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

a martingale approach to metastability we presented in beltran and landim j stat phys 140 1065 1114 2010 beltran and landim j stat phys 149 598 618 2012 an approach to derive the metastable behavior of continuous time markov chains we assumed in these articles that the markov chains visit points in the time scale in which it jumps among the metastable sets we replace this condition here by assumptions on the mixing times and on the relaxation times of the chains reflected at the boundary of the metastable sets

asymptotic evolution of certain observables and completeness in coulomb scattering i the behavior of the dilation operator in the distant past and future is obtained when it evolves under a unitary group whose generator is a schrodinger operator the total and free evolutions are then compared by their actions on the dilation and position observable this result is employed to give a simple proof of completeness for scattering by coulomb like potentials

life span of solutions for a semilinear parabolic problem with small diffusion abstract this paper is concerned with the initial boundary value problem formula where p 1 e 0 is a bounded domain in rn and is a continuous function on it is shown that the blowup time t e of the solution of this problem satisfies t e 1 p 1 1 p as e 0 moreover when the maximum of x is attained at one point we determine the higher order term of t e which reflects the pointedness of the peak of the proof is based on a careful construction of super and subsolutions

operateurs conjugues et proprietes de propagation we use an algebraic criteria based on local positivity of a commutator which asserts the existence of a direction of propagation for the flowe iht associated to a self adjoint operatorh

life span of solutions with large initial data for a superlinear heat equation abstract we investigate the initial boundary problem u t u f u in 0 u 0 on 0 u x 0 x in where is a bounded domain in r n with a smooth boundary 0 x is a nonnegative continuous function on f u is a nonnegative superlinear continuous function on 0 we show that the life span or blow up time of the solution of this problem denoted by t satisfies t d u f u h o t as moreover when the maximum of is attained at a finite number of points in we can determine the higher order term of t which depends on the minimal value of at the maximal points of the proof is based on a careful construction of a supersolution and a subsolution

global existence and blow up phenomena for nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions abstract in this paper we consider nonlinear divergence form parabolic equations with inhomogeneous neumann boundary conditions we establish respectively the conditions on nonlinearities to guarantee that u x t exists globally or blows up at some finite time if blow up occurs we obtain upper and lower bounds of the blow up time

metastability of reversible random walks in potential fields let xi be an open and bounded subset of mathbb r d and let f xi rightarrow mathbb r be a twice continuously differentiable function denote by xi n the discretization of xi xi n xi cap n 1 mathbb z d and denote by x n t the continuous time nearest neighbor random walk on xi n which jumps from varvec x to varvec y at rate e 1 2 n f varvec y f varvec x we examine in this article the metastable behavior of x n t among the wells of the potential f

on long range scattering even if special cases are not well understood it pays sometimes to consider a problem at hand in the most general framework thus addressing the existence of wave operators for the most general potentials hormander ho1 has discovered a natural asymptotic evolution for the long range scattering it is determined by a solution s k t of a certain hamilton jacobi equation

convolution and hypergroup structures associated with a class of sturm liouville systems product formulas of the type u k u k 0 u k d are obtained for the eigenfunctions of a class of second order regular and regular singular sturm liouville problems on 0 by using the riemann integration method to solve a cauchy problem for an associated hyperbolic differential equation when d is nonnegative which can be guaranteed by a simple restriction on the differential operator of the sturm liouville problem it is possible to define a convolution with respect to which m 0 becomes a banach algebra with the functions u k u 0 as its characters

slowly changing potential problems in quantum mechanics adiabatic theorems ergodic theorems and scattering we employ the recently developed multi time scale averaging method to study the large time behavior of slowly changing in time hamiltonians we treat some known cases in a new way such as the zener problem and we give another proof of the adiabatic theorem in the gapless case we prove a new uniform ergodic theorem for slowly changing unitary operators this theorem is then used to derive the adiabatic theorem do the scattering theory for such hamiltonians and prove some classical propagation estimates and asymptotic completeness

