srde partial t nu varepsilon mathscr l nu varepsilon f x nu varepsilon varepsilon sigma x nu varepsilon ddot w tx as a random perturbation of the deterministic rde partial t nu 0 mathscr l nu 0 f x nu 0 here the space variable takes values on the unit circle s 1 and mathscr l is a strongly elliptic second order operator with constant coefficients the functions f and sigma are sufficiently regular so that there is a unique solution to the above srde for any continuous initial condition we also assume that there are positive constants m and m such that m leq sigma x y leq m for all x in s 1 and all y in mathbb r the perturbation ddot w tx is the formal derivative of a brownian sheet it is known that if the initial condition is continuous then the solution will also be continuous and moreover if the initial condition is assumed to be holder continuous of exponent kappa for some 0 kappa frac 1 2 then the solution will be holder continuous of exponent kappa 2 as a function of t x in this paper we establish the large deviations principle for nu varepsilon in the holder norm of exponent kappa 2 when the initial condition is holder continuous of exponent kappa for any 0 kappa frac 1 2 and when the initial condition is assumed only to be continuous we establish the large deviations principle for nu varepsilon in the supremum norm moreover we prove that these large deviations principles are uniform with respect to the initial condition

on minimum energy problems a stochastic system described by a semilinear equation with a small noise is considered under suitable hypotheses the rate functionals for the family of distributions associated to the solution and the exit time and exit place of the solution are computed

approximation of quasi potentials and exit problems for multidimensional rde s with noise we deal with a class of reaction diffusion equations in space dimension d 1 perturbed by a gaussian noise w t which is white in time and colored in space we assume that the noise has a small correlation radius so that it converges to the white noise w t as 0 by using arguments of convergence we prove that under suitable assumptions the quasi potential v converges to the quasi potential v corresponding to space time white noise in spite of the fact that the equation perturbed by space time white noise has no solution we apply these results to the asymptotic estimate of the mean of the exit time of the solution of the stochastic problem from a basin of attraction of an asymptotically stable point for the unperturbed problem

stochastic euler bernoulli beam driven by additive white noise global random attractors and global dynamics abstract this paper concerns with the long time dynamical behavior of a stochastic euler bernoulli beam driven by additive white noise by verifying the existence of absorbing set and obtaining the stabilization estimation of the dynamical system induced by the beam the existence of global random attractors that attracts all bounded sets in phase space is proved furthermore the finite hausdorff dimension for the global random attractors is attained in light of the relationship between global random attractor and random invariant probability measure the global dynamics of the beam are analyzed according to numerical simulation on global random basic attractors and global random point attractors

explosive solutions of stochastic wave equations with damping on r d a class of strong damped stochastic wave equations driven by multiplicative noises is considered in this article by using an appropriate energy inequality we shall give sufficient conditions such that the local solutions of the stochastic equations are blowup with positive probability or explosive in l2 sense

strong solution for a stochastic model of two dimensional second grade fluids existence uniqueness and asymptotic behavior abstract we investigate a stochastic evolution equation for the motion of a second grade fluid filling a bounded domain of r 2 global existence and uniqueness of strong probabilistic solution is established in contrast to previous results on this model we show that the sequence of galerkin approximation converges in mean square to the exact strong probabilistic solution of the problem we also give two results on the long time behavior of the solution mainly we prove that the strong solution of our stochastic model converges exponentially in mean square to the stationary solution of the time independent second grade fluids equations if the deterministic part of the external force does not depend on time if the deterministic forcing term explicitly depends on time then the strong probabilistic solution decays exponentially in mean square

large deviation for a 2d cahn hilliard navier stokes model under random influences abstract in this article we derive a large deviation principle for a 2d cahn hilliard navier stokes model under random influences the model consists of the navier stokes equations for the velocity coupled with a cahn hilliard equation for the order phase parameter the proof relies on the weak convergence method that was introduced in 3 4 5 and based on a variational representation on infinite dimensional brownian motion

