irreversibility implies the occurrence of nonmonotonic power spectra both continuous time and discrete time systems driven by l2 exponential ergodic stationary markov processes are investigated we prove that in both cases irreversibility implies the occurrence of nonmonotonic power spectra of some observables this result combined with the work of jiang and zhang j math phys 44 4681 4689 2003 tells us that in the continuous time case reversibility can be characterized by monotonicity in all power spectra while this is not true in the discrete time case

clustering of linearly interacting diffusions and universality of their long time limit distribution let k d d 1 be a compact convex set and a countable abelian group we study a stochastic process x in k equipped with the product topology where each coordinate solves a sde of the form dx i t j a j i x j t x i t dt x i t db i t here a is the kernel of a continuous time random walk on and is a continuous root of a diffusion matrix w on k if x t converges in distribution to a limit x and the symmetrized random walk with kernel a s i a i a i is recurrent then each component x i is concentrated on x k x 0 and the coordinates agree i e the system clusters both these statements fail if a s is transient under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of k we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components this distribution turns out to be universal in all recurrent kernels a s on abelian groups

the bogomolov conjecture for totally degenerate abelian varieties we prove the bogomolov conjecture for a totally degenerate abelian variety a over a function field we adapt zhang s proof of the number field case replacing the complex analytic tools by tropical analytic geometry a key step is the tropical equidistribution theorem for a at the totally degenerate place v as a corollary we obtain finiteness of torsion points with coordinates in the maximal unramified algebraic extension over v

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

interacting fisher wright diffusions in a catalytic medium we study the longtime behaviour of interacting systems in a randomly fluctuating space time medium and focus on models from population genetics there are two prototypes of spatial models in population genetics spatial branching processes and interacting fisher wright diffusions quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment catalytic medium

multiple time scale analysis of interacting diffusions we consider the questions how can the long term behavior of large systems of interacting components be described in terms of infinite systems on what time scale does the infinite system give a qualitatively correct description and what happens at large resp critical time scales lety n t be a solution y i n t i n n of the system of stochastic differential equations w i t are i i d brownian motions dy i n t left frac 1 2n 1 sum limits j n n y j n t y i n t right dt sqrt 2g y i n t dw i t in the mckean vlasov limit n we obtain the infinite independent system dy i infty t e y i infty t y i infty t dt sqrt 2g y i infty t dw i t i in z this infinite system has a one parameter set of invariant measures v theta mathop otimes limits x in z gamma theta with the equilibrium measure of dx t theta x t dt sqrt 2g x t dw t letq s be the transition kernel of the diffusion with generator u g x left frac partial partial x right 2 with u g x int g y gamma x dy then one main result is that asn mathcal l y n s 2n 1 rightarrow int q s theta d theta v theta theta e y 0 this provides a new example of a phenomenon also exhibited by the voter model and branching random walk in particular we are also able to modify our model by adding the termcn 1 a y i n t dt to obtain the first example in which the analog ofq s converges to an honest equilibrium instead of absorption in traps as in all models previously studied in the literature finally we discuss a hierarchical model with two levels from the point of view discussed above but now in two fast time scales

renormalization of the voter model in equilibrium we consider the d dimensional voter model for d 3 our interest is the large scale limit of the equilibrium state of the voter model where we prove the d 3 results of 1 for d 4 which turn out to be of a different nature than for d 3 for this purpose we use the historical process we establish some surprising facts about the green s function of random walks in dimension d 4 which lead to the different features in d 3 versus d 4 secondly we prove an analogous result for the voter model on the hierarchical group

a phase transition for the coupled branching process part i the ergodic theory in the range of finite second moments we consider a particular markov process tu on s s n the random variable tu x is interpreted as the number of particles atx at timet the initial distribution of this process is a translation invariant measure withf x d the evolution is as follows at rateb x a particle is born atx but moves instantaneously toy chosen with probabilityq x y all particles at a site die at ratepd withp 0 1 d and individual particles die independently from each other at rate 1 p d every particle moves independently of everything else according to a continuous time random walk

on the intersection of a clarke cone with a boltyanskii cone we provide an example of two closed sets s 1 s 2 subset r 4 such that s 1 cap s 2 0 yet at the origin a boltyanskii tangent cone c 1 to s 1 and the clarke tangent cone c 2 to s 2 are strongly transversal this settles a question originally proposed by h sussmann

