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

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

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

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

stability and traveling waves of a vaccination model with nonlinear incidence abstract in this paper we propose a spatial vaccination model with nonlinear incidence first we consider the well posedness of solutions of the model second in the case of the bounded spatial habitat r n we investigate the global stability of the model more precisely it is shown that if the threshold value r 0 1 then the disease free equilibrium e 0 is globally asymptotically stable if r 0 1 then there exists a unique disease equilibrium e which is globally asymptotically stable third in the case of the unbounded spatial habitat r n we study the existence of traveling wave solutions of the model here we show that when the threshold value r 0 1 then there exists c 0 such that there exist positive traveling wave solutions of the model connecting the two equilibria e 0 and e with speed c c and when r 0 1 there is not such a traveling wave solution with speed c c numerical simulations are performed to illustrate our analytic results our results indicate that the global dynamics of the model are completely determined by the threshold value r 0

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

dynamics of a diffusive vaccination model with nonlinear incidence abstract a recent paper xu et al 2018 studied the global stability and traveling wave solution of a vaccination model with nonlinear incidence furthermore in this paper we first study the local stability of this model and then establish the uniform persistence result our results are a supplement to above paper

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

ideals in computable rings abstract we show that the existence of a nontrivial proper ideal in a commutative ring with identity which is not a field is equivalent to wkl 0 over rca 0 and that the existence of a nontrivial proper finitely generated ideal in a commutative ring with identity which is not a field is equivalent to aca 0 over rca 0 we also prove that there are computable commutative rings with identity where the nilradical is 1 0 complete and the jacobson radical is 2 0 complete respectively

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

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

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

chain conditions in computable rings friedman simpson and smith showed that over rca 0 the statements every ring has a maximal ideal and every ring has a prime ideal are equivalent to aca 0 and wkl 0 respectively more recently downey lempp and mileti have shown that over rca 0 the statement every ring that is not a field contains a nontrivial ideal is equivalent to wklo in this article we explore the reverse mathematical strength of the classic theorems from commutative algebra which say that every artinian ring is noetherian and every artinian ring is of finite length in particular we show that over rca 0 the former implies wkl 0 and is implied by aca 0 while over rca 0 b 2 the latter is equivalent to aca 0

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

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

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

proper divisibility in computable rings abstract we study divisibility in computable integral domains we develop a technique for coding 2 0 binary trees into the divisibility relation of a computable integral domain we then use this technique to prove two theorems about non atomic integral domains in every atomic integral domain the divisibility relation is well founded we show that this classical theorem is equivalent to aca 0 over rca 0 in every computable non atomic integral domain there is a 3 0 infinite sequence of proper divisions we show that this upper bound cannot be improved to 2 0 in general

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

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 computability definability and proof theory of artinian rings abstract we show that in the context of reverse mathematics wk l 0 weak konig s lemma implies the statement ar t 0 which says that every artinian ring is noetherian over rc a 0 recursive comprehension axiom to achieve this goal we prove a general computable full structure theorem for computable artinian rings similar to the classical version found in most algebra texts

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

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

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

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

infinite dimensional proper subspaces of computable vector spaces abstract this article examines and distinguishes different techniques for coding incomputable information into infinite dimensional proper subspaces of a computable vector space and is divided into two main parts in the first part we describe different methods for coding into infinite dimensional subspaces more specifically we construct several computable infinite dimensional vector spaces each of which satisfies one of the following 1 every infinite coinfinite dimensional subspace computes turing s halting set 2 every infinite cofinite dimensional proper subspace computes turing s halting set 3 there exists x v such that every infinite dimensional proper subspace not containing x computes turing s halting set 4 every infinite dimensional proper subspace computes turing s halting set vector space 4 generalizes vector spaces 1 and 2 and its construction is more complicated the same simple and natural technique is used to construct vector spaces 1 3 finally we examine the reverse mathematical implications of our constructions 1 4 in the second part we examine the limitations of our simple and natural method for coding into infinite dimensional subspaces described in the previous paragraph in particular we prove that our simple and natural coding technique cannot produce a vector space of type 4 above and that any vector space of type 4 must have densely many from a certain point of view finite dimensional computable subspaces in other words the construction of a vector space of type 4 is necessarily more complicated than the construction of vector spaces of types 1 3 we also introduce a new statement in second order arithmetic about the existence of infinite dimensional proper subspaces in a restricted class of vector spaces related to 1 3 above and show that it is implied by weak konig s lemma in the context of reverse mathematics in the context of reverse mathematics this gives rise to two statements from effective algebra about the existence of infinite dimensional proper subspaces for a certain class of vector spaces of the form v x v a v and v x v b v that each imply a c a 0 over r c a 0 but such that the seemingly weaker statement v x v a v b v is provable via w k l 0 over r c a 0 furthermore we highlight some general similarities between constructing of infinite dimensional proper subspaces of computable vector spaces and constructing solutions to computable instances of various combinatorial principles such as ramsey s theorem for pairs

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

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

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

subspaces of computable vector spaces abstract we show that the existence of a nontrivial proper subspace of a vector space of dimension greater than one over an infinite field is equivalent to wkl 0 over rca 0 and that the existence of a finite dimensional nontrivial proper subspace of such a vector space is equivalent to aca 0 over rca 0

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

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

dynamics of a reaction diffusion waterborne pathogen model with direct and indirect transmission in this paper a reaction diffusion waterborne pathogen model which incorporates both direct and indirect transmission pathways is proposed then the basic reproduction number r 0 defined as the spectral radius of the next generation operator is established for the model system and an equivalent characterization of r 0 is obtained in terms of an elliptic eigenvalue problem furthermore we show that r 0 serves as a threshold parameter which predicts whether the disease will persist by using numerical simulations we study the influence of diffusion coefficients spatial heterogeneity and multiple transmission pathways

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

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

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

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

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

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

a nonlocal and time delayed reaction diffusion model of dengue transmission a nonlocal and time delayed reaction diffusion model of dengue fever is first proposed that incorporates the aquatic stage the winged stage and the incubation periods of the dengue virus within mosquitos and hosts then the basic reproduction number r 0 is established for the model system and an explicit formula of r 0 is obtained in the case of spatially homogeneous infections it is shown that this r 0 gives the threshold dynamics in the sense that the disease free equilibrium is asymptotically stable if r 0 1 the influences of diffusion coefficients time delays and infection heterogeneity on the spread of the disease are also studied via numerical simulations it turns out that the infection risk may be underestimated if the spatially averaged parameters are used to compute the basic reproduction number for spatially heterogeneous infections

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

on the complexity of radicals in noncommutative rings this article expands upon the recent work by downey et al 2007 3 who classified the complexity of the nilradical and jacobson radical of commutative rings in terms of the arithmetical hierarchy let r be a computable not necessarily commutative ring with identity then it follows from the definitions that the prime radical of r is 11 and the levitzki radical of r is 20 we show that these upper bounds for the complexity of the prime and levitzki radicals are optimal by constructing two noncommutative computable rings with identity such that the prime radical of one is 11 complete while the levitzki radical of the other is 20 complete