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  • Articles  (12)
  • 60K35  (12)
  • Springer  (12)
  • American Geophysical Union (AGU)
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  • 1995-1999  (12)
  • 1980-1984
  • 1935-1939
  • 1995  (12)
  • Mathematics  (12)
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  • Articles  (12)
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  • Springer  (12)
  • American Geophysical Union (AGU)
  • Annual Reviews
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  • 1995-1999  (12)
  • 1980-1984
  • 1935-1939
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  • Mathematics  (12)
  • 1
    Electronic Resource
    Electronic Resource
    The scaling limit for a stochastic PDE and the separation of phases (1995)
    Springer
    Probability theory and related fields 102 (1995), S. 221-288 
    Details
    ISSN: 1432-2064
    Keywords: 60H15 ; 60K35 ; 35R60 ; 82C24
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We investigate the problem of singular perturbation for a reaction-diffusion equation with additive noise (or a stochastic partial differential equation of Ginzburg-Landau type) under the situation that the reaction term is determined by a potential with double-wells of equal depth. As the parameter ε (the temperature of the system) tends to 0, the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit. We derive a stochastic differential equation which describes the random movement of the phase separation point. The proof consists of two main steps. We show that the solution stays near a manifoldM ε of minimal energy configurations based on a Lyapunov type argument. Then, the limit equation is identified by introducing a nice coordinate system in a neighborhood ofM ε.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01213390
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  • 2
    Electronic Resource
    Electronic Resource
    Onset and structure of interfaces in a Kawasaki+Glauber interacting particle system (1995)
    Springer
    Probability theory and related fields 103 (1995), S. 1-24 
    Details
    ISSN: 1432-2064
    Keywords: 60K35 ; 60G17 ; 82C24 ; 35K57
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01199029
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  • 3
    Electronic Resource
    Electronic Resource
    Time dependent critical fluctuations of a one dimensional local mean field model (1995)
    Springer
    Probability theory and related fields 103 (1995), S. 381-407 
    Details
    ISSN: 1432-2064
    Keywords: 60K35 ; 82A05 ; 82B40
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary One-dimensional stochastic Ising systems with a local mean field interaction (Kac potential) are investigated. It is shown that near the critical temperature of the equilibrium (Gibbs) distribution the time dependent process admits a scaling limit given by a nonlinear stochastic PDE. The initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way. Earlier results of Bertini-Presutti-Rüdiger-Saada are improved, the proof is based on an energy inequality obtained by coupling the Glauber dynamics to its voter type, linear approximation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01195480
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  • 4
    Electronic Resource
    Electronic Resource
    On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations (1995)
    Springer
    Probability theory and related fields 101 (1995), S. 1-44 
    Details
    ISSN: 1432-2064
    Keywords: 60F10 ; 60G57 ; 60K35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01192194
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  • 5
    Electronic Resource
    Electronic Resource
    Homogenization and propagation of chaos to a nonlinear diffusion with sticky reflection (1995)
    Springer
    Probability theory and related fields 101 (1995), S. 291-302 
    Details
    ISSN: 1432-2064
    Keywords: 60F17 ; 60K35 ; 35K60
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study a process reflecting in a domain. The process follows Wentzell non-sticky boundary conditions while being adsorbed at the boundary at a certain rate with respect to local time and desorbed at a rate with respect to natural time. We show that when the rates go to infinity with a converging ratio, the process converges to a process with sticky reflection having the limit ratio as the sojourn coefficient. We then study a mean-field interacting system of such particles. We show propagation of chaos to a nonlinear diffusion with sticky reflection when we perform this homogenization simultaneously as the number of particles goes to infinity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01200497
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  • 6
    Electronic Resource
    Electronic Resource
    The diffusive phase of a model of self-interacting walks (1995)
    Springer
    Probability theory and related fields 103 (1995), S. 285-315 
    Details
    ISSN: 1432-2064