comparisons for measure valued processes with interactions this paper considers some measure valued processes xt dvtxt 0 t based on an underlying critical branching particle structure with random branching rates in the case of constant branching these processes are dawson watanabe processes sufficient conditions on functionals of the process are given that imply that the expectations e xt are comparable to the constant branching case applications to hitting estimates and regularity of solutions are discussed the result is established via the martingale optimality principle of stochastic control theory key steps which are of independent interest are the proof of a version of ito s lemma for xt suitable for a large class of functions of measures theorem 3 and the proof of various smoothing properties of the dawson watanabe transition semigroup section 3

constrained optimization for average cost continuous time markov decision processes to the best of our knowledge this note is a first study on a multiconstrained optimization problem for average cost continuous time markov decision processes in polish spaces the optimality criterion is the long run expected average cost and the constraints are imposed on similar average costs we give suitable conditions under which the existence of a constrained optimal stationary policy is shown by introducing occupation measures and using a martingale technique also we illustrate our conditions with controlled potlach processes

coalescing random walks and voter model consensus times on the torus in mathbb z d voter models on z d and on the torus of site n in z d are considered the second ones get trapped with probability 1 as t tends to infinity at all 0 s or all 1 s the asymptotic growth of these trapping times are examined as n tends to infinity this is done with limit thewems for coalescing random walk systems on the torus of side n in z d

the distribution of galois orbits of points of small height in toric varieties we study the distribution of galois orbits of points of small height on proper toric varieties and its application to the bogomolov problem we introduce the notion of monocritical toric metrized divisor we prove that a toric metrized divisor d on a proper toric variety x over a global eld k is monocritical if and only if for every generic d small sequence of algebraic points of x and every place v of k the sequence of their galois orbits on the analytic space x an converges to a measure when this is the case the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of x the key ingredient is the study of the v adic modulus distribution of ga lois orbits of generic d small sequences of algebraic points in particular we characterize all their cluster measures we generalize the bogomolov problem by asking when a subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the ambient variety must be a translate of a subtorus we prove that the generalized bogomolov problem has a positive answer for monocritical toric metrized divisors and we give several examples of toric metrized divisors for which the bogomolov problem has a negative answer

exclusion processes with degenerate rates convergence to equilibrium and tagged particle stochastic lattice gases with degenerate rates namely conservative particle systems where the exchange rates vanish for some configurations have been introduced as simplified models for glassy dynamics we introduce two particular models and consider them in a finite volume of size l in contact with particle reservoirs at the boundary we prove that as for non degenerate rates the inverse of the spectral gap and the logarithmic sobolev constant grow as l2 it is also shown how one can obtain via a scaling limit from the logarithmic sobolev inequality the exponential decay of a lyapunov functional for a degenerate parabolic differential equation porous media equation we analyze finally the tagged particle displacement for the stationary process in infinite volume in dimension larger than two we prove that in the diffusive scaling limit it converges to a brownian motion with non degenerate diffusion coefficient

phase transitions for the long time behavior of interacting diffusions let xi t i zd t 0 be the system of interacting diffusions on 0 defined by the following collection of coupled stochastic differential equations dxi t j zd a i j xj t xi t dt bxi t dwi t i z t 0 here a is an irreducible random walk transition kernel on z z b 0 is a diffusion parameter and wi t i zd t 0 is a collection of independent standard brownian motions on r the initial condition is chosen such that xi 0 i zd is a shift invariant and shift ergodic random field on 0 with mean 0 the evolution preserves the mean we show that the long time behaviour of this system is the result of a delicate interplay between a and b in contrast to systems where the diffusion function is subquadratic in particular let i j 12 a i j a j i i j z denote the symmetrised transition kernel we show that a if is recurrent then for any b 0 the system locally dies out b if is transient then there exist b b2 0 such that b1 the system converges to an equilibrium with mean if 0 b b3 has a finite 2 nd moment if and only if 0 b2 the equilibrium is shown to be associated and mixing for all 0 b2 we further conjecture that the system locally dies out at b b for the case where a is symmetric and transient we further show that c there exists a sequence b2 b3 b4 0 such that c1 has a finite m th moment if and only if 0 bm c3 b2 m 1 bm 2 c4 limm m 1 bm c supm 2 m 1 bm mathematisches institut universitat erlangen nurnberg bismarckstrasse 1 1 2 d 91504 erlangen germany greven mi uni erlangen de mathematical institute leiden university p o box 9512 2300 ra leiden the netherlands denholla math leidenuniv nl eurandom p o box 513 5600 mb eindhoven the netherlands

different clustering regimes in systems of hierarchically interacting diffusions we study a system of interacting diffusions dx t a x t x t dt g x t dw t indexed by the hierarchical group as a genealogical two genotype model where x t denotes the frequency of say type a with hierarchically determined degrees of relationship between colonies in the case of short interaction range it is known that the system clusters that is locally one genotype dies out we focus on the description of the different regimes of cluster growth which is shown to depend on the interaction kernel a via its recurrent potential kernel one of these regimes will be further investigated by mean field methods for general interaction range we shall also relate the behavior of large finite systems indexed by finite subsets n of to that of the infinite system on the way we will establish relations between hitting times of random walks and their potentials

central limit theorem for an infinite lattice system of interacting diffusion processes on donne une demonstration basee sur une representation par integrale stochastique de dimension infinie des fonctionnelles lisses des processus de diffusion

finite and infinite systems of interacting diffusions we study the problem of relating the long time behavior of finite and infinite systems of locally interacting components we consider in detail a class of lincarly interacting diffusionsx t x i t i d in the regime where there is a one parameter family of nontrivial invariant measures for these systems there are naturally defined corresponding finite systems x n t left x i n t i in lambda n right with lambda n n n d cap mathbb z d our main result gives a comparison between the laws ofx t n andx n t n for timest n asn the comparison involves certain mixtures of the invariant measures for the infinite system

second variation of zhang s λ invariant on the moduli space of curves we compute the second variation of the invariant recently introduced by s zhang on the complex moduli space mg of curves of genus g 2 using work of n kawazumi as a result we prove that 8g 4 is equal up to a constant to the invariant introduced some time ago by r hain and d reed we deduce some consequences for example we calculate the invariant for each hyperelliptic curve expressing it in terms of the petersson norm of the discriminant modular form 1 introduction recently independently s zhang 27 and n kawazumi 16 introduced a new interesting real valued function on the moduli space mg of complex curves of genus g 2 its value at a curve x m g is given as follows let h 0 x x be the space of holomorphic differentials on x equipped with the

controllability along a trajectory a variational approach this paper unifies and improves most sufficient conditions of local controllability both along a trajectory and at a point this is accomplished by defining high order variations that can be continuously summed the peculiar property of these variations is that they may be generated by thin conditions as relations in the lie algebra associated with a control system this property leads to a variational interpretation of sussmann type sufficient conditions of local controllability neutralization of obstructions and it allows the use of different weights for neutralization

on the long term behavior of some finite particle systems we consider the problem of comparing large finite and infinite systems with locally interacting components and present a general comparison scheme for the case when the infinite system is nonergodic we show that this scheme holds for some specific models one of these is critical branching random walk onz d let t denote this system and let denote a finite version of t defined on the torus n n d z d ford 3 we prove that for stationary shift ergodic initial measures with density that ift n andt n 2n 1 d s 0 asn then v 0 is the set of extremal invariant measures for the infinite system t andq s is the transition function of feller s branching diffusion we prove several extensions and refinements of this result the other systems we consider are the voter model and the contact process

fibered algebraic surfaces with low slope the background of this paper is the following conjecture on complex surfaces of general type let s be a smooth projective minimal surface of general type over ti with c 2 throughout the paper

ergodicity of reversible reaction diffusion processes reaction diffusion processes were introduced by nicolis and prigogine and haken existence theorems have been established for most models but not much is known about ergodic properties in this paper we study a class of models which have a reversible measure we show that the stationary distribution is unique and is the limit starting from any initial distribution

on the dirichlet problem for asymmetric zero range process on increasing domains in order to obtain hitting time estimates for the asymmetric zero range process azrp on d in dimensions d 3 we characterize the principal eigenvalue of the generator of the azrp with dirichlet boundary on special domains we obtain a donsker varadhan variational representation and show that the corresponding eigenfunction is unique in a natural class of functions

zhang s conjecture and the effective bogomolov conjecture over function fields we prove the effective bogomolov conjecture and so the bogomolov conjecture over a function field of characteristic 0 by proving zhang s conjecture about certain invariants of metrized graphs in the function field case these conjectures were previously known to be true only for curves of good reduction for curves of genus at most 4 and a few other special cases we also either verify or improve the previous results we relate the invariants involved in zhang s conjecture to the tau constant of metrized graphs then we use and extend our previous results on the tau constant by proving another conjecture of zhang we obtain a new proof of the slope inequality for faltings heights on moduli space of curves

strong invariance and noise comparison principles for some parabolic stochastic pdes we consider a system of interacting diffusions on the integer lattice by letting the mesh size go to zero and by using a suitable scaling we show that the system converges in a strong sense to a solution of the stochastic heat equation on the real line as a consequence we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities

optimization and controllability without differentiability assumptions let mathcal x and mathcal y be real normed vector spaces k subset mathcal x convex and compact c subset mathcal y a convex body mathcal u subset k and phi phi k to mathbb r m times mathcal y a function that can be appropriately approximated by functions phi i phi i whose compositions with linear maps of small finite dimensional simplices are c 1 we derive sufficient conditions for phi to be locally controllable on mathcal u subject to the restriction phi u in c and obtain as a corollary corresponding necessary conditions for a related restricted minimum these conditions are formulated in terms of directional derivate containers which are a type of set valued derivatives of phi phi and they improve on and extend previously obtained results they are used elsewhere to obtain new conditions for controllability and restricted minimum in nonsmooth optimal control problems defined by differential or functional integral equatio

renormalization of hierarchically interacting isotropic diffusions we study a renormalization transformation arising in an infinite system of interacting diffusions the components of the system are labeled by the n dimensional hierarchical lattice n 2 and take values in the closure of a compact convex set bar d subset mathbb r d d geqslant 1 each component starts at some d and is subject to two motions 1 an isotropic diffusion according to a local diffusion rate g bar d to 0 infty chosen from an appropriate class 2 a linear drift toward an average of the surrounding components weighted according to their hierarchical distance in the local mean field limit n block averages of diffusions within a hierarchical distance k on an appropriate time scale are expected to perform a diffusion with local diffusion rate f k g where f k g f c k circ circ f c 1 g is the kth iterate of renormalization transformations fc c 0 applied to g here the ck measure the strength of the interaction at hierarchical distance k we identify fc and study its orbit f k g k 0 we show that there exists a fixed shape g such that limk kf k g g for all g where the k are normalizing constants in terms of the infinite system this property means that there is complete universal behavior on large space time scales our results extend earlier work for d 1 and bar d 0 1 resp 0 the renormalization transformation fc is defined in terms of the ergodic measure of a d dimensional diffusion in d 1 this diffusion allows a yamada watanabe type coupling its ergodic measure is reversible and the renormalization transformation fc is given by an explicit formula all this breaks down in d 2 which complicates the analysis considerably and forces us to new methods part of our results depend on a certain martingale problem being well posed

diffusive clustering in an infinite system of hierarchically interacting diffusions we study a countable system of interacting diffusions on the interval 0 1 indexed by a hierarchical group a particular choice of the interaction guaranties we are in the diffusive clustering regime this means clusters of components with values either close to 0 or close to 1 grow on various different scales however single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary the processes in the whole class considered and starting with a shift ergodic initial law have the same qualitative properties universality

comparison of interacting diffusions and an application to their ergodic theory a general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived as an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent and the components take values in an interval bounded on one side the technique also gives an alternative proof of clustering in the case of compact intervals

homeomorphisms and local c 1 approximations soient 7b h et 7b y des espaces de banach v 7b h ouvert et f v 7b y continue on demontre que les extensions de john du theoreme de hadamard sont applicables a des fonctions entre des espaces de banach avec des conteneurs de derivees locaux inversibles

generalized potlatch and smoothing processes we consider simple generalizations of the potlatch and smoothing processes which were introduced in 8 and studied in 5 these generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition the original potlatch and smoothing processes do not exhibit phase transition our results show that for the generalized processes phase transition does not usually occur in one or two dimensions but usually does occur in higher dimensions upper and lower bounds for the relevant critical values are obtained as one application of our results we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d thus answering a question we raised in 3 this is done by comparing the contact process with an appropriate generalized smoothing process

recurrence and ergodicity of interacting particle systems an important technical result proposition 2 3 and its proof present in the original submission was erroneously omitted when this paper was published in 2000 the missing text which should have appeared on page 243 directly before section 3 is included in this erratum together with a short account of its context simultaneously with the publication of this erratum the electronic version of the paper in vol 116 no 2 2000 will be completed by insertion of the missing text readers of probability theory and related fields who have access to the electronic version of this erratum will also have access via a url to the full intact paper