blowup for the heat equation with a noise term in this paper we study blow up of the equation u t u xx u gamma dot w tx where dot w tx is a two dimensional white noise field and where dirichlet boundary conditions are enforced it is known that if 3 2 then the solution exists for all time in this paper we show that if is much larger than 3 2 then the solution blows up in finite time with positive probability we prove this by considering how peaks in the solution propagate if a peak of high mass forms we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass and these peaks evolve almost independently in this way we compare the evolution ofu to a branching process large peaks are regarded as particles in this branching process offspring are peaks which are higher by some factor we show that the expected number of offspring is greater than one when is much larger than 3 2 and thus the branching process survives with positive probability corresponding to blowup in finite time

on the stability of the abramov transfer for differential algebraic equations of index 1 the transfer of boundary conditions for ordinary differential equations developed by abramov zh vychisl mat mat fiz 1 1961 pp 542 545 is a stable method for representing the solution spaces of linear boundary value problems instead of boundary value problems matrix valued initial value problems are solved when integrating these differential equations the inner independence of the columns of the solution matrix and hence of the solutions of the resulting linear system of equations remains valid balla and marz have generalized abramov s transfer for homogenized index 1 differential algebraic equations siam j numer anal 33 1996 pp 2318 2332 in this article a direct version of the abramov transfer for inhomogeneous linear mbox index 1 differen linebreak 4 tial algebraic equations is developed and the numerical stability of this method is proved

exponential ergodicity of stochastic burgers equations driven by α stable processes in this work we prove the strong feller property and the exponential ergodicity of stochastic burgers equations driven by 2 subordinated cylindrical brownian motions with 1 2 to prove the results we truncate the nonlinearity and use the derivative formula for sdes driven by stable noises established in zhang in stoch process appl 123 4 1213 1228 2013

almost sure and moment lyapunov exponents for a stochastic beam equation the transverse vibrations of an euler bernoulli beam with axial tension p and axial white noise forcing are given by mytt m yt eiyxxxx pyxx 0yxxw t 0 x l with hinged endpoints y 0 t yxx 0 t y l t yxx l t 0 the solution may be written y x t n 1 sin n xl cn t where the amplitude cn t of the nth mode of vibration is given by a two dimensional linear stochastic ordinary differential equation the main results give formulas for the almost sure and moment lyapunov exponents for the spde in terms of the almost sure and moment lyapunov exponents for each of the sdes corresponding to modes which are present in the initial condition it is shown that the almost sure and moment lyapunov exponents for the spde depend sensitively on the initial distribution of energy amongst the infinitely many modes of vibration and that they cannot be well approximated using only finitely many modes of vibration

long time existence for the heat equation with a noise term we consider the equationu t u xx u w forx on a finite interval with dirichlet boundary conditions w is spacetime white noise the initial condition is continuous and nonnegative we show existence and uniqueness for all time provided 1 3 2

analysis of equilibrium states of markov solutions to the 3d navier stokes equations driven by additive noise we prove that every markov solution to the three dimensional navier stokes equations with periodic boundary conditions driven by additive gaussian noise is uniquely ergodic the convergence to the unique invariant measure is exponentially fast

large deviations for the invariant measure of a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the invariant measure of the non gaussian stochastic partial differential equation spde t v e v e f x v e e x v e ddot w tx here is a strongly elliptic second order operator with constant coefficients h dh xx h and the space variablex takes values on the unit circles 1 the functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic pde and in particular we require that 0 m m wherem andm are some finite positive constants the perturbationw is a brownian sheet it is well known that under some simple assumptions the solutionv 2 is ac k s 1 valued markov process for each 0 1 2 wherec s 1 is the banach space of real valued continuous functions ons 1 which are holder continuous of exponent we prove under some further natural assumptions onf and which imply that the zero element ofc s 1 is a globally exponentially stable critical point of the unperturbed equation t 0 0 f x 0 that e has a unique stationary distributionv k on c s 1 c k s 1 when the perturbation parameter e is small enough some further calculations show that as e tends to zero v k tends tov k 0 the point mass centered on the zero element ofc s 1 the main goal of this paper is to show that in factv k is governed by a large deviations principle ldp our starting point in establishing the ldp forv k is the ldp for the process e which has been shown in an earlier paper our methods of deriving the ldp forv k based on the ldp for e are slightly non standard compared to the corresponding proofs for finite dimensional stochastic differential equations since the state spacec s 1 is inherently infinite dimensional