the blow up problem for a semilinear parabolic equation with a potential abstract let be a bounded smooth domain in r n we consider the problem u t u v x u p in 0 t with dirichlet boundary conditions u 0 on 0 t and initial datum u x 0 m x where m 0 is positive and compatible with the boundary condition we give estimates for the blow up time of solutions for large values of m as a consequence of these estimates we find that for m large the blow up set concentrates near the points where p 1 v attains its maximum

blow up set of type i blowing up solutions for nonlinear parabolic systems we consider the blow up problem for systems of nonlinear parabolic inequalities and establish a criterion for the location of the blow up set our criterion enables us to obtain sufficient conditions for the boundedness of the blow up set and no boundary blow up for type i blowing up solutions

threshold and generic type i behaviors for a supercritical nonlinear heat equation we study blow up of radially symmetric solutions of the nonlinear heat equation ut u u p 1u either on rn or on a finite ball under the dirichlet boundary conditions we assume that n 3 and p ps n 2n 2 our first goal is to analyze a threshold behavior for solutions with initial data u0 v where v c h1 and v 0 v 0 it is known that there exists 0 such that the solution converges to 0 as t if 0 with j j 1 2 k the blow up is complete and of type i with a flat local profile our method is based on a combination of the zero number principle and energy estimates in the second part of the paper we employ the very same idea to show that the constant solution attains the smallest rescaled energy among all non zero stationary solutions of the rescaled equation using this result we derive a sharp criterion for no blow up

some blow up problems for a semilinear parabolic equation with a potential the blow up rate estimate for the solution to a semilinear parabolic equation ut u v x u p 1u in 0 t with 0 dirichlet boundary condition is obtained as an application it is shown that the asymptotic behavior of blow up time and blow up set of the problem with nonnegative initial data u x 0 m x as m goes to infinity which have been found in c cortazar m elgueta j d rossi the blow up problem for a semilinear parabolic equation with a potential preprint arxiv math ap 0607055 july 2006 is improved under some reasonable and weaker conditions compared with c cortazar m elgueta j d rossi the blow up problem for a semilinear parabolic equation with a potential preprint arxiv math ap 0607055 july 2006

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

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

metastability and low lying spectra in reversible markov chains we study a large class of reversible markov chains with discrete state space and transition matrix pn we define the notion of a set of metastable points as a subset of the state space n such that i this set is reached from any point x n without return to x with probability at least bn while ii for any two points x y in the metastable set the probability t 1x y to reach y from x without return to x is smaller than an 1 bn under some additional non degeneracy assumption we show that in such a situation

homogeneous nucleation for glauber and kawasaki dynamics in large volumes at low temperatures in this paper we study metastability in large volumes at low temperatures we consider both ising spins subject to glauber spin flip dynamics and lattice gas particles subject to kawasaki hopping dynamics let denote the inverse temperature and let c 2 be a square box with periodic boundary conditions such that lim we run the dynamics on starting from a random initial configuration where all of the droplets clusters of plus spins and clusters of particles respectively are small for large and for interaction parameters that correspond to the metastable regime we investigate how the transition from the metastable state with only small droplets to the stable state with one or more large droplets takes place under the dynamics this transition is triggered by the appearance of a single critical droplet somewhere in using potential theoretic methods we compute the average nucleation time the first time a critical droplet appears and starts growing up to a multiplicative factor that tends to 1 as it turns out that this time grows as ke for glauber dynamics and as k e for kawasaki dynamics where is the local canonical resp grand canonical energy to create a critical droplet and k is a constant reflecting the geometry of the critical droplet provided these times tend to infinity which puts a growth restriction on the fact that the average nucleation time is inversely proportional to is referred to as homogeneous nucleation because it says that the critical droplet for the transition appears essentially independently in small boxes that partition

propagation of states in dilation analytic potentials and asymptotic completeness we estimate the space time behavior of scattering states for two body schrodinger operators with smooth dilation analytic potentials we use our estimates to give a simple proof of asymptotic completeness for a class of long range potentials including the coulomb potential plus a fairly general short range perturbation

asymptotic completeness for n body short range quantum systems a new proof we give an alternative geometrical proof of asymptotic completeness for an arbitrary number of quantum particles interacting through shortrange pair potentials it relies on an estimate showing that the intercluster motion concentrates asymptotically on classical trajectories