cluster formation in a stepping stone model with continuous hierarchically structured sites a stepping stone model with site space a continuous hierarchical group is constructed via duality with a system of delayed coalescing stable levy processes this model can be understood as a continuum limit of discrete state space two allele genetics models with hierarchically structured resampling and migration the existence of a process rescaling limit on suitably related large space and time scales is established and interpreted in terms of the dynamics of cluster formation this paper was inspired by recent work of klenke

the green kubo formula for general markov processes with a continuous time parameter for general markov processes the green kubo formula is shown to be valid under a mild condition a class of stochastic evolution equations on a separable hilbert space and three typical infinite systems of locally interacting diffusions on irreversible in most cases are shown to satisfy the green kubo formula and the einstein relations for these stochastic evolution equations are shown explicitly as a corollary

continuous time markov decision processes with discounted rewards the case of polish spaces this paper deals with continuous time markov decision processes in polish spaces under an expected discounted reward criterion the transition rates of underlying continuous time jump markov processes are allowed to be unbounded and the reward rates may have neither upper nor lower bounds we first give conditions on the controlled system s primitive data under these conditions we prove that the transition functions of possibly nonhomogeneous continuous time markov processes are regular by using feller s construction approach to such transition functions then under additional continuity and compactness conditions we ensure the existence of optimal stationary policies by using the technique of extended infinitesimal operators associated with the transition functions and also provide a recursive way to compute or at least to approximate the optimal reward values finally we use examples to illustrate our results and the gap between our conditions and those in the previous literature

lower bounds on the arithmetic self intersection number of the relative dualizing sheaf on arithmetic surfaces we give an explicitly computable lower bound for the arithmetic self intersection number of the dualizing sheaf on a large class of arithmetic surfaces if some technical conditions are satisfied then this lower bound is positive in particular these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non reduced fibers

set separation approximating multicones and the lipschitz maximum principle abstract we present a general necessary condition for separation of the reachable set of a lipschitz control system from another given set of states expressed in terms of an approximating multicone to the set in a sense that contains as special cases the clarke and mordukhovich cones we then show how this separation result implies a strengthened form of the usual sufficient condition for local controllability along the reference curve and the necessary condition for optimality

diffusive clustering in the two dimensional voter model on donne des resultats de simulation et une enquete concernant les resultats connus sur le modele du votant dans trois cas distincts trois ou plusieurs dimensions une dimension et deux dimensions

infinite rate mutually catalytic branching in infinitely many colonies the longtime behavior consider the infinite rate mutually catalytic branching process imub constructed in infinite rate mutually catalytic branching in infinitely many colonies construction characterization and convergence 2008 preprint and ann probab 38 2010 479 497 for finite initial conditions we show that only one type survives in the long run if the interaction kernel is recurrent on the other hand under a slightly stronger condition than transience we show that both types can coexist

large deviations and variational principle for harmonic crystals we consider massless gaussian fields with covariance related to the green function of a long range random walk on d these are viewed as gibbs measures for a linear quadratic interaction we establish thermodynamic identities and prove a version of gibbs variational principle showing that translation invariant gibbs measures are characterized as minimizers of the relative entropy density we then study the large deviations of the empirical field of a gibbs measure we show that a weak large deviation principle holds at the volume order with rate given by the relative entropy density

ergodic theorems for coupled random walks and other systems with locally interacting components in 5 the second author introduced a variety of new infinite systems with locally interacting components on the basis of computations for the finite analogues of these systems he made conjectures ragarding their limiting behavior as t this paper is devoted to the construction of these processes and to the proofs of these conjectures we restrict ourselves primarily to spatially homogeneous situations interesting problems remain unsolved in inhomogeneous cases two features distinguish these processes from most other infinite particle systems which have been studied one is that the state spaces of these systems are noncompact the other that even though the invariant measures are not generally of product form one can nevertheless compute explicitly the first and second moments of the number of particles per site in equilibrium the second moment computations are of inherent interest of course and they play an important role in the proofs of the ergodic theorems as well

