dynamics of the nonlinear stochastic heat equation with singular perturbation the approximation in probability for a singular perturbed nonlinear stochastic heat equation is studied first the approximation result in the sense of probability is obtained for solutions defined on any finite time interval furthermore it is proved that the long time behavior of the stochastic system is described by a global random attractor which is upper semi continuous with respect to the singular perturbed parameter this also means the long time effectivity of the approximation with probability one

regular random attractors for non autonomous stochastic evolution equations with time varying delays on thin domains this paper deals with the limiting dynamical behavior of non autonomous stochastic reaction diffusion equations with time varying delays on thin domains first we prove the existence and uniqueness of the regular random attractor then we prove the upper semicontinuity of the regular random attractors for the equations on a family of n 1 dimensional thin domains that collapses onto an n dimensional domain

approximate the dynamical behavior for stochastic systems by wong zakai approaching abstract random invariant manifolds and foliations play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations in a general way these random objects are difficult to be visualized geometrically or computed numerically the current work provides a perturbation approach to approximate these random invariant manifolds and foliations after briefly discussing the existence of random invariant manifolds and foliations for a class of stochastic systems driven by additive noises the corresponding wong zakai type of convergence result in path wise sense is established

an orthogonal spline collocation alternating direction implicit crank nicolson method for linear parabolic problems on rectangles we formulate an alternating direction implicit crank nicolson scheme for solving a general linear variable coefficient parabolic problem in nondivergence form on a rectangle with the solution subject to nonhomogeneous dirichlet boundary condition orthogonal spline collocation with piecewise hermite bicubics is used for spatial discretization we show that for sufficiently small time stepsize the scheme is stable and of optimal order accuracy in time and the h1 norm in space we also describe an efficient implementation of the scheme and present numerical results demonstrating the accuracy and convergence rates in various norms

dynamics of stochastic 2d navier stokes equations abstract in this paper we study the dynamics of a two dimensional stochastic navier stokes equation on a smooth domain driven by linear multiplicative white noise we show that solutions of the 2d navier stokes equation generate a perfect and locally compacting c 1 1 cocycle using multiplicative ergodic theory techniques we establish the existence of a discrete non random lyapunov spectrum for the cocycle the lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium stationary solution we give sufficient conditions on the parameters of the navier stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium uniqueness of the stationary solution viz ergodicity local almost sure asymptotic stability of the cocycle and the existence of global invariant foliations of the energy space

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

existence and upper semicontinuity of random attractors for non autonomous stochastic strongly damped sine gordon equation on unbounded domains abstract in this paper we study the asymptotic behavior of solutions of the non autonomous stochastic strongly damped sine gordon equation driven by multiplicative noise defined on an unbounded domain first we introduce a continuous cocycle for the equation and establish the pullback asymptotic compactness of solutions second we consider the existence and upper semicontinuity of random attractors for the equation

alternating direction collocation for separable elliptic partial differential equations a complete theory is given for the convergence of alternating direction collocation adc applied to a large class of linear separable elliptic partial differential equations on a rectangular domain it is shown that adc using bicubic hermite splines converges in this setting and converges to the full collocation solution moreover it is shown that a complex embedding of the problem is essential to the theory and that the associated eigensplines and generalized eigensplines can be complex with complex eigenvalues whose geometric and algebraic multiplicities need not be equal nonetheless only real computations are required

an impact of noise on invariant manifolds in nonlinear dynamical systems invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems however these nonlinear systems are often subject to random fluctuations or noises it is thus desirable to quantify the impact of noises on the invariant manifolds when the noise intensity is small this impact is estimated via asymptotic analysis in the context of liapunov perron formulation namely the random invariant manifold is represented as a perturbation of the deterministic invariant manifold with a well defined bound for the deviation

existence stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations abstract in this paper we study the existence stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise we first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions we then examine the existence and stability of random complete quasi solutions and establish the relations of these solutions and the structures of tempered attractors when the stochastic equations are incorporated with periodic forcing we obtain the existence and stability of random periodic solutions for the stochastic chafee infante equation we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions

analyses of spline collocation methods for parabolic and hyperbolic problems in two space variables optimal order error estimates are derived for the continuous time collocation method and a discrete time collocation method the crank nicolson collocation method for approximating the solution of a semilinear parabolic initial boundary value problem in r times 0 t where r is the unit square in each case at each time level the approximation is a c 1 piecewise polynomial of degree r geqq 3 defined by collocation at gauss points in r optimal order error estimates are also derived for a family of discrete time collocation methods for a semilinear second order hyperbolic initial boundary value problem in r times 0 t