glauber dynamics for the mean field ising model cut off critical power law and metastability we study the glauber dynamics for the ising model on the complete graph also known as the curie weiss model for 1 we study metastability in particular we show that the glauber dynamics restricted to states of non negative magnetization has mixing time o n log n

blow up set for a semilinear heat equation with small diffusion abstract we consider the blow up problem for a semilinear heat equation t u u u p in 0 t u x t 0 on 0 t if u x 0 x 0 in where is a domain in r n n 1 0 p 1 and t 0 in this paper under suitable assumptions on we prove that if the family of the solutions u satisfies a uniform type i blow up estimate with respect to then the solution u blows up only near the maximum points of the initial datum for any sufficiently small 0 this is proved without any conditions on the exponent p and the domain such as n 2 p n 2 and the convexity of the domain

blow up problems for a semilinear heat equation with large diffusion abstract we consider the blow up problem of a semilinear heat equation u t d u u p in 0 t d u x t 0 on 0 t d u x 0 x 0 in where is a bounded smooth domain in r n t d 0 d 0 and p 1 we study the blow up time the location of the blow up set and the blow up profile of the blow up solution for sufficiently large d in particular we prove that for almost all initial data if d is sufficiently large then the solution blows up only near the maximum points of the orthogonal projection of the initial data from l 2 onto the second neumann eigenspace

the n particle scattering problem asymptotic completeness for short range systems on demontre la completude asymptotique pour des systemes de mecanique quantique a courte portee constitues d un nombre arbitraire de particules

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

propagation of singularities in many body scattering in the presence of bound states abstract in this paper we describe the propagation of singularities of tempered distributional solutions u s of h u 0 where h is a many body hamiltonian h v 0 v a v a and is not a threshold of h under the assumption that the inter particle e g two body interactions v a are real valued polyhomogeneous symbols of order 1 e g coulomb type with the singularity at the origin removed here the term singularity refers to a microlocal description of the lack of decay at infinity thus we prove that the set of singularities of u is a union of maximally extended broken bicharacteristics of h these are curves in the characteristic variety of h which can be quite complicated due to the existence of bound states we use this result to describe the wave front relation of the s matrices we also analyze lagrangian properties of this relation which shows that the relation is not too large in terms of its dimension

blow up for a semilinear parabolic equation with large diffusion on rn ii we are concerned with the cauchy problem for a semilinear heat equation p tu d u u p 1u x rn t 0 u x 0 x x rn where d 0 p 1 n 3 0 and l rn l1 rn 1 x 2dx in the paper of fujishima and ishige 2011 8 the authors of this paper studied the behavior of the blow up time and the blow up set of the solution of p as d for the case rn x dx 0 in this paper as a continuation of fujishima and ishige 2011 8 we consider the case rn x dx 0 and study the behavior of the blow up time and the blow up set of the solution of p as d the behavior in the case rn x dx 0 is completely different from the one in the case rn x dx 0

determination of the curvature of the blow up set and refined singular behavior for a semilinear heat equation we consider u x t a solution of ut u u p 1u which blows up at some time t 0 where u rn 0 t r p 1 and n 2 p n 2 under a nondegeneracy condition we show that the mere hypothesis that the blow up set s is continuous and n 1 dimensional implies that it is c2 in particular we compute the n 1 principal curvatures and directions of s moreover a much more refined blow up behavior is derived for the solution in terms of the newly exhibited geometric objects refined regularity for s and refined singular behavior of u near s are linked through a new mechanism of algebraic cancellations that we explain in detail

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

asymptotic completeness for 3 particle long range systems we prove asymptotic completeness for 3 particle systems with pair potentials satisfying the virial condition and decreasing at infinity as for 1 2

metastability in stochastic dynamics of disordered mean field models we study a class of markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding gibbs measures we derive the analog of the wentzell freidlin theory in this case showing that any transition can be decomposed with probability exponentially close to one into a deterministic sequence of admissible transitions for these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor the distributions of the rescaled transition times are shown to converge to the exponential distribution we exemplify our results in the context of the random field curie weiss model

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk