h c1 maps and elliptic spdes with polynomial and exponential perturbations of nelson s euclidean free field abstract elliptic stochastic partial differential equations spde with polynomial and exponential perturbation terms defined in terms of nelson s euclidean free field on rd are studied using results by s kusuoka and a s ustunel and m zakai concerning transformation of measures on abstract wiener space spdes of this type arise in particular in euclidean quantum field theory with interactions of the polynomial or exponential type the probability laws of the solutions of such spdes are given by girsanov probability measures that are non linearly transformed measures of the probability law of nelson s free field defined on subspaces of schwartz space of tempered distributions

metastable behavior of stochastic dynamics a pathwise approach in this paper a new approach to metastability for stochastic dynamics is proposed the basic idea is to study the statistics of each path performing time averages along the evolution metastability would be characterized by the fact that the process of these time averages converges under a suitable rescaling to a measure valued markov jump process here this convergence is shown for the curie weiss mean field dynamics and also for a model with spatial structure harris contact process

reduction formula for moments of stochastic integrals we show how moments of stochastic integrals can be expressed explicitly as expectation values of ordinary lebesgue integrals this result generalizes the well known relationship for the second moment

pointwise estimates and exponential laws in metastable systems via coupling methods we show how coupling techniques can be used in some metastable systems to prove that mean metastable exit times are almost constant as functions of the starting microscopic configuration within a meta stable set in the example of the random field curie weiss model we show that these ideas can also be used to prove asymptotic exponentiallity of normalized metastable escape times

on inverse spectral theory for self adjoint extensions mixed types of spectra abstract lethbe a symmetric operator in a separable hilbert space h suppose thathhas some gapj we shall investigate the question about what spectral properties the self adjoint extensions ofhcan have inside the gapjand provide methods on how to construct self adjoint extensions ofhwith prescribed spectral properties insidej under some weak assumptions about the operatorhwhich are satisfied e g provided the deficiency indices ofhare infinite and the operator h 1is compact for one regular point ofh we shall show that for every auxiliary self adjoint operatorm in the hilbert space h and every open subsetj0of the gapjofhthere exists a self adjoint extensionhofhsuch that insidejthe self adjoint extensionhofhhas the same absolutely continuous and the same point spectrum as the given operatorm and the singular continuous spectrum ofhinjequals the closure ofj0inj moreover we shall present a method of how to construct such a self adjoint extensionh via our methods it is possible to construct new kinds of self adjoint realizations of the laplacian on a bounded domain in r d d 1 with spectral properties very different from the spectral properties of the self adjoint realizations known before mathematics subject classification 1991 47a10 47a60 47b25 47e05 47f05

constrained rough paths we introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural on the way we develop a notion of rough integration and an ecient and intrinsic theory of rough dierential equations rdes on manifolds the theory of rdes is then used to construct parallel translation along manifold valued rough paths finally this framework is used to show there is a one to one correspondence between rough paths on a d dimensional manifold and rough paths on d dimensional euclidean space this last result is a rough path analogue of cartan s development map and its stochastic version which was developed by eells and elworthy and malliavin

global well posedness of the dynamic φ4 model in the plane we show global well posedness of the dynamic 4 4 model in the plane the model is a nonlinear stochastic pde that can only be interpreted in a renormalised sense solutions take values in suitable weighted besov spaces of negative regularity

a spatial version of the ito stratonovich correction we consider a class of stochastic pdes of burgers type in spatial dimension 1 driven by space time white noise even though it is well known that these equations are well posed it turns out that if one performs a spatial discretization of the nonlinearity in the wrong way then the sequence of approximate equations does converge to a limit but this limit exhibits an additional correction term this correction term is proportional to the local quadratic cross variation in space of the gradient of the conserved quantity with the solution itself this can be understood as a consequence of the fact that for any fixed time the law of the solution is locally equivalent to wiener measure where space plays the role of time in this sense the correction term is similar to the usual ito stratonovich correction term that arises when one considers different temporal discretizations of stochastic odes