consolidation rates for two interacting systems in the plane this paper is a sequel of a paper of cox and griffeath diffusive clustering in the two dimensional voter model we continue our study of the voter model and coalescing random walks on the two dimensional integer lattice some exact asymptotics concerning the rate of clustering in the former process and the coalescence rate of the latter are derived we use these results to prove a limit law announced in that earlier paper concerning the size of the largest square centered at the origin which is of solid color at a large time t

relative bogomolov s inequality and the cone of positive divisors on the moduli space of stable curves an interesting point of the above theorem is that even if the weak positivity of disx y e at y is a global property on y it can be derived from the local assumption the goodness of xg and the semistability of eg this gives a great advantage to our applications in order to understand the intuition underlying the theorem let us consider a simulated case namely we suppose that f x y is a smooth surface fibred over a curve and the fiber is general bogomolov s instability theorem 1 says that if e is semistable then the codimension two cycle 2rc2 e r 1 cl e 2 has nonnegative degree so if we push it down to a codimension one cycle on y then one can rephrase bogomolov s theorem as saying that the semistability of ey implies the non negativity of disx y e

coalescing systems of non brownian particles a well known result of arratia shows that one can make rigorous the notion of starting an independent brownian motion at every point of an arbitrary closed subset of the real line and then building a set valued process by requiring particles to coalesce when they collide arratia noted that the value of this process will be almost surely a locally finite set at all positive times and a finite set almost surely if the initial value is compact the key to both of these facts is the observation that because of the topology of the real line and the continuity of brownian sample paths at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them we investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous we give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of brownian motions on the sierpinski gasket and stable processes on the real line with stable index greater than one

infinite systems with locally interacting components in 1963 glauber analyzed a one dimensional model for magnetism it was the first study of the markovian time evolution of a system with infinitely many interacting components in section 1 it will be discussed in the light of recent progress in this field the remaining sections 2 3 and 4 form an introduction to recent joint work with thomas m liggett they concern new types of systems where each component takes on a real value which fluctuates in a way depending linearly on the values of neighboring components the ergodic theory for such systems with finitely many components is the subject of section 2 the results suggest conjectures for the case of infinitely many components stated in section 3 and proved in the joint paper with t liggett ibid section 4 introduces another class of time evolutions whose ergodic behavior may be analyzed by similar methods

stochastic processes with gaussian interaction of components letu x xe d 0 be a nonnegative even function such that x 0u x 1 in this paper we consider an infinite system of stochastic process t x xe d with the following mechanism at each sitex after mean 1 exponential waiting time t x is replaced by a gaussian random variable with mean y x t y u y x and variance 1 it is understood here that all the interactions are independent of one another the behavior of this system will be investigated and some ergodic theorems will be derived the results strongly depend whether x 0u x 1 or 1

ergodic theorems for infinite systems of locally interacting diffusions let x t x i t i in mathbb z d be the solution of the system of stochastic differential equations dx i t bigg sum j in mathbb z d a i j x j t x i t bigg dt sqrt 2g x i t dw i t quad i in mathbb z d here g lbrack 0 1 rbrack rightarrow mathbb r satisfies g 0 on 0 1 g 0 g 1 0 g is lipschitz a i j is an irreducible random walk kernel on mathbb z d and w i t i in mathbb z d is a family of standard independent brownian motions on mathbb r x t is a markov process on x lbrack 0 1 rbrack mathbb z d this class of processes was studied by notohara and shiga the special case g v v 1 v has been studied extensively by shiga we show that the long term behavior of x t depends only on hat a i j a i j a j i 2 and is universal for the entire class of g considered if hat a i j is transient then there exists a family nu theta theta in lbrack 0 1 rbrack of extremal translation invariant equilibria each nu theta is mixing and has density theta int x 0 d nu theta if hat a i j is recurrent then the set of extremal translation invariant equilibria consists of the point masses delta 0 delta 1 the process starting in a translation invariant shift ergodic measure mu on x with int x 0 d mu theta converges weakly as t rightarrow infty to nu theta if hat a i j is transient and to 1 theta delta 0 theta delta 1 if hat a i j is recurrent our results in the recurrent case remove a mild assumption on g imposed by notohara and shiga for the case hat a i j transient we use methods developed for infinite particle systems by liggett and spitzer for the case hat a i j recurrent we use a duality comparison argument

