Stochastic approximations to curve-shortening flows via particle systems

Curvature-driven flows have been extensively considered from a deterministic point of view. Besides their mathematical interest, they have been shown to be useful for a number of applications including crystal growth, flame propagation, and computer vision. In this paper, we describe a random particle system, evolving on the discretized unit circle, whose profile converges toward the Gauss-Minkowsky transformation of solutions of curve-shortening flows initiated by convex curves. Our approach may be considered as a type of stochastic crystalline algorithm. Our proofs are based on certain techniques from the theory of hydrodynamical limits.     

Shock fluctuations in asymmetric simple exclusion

Summary The one dimensional nearest neighbors asymmetric simple exclusion process in used as a microscopic approximation to the Burgers equation. We study the process with rates of jumpsp>q to the right and left, respectively, and with initial product measure with densities < to the left and right of the origin, respectively (with shock initial conditions). We prove that a second class particle added to the system at the origin at time zero identifies microscopically the shock for all later times. If this particle is added at another site, then it describes the behavior of a characteristic of the Burgers equation. For vanishing left density (=0) we prove, in the scale t^1/2, that the position of the shock at timet depends only on the initial configuration in a region depending ont. The proofs are based on laws of large numbers for the second class particle.     

The divergence of fluctuations for shape in first passage percolation

We consider the first passage percolation model on Z ^d for d ≥ 2. In this model, we assign independently to each edge the value zero with probability p and the value one with probability 1−p. We denote by T(0, ν) the passage time from the origin to ν for ν ∈ R ^d and It is well known that if p < p _c , there exists a compact shape B _d ⊂ R ^d such that for all > 0, t B _d (1 − ) ⊂ B(t) ⊂ tB _d (1 + ) and G(t)(1 − ) ⊂ B(t) ⊂ G(t)(1 + ) eventually w.p.1. We denote the fluctuations of B(t) from tB _d and G(t) by In this paper, we show that for all d ≥ 2 with a high probability, the fluctuations F(B(t), G(t)) and F(B(t), tB _d ) diverge with a rate of at least C log t for some constant C. The proof of this argument depends on the linearity between the number of pivotal edges of all minimizing paths and the paths themselves. This linearity is also independently interesting. Research supported by NSF grant DMS-0405150     

On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations

Summary We consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous Boltzmann type equation, as the number of particles tends to infinity. These laws of large numbers were proved for the Maxwellian molecules by H. Tanaka [Tal] and for the hard spheres by A.S. Sznitman [Szl]. In the present paper we investigate the corresponding large deviations: the large deviation upper bound is obtained and, using convex analysis, a non-variational formulation of the rate function is given. Our results hold for Maxwellian molecules with a cutoff potential and for hard spheres.     

Recurrence and ergodicity of interacting particle systems

Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s.the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that there exists a convergence–determining class of distributions that is (strongly) preserved under the dynamics, we show that the system is in fact recurrent in the above sense. We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we answer a question raised by Ed Perkins concerning the change of the locally predominant type in a model of mutually catalytic branching. Received: 22 January 1999 / Revised version: 24 May 1999     

Central limit theorem for a tagged particle in asymmetric simple exclusion

We prove a functional central limit theorem for the position of a tagged particle in the one-dimensional asymmetric simple exclusion process for hyperbolic scaling, starting from a Bernoulli product measure conditioned to have a particle at the origin. We also prove that the position of the tagged particle at time t$t$ depends on the initial configuration, through the number of empty sites in the interval [0,(p−q)αt]$[0,(p-q)\alpha t]$ divided by α$\alpha$, on the hyperbolic time scale and on a longer time scale, namely N^4/3$N^{4/3}$.     

Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems

We study the following class of scalar hyperbolic conservation laws with discontinuous fluxes:

(0.1)     

Phase transition in equilibrium fluctuations of symmetric slowed exclusion

We analyze the equilibrium fluctuations of density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is αn^−β$\alpha n^{-\beta }$, with α>0$\alpha >0$, β∈[0,+∞]$\beta \in [0,+\infty ]$ and n$n$ is the scaling parameter. Depending on the regime of β$\beta$, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value β=1$\beta =1$, starting a tagged particle near the slow bond, we obtain a family of Gaussian processes indexed in α$\alpha$, interpolating a fractional Brownian motion of Hurst exponent 1/4$1/4$ and the degenerate process equal to zero.     