weyl solutions and j selfadjointness for dirac operators we consider a non selfadjoint dirac type differential expression 0 1 d q y j n dy over dx q x y with a non selfadjoint potential matrix q epsilon l 1 loc scr j bbb c n times n and a signature matrix j n j j n j n epsilon bbb c n times n here scr j denotes either the line bbb r or the half line bbb r with this differential expression one associates in l 2 scr j bbb c n the closed maximal and minimal operators d max q and d min q respectively one of our main results for the whole line case states that d max q d min q in l 2 bbb r bbb c n moreover we show that if the minimal operator d min q in l 2 bbb r bbb c n is j symmetric with respect to an appropriate involution j then it is j selfadjoint similar results are valid in the case of the semiaxis bbb r in particular we show that if n 2p and the minimal operator d min q in l 2 bbb r bbb c 2p is j symmetric then there exists a 2p times p weyl type matrix solution psi z cdot epsilon l 2 bbb r bbb c 2p times p of the equation d max q psi z cdot z psi z cdot a similar result is valid for the expression 0 1 whenever there exists a proper extension tilde a with dim dom tilde a dom d min q p and nonempty resolvent set in particular it holds if a potential matrix q has a bounded imaginary part this leads to the existence of a unique weyl function for the express 0 1 the main results are proven by means of a reduction to the self adjoint case by using the technique of dual pairs of operators the differential expression 0 1 is of significance as it appears in the lax formulation of the vector valued nonlinear schrodinger equation

new approach to the semiclassical limit of quantum mechanics i multiple tunnelings in one dimension we propose a new approach for the estimate of the rate of degeneracy of the lowest eigenvalues of the schrodinger operator in the presence of tunneling based on the theory of diffusion processes our method provides lower and upper bounds for the energy splittings and the rates of localization of the wave functions and enables us to discuss cases which as far as we know have never been treated rigorously in the literature in particular we give an analysis of the effect on eigenvalues and eigenfunctions of localized deformations of 1 symmetric double well potentials 2 potentials periodic and symmetric over a finite interval theses situations are characterized by a remarkable dependence on such deformations our probabilistic techniques are inspired by the theory of small random perturbations of dynamical systems

action minimization and sharp interface limits for the stochastic allen cahn equation we study the action minimization problem which is formally associated to phase transformation in the stochastically perturbed allen cahn equation the sharp interface limit is related to but di erent from the sharp interface limits of the related energy functional and deterministic gradient flows in the sharp interface limit of the action minimization problem we find distinct most likely switching pathways depending on the relative costs of nucleation and propagation of interfaces this competition is captured by the limit of the action functional which we derive formally and propose as the natural candidate for the limit guided by the reduced functional we prove upper and lower bounds for the minimal action which agree on the level of scaling this is a preprint of an article accepted for publication in comm pure app math october 2005

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

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

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

exponential loss of memory for the 2 dimensional allen cahn equation with small noise we prove an asymptotic coupling theorem for the 2 dimensional allen cahn equation perturbed by a small space time white noise we show that with overwhelming probability two profiles that start close to the minimisers of the potential of the deterministic system contract exponentially fast in a suitable topology in the 1 dimensional case a similar result was shown in martinelli et al commun math phys 120 1 25 69 1988 j stat phys 55 3 4 477 504 1989 it is well known that in two or more dimensions solutions of this equation are distribution valued and the equation has to be interpreted in a renormalised sense formally this renormalisation corresponds to moving the minima of the potential infinitely far apart and making them infinitely deep we show that despite this renormalisation solutions behave like perturbations of the deterministic system without renormalisation they spend large stretches of time close to the minimisers of the un renormalised potential and the exponential contraction rate of different profiles is given by the second derivative of the potential in these points as an application we prove an eyring kramers law for the transition times between the stable solutions of the deterministic system for fixed initial conditions

behavior of droplets for a class of glauber dynamics at very low temperature we consider a class of glauber dynamics for the two dimensional nearest neighbor ferromagnetic ising model in which the rate with which each spin flips depends only on the increment in energy caused by its flip in a monotonic non increasing fashion