using a new zero forcing process to guarantee the strong arnold property abstract the maximum nullity m g and the colin de verdiere type parameter g both consider the largest possible nullity over matrices in s g which is the family of real symmetric matrices whose i j entry i j is nonzero if i is adjacent to j and zero otherwise however g restricts to those matrices a in s g with the strong arnold property which means x o is the only symmetric matrix that satisfies a x o i x o and a x o this paper introduces zero forcing parameters z sap g and z vc g and proves that z sap g 0 implies every matrix a s g has the strong arnold property and that the inequality m g z vc g g holds for every graph g finally the values of g are computed for all graphs up to 7 vertices establishing g z g for these graphs

quasipotential for the ferromagnetic wire governed by the 1d landau lifshitz gilbert equations abstract we consider 1d landau lifshitz gilbert equations with an external force we first prove the existence and uniqueness of the strong solution and then give a definition of the quasipotential and prove that the quasipotential is equal to the potential energy of the system

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

uniqueness for stochastic non linear wave equations abstract we prove uniqueness of solutions of the equation u t t u f u g u w u 0 u 0 u t 0 v 0 on r d in the class of processes with paths in h 1 here u 0 v 0 is a random variable in h 1 l 2 w is a spatially homogeneous wiener process with finite spectral measure and f g are locally lipschitz functions of subcritical growth

the inverse inertia problem for graphs cut vertices trees and a counterexample abstract let g be an undirected graph on n vertices and let s g be the set of all real symmetric n n matrices whose nonzero off diagonal entries occur in exactly the positions corresponding to the edges of g the inverse inertia problem for g asks which inertias can be attained by a matrix in s g we give a complete answer to this question for trees in terms of a new family of graph parameters the maximal disconnection numbers of a graph we also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs finally we give an example of a graph that is not inertia balanced which settles an open problem from the october 2006 aim workshop on spectra of families of matrices described by graphs digraphs and sign patterns we also determine some restrictions on the inertia set of any graph

exponential mixing for stochastic pdes the non additive case we establish a general criterion which ensures exponential mixing of parabolic stochastic partial differential equations spde driven by a non additive noise which is white in time and smooth in space we apply this criterion on two representative examples 2d navier stokes ns equations and complex ginzburg landau cgl equation with a locally lipschitz noise due to the possible degeneracy of the noise doob theorem cannot be applied hence a coupling method is used in the spirit of kuksin and shirikyan j math pures appl 1 567 602 2002 and mattingly commun math phys 230 421 462 2002

stability in mean of partial variables for stochastic reaction diffusion systems abstract it is with the aid of the lyapunov method that stochastic ordinary differential equations and stochastic functional differential equations have been investigated intensively in the past decades however for stochastic reaction diffusion equations this useful technique seems to feel helpless on account of the emptiness of its own ito s formula to get over this difficulty we will manage to regard the integral of the considered trajectory with respect to spatial variables as the solution of the corresponding stochastic ordinary differential equations and then employ the ito formula to study stability in mean of partial variables for ito stochastic reaction diffusion equations some sufficient conditions for uniform stability in mean asymptotic stability in mean uniformly asymptotic stability in mean exponential stability in mean of partial variables are presented

stochastic wave equation of pure jumps existence uniqueness and invariant measures abstract in this paper we investigate a class of nonlinear damped stochastic hyperbolic equations with jumps the jump component considered here is described as a poisson point process this paper is divided into two parts the first part deals with existence and uniqueness of global weak and strong solutions to this type of equations based on the energy approach the second part devotes to the existence and support of invariant measures corresponding to the weak solution semi group based on markov property of the solution

explosive solutions of stochastic nonlinear beam equations with damping abstract this study considers a class of damped stochastic nonlinear beam equations driven by multiplicative noise by an appropriate energy inequality we provide sufficient conditions such that the local solutions of the stochastic equations blow up with a positive probability or are explosive in an l 2 sense we also derive estimates of the upper bound of the blow up time

stochastic boundary control design for extensible marine risers in three dimensional space this paper presents a new design of boundary controllers for global practical k exponential p stabilization of vibration motions of extensible marine risers in three dimensional 3d space under both stochastic and deterministic sea loads the control design and analysis of well posedness and stability of the closed loop system are carried out based on a new lyapunov type theorem which is developed for studying well posedness and p stability of a class of stochastic evolution systems sess in hilbert space since this theorem eases difficulties in verification of the coercivity condition but requires conditions of a form similar to lyapunov type theorems for stochastic lumped parameter systems it has a potential application to other stochastic distributed parameter systems