the contact process on a finite set ii in this paper we complete the work started in part i we show that if sigma n is the time that the contact process on 1 ldots n first hits the empty set then for lambda lambda c the critical value for the process on z there is a positive constant gamma lambda so that log sigma n n rightarrow gamma lambda in probability as n rightarrow infty we also give a new simple proof that sigma n e sigma n converges to a mean one exponential the keys to the proof of the first result are a planar graph duality for the contact process and an observation of j chayes and l chayes that exponential decay rates for connections in strips approach the decay rates in the plane as the width of the strip goes to infty

ergodicity for generalized kawasaki dynamics we give a necessary and sufficient condition for a gibbs measure on the product space s1 d to satisfy the spectral gap or the logarithmic sobolev inequality with the following quadratic form where y is a finite set and al are integers as a consequence we prove that the generalized kawasaki dynamics decays exponentially to equilibrium in the supremum norm in a strong mixing region

hierarchical models of interacting diffusions multiple time scale phenomena phase transition and pattern of cluster formation the purpose of this paper is to explore the connection between multiple space time scale behaviour for block averages and phase transitions respectively formation of clusters in infinite systems with locally interacting components the essential object is the associated markov chain which describes the joint distribution of the block averages at different time scales a fixed point and stability property of a particular dynamical system under a renormalisation procedure is used to explain this pattern of cluster formation and the fact that the longtime behaviour is universal in entire classes of evolutions

some local approximations of dawson watanabe superprocesses let be a dawson watanabe superprocess in rd such that t is a s locally finite for every t 0 then for d 2 and fixed t 0 the singular ran dom measure t can be a s approximated by suitably normalized restrictions of lebesgue measure to the neighborhoods of suppff when d 3 the local distributions of near a hitting point can be approximated in total vari ation by those of a stationary and self similar pseudo random measure f by contrast the corresponding distributions for d 2 are locally invariant fur ther results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities our main proofs are based on a detailed analysis of the historical structure of f

stationary states and their stability of the stepping stone model involving mutation and selection on considere un modele de diffusion qui est un processus de diffusion de dimension infinie on etudie la regularite des distributions marginales de dimension finie des etats stationnaires

a functional clt for the occupation time of a state dependent branching random walk we show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension d 3 started off either from a poisson field or in equilibrium when suitably normalized converges to a brownian motion in d 4 in d 3 the limit process is a fractional brownian motion with hurst parameter 3 4 when starting in equilibrium and a related gaussian process when starting from a poisson field for dependent branching random walks with state dependent branching rate we obtain convergence in f d d to the same limit process and for d 3 also a functional limit theorem

multiple scale analysis of clusters in spatial branching models in this paper we will investigate the long time behaviour of critical branching brownian motion and finite variance super brownian motion the so called dawson watanabe process on r d these processes are known to be persistent if d 3 that is there exist non trivial equilibrium measures if d 2 they cluster i e the processes converge to the 0 configuration while the surviving mass piles up in so called clusters we study the spatial profile of the clusters in the critical dimension d 2 via multiple space scale analysis we will also investigate the long time behaviour of these models restricted to finite boxes in d 2 on the way we develop coupling and comparison methods for spatial branching models

branching coalescing particle systems we study the ergodic behavior of systems of particles performing independent random walks binary splitting coalescence and deaths such particle systems are dual to systems of linearly interacting wright fisher diffusions used to model a population with resampling selection and mutations we use this duality to prove that the upper invariant measure of the particle system is the only homogeneous nontrivial invariant law and the limit started from any homogeneous nontrivial initial law

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

dissipation and high disorder given a fieldfb x gx2zd of independent standard brownian motions indexed by z d the generator of a suitable markov process on z d g and suciently nice function 0 1 0 1 we consider the influence of the parameter on the behavior of the system dut x gut x dt ut x dbt x t 0 x2 z d u0 x c0 0 x we show that for any

continuum sites stepping stone models coalescing exchangeable partitions and random trees analogues of stepping stone models are considered where the sitespace is continuous the migration process is a general markov process and the type space is infinite such processes were defined in previous work of the second author by specifying a feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing markov processes an alternative representation is obtained here in terms of a limit of interacting particle systems it is shown that under a mild condition on the migration process the continuum sites stepping stone process has continuous sample paths the case when the migration process is brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating brownian motion this duality relation is also used to show that a tree like random compact metric space that is naturally associated to an in nite family of coalescing brownian motions on the circle has hausdorff and packing dimension both almost surely equal to 1 2 and moreover this space is capacity equivalent to the middle 1 2 cantor set and hence also to the brownian zero set