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

enhancing the convergence efficiency of a self propelled agent system via a weighted model we investigate a weighted self propelled agent system wherein each agent s direction is determined by its spatial neighbors directions with exponential weights according to the neighbor numbers in order to describe the fact that some agents with more neighbors might have larger influence on its neighbors we introduce a scaling exponent of the neighbor number between 0 and ensuremath infty when the exponent is equal to 1 the convergence efficiency is enhanced in our simulation furthermore as the exponent increases i e the effect of weight becomes stronger the network of agents becomes easier to achieve direction consensus

temporal integrators for fluctuating hydrodynamics including the effect of thermal fluctuations in traditional computational fluid dynamics requires developing numerical techniques for solving the stochastic partial differential equations of fluctuating hydrodynamics these langevin equations possess a special fluctuation dissipation structure that needs to be preserved by spatio temporal discretizations in order for the computed solution to reproduce the correct long time behavior in particular numerical solutions should approximate the gibbs boltzmann equilibrium distribution and ideally this will hold even for large time step sizes we describe finite volume spatial discretizations for the fluctuating burgers and fluctuating incompressible navier stokes equations that obey a discrete fluctuation dissipation balance principle just like the continuum equations we develop implicit explicit predictor corrector temporal integrators for the resulting stochastic method of lines discretization these stochastic runge kutta schemes treat diffusion implicitly and advection explicitly are weakly second order accurate for additive noise for small time steps and give a good approximation to the equilibrium distribution even for very strong fluctuations numerical results demonstrate that a midpoint predictor corrector scheme is very robust over a broad range of time step sizes

zero white noise limit through dirichlet forms with application to diffusions in a random medium we study the zero white noise limit for diffusions in a continuous multidimensional medium given a continuous function on n w we consider diffusions whose drift term is the gradient ofw and whose diffusion coefficient is constant equal to e we describe the asymptotics of the exit time from a domain and of the law of the process when e tends to zero by applying these results to a random self similar mediumw we prove limit theorems for a diffusion in a random medium our theorems agree with results usually proved through the large deviation principle although in our setup this last tool is not available we extend to the multidimensional case properties of diffusions in a random medium already known in one dimension

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

rough evolution equations we generalize lyons rough paths theory in order to give a pathwise meaning to some nonlinear infinite dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise as an illustration we discuss a class of linear and nonlinear 1d spdes driven by a space time gaussian noise with singular space covariance and brownian time dependence

small random perturbations of a dynamical system with blow up abstract we study small random perturbations by additive white noise of a spatial discretization of a reaction diffusion equation with a stable equilibrium and solutions that blow up in finite time we prove that the perturbed system blows up with total probability and establish its order of magnitude and asymptotic distribution for initial data in the domain of explosion we prove that the explosion time converges to the deterministic one while for initial data in the domain of attraction of the stable equilibrium we show that the system exhibits metastable behavior

inverse spectral theory for symmetric operators with several gaps scalar type weyl functions let s be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non zero deficiency indices let j be an open subset of r if there exists a self adjoint extension s0 of s such that j is contained in the resolvent set of s0 and the associated weyl function of the pair s s0 is monotone with respect to j then for any self adjoint operator r there exists a self adjoint extension s such that the spectral parts s j and rj are unitarily equivalent it is shown that for any extension s of s the absolutely continuous spectrum of s0 is contained in that one of s moreover for a wide class of extensions the absolutely continuous parts of s and s are even unitarily equivalent

couplings and asymptotic exponentiality of exit times the goal of this note is simply to call attention to the resulting simplification in the proof of asymptotic exponentiality of exit times in the freidlin wentzell regime as proved by f martinelli et al by using the coupling proposed by t lindvall and c rogers

adaptive velocity strategy for swarm aggregation collective behaviors of biological swarms have attracted significant interest in recent years but much attention and efforts have been focused on constant speed models in which all agents are assumed to move with the same constant speed one limitation of the constant speed assumption without attraction function is that global convergence is quite difficult or even practically impossible to achieve if the speed is relatively fast in this paper we propose an adaptive velocity model in which each agent not only adjusts its moving direction but also adjusts its speed according to the degree of direction consensus among its local neighbors one important feature of the adaptive velocity model is that the speeds of all agents are adaptively tuned to the same maximum constant speed after a short transient process the adaptive velocity strategy can greatly enhance the global convergence probability and thus provides a powerful mechanism for coordinated motion in biological and technological multiagent systems

small random perturbations of finite and infinite dimensional dynamical systems unpredictability of exit times we apply previous results on the pathwise exponential loss of memory of the initial condition for stochastic differential equations with small diffusion to the problem of the asymptotic distribution of the first exit times from an attracted domain we show under general hypotheses that the suitably rescaled exit time converges in the zero noise limit to an exponential random variable then we extend the results to an infinite dimensional case obtained by adding a small random perturbation to a nonlinear heat equation

erratum to rough burgers like equations with multiplicative noise unfortunately the proof of proposition 4 8 in hw13 is not correct in fact equation 4 60 wherer is rewritten as a stochastic integral is not correct because the integrand is not adapted to the filtration hence in 4 61 we are not allowed to apply the burkholder davies gundy inequality the argument was corrected in hmw12 lemma 3 6 and corollary 3 7 in a more complicated situation in order to transfer these statement to the situation in hw13 some smaller changes are needed first of all the regularity of the linearised process x should be measured in a holder norm with slightly bigger index than the solution u this can be done without further problems then the definition of the stopping time in 4 53 becomes for k 0 and for an 1 2 we introduce the stopping time

small random perturbations of dynamical systems exponential loss of memory of the initial condition we consider markov processes arising from small random perturbations of non chaotic dynamical systems under rather general conditions we prove that with large probability the distance between two arbitrary paths starting close to a same attractor of the unperturbed system decreases exponentially fast in time the case of paths starting in different basins of attraction is also considered as well as some applications to the analysis of the invariant measure and to elliptic problems with small parameter in front to the second derivatives the proof is based on a multiscale analysis of the typical trajectories of the markov process this analysis is done using techniques involved in the proof of anderson localization for disordered quantum systems

on the swendsen wang dynamics i exponential convergence to equilibrium we present rigorous results on the exponential convergence to equilibrium for the swendsen wang stochastic dynamics for thed dimensional ising ferromagnet with external magnetic fieldh in the thermodynamic limit we consider various situations mainly in the low temperature regime in which boundary conditions are homogeneous and parallel or opposite to the external field in the latter case we relate directly the tunneling from the metastable phase to the stable one with the exponential convergence to equilibrium

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

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

accelerating consensus of self driven swarm via adaptive speed abstract in recent years the well developed vicsek model has attracted more and more attention unfortunately in depth research on its convergence speed is not yet completed in this paper we investigate some key factors governing the convergence speed of the vicsek model with the assistance of extensive numerical simulations a significant phenomenon surfaces that the convergence time scales obeys a power law with r 2 ln n with r and n being the horizon radius and the number of particles respectively to further accelerate the convergence procedure we propose a kind of improved vicsek model with self driven particles governed by variational speeds which can remarkably shorten the convergence time of the standard vicsek model

heat semigroup and singular pdes we provide in this work a semigroup approach to the study of singular pdes in the line of the paracontrolled approach developed recently by gubinelli imkeller and perkowski starting from a heat semigroup we develop a functional calculus and introduce a paraproduct based on the semigroup for which commutator estimates and schauder estimates are proved together with their paracontrolled extensions this machinery allows us to investigate singular pdes in potentially unbounded riemannian manifolds under mild geometric conditions as an illustration we study the generalized parabolic anderson model equation and prove under mild geometric conditions its well posed character in small time on a potentially unbounded 2 dimensional riemannian manifold for an equation driven by a coloured noise and for all times for the linear parabolic anderson model equation in 2 dimensional unbounded manifolds this machinery can be extended to an even more singular setting and deal with sobolev spaces rather than holder spaces

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

renormalization group for markov chains and application to metastability in this paper we introduce a new renormalization group method for the study of the long time behavior of markov chains with finite state space and with transition probabilities exponentially small in an external parameter a general approach of metastability problems emerges from this analysis and is discussed in detail in the case of a two dimensional ising system at low temperature

metastability for a class of dynamical systems subject to small random perturbations we consider dynamical systems in mathbb r d driven by a vector field b x nabla a x where a is a double well potential with some smoothness conditions we show that these dynamical systems when subjected to a small random disturbance exhibit metastable behavior in the sense defined in 2 more precisely we prove that the process of moving averages along a path of such a system converges in law when properly normalized to a jump markov process the main tool for our analysis is the theory of freidlin and wentzell 7