stabilization by noise for a class of stochastic reaction diffusion equations we prove uniqueness ergodicity and strongly mixing property of the invariant measure for a class of stochastic reaction diffusion equations with multiplicative noise in which the diffusion term in front of the noise may vanish and the deterministic part of the equation is not necessary asymptotically stable to this purpose we show that the l1 norm of the difference of two solutions starting from any two different initial data converges a s to zero as time goes to infinity

elliptic equations for measures on infinite dimensional spaces and applications we introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces this concept allows one to consider equations whose coefficients are not globally integrable by using a suitably extended lyapunov function technique we derive a priori estimates for the solutions of such equations and prove new existence results as an application we consider stochastic burgers reaction diffusion and navier stokes equations and investigate the elliptic equations for the corresponding invariant measures our general theorems yield a priori estimates and existence results for such elliptic equations we also obtain moment estimates for gibbs distributions and prove an existence result applicable to a wide class of models

stochastic conformal schemes for damped stochastic klein gordon equation with additive noise abstract in this article stochastic conformal schemes of the damped stochastic klein gordon equation with additive noise are studied it is shown that this equation possesses the stochastic conformal multi symplectic conservation law under appropriate boundary conditions the global momentum evolution law and the global energy evolution law are proposed we chiefly develop the stochastic conformal preissman scheme the stochastic conformal discrete gradient scheme and the stochastic conformal euler box scheme to preserve the geometric structures of the original system specifically we make theoretical discussions on the three proposed schemes to obtain corresponding discrete conservation laws or discrete evolution laws then the damped stochastic linear klein gordon equation and the damped stochastic nonlinear klein gordon equation with cubic nonlinearity are taken as examples to demonstrate the validity of the proposed schemes through numerical experiments and comparisons the superiorities of the proposed schemes are fully shown which are consistent with our theoretical analysis moreover the mean square convergence orders of the three stochastic conformal schemes in time direction and space direction are tested numerically

global existence and explosion of the stochastic viscoelastic wave equation driven by multiplicative noises in this paper we discuss an initial boundary value problem of stochastic viscoelastic wave equation driven by multiplicative noise involving the nonlinear damping term utq 2ut and a source term of the type up 2u we first establish the local existence and uniqueness of solution by the iterative technique truncation function method moreover we also show that the solution is global for q p lastly by modifying the energy functional we give sufficient conditions such that the local solution of the stochastic equations will blow up with positive probability or explode in energy sense for p q

asymptotic compactness and absorbing sets for 2d stochastic navier stokes equations on some unbounded domains we introduce a notion of an asymptotically compact ac random dynamical system rds we prove that for an ac rds the limit set b of any bounded set b is nonempty compact strictly invariant and attracts the set b we establish that the 2d navier stokes equations nses in a domain satisfying the poincare inequality perturbed by an additive irregular noise generate an ac rds in the energy space h as a consequence we deduce existence of an invariant measure for such nses our study generalizes on the one hand the earlier results by flandoli crauel 1994 and schmalfuss 1992 obtained in the case of bounded domains and regular noise and on the other hand the results by rosa 1998 for the deterministic nses

stochastic compressible euler equations and inviscid limits abstract we prove the existence of a unique local strong solution to the stochastic compressible euler system with nonlinear multiplicative noise this solution exists up to a positive stopping time and is strong in both the pde and probabilistic sense based on this existence result we study the inviscid limit of the stochastic compressible navier stokes system as the viscosity tends to zero any sequence of finite energy weak martingale solutions converges to the compressible euler system

lower estimates of transition densities and bounds on exponential ergodicity for stochastic pde s a formula for the transition density of a markov process defined by an infinite dimensional stochastic equation is given in terms of the ornstein uhlenbeck bridge and a useful lower estimate on the density is provided as a consequence uniform exponential ergodicity and v ergodicity are proved for a large class of equations we also provide computable bounds on the convergence rates and the spectral gap for the markov semigroups defined by the equations the bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients examples of finite dimensional stochastic equations and semilinear parabolic equations are given

invariant measures for stochastic nonlinear beam and wave equations existence of an invariant measure for a stochastic extensible beam equation and for a stochastic damped wave equation with polynomial nonlinearities is proved it is shown first that the corresponding transition semigroups map the space of all bounded sequentially weakly continuous functions on the state space into itself and then by a lyapunov functions approach solutions bounded in probability are found