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

a random fixed point theorem and the random graph transformation in the first part of this article we formulate a random fixed point theorem for random dynamical systems where a contraction condition is formulated as an expectation of a particular expression in the following we are going to formulate a random graph transformation this transformation defines a random dynamical system under certain conditions this random dynamical system has a random fixed point which can be found by the mentioned random fixed point theorem in one application we show that random dynamical systems given by particular differential equations have a random unstable invariant manifold

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

pullback attractors for reaction diffusion delay equations on unbounded domains with non autonomous deterministic and stochastic forcing terms the existence of a pullback attractor in a space of higher regularity is established for the non autonomous non compact dynamical system associated with the reaction diffusion delay equation with both non autonomous deterministic and stochastic forcing terms defined on all n dimensional space the asymptotical compactness of the solutions is demonstrated by using the uniform estimates for far field values of solutions and a new method

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

slow foliation of a slow fast stochastic evolutionary system abstract this work is concerned with the dynamics of a slow fast stochastic evolutionary system quantified with a scale parameter an invariant foliation decomposes the state space into geometric regions of different dynamical regimes and thus helps understand dynamics a slow invariant foliation is established for this system it is shown that the slow foliation converges to a critical foliation i e the scale parameter is zero in probability distribution as the scale parameter tends to zero the approximation of slow foliation is also constructed with error estimate in distribution furthermore the geometric structure of the slow foliation is investigated every fiber of the slow foliation parallels each other with the slow manifold as a special fiber in fact when an arbitrarily chosen point of a fiber falls in the slow manifold the fiber must be the slow manifold itself

geometric shape of invariant manifolds for a class of stochastic partial differential equationsa invariant manifolds play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations however the geometric shape of these manifolds is largely unclear the purpose of the present paper is to try to describe the geometric shape of invariant manifolds for a class of stochastic partial differential equations with multiplicative white noises the local geometric shape of invariant manifolds is approximated which holds with significant likelihood furthermore the result is compared with that for the corresponding deterministic partial differential equations

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

wong zakai approximations and center manifolds of stochastic differential equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of the stochastic differential equation driven by a l dimensional brownian motion we prove that the solutions of wong zakai approximations converge in the mean square to the solutions of the stratonovich stochastic differential equation we also show that for a simple multiplicative noise the center manifold of the wong zakai approximations converges to the center manifold of the stratonovich stochastic differential equation

slow manifold for a nonlocal stochastic evolutionary system with fast and slow components abstract this work is devoted to investigating invariant manifolds for a fast slow stochastic evolutionary system with nonlocal diffusion we establish the slow reduction via a random slow manifold which captures slow dynamics of the original stochastic fast slow system a simple example is shown to illustrate this slow reduction method

on growth rates of sub additive functions for semi flows determined and random cases abstract let r m m be a measurable random dynamical systems on the compact metric space m over f p t t r with time r let m p and e p denote the set of all invariant measures on m and the set of all ergodic invariant measures whose marginal on coincide with p respectively a function f r m r is sub additive with respect to if f t s x f t x f s t t x we define the maximal growth rate of f to be lim sup t 1 t sup x m f t x for p a e it is shown that it is equal to max m p where lim t 1 t m f t x d and there exists e p such that max m p the result may have some applications in the study of the dynamical spectrum of infinite dimension random dynamical systems and robust permanence for differential equations

autonomous and non autonomous attractors for differential equations with delays abstract the asymptotic behaviour of some types of retarded differential equations with both variable and distributed delays is analysed in fact the existence of global attractors is established for different situations with and without uniqueness and for both autonomous and non autonomous cases using the classical notion of attractor and the recently new concept of pullback one respectively

random periodic solutions of random dynamical systems abstract in this paper we give the definition of the random periodic solutions of random dynamical systems we prove the existence of such periodic solutions for a c 1 perfect cocycle on a cylinder using a random invariant set the lyapunov exponents and the pullback of the cocycle