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

strong solutions to the stochastic quantization equations we prove the existence and uniqueness of a strong solution of the stochastic quantization equation in dimension 2 for almost all initial data with respect to the invariant measure the method is based on a fixed point result in suitable besov spaces

weighted stochastic sobolev spaces and bilinear spdes driven by space time white noise in this paper we develop basic elements of malliavin calculus on a weightedl2 this class of generalized wiener functionals is a hilbert space it turns out to be substantially smaller than the space of hida distributions while large enough to accommodate solutions of bilinear stochastic pdes as an example we consider a stochastic advection diffusion equation driven by space time white noise in rd it is known that ford 1 this equation has no solutions inl2 in contrast it is shown in the paper that in an appropriately weightedl2 there is a unique solution to the stochastic advection diffusion equation for anyd 1 in addition we present explicit formulas for the hermite fourier coefficients in the wiener chaos expansion of the solution

large deviation estimates in the stochastic quantization ofϕ 2 4 we study in the small noise limit the behaviour of field trajectories for the process constructed by the authors in connection with the stochastic quantization of 2 4 due to the presence of infinite renormalization the usual large deviation techniques do not apply immediately and a new strategy has to be developed we prove some estimates analogous to the freidlin ventzel inequalities from these it follows that the field trajectories suitably smeared in space over a scaler0 behave when the noise is small as the projection on the same scale of a field obeying a regularized stochastic equation with a large cut off however the estimates are not uniform in the cut off and an interesting feature of the problem is that the scale over which the field is smeared determines whether the noise is sufficiently small for the estimates to apply

stochastic quantization of field theory in finite and infinite volume abstract using the theory of dirichlet forms a distribution valued symmetric markov process is constructed so that it has the 4 2 measure as its invariant probability measure several properties of this process are proved and a finite to infinite volume limit theorem is established

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

decentralized control of multi agent systems for swarming with a given geometric pattern this paper addresses a problem of cooperative formation control of a network of self deployed autonomous agents we propose a decentralized motion coordination control for the agents so that they collectively move in a desired geometric pattern from any initial position there are no predefined leaders in the group and only local information is required for the control the control algorithm is developed using the ideas of information consensus and its effectiveness is illustrated via numerical simulations

large deviations for the macroscopic motion of an interface we study the most probable way an interface moves on a macroscopic scale from an initial to a final position within a fixed time in the context of large deviations for a stochastic microscopic lattice system of ising spins with kac interaction evolving in time according to glauber non conservative dynamics such interfaces separate two stable phases of a ferromagnetic system and in the macroscopic scale are represented by sharp transitions we derive quantitative estimates for the upper and the lower bound of the cost functional that penalizes all possible deviations and obtain explicit error terms which are valid also in the macroscopic scale furthermore using the result of a companion paper about the minimizers of this cost functional for the macroscopic motion of the interface in a fixed time we prove that the probability of such events can concentrate on nucleations should the transition happen fast enough

on a modelled rough heat equation we use the formalism of hairer s regularity structures theory to study a heat equation model with non linear perturbation driven by a space time fractional noise different regimes are observed depending on the global pathwise roughness of the noise to this end and following the procedure exhibited in hairer invent math 198 2 269 504 2014 the equation is first lifted into some abstract regularity structure and therein solved through a fixed point argument then we construct a consistent model above the fractional noise by relying on a smooth fourier type approximation of the process

synchronization by noise we provide sufficient conditions for synchronization by noise i e under these conditions we prove that weak random attractors for random dynamical systems consist of single random points in the case of sde with additive noise these conditions are also essentially necessary in addition we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point as a result synchronization by noise is proven for a large class of sde with additive noise in particular we prove that the random attractor for an sde with drift given by a multidimensional double well potential and additive noise consists of a single random point all examples treated in tearne probab theory relat fields 141 1 2 1 18 2008 are also included