convergence in distribution of products of random matrices we consider a sequence a2 a2 of i i d nonnegative matrices of size d d and investigate convergence in distribution of the product mn a1 an when d 2 it is possible for mn to converge in distribution without normalization to a distribution not concentrated on the zero matrix several equivalent conditions for this to happen are given these lead to a fairly general family of examples these conditions can also be used to determine when the a s limit of 1 nlog mn equals the logarithm of the largest eigenvalue of e a1

an euler lagrange inclusion for optimal control problems a new first order necessary condition is proved for nonsmooth nonlinear optimal control problems with general endpoint constraints and for which the velocity set may be possibly nonconvex it is in the nature of a generalization of the euler lagrange equation of the calculus of variations to optimal control it resembles the weak form of the maximum principle but it is distinct from it because it employs a total generalized gradient instead of the customary product of partial generalized gradients the optimality condition is shown to be sufficient for optimality when it is specialized to apply to normal convex problems a counterexample illustrates that for such problems the maximum principle is not a sufficient condition

on the attracting orbit of a nonlinear transformation arising from renormalization of hierarchically interacting diffusions abstract this paper analyzes the n fold composition of a non linear integral operator acting on a class of functions on 0 attracting orbits and attracting fixed points are identified various results of convergence to these orbits and to these fixed points are derived the proofs are based on order preserving properties and comparison techniques a key role is played by the eigenfunctions of the operator which are used as comparison objects the results imply that the space time scaling limit of an infinite system of interacting diffusions has universal behavior independent of model parameters the paper can be read independently of part i

coupling concentration inequalities and stochastic dynamics in the context of interacting particle systems we study the influence of the action of the semigroup on the concentration property of lipschitz functions as an application this gives a new approach to estimate the relaxation speed to equilibrium of interacting particle systems we illustrate our approach in a variety of examples for which we obtain several new results with short and nontechnical proofs these examples include the symmetric and asymmetric exclusion processes and high temperature spin flip dynamics glauber dynamics we also give a new proof of the poincare inequality based on coupling in the context of one dimensional gibbs measures in particular we cover the case of polynomially decaying potentials where the log sobolev inequality does not hold

occupation statistics of critical branching random walks in two or higher dimensions consider a critical nearest neighbor branching random walk on the d dimensional integer lattice initiated by a single particle at the origin let g n be the event that the branching random walk survives to generation n we obtain the following limit theorems conditional on the event g n for a variety of occupation statistics 1 let v n be the maximal number of particles at a single site at time n if the offspring distribution has finite th moment for some integer a 2 then in dimensions 3 and higher v n o p n 1 if the offspring distribution has an exponentially decaying tail then v n o p log n in dimensions 3 and higher and v n o p log n 2 in dimension 2 furthermore if the offspring distribution is nondegenerate then p vn log n g n 1 for some 0 2 let m n j be the number of multiplicity j sites in the nth generation that is sites occupied by exactly j particles in dimensions 3 and higher the random variables m n j n converge jointly to multiples of an exponential random variable 3 in dimension 2 the number of particles at a typical site i e at the location of a randomly chosen particle of the nth generation is of order o p log n and the number of occupied sites is o p n log n we also show that in dimension 2 there is particle clustering around a typical site

a high order test for optimality of bang bang controls for control systems of the form dot x x x sum i 1 m y i x u i a strengthened version of the classical pontryagin maximum principle is proved the necessary condition for optimality given here is obtained using functional analytic techniques and quite general high order perturbations of the reference control as shown by an example this test is particularly effective when applied to bang bang controls a case where other high order tests do not provide additional information

asymptotic density in a threshold coalescing and annihilating random walk we consider an interacting random walk on z d where particles interact only at times when a particle jumps to a site at which there are at least n 1 other particles present if there are i n 1 particles present then the particle coalesces is removed from the system with probability c i and annihilates is removed along with another particle with probability a i we call this process the n threshold randomly coalescing and annihilating random walk we show that for n 3 if both a i and a i c i are increasing in i and if the dimension d is at least 2n 4 then p the origin is occupied at time t c d n t 1 n 1 e number of particles at the origin at time t c d n t 1 the constants c d n are explicitly identified the proof is an extension of a result obtained by kesten and van den berg for the 2 threshold coalescing random walk and is based on an approximation for de t dt