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

global well posedness of stochastic 2d primitive equations with random initial conditions abstract in this article we consider 2d stochastic primitive equations pes driven by affine linear multiplicative white noise and with random initial conditions we obtain the global well posedness of the stochastic pes when the random initial condition satisfies sufficient malliavin regularity in the proof process of the global existence of solutions to 2d stochastic pes the malliavin calculus plays a key role

unstable invariant manifolds for stochastic pdes driven by a fractional brownian motion abstract in this paper we consider a class of stochastic partial differential equations spdes driven by a fractional brownian motion fbm with the hurst parameter bigger than 1 2 the existence of local random unstable manifolds is shown if the linear parts of these spdes are hyperbolic for this purpose we introduce a modified lyapunov perron transform which contains stochastic integrals by the singularities inside these integrals we obtain a special lyapunov perron s approach by treating a segment of the solution over time interval 0 1 as a starting point and setting up an infinite series equation involving these segments as time evolves using this approach we establish the existence of local random unstable manifolds in a tempered neighborhood of an equilibrium

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

flattening squeezing and the existence of random attractors the study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems in the deterministic case the properties of flattening and squeezing in infinite dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor the flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories it is shown here that the flattening property is implied by the squeezing property and is in fact weaker since the attractor in a system with the flattening property can be infinite dimensional whereas it is always finite dimensional in a system with the squeezing property the flattening property is then generalized to random dynamical systems for which it is called the pullback flattening property it is shown to be weaker than the random squeezing property but equivalent to pullback asymptotic compactness and pullback limit set compactness and thus implies the existence of a random attractor the results are also valid for deterministic non autonomous dynamical systems formulated as skew product flows

finite difference preconditioning for solving orthogonal collocation equations for boundary value problems a technique to construct a low order finite difference preconditioner for solving orthogonal collocation equations for boundary value problems is presented it is shown numerically and theoretically that the spectral condition numbers of the preconditioned collocation matrices are bounded by constants independent of the number of mesh nodes when certain exact low order finite difference preconditionings are used preconditioners based on incomplete lu factorization are also discussed numerical experiments show the efficiency and robustness of the preconditioning

invariant manifolds for stochastic partial differential equations invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems the theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature in this paper we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations we first introduce a random graph transform and a fixed point theorem for nonautonomous systems then we show the existence of generalized fixed points which give the desired invariant manifolds

anticipating random periodic solutions i sdes with multiplicative linear noise in this paper we study the existence of random periodic solutions for semilinear stochastic differential equations we identify them as solutions of coupled forward backward infinite horizon stochastic integral equations ihsies using the substitution theorem of stochastic differential equations with anticipating initial conditions in general random periodic solutions and the solutions of ihsies are anticipating for the linear noise case with the help of the exponential dichotomy given in the multiplicative ergodic theorem we can identify them as the solutions of infinite horizon random integral equations ihsies we then solve a localised forward backward ihrie in c r lloc2 using an argument of truncations the malliavin calculus the relative compactness of wiener sobolev spaces in c 0 t l2 c 0 t l2 and schauder s fixed point theorem we finally measurably glue the local solutions together to obtain a global solution in c r l2 c r l2 thus we obtain the existence of a random periodic solution and a periodic measure

upper semicontinuity of attractors of stochastic delay reaction diffusion equations in the delay a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero

a wong zakai approximation for random invariant manifolds random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences but these random objects are difficult to be visualized geometrically or computed numerically the current work provides a perturbation approach to approximate these random invariant manifolds we first discuss the existence of a random invariant manifold for a class of stochastic evolutionary equations then we approximate the random invariant manifold by the invariant manifold of a new system with smooth colored noise i e integrated ornstein uhlenbeck processes the convergence in a pathwise wong zakai sense is shown