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

stochastic nonlinear beam equations an extensible beam equation with a stochastic force of a white noise type is studied lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution

stochastic exponential integrator for finite element spatial discretization of stochastic elastic equation we investigate the strong convergence of a full discretization for the stochastic elastic equation driven by multiplicative noise finite element method is used for the spatial approximation and an exponential integrator scheme is proposed for the temporal discretization we show that the fully discrete scheme strongly converges to the mild solution of the continuous problem with precise convergence rates given as shown in our result the exponential integrator scheme achieves a higher strong convergence rate than the backward euler method does in the literature

fractional stochastic volterra equation perturbed by fractional brownian motion in this paper we consider a class of fractional stochastic volterra equation of convolution type driven by infinite dimensional fractional brownian motion with hurst index h 0 1 base on the explicit formula for the scalar resolvent function and the properties of the mittag leffler s function the existence and regularity results of the stochastic convolution process are established separate proofs are required for the cases of hurst parameter above and below 1 2 and it will turn out that the regularity of the solution increases with hurst parameter h

kolmogorov equation associated to a stochastic navier stokes equation abstract a direct solution of the kolmogorov equation associated to a stochastic navier stokes equation is given with restriction to two space dimensions and periodic boundary conditions the existence of a variational solution is proved using a special property of the nonlinear operator

well posedness for stochastic camassa holm equation abstract the camassa holm equation describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity due to uncertainty in the modelling and external environment this modelling could be subject to random fluctuations in this article the stochastic camassa holm equation with additive noise is considered using regularization a local existence and uniqueness result in the sobolev space h s r with s 3 2 of stochastic camassa holm equation is obtained with the help of priori estimates the local solution will blow up in h q r in finite time for any q 3 2 when initial value satisfies some conditions

a coupling approach to randomly forced nonlinear pde s ii we consider a class of discrete time random dynamical systems and establish the exponential convergence of its trajectories to a unique stationary measure the result obtained applies in particular to the 2d navier stokes system and multidimensional complex ginzburg landau equation with random kick force

asymptotic behavior of stochastic discrete wave equations with nonlinear noise and damping in this article we study the asymptotic behavior for a class of discrete wave equations with nonlinear noise and damping defined on a k dimensional integer set the well posedness of the system is established when the nonlinear drift function and the nonlinear diffusion term are only locally lipschitz continuous the mean random dynamical system associated with the non autonomous system is shown to possess a unique tempered weak pullback mean random attractor in l2 f l2 l2 the existence of invariant measures for the autonomous system is also derived by using the krylov bogolyubov method the difficulty in proving the tightness of a family of distribution laws of the solutions is overcome by using the idea of uniform estimates on the tails of solutions

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

the inertia set of a signed graph abstract a signed graph is a pair g where g v e is a graph in which parallel edges are permitted but loops are not with v 1 n and e the edges in are called odd edges and the other edges of e even by s g we denote the set of all symmetric v v matrices a a i j with a i j 0 if i and j are adjacent and all edges between i and j are even a i j 0 if i and j are adjacent and all edges between i and j are odd a i j r if i and j are connected by even and odd edges a i j 0 if i j and i and j are non adjacent and a i i r for all vertices i the stable inertia set of a signed graph g is the set of all pairs p q for which there exists a matrix a s g with p positive and q negative eigenvalues which has the strong arnold property in this paper we study the stable inertia set of signed graphs

stability and stabilizability of infinite dimensional systems in this paper we are concerned with stability problems for infinite dimensional systems first we review the theory for linear systems where the dynamics are governed by strongly continuous semigro

ergodicity and exponential mixing of the real ginzburg landau equation with a degenerate noise abstract in this paper we establish the existence uniqueness and attraction properties of an invariant measure for the real ginzburg landau equation in the presence of a degenerate stochastic forcing acting only in four directions the main challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system to achieve this we propose a condition which only requires four noises