critical behavior of the kramers escape rate in asymmetric classical field theories we introduce an asymmetric classical ginzburg landau model in a bounded interval and study its dynamical behavior when perturbed by weak spatiotemporal noise the kramers escape rate from a locally stable state is computed as a function of the interval length an asymptotically sharp second order phase transition in activation behavior with corresponding critical behavior of the rate prefactor occurs at a critical length l c similar to what is observed in symmetric models the weak noise exit time asymptotics to both leading and subdominant orders are analyzed at all interval lengthscales the divergence of the prefactor as the critical length is approached is discussed in terms of a crossover from non arrhenius to arrhenius behavior as noise intensity decreases more general models without symmetry are observed to display similar behavior suggesting that the presence of a phase transition in escape behavior is a robust and widespread phenomenon

scaling limits of wick ordered kpz equation consider the kpz equation x d where w t x is a space time white noise this paper investigates the question of whether for some exponents and z k u k z t kx converges in some sense as k to infty and if so what are the values of these exponents the non linear term in the kpz equation is interpreted as a wick product and the equation is solved in a suitable space of stochastic distributions the main tools for establishing the scaling properties of the solution are those of white noise analysis in particular the wiener chaos expansion a notion of convergence in law in the sense of wiener chaos is formulated and convergence in this sense of k u k z t kx as k is established for various values of and z depending on the dimension d

a priori estimates for rough pdes with application to rough conservation laws we introduce a general framework to study pdes driven by rough paths we develop new a priori estimates based on a rough gronwall lemma argument for weak solutions to rough pdes this will allow us to follow standard pde strategies to obtain existence and uniqueness results in particular our approach does not rely on any sort of transformation formula flow transformation feynman kac representation formula etc and is therefore rather flexible as an application we study conservation laws driven by rough paths establishing well posedness for the corresponding kinetic formulation

tunneling in two dimensions tunneling is studied here as a variational problem formulated in terms of a functional which approximates the rate function for large deviations in ising systems with glauber dynamics and kac potentials 9 the spatial domain is a two dimensional square of side l with reflecting boundary conditions for l large enough the penalty for tunneling from the minus to the plus equilibrium states is determined minimizing sequences are fully characterized and shown to have approximately a planar symmetry at all times thus departing from the wulff shape in the initial and final stages of the tunneling in a final section sect 11 we extend the results to d 3 but their validity in d 3 is still open

singular perturbations to semilinear stochastic heat equations we consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof the interesting feature of these perturbations is that as the small parameter e tends to zero their solutions converge to the wrong limit i e they do not converge to the solution obtained by simply setting e 0 a similar effect is also observed for some formally small stochastic perturbations of a deterministic semilinear parabolic pde our proofs are based on a detailed analysis of the spatially rough component of the equations combined with a judicious use of gaussian concentration inequalities

rough burgers like equations with multiplicative noise we construct solutions to vector valued burgers type equations perturbed by a multiplicative space time white noise in one space dimension due to the roughness of the driving noise solutions are not regular enough to be amenable to classical methods we use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution subject to the choice of the correct reference rough path we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise

rare events in stochastic partial differential equations on large spatial domains a methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply the idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another the methodology is illustrated in the context of a simple one dimensional stochastic partial differential equation the application reveals a connection between the dynamics of the partial differential equation and the classical johnson mehl avrami kolmogorov nucleation and growth model it also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space

large density fluctuations for the one dimensional supercritical contact process we consider the one dimensional supercritical contact process lettv be the first time the process reaches a densityq larger than the equilibrium one in the region 1 n we prove that starting from equilibrium tn e tn converges to an exponential random time of mean one in this way we extend previous results of lebowitz and schonmann

concept of complexity in random dynamical systems we introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system in dynamical systems with small random perturbations this indicator coincides with the rate k of divergence of nearby trajectories evolving under two different noise realizations the meaning of k is discussed in the context of the information theory and it is shown that it can be determined from real experimental data in the presence of strong dynamical intermittency the value of k is very different from the standard lyapunov exponent ensuremath lambda mathrm ensuremath sigma computed considering two nearby trajectories evolving under the same realization of the randomness however the former is much more relevant than the latter from a physical point of view as illustrated by some numerical computations for noisy maps and sandpile models textcopyright 1996 the american physical society