Coupling in the theory of interacting systems

In this paper we consider the construction of couplings for Markovian evolutions on a state space of the formE , with (measurable) and a countable group (^d for example). The evolutions we focus on are mainly systems of linearly interacting diffusions, withE compact. We explain and state properties of such couplings and show how they are used to obtain information on the behaviour of the evolution in finite time and as time tends to infinity. An important property of a coupling is to be a successful coupling. The latter concept is introduced here in the context of interacting systems, which is different from the classical concept for Markov chains or processes with state space ^d. The analysis of the question when a coupling is successful depends heavily on the structure of the interaction term and is investigated in detail. We formulate some open problems and conjectures.The paper puts in perspective the coupling statements appearing in the proofs of various results and is largely based on the works of Cox and Greven, Fleischmann and Greven, Dawson and Greven, Greven, and Cox, Greven and Shiga.     

Finite and infinite systems of interacting diffusions

Summary 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 ∈ ℤ ^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, , with . 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. Partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, by the National Science Foundation, and by the National Security Agency Research supported in part by the DFG Partly supported by S.R.63540155 of Japan Ministry of Education     

N/V-limit for stochastic dynamics in continuous particle systems

We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on ℝ ^d ,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ ^d with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.     

Kinetic limits for a class of interacting particle systems

Summary We study one dimensional particle systems in which particles travel as independent random walks and collide stochastically. The collision rates are chosen so that each particle experiences finitely many collisions per unit time. We establish the kinetic limit and derive the discrete Boltzmann equation for the macroscopic particle density.     

Hammersley's interacting particle process and longest increasing subsequences

Summary In a famous paper [8] Hammersley investigated the lengthL _n of the longest increasing subsequence of a randomn-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly “soft” arguments that limn ^′1/2 EL _n =2. This is a known result, but previous proofs [14, 11] relied on hard analysis of combinatorial asymptotics. Research supported by NSF Grant MCS 92-24857 and the Miller Institute for Basic Research in Science Research supported by NSF Grant DMS92-04864     

Convergence to equilibrium for classical and quantum spin systems

Summary The paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on Euclidean spaces. Existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted Hilbert and Banach spaces. This paper is an enlarged and rewritten version of the paper [7].Partially supported by the Italian National Project MURST Problemi nonlinearinell' Analisi... and by DRET under contract 901636/A000/DRET/DSISR.Partially sponsored by the KBN grant 2 2003 91 02 and by the KBN grant 2PO3A 082 08     

Onset and structure of interfaces in a Kawasaki+Glauber interacting particle system

Summary We investigate the spatial structure of typical configurations of a reaction-diffusion spin system (Kawasaki+Glauber model), following the noise induced escape from an unstable spatially homogeneous state. After the escape, the system will be locally in a stationary phase, but will display a globally nontrivial spatial behavior, characterized by large clusters of the (two) different phases. The system can be spatially rescaled according to the typical linear dimension of the clusters and, on this space scale, regions of the opposite phases are separated by smooth (hyper) surfaces, called interfaces. The location of the interfaces is determined by means of the zero-level set of the trajectories of a Gaussian random field. This paper is devoted primarily to the characterization of the structure which appears on a finer scale (the hydrodynamical one) at the interface. A better understanding of the dynamics of the escape (especially in its last and nonlinear stage) leads to substantial improvements of the results in [7, 12].This research has been partly supported by NSF grant DMR 92-13424 and by a CNR fellowship     

Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems

Global asymptotic dynamics of a representative cubic-autocatalytic reaction-diffusion system, the reversible Selkov equations, are investigated. This system features two pairs of oppositely signed nonlinear terms so that the asymptotic dissipative condition is not satisfied, which causes substantial difficulties in an attempt to attest that the longtime dynamics are asymptotically dissipative. An L² to H¹ global attractor of finite fractal dimension is shown to exist for the semiflow of the weak solutions of the reversible Selkov equations with the Dirichlet boundary condition on a bounded domain of dimension n≤3. A new method of rescaling and grouping estimation is used to prove the absorbing property and the asymptotical compactness. Importantly, the upper semicontinuity (robustness) in the H¹ product space of the global attractors for the family of solution semiflows with respect to the reverse reaction rate as it tends to zero is proved through a new approach of transformative decomposition to overcome the barrier of the perturbed singularity between the reversible and non-reversible systems by showing the uniform dissipativity and the uniformly bounded evolution of the union of global attractors under the bundle of reversible and non-reversible semiflows.     

Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

Summary For Gibbsian systems of particles inR ^d , we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable, regular pair potential. The large deviation principle is established for systems with free or periodic boundary conditions and, under a stronger stability hypothesis on the potential, for systems with tempered boundary conditions, and for tempered (infinite-volume) Gibbs measures. As a by-product we obtain the Gibbs variational formula for the pressure. We also prove the asymptotic equivalence of microcanonical and grand canonical Gibbs distributions and establish a variational expression for the thermodynamic entropy density.