periodic nonlinear schrodinger equation and invariant measures in this paper we continue some investigations on the periodic nlseiu u iu xx u u p 2 p 6 started in lrs we prove that the equation is globally wellposed for a set of data of full normalized gibbs measrue e beta h phi hd phi x h phi tfrac 1 2 int left phi right 2 tfrac 1 p int left phi right p after suitablel 2 truncation the set and the measure are invariant under the flow the proof of a similar result for the kdv and modified kdv equations is outlined the main ingredients used are some estimates from b1 on periodic nls and kdv type equations

unique ergodicity for a class of stochastic hyperbolic equations with additive space time white noise in this paper we consider a certain class of second order nonlinear pdes with damping and space time white noise forcing posed on the d dimensional torus this class includes the wave equation for d 1 and the beam equation for d le 3 we show that the gibbs measure is the unique invariant measure for this system since the flow does not satisfy the strong feller property we introduce a new technique for showing unique ergodicity this approach may be also useful in situations in which finite time blowup is possible

a microscopic derivation of time dependent correlation functions of the 1d cubic nonlinear schrodinger equation abstract we give a microscopic derivation of time dependent correlation functions of the 1d cubic nonlinear schrodinger equation nls from many body quantum theory the starting point of our proof is 11 on the time independent problem and 16 on the corresponding problem on a finite lattice an important new obstacle in our analysis is the need to work with a cutoff in the number of particles which breaks the gaussian structure of the free quantum field and prevents the use of the wick theorem we overcome it by means of complex analytic methods our methods apply to the nonlocal nls with bounded convolution potential in the periodic setting we also consider the local nls arising from short range interactions in the many body setting to that end we need the dispersion of the nls in the form of periodic strichartz estimates in x s b spaces

modulation spaces wiener amalgam spaces and brownian motions abstract we study the local in time regularity of the brownian motion with respect to localized variants of modulation spaces m s p q and wiener amalgam spaces w s p q we show that the periodic brownian motion belongs locally in time to m s p q t and w s p q t for s 1 q 1 and the condition on the indices is optimal moreover with the wiener measure on t we show that m s p q t and w s p q t form abstract wiener spaces for the same range of indices yielding large deviation estimates we also establish the endpoint regularity of the periodic brownian motion with respect to a besov type space b p s t specifically we prove that the brownian motion belongs to b p s t for s 1 p 1 and it obeys a large deviation estimate finally we revisit the regularity of brownian motion on usual local besov spaces b p q s and indicate the endpoint large deviation estimates

optimal local well posedness for the periodic derivative nonlinear schrodinger equation we prove local well posedness for the periodic derivative nonlinear schrodinger equation which is l 2 critical in fourier lebesgue spaces which scale like h s mathbb t for s 0 our result is optimal in the sense that it covers the full subcritical regime in particular we close the existing gap in the subcritical theory by improving the result of grunrock and herr siam j math anal 39 6 1890 1920 2008 which established local well posedness in fourier lebesgue spaces which scale like h s mathbb t for s frac 1 4 we achieve this result by a delicate analysis of the structure of the solution and the construction of an adapted nonlinear submanifold of a suitable function space together these allow us to construct the unique solution to the given subcritical data this constructive procedure is inspired by the theory of para controlled distributions developed by gubinelli et al forum math pi 3 75 2015 and catellier and chouk ann probab 46 5 2621 2679 2018 in the context of stochastic pde our proof and results however are purely deterministic

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

local times for solutions of the complex ginzburg landau equation and the inviscid limit abstract we consider the behaviour of the distribution for stationary solutions of the complex ginzburg landau equation perturbed by a random force it was proved in s kuksin and a shirikyan 2004 4 that if the random force is proportional to the square root of the viscosity 0 then the family of stationary measures possesses an accumulation point as 0 we show that if is such a point then the distributions of the l 2 norm and of the energy possess a density with respect to the lebesgue measure the proofs are based on ito s formula and some properties of local time for semimartingales

probabilistic well posedness of the mass critical nls with radial data below l2 rd abstract in this paper we consider the cauchy problem of the mass critical nonlinear schrodinger equation nls with radial data below l 2 r d we prove almost sure local well posedness along with small data global existence and scattering furthermore we also derive conditional almost sure global well posedness of the defocusing nls under the assumption of a probabilistic a priori energy bound the main ingredient is to establish the probabilistic radial strichartz estimates

statistical mechanics of the 2 dimensional focusing nonlinear schrodinger equation we study a natural construction of an invariant measure for the 2 dimensional periodic focusing nonlinear schrodinger equation with the critical cubic nonlinearity we find that a phase transition occurs as the coupling constant defining the strength of the nonlinearity is increased but that the natural construction successful for the 1 dimensional case and for the 2 dimensional defocusing case cannot produce an invariant measure our methods rely on an analysis of a statistical mechanical model closely related to the spherical model of berlin and kac