a white noise approach to a class of non linear stochastic heat equations abstract cauchy problems for a class of non linear stochastic evolution equations are studied they are formulated as integral equations for generalized random fields by methods of white noise analysis s transformation characterization theorem etc these problems are reduced to fixed point problems in appropriatly constructed banach spaces this technique provides a systematic treatment of existence and uniqueness questions for a variety of equations the method is applied to non linear heat equations non linear volterra equations and non linear ordinary differential equations which may all be anticipating for each case examples are given such as the stochastic burgers equation and stochastic reaction diffusion equations

rough stochastic pdes in this article we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic pdes of burgers type that exhibit too high spatial roughness for classical analytical methods to apply in fact the class of spdes that we consider is genuinely ill posed in the sense that different approximations to the nonlinearity may converge to different limits using rough path theory a pathwise notion of solution to these spdes is formulated and we show that this yields a well posed problem that is stable under a large class of perturbations including the approximation of the rough driving noise by a mollified version and the addition of hyperviscosity we also show that under certain structural assumptions on the coefficients the spdes under consideration generate a reversible markov semigroup with respect to a diffusion measure that can be given explicitly

on unbiased stochastic navier stokes equations a random perturbation of a deterministic navier stokes equation is considered in the form of an spde with wick type nonlinearity the nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term u nabla u this perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic navier stokes equation the perturbed equation is solved in the space of generalized stochastic processes using the cameron martin version of the wiener chaos expansion it is shown that the generalized solution is a markov process and scales effectively by catalan numbers

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

study of stochastic differential equations by constructive methods i in this work we give an algorithm to express as a convergent series the stationary averages for a class of gradient perturbations of a nonsymmetric nongradient ornstein uhlenbeck process the method relies on a cluster expansion in time of the girsanov cameron martin formula for the density of the perturbed measure with respect to the ornstein uhlenbeck measure in the second paper of this series the approach is extended to more general perturbations

on a class of stochastic reaction diffusion equations in two space dimensions the authors prove for any finite time the existence of a weak dynamics for a class of reaction diffusion systems in two space dimensions perturbed by a white noise this class includes models which may exhibit complex behaviour in space and time their approach based on the methods of constructive quantum field theory extends previous results established for gradient type systems

random composition of two rational maps singularity of the invariant measure we study the invariant measure of a markov chain obtained by randomly composing two rational maps related to the anderson model with a bernoulli potential for a certain range of the parameters we show that the invariant measure is singular continuous in certain cases the support turns out to be a cantor set with a multifractal structure

precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime we investigate the close connection between metastability of the reversible diffusion process x defined by the stochastic differential equation d xt f xt d t 2e d wt e 0 and the spectrum near zero of its generator l f where f d and w denotes brownian motion on d for generic f to each local minimum of f there corresponds a metastable state we prove that the distribution of its rescaled relaxation time converges to the exponential distribution as 0 with optimal and uniform error estimates each metastable state can be viewed as an eigenstate of l with eigenvalue which converges to zero exponentially fast in 1 modulo errors of exponentially small order in 1 this eigenvalue is given as the inverse of the expected metastable relaxation time the eigenstate is highly concentrated in the basin of attraction of the corresponding trap

the jain monrad criterion for rough paths and applications to random fourier series and non markovian hormander theory we discuss stochastic calculus for large classes of gaussian processes based on rough path analysis our key condition is a covariance measure structure combined with a classical criterion due to jain and monrad ann probab 11 1983 46 57 this condition is verified in many examples even in absence of explicit expressions for the covariance or volterra kernels of special interest are random fourier series with covariance given as fourier series itself and we formulate conditions directly in terms of the fourier coefficients we also establish convergence and rates of convergence in rough path metrics of approximations to such random fourier series an application to spde is given our criterion also leads to an embedding result for cameron martin paths and complementary young regularity cyr of the cameron martin space and gaussian sample paths cyr is known to imply malliavin regularity and also ito like probabilistic estimates for stochastic integrals resp stochastic differential equations despite their rough pathwise construction at last we give an application in the context of non markovian hormander theory

a two space dimensional semilinear heat equation perturbed by gaussian white noise a two space dimensional heat equation perturbed by a white noise in a bounded volume is considered the equation is perturbed by a non linearity of the type f au where means wick re ordering with respect to the free solution a are small parameters u denotes a solution f is the fourier transform of a complex measure with compact support