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

on dichotomy and well conditioning in bvp we investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand it is shown that if these stability bounds are moderate i e if the problem is well conditioned then the homogeneous solution space is dichotomic which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes this is done by carefully examining the green s functions if these exhibit an exponential behaviour then the solution space is also exponentially dichotomic on the other hand we also show that exponential dichotomy implies moderate stability constants i e well conditioning from this it follows that both concepts are more or less equivalent

forbidden minors for the class of graphs g with ξ g 2 abstract for a given simple graph g s g is defined to be the set of real symmetric matrices a whose i j th entry is nonzero whenever i j and ij is an edge in g in f barioli s fallat l hogben a variant on the graph parameters of colin de verdiere implications to the minimum rank of graphs electron j linear algebra 13 2005 387 404 g is defined to be the maximum corank i e nullity among a s g having the strong arnold property is used to study the minimum rank maximum eigenvalue multiplicity problem for g since is minor monotone the graphs g such that g k can be described by a finite set of forbidden minors we determine the forbidden minors for g 2 and present an application of this characterization to computation of minimum rank among matrices in s g

random perturbations of reaction diffusion equations the quasi deterministic approximation on considere des champs aleatoires u e t x u 1 e t x u n e t x definis comme les solutions d un systeme d equations aux derivees partielles u k e t l k u k e f k x u 1 e u n e e k t x les l k sont des operateurs elliptiques lineaires d ordre 2 k sont des champs gaussiens bruit blanc

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

a dynamic solver consistent minimum action method this paper discusses the necessity and strategy to unify the development of a dynamic solver and a minimum action method mam for a spatially extended system when employing the large deviation principle ldp to study the effects of small random perturbations a dynamic solver is used to approximate the unperturbed system and a minimum action method is used to approximate the ldp which corresponds to solving an euler lagrange equation related to but more complicated than the unperturbed system we will clarify possible inconsistencies induced by independent numerical approximations of the unperturbed system and the ldp based on which we propose to define both the dynamic solver and the mam on the same approximation space for spatial discretization the semi discrete ldp can then be regarded as the exact ldp of the semi discrete unperturbed system which is a finite dimensional ode system we achieve this methodology for the two dimensional navier stokes equations using a divergence free approximation space the method developed can be used to study the nonlinear instability of wall bounded parallel shear flows and be generalized straightforwardly to three dimensional cases numerical experiments are presented

large deviations for stochastic nonlinear beam equations abstract we establish a large deviation principle for the solutions of stochastic partial differential equations for nonlinear vibration of elastic panels also called stochastic nonlinear beam equations

stochastic equations in hilbert space with application to navier stokes equations in any dimension abstract we give an existence theorem for an abstract nonlinear stochastic evolution equation in a hilbert space the result is applicable to the stochastic navier stokes equation in any dimension with a nonlinear noise term

two dimensional navier stokes equations driven by a space time white noise abstract we study the two dimensional navier stokes equations with periodic boundary conditions perturbed by a space time white noise it is shown that although the solution is not expected to be smooth the nonlinear term can be defined without changing the equation we first construct a stationary martingale solution then we prove that for almost every initial data with respect to a measure supported by negative spaces there exists a unique global solution in the strong probabilistic sense

stationary solutions to the compressible navier stokes system driven by stochastic forces we study the long time behavior of solutions to a stochastically driven navier stokes system describing the motion of a compressible viscous fluid driven by a temporal multiplicative white noise perturbation the existence of stationary solutions is established in the framework of lebesgue sobolev spaces pertinent to the class of weak martingale solutions the methods are based on new global in time estimates and a combination of deterministic and stochastic compactness arguments an essential tool in order to obtain the global in time estimate is the stationarity of solutions on each approximation level which provides a certain regularizing effect in contrast with the deterministic case where related results were obtained only under rather restrictive constitutive assumptions for the pressure the stochastic case is tractable in the full range of constitutive relations allowed by the available existence theory due to the underlying martingale structure of the noise

variational solutions of dissipative jump type stochastic evolution equations we deal with existence and uniqueness of variational solutions to a class of dissipative stochastic evolution equations driven by general levy processes furthermore we prove existence and uniqueness of the invariant measure and the existence of an invariant set associated with the solutions under mild conditions respectively