pullback attractors for multi valued non compact random dynamical systems generated by reaction diffusion equations on an unbounded domain in this paper we study pullback attractors of reaction diffusion equations on an unbounded domain with non autonomous deterministic as well as stochastic forcing terms for which the uniqueness of solutions need not hold we first present the existence and structure of pullback attractors of multi valued non compact random dynamical systems then we prove the existence of pullback attractors in l2 rn lp rn and h1 rn for multi valued non compact random dynamical systems associated with the reaction diffusion equations on rn and the identical relation of pullback attractors in different spaces is also provided in particular the measurability of pullback attractors is established for reaction diffusion equations on rn

weak solutions to the heat flow for surfaces of prescribed mean curvature in this talk we establish the existence of global weak solutions to the heat flow for surfaces of prescribed mean curvature i e the existence for the cauchy dirichlet problem to parabolic systems of the type tu u 2 h u d1u d2u in b 0 u uo on par b 0 where h r3 r is a bounded continuous function satisfying an isoperimetric condition b the unit ball in r2 and u b 0 r3 as one of the possible applications we show that the problem has a solution with values inbr r3 whenever uo b br and furthermore there holds br h 3 2r h d 9 2 h a 1 r for a br the results that will be presented in the talk are joint work with verena bogelein und chrsitoph scheven from erlangen frank duzaar department of mathematics university erlangen nurnberg cauerstr 11 91056 erlangen germany e mail address frank duzaar zuv uni erlangen de date december 29 2011 1

an elementary partial regularity proof for vector valued obstacle problems soit r n et m r n des ensembles ouverts bornes on suppose n 3 n 2 et on requiert que la frontiere de m est une hyper surface lisse de classe c 2 l espace de fonction h 1 2 m des fonctions de comparaison consiste en toutes les fonctions de sobolev v h 1 2 r n avec la propriete v x m pour presque tous les points x dans si u h 1 2 m minimise localement l energie alors il y a un sous ensemble ouvert 0 de tel que u c 1 0 r n pour tout 0 1 h n 2 0 0 et x 0 si et seulement si lim 1 0 r 2 n br x du 2 dz 0

smooth convergence of random center manifolds for spdes in varying phase spaces abstract in this paper we study the limiting behavior of center manifolds for a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces we first prove the existence and smoothness of random center manifolds for these equations then we establish the smooth convergence of these random center manifolds as the phase spaces approach to their singular limit

a dynamical approximation for stochastic partial differential equations random invariant manifolds provide geometric structures for understanding stochastic dynamics in this paper a dynamical approximation estimate is derived for a class of stochastic partial differential equations by showing that the random invariant manifold is almost surely asymptotically complete the asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold under suitable conditions as an application stationary states invariant measures are considered for a class of stochastic hyperbolic partial differential equations

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

alternating direction collocation for rectangular regions abstract an alternating direction procedure for finite element collocation equations can be formulated on rectangular regions finite element collocation offers a great savings over finite element techniques in the matrix formation stage of the numerical process moreover with an alternating direction procedure multidimensional problems can be solved as a series of one dimensional problems this greatly reduces the amount of computer time and storage required to solve large problems

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

invariant manifolds for stochastic wave equations abstract in this paper we consider a class of stochastic wave equations with nonlinear multiplicative noise we first show that these stochastic wave equations generate random dynamical systems or stochastic flows by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism then we establish the existence of random invariant manifolds for the random wave equations due to the temperedness of the nonlinearity we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases based on these random dynamical systems we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium finally we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations

concentration and limit behaviors of stationary measures abstract in this paper we study limit behaviors of stationary measures of the fokker planck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative noise including additive white noise case as the noises are vanishing various results on the invariance and concentration of the limit measures are obtained in particular we show that if the noise perturbed systems admit a uniform lyapunov function then the stationary measures form a relatively sequentially compact set whose weak limits are invariant measures of the unperturbed system concentrated on its global attractor in the case that the global attractor contains a strong local attractor we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor and on the contrary in the presence of a strong local repeller in the global attractor there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller moreover we show that if there is a strongly repelling equilibrium in the global attractor then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium as applications of these results an example of stochastic hopf bifurcation and an example with non decomposable limit sets are provided our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system

anticipating stochastic 2d navier stokes equations abstract in this article we consider a two dimensional stochastic navier stokes equation snse on a smooth bounded domain driven by affine linear multiplicative white noise and with random initial conditions and dirichlet boundary conditions the random initial condition is allowed to anticipate the forcing noise our main objective is to prove the existence and uniqueness of the solution to the snse under sufficient malliavin regularity of the initial condition to this end we employ anticipating calculus ideas

random attractor for stochastic reaction diffusion equation with multiplicative noise on unbounded domains abstract in this paper we study the asymptotic dynamics for stochastic reaction diffusion equation with multiplicative noise defined on unbounded domains we investigate the existence of a random attractor for the random dynamical system associated with the equation the asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far field values of solutions and a cut off technique

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory this series of papers is concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part of the series focuses on the development of general theory first the notions of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations for general positive random dynamical systems in ordered banach spaces are introduced which extend the classical notions of principal floquet subspaces principal lyapunov exponents and exponential separations for strongly positive deterministic systems in strongly ordered banach to general positive random dynamical systems in ordered banach spaces under some quite general assumptions it is then shown that a positive random dynamical system in an ordered banach space admits a family of generalized principal floquet subspaces a generalized principal lyapunov exponent and a generalized exponential separation we will consider in the forthcoming part s the applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations including random leslie matrix models random cooperative systems of ordinary differential equations and random parabolic equations institute of mathematics and computer science wroc law university of technology wybrzeze wyspianskiego 27 pl 50 370 wroc law poland url http www im pwr wroc pl mierczyn index html e mail address mierczyn pwr wroc pl department of mathematics and statistics auburn university auburn university al 36849 usa e mail address wenxish auburn edu 2010 mathematics subject classification primary 37h15 37l55 37a30 secondary 15b52 34f05 35r60

a stochastic pitchfork bifurcation in a reaction diffusion equation we study in some detail the structure of the random attractor for the chafee infante reaction diffusion equation perturbed by a multiplicative white noise d u u u u 3 d t u o d w t x d r m first we prove for m 5 a lower bound on the dimension of the random attractor which is of the same order in as the upper bound we derived in an earlier paper and is the same as that obtained in the deterministic case then we show for m 1 that as passes through 1 the first eigenvalue of the negative laplacian from below the system undergoes a stochastic bifurcation of pitchfork type we believe that this is the first example of such a stochastic bifurcation in an infinite dimensional setting central to our approach is the existence of a random unstable manifold

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

p attracting and p invariant sets for a class of impulsive stochastic functional differential equations in this paper a nonlinear and nonautonomous impulsive stochastic functional differential equation is considered by establishing an l operator differential inequality and stochastic analysis technique we obtain the p attracting set and p invariant set of the impulsive stochastic functional differential equation an example is also discussed to illustrate the efficiency of the obtained results

asymptotic behaviour of nonautonomous difference inclusions abstract nonautonomous difference inclusions are formulated as difference inclusion cocycles which generalize set valued semigroups corresponding to autonomous difference inclusions and the existence of a difference inclusion cocycle attractor is established under the assumption of upper semi continuity of the cocycle mappings if the difference inclusion cocycle has an absorbing set several illustrative examples are provided

the wong zakai approximations of invariant manifolds and foliations for stochastic evolution equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of a class of stochastic evolution equations with a multiplicative white noise we prove that the solutions of wong zakai approximations almost surely converge to the solutions of the stratonovich stochastic evolution equation we also show that the invariant manifolds and stable foliations of the wong zakai approximations converge to the invariant manifolds and stable foliations of the stratonovich stochastic evolution equation respectively