probabilistic methods for discrete nonlinear schrodinger equations we show that the thermodynamics of the focusing cubic discrete nonlinear schrodinger equation are exactly solvable in dimension 3 and higher a number of explicit formulas are derived

the dynamic phi 4_3 φ 3 4 model comes down from infinity we prove an a priori bound for the dynamic phi 4 3 model on the torus which is independent of the initial condition in particular this bound rules out the possibility of finite time blow up of the solution it also gives a uniform control over solutions at large times and thus allows one to construct invariant measures via the krylov bogoliubov method it thereby provides a new dynamic construction of the euclidean phi 4 3 field theory on finite volume our method is based on the local in time solution theory developed recently by gubinelli imkeller perkowski and catellier chouk the argument relies entirely on deterministic pde arguments such as embeddings of besov spaces and interpolation which are combined to derive energy inequalities

local existence of solutions to randomized gross pitaevskii hierarchies in this paper we study the local in time existence of solutions to randomized forms of the gross pitaevskii hierarchy on periodic domains in particular we study the independently randomized gross pitaevskii hierarchy and the dependently randomized gross pitaevskii hierarchy which were first introduced in the author s joint work with staffilani 2013 for these hierarchies we construct local in time low regularity solutions in spaces which contain a random component the constructed density matrices will solve the full randomized hierarchies thus extending the results from the author and staffilani s joint work where solutions solving arbitrarily long subhierarchies were given our analysis will be based on the truncation argument which was first used in the deterministic setting in the work of t chen and pavlovic 2013 the presence of randomization in the problem adds additional difficulties most notably to estimating the duhamel expansions that are crucial in the truncation argument these difficulties are overcome by a detailed analysis of the duhamel expansions in the independently randomized case we need to keep track of which randomization parameters appear in the duhamel terms whereas in the dependently randomized case we express the duhamel terms directly in terms of the initial data in both cases we can obtain stronger results with respect to the time variable if we assume additional regularity on the initial data

random data cauchy theory for supercritical wave equations ii a global existence result we prove that the subquartic wave equation on the three dimensional ball with dirichlet boundary conditions admits global strong solutions for a large set of random supercritical initial data in bigcap s 1 2 h s theta times h s 1 theta we obtain this result as a consequence of a general random data cauchy theory for supercritical wave equations developed in our previous work 6 and invariant measure considerations inspired by earlier works by bourgain 2 3 on the non linear schrodinger equation which allow us to obtain also precise large time dynamical informations on our solutions

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

coupled map lattices some topological and ergodic properties abstract lattice dynamical systems ldss form the class of extended systems that is the intermediate one between partial differential equations pdes and cellular automata the most popular class of ldss is formed by coupled map lattices cmls while being introduced rather recently ldss allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio temporal dynamics and to obtain some new and even rigorous results on space time chaos intermittency and pattern formation we discuss the rigorous results that were obtained in this area but eventually give also some applications

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

the number of integral points on arcs and ovals integral lattice points and that the exponent and constant are best possible however swinnerton dyer 10 showed that the preceding result can be substantially improved if we start with a fixed c strictly convex arc and consider the number of lattice points on t the dilation of by a factor t t 1 this of course is the same as asking for rational points mn n n on as n in fact swinnerton dyer proves a bound of type t zz c e t 3 5 e

invariant measures and the soliton resolution conjecture the soliton resolution conjecture for the focusing nonlinear schrodinger equation nls is the vaguely worded claim that a global solution of the nls for generic initial data will eventually resolve into a radiation component that disperses like a linear solution plus a localized component that behaves like a soliton or multisoliton solution considered to be one of the fundamental open problems in the area of nonlinear dispersive equations this conjecture has eluded a proof or even a precise formulation to date this paper proves a statistical version of this conjecture at mass subcritical nonlinearity in the following sense the uniform probability distribution on the set of all functions with a given mass and energy if such a thing existed would be a natural invariant measure for the nls flow and would reflect the long term behavior for generic initial data with that mass and energy unfortunately such a probability measure does not exist we circumvent this problem by constructing a sequence of discrete measures that in principle approximate this fictitious probability distribution as the grid size goes to 0 we then show that a continuum limit of this sequence of probability measures does exist in a certain sense and in agreement with the soliton resolution conjecture the limit measure concentrates on the unique ground state soliton combining this with results from ergodic theory we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting the above results following in the footsteps of a program of studying the long term behavior of nonlinear dispersive equations through their natural invariant measures initiated by lebowitz rose and speer and carried forward by bourgain mckean tzvetkov oh and others are proved using a combination of techniques from large deviations pdes harmonic analysis and bare hands probability theory it is valid in any dimension 2014 wiley periodicals inc