on a global solution and asymptotic behaviour for the generalized damped extensible beam equation abstract we prove the existence and uniqueness of global solutions for the cauchy problem concerning the evolution equation u a 2 u m a 1 2 u 2 h au g u 0 suggested by the study of plates and beams where a is a linear operator in a hilbert space h and m and g are real functions we also study the asymptotic behaviour of the solutions under suitable growth assumptions on the nonlinear term g

field theoretic thermodynamic uncertainty relation general formulation exemplified with the kardar parisi zhang equation we propose a field theoretic thermodynamic uncertainty relation as an extension of the one derived so far for a markovian dynamics on a discrete set of states and for overdamped langevin equations we first formulate a framework which describes quantities like current entropy production and diffusivity in the case of a generic field theory we will then apply this general setting to the one dimensional kardar parisi zhang equation a paradigmatic example of a non linear field theoretic langevin equation in particular we will treat the dimensionless kardar parisi zhang equation with an effective coupling parameter measuring the strength of the non linearity it will be shown that a field theoretic thermodynamic uncertainty relation holds up to second order in a perturbation expansion with respect to a small effective coupling constant the calculations show that the field theoretic variant of the thermodynamic uncertainty relation is not saturated for the case of the kardar parisi zhang equation due to an excess term stemming from its non linearity

large deviations principle for the invariant measures of the 2d stochastic navier stokes equations on a torus abstract we prove here the validity of a large deviation principle for the family of invariant measures associated to a two dimensional navier stokes equation on a torus perturbed by a smooth additive noise

undamped nonlinear beam excited by additive l2 regular noise abstract stability of a nonlinear elastic beam excited by a space time dependent nonrandom force or a l 2 regular random noise f x t white in time with general spatial covariance is considered it is supposed that the transversal displacement y of the beam with length l is governed by the quasilinear partial differential equation pde 0 h y t t d y x x x x n 0 e 0 l y x 2 d x y x x k y f x t 0 x l t 0 as a main result we show that its expected energy is linearly bounded at time t in spite of the presence of additive l 2 regular space time noise and cubic type nonlinearities appropriate partial implicit discretization with similar qualitative behavior consistency and stability are discussed as well the technique of lyapunov type functionals processing the information on the related energy e is exploited we compare our results to those for the linear beam in both cases the expected energy e e t is nondecreasing and growing linearly in time t in fact for all nonrandom t s and additive l 2 regular noise f with covariance operator q f with finite trace it satisfies the trace formula e e t e e s trace q f t s 2

exponential mixing of the 2d stochastic navier stokes dynamics we consider the navier stokes equation on a two dimensional torus with a random force which is white noise in time and excites only a finite number of modes the number of excited modes depends on the viscosity v and grows like v 3 when v goes to zero we prove that this markov process has a unique invariant measure and is exponentially mixing in time

on numerical differential algebraic problems with application to semiconductor device simulation this paper considers questions regarding conditioning of and numerical methods for certain differential algebraic equations subject to initial and boundary conditions the approach taken is that of separating differential and algebraic solution components at least theoretically this yields conditioning results for differential algebraic boundary value problems in terms of pure differential problems for which existing theory is well developed the process is carried out for problems with global index 1 or 2 for semi explicit boundary value problems of index 1 where solution components are separated a convergence theorem is given for a special class of collocation methods for general index 1 problems advantages and disadvantages of certain symmetric difference schemes are discussed for initial value problems with index 2 the use of bdf schemes is examined with a summary of conditions for their successful and stable utilization finally the present considerations and analysis are applied to

asymptotic behavior for second order stochastic evolution equations with memory abstract in this paper a general second order integro differential evolution equation with memory driven by multiplicative noise is considered we prove the existence of global mild solution and asymptotic stability of the zero solution using lyapunov function techniques moreover we discuss three examples to show that the asymptotic stability results can be applied to various partial differential equations

exponential ergodicity for stochastic burgers and 2d navier stokes equations abstract it is shown that transition measures of the stochastic navier stokes equation in 2 d converge exponentially fast to the corresponding invariant measures in the distance of total variation as a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state transformation analogous results are proved for the stochastic burgers equation