on the principle of reduced stability abstract the principle of reduced stability says that the stability of bifurcating stationary or periodic solutions is given by the finite dimensional bifurcation equation obtained by the method of lyapunov schmidt to be more precise the linearized stability is governed by the linearization of the bifurcation equation about the bifurcating branch of solutions and in particular by the signs of the real parts of the perturbation of the eigenvalues along this branch this principle is true for simple eigenvalue bifurcation whereas it may be false for higher dimensional bifurcation equations a condition for the validity of that principle is given a counterexample shows that it cannot be dropped in general

the weakly nonlinear large box limit of the 2d cubic nonlinear schrodinger equation we consider the cubic nonlinear schrodinger nls equation set on a two dimensional box of size l with periodic boundary conditions by taking the large box limit l to infty in the weakly nonlinear regime characterized by smallness in the critical space we derive a new equation set on r 2 that approximates the dynamics of the frequency modes this nonlinear equation turns out to be hamiltonian and enjoys interesting symmetries such as its invariance under fourier transform as well as several families of explicit solutions a large part of this work is devoted to a rigorous approximation result that allows to project the long time dynamics of the limit equation into that of the cubic nls equation on a box of finite size

almost sure local well posedness and scattering for the 4d cubic nonlinear schrodinger equation abstract we consider the cauchy problem for the defocusing cubic nonlinear schrodinger equation in four space dimensions and establish almost sure local well posedness and conditional almost sure scattering for random initial data in h x s r 4 with 1 3 s 1 the main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear schrodinger equation which is inspired by certain function spaces used in the study of the schrodinger maps problem and is based on strichartz spaces as well as variants of local smoothing inhomogeneous local smoothing and maximal function spaces additionally we prove an almost sure scattering result for randomized radially symmetric initial data in h x s r 4 with 1 2 s 1

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

quasi invariant gaussian measures for the cubic fourth order nonlinear schrodinger equation we consider the cubic fourth order nonlinear schrodinger equation on the circle in particular we prove that the mean zero gaussian measures on sobolev spaces h s mathbb t s frac 3 4 are quasi invariant under the flow

invariant gibbs measure evolution for the radial nonlinear wave equation on the 3d ball abstract we establish new global well posedness results along gibbs measure evolution for the nonlinear wave equation posed on the unit ball in r 3 via two distinct approaches the first approach invokes the method established in the works bourgain 1994 1996 3 4 5 based on a contraction mapping principle and applies to a certain range of nonlinearities the second approach allows to cover the full range of nonlinearities admissible to treatment by gibbs measure working instead with a delicate analysis of convergence properties of solutions the method of the second approach is quite general and we shall give applications to the nonlinear schrodinger equation on the unit ball in subsequent works bourgain and bulut 2013 10 11

random data cauchy problem for the fourth order schrodinger equation with the second order derivative nonlinearities abstract we consider the cauchy problem of the fourth order nonlinear schrodinger equation 4nls i t e 2 u p m x u 2 x u 2 m 3 on r d d 2 with random initial data here p m is a homogeneous polynomial of degree m containing the second order derivative we show the almost sure local well posedness and small data global existence in h s r d with some range of s s c where s c is the scaling critical regularity i e s c d 2 2 m 1 our results contain the cases s s c 1 2 s c when d 3 and m 5 similar supercritical well posedness results also hold for d 2 m 4 and d 3 3 m 5

invariant measures for the 2d defocusing nonlinear schrodinger equation consider the2d defocusing cubic nlsiu t u u u 2 0 with hamiltonian smallint left left nabla phi right 2 tfrac 1 2 left phi right 4 right it is shown that the gibbs measure constructed from the wick ordered hamiltonian i e replacing 4 by 4 is an invariant measure for the appropriately modified equationiu t u u u 2 2 u 2 dx u 0 there is a well defined flow on thesupport of the measure in fact it is shown that for almost all data the solutionu u 0 satisfiesu t e it c hs for somes 0 first a result local in time is established and next measure invariance considerations are used to extend the local result to a global one cf b2

classification and unfoldings of 1 2 resonant hopf bifurcation in this paper we study the bifurcations of periodic solutions from an equilibrium point of a differential equation whose linearization has two pairs of simple pure imaginary complex conjugate eigenvalues which are in 1 2 ratio this corresponds to a hopf hopf mode interaction with 1 2 resonance as occurs in the context of dissipative mechanical systems using an approach based on liapunov schmidt reduction and singularity theory we give a framework in which to study these problems and their perturbations in two cases no distinguished parameter and one distinguished bifurcation parameter we give a complete classification of the generic cases and their unfoldings