    Keywords: 82B41 ; 60K35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We consider simple random walk onZ d perturbed by a factor exp[βT −P J T], whereT is the length of the walk and $$J_T = \sum\nolimits_{0 \leqslant i〈 j \leqslant T} \delta _{\omega (i),\omega (j)} $$ . Forp=1 and dimensionsd≥2, we prove that this walk behaves diffusively for all − ∞ 〈 β 〈0, with β0 〉 0. Ford〉2 the diffusion constant is equal to 1, but ford=2 it is renormalized. Ford=1 andp=3/2, we prove diffusion for all real β (positive or negative). Ford〉2 the scaling limit is Brownian motion, but ford≤2 it is the Edwards model (with the “wrong” sign of the coupling when β〉0) which governs the limiting behaviour; the latter arises since for $$p = \frac{{4 - d}}{2}$$ ,T −p J T is the discrete self-intersection local time. This establishes existence of a diffusive phase for this model. Existence of a collapsed phase for a very closely related model has been proven in work of Bolthausen and Schmock.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01195476
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  • 7
    Electronic Resource
    Electronic Resource
    Convergence to equilibrium for classical and quantum spin systems (1995)
    Springer
    Probability theory and related fields 103 (1995), S. 529-552 
    Details
    ISSN: 1432-2064
    Keywords: 60H15 ; 60K35 ; 35K55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01246338
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  • 8
    Electronic Resource
    Electronic Resource
    Hypercontractivité pour des systèmes de spins de portée infinie (1995)
    Springer
    Probability theory and related fields 101 (1995), S. 89-132 
    Details
    ISSN: 1432-2064
    Keywords: 60K35 ; 82C20 ; 82C26
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Summary We consider a spin system onℤ d . We prove the equivalence between first a weak mixing condition, secondly the controle of spectral gap and thirdly the controle of logarithmic Sobolev constants for non necessarily finite range Gibbs potentials. Hence we draw consequences concerning theL 2 decay to equilibrium and the correlations decay: there is no transitory rate between an algebraic decay ast −2d (resp. |j-k|−2d)and exponential decay. The general results are obtained for both continuous and discrete compact spins.
    Notes: Résumé Nous considérons un système de spins surℤ d . Nous prouvons l'équivalence entre premièrement une condition faible de mélange deuxièmement le contrôle du trou dans le spectre et troisièmement celui de la constante de Sobolev logarithmique pour des potentiels de Gibbs de portée non nécessairement finie. Nous en tirons des conséquences sur la vitesse de convergence des semi-groupes dansL 2 et sur la décroissance des corrélations: il n'y a pas de régime intermédiaire entre un taux algébrique ent −2d (resp. |j-k|−2d) et un taux exponentiel. Les résultats généraux sont montrés pour des spins à valeur dans une variété riemannienne compacte ou dans un espace fini.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01192197
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  • 9
    Electronic Resource
    Electronic Resource
    Maximum entropy principles for disordered spins (1995)
    Springer
    Probability theory and related fields 101 (1995), S. 547-576 
    Details
    ISSN: 1432-2064
    Keywords: 60F10 ; 60K35 ; 82B44
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We transform nonstationary independent random fields with exponential Radon-Nikodym factors and study the asymptotics of the transformed processes. As applications we deduce conditional limit theorems for such random fields, and we study a Curie-Weiss-type mean-field model of a quenched mixed magnetic crystal. This model has quenched site disorder and frustration but non-random coupling constants. We find a continuous phase transition with critical exponents equal to those of the classical Curie-Weiss theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01202784
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  • 10
    Electronic Resource
    Electronic Resource
    On the relation between finite range potentials and subshifts of finite type (1995)
    Springer
    Probability theory and related fields 101 (1995), S. 469-478 
    Details
    ISSN: 1432-2064
    Keywords: 60K35 ; 28D15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Consider a Gibbs potential on the integer lattice ind dimensions for a system with a finite state space. Suppose the interactions are translation invariant and have bounded range (the Ising and Potts models fit into this description). If the parameters of the potential are rational, we show how to construct an equivalent subshift of finite type, in the sense that there is a canonical bijection between Gibbs states for the potential and measures of maximal entropy for the subshift of finite type.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01202780
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  • 11
    Electronic Resource
    Electronic Resource
    Large deviation principles for the Hopfield model and the Kac-Hopfield model (1995)
    Springer
    Probability theory and related fields 101 (1995), S. 511-546 
    Details
    ISSN: 1432-2064
    Keywords: 60K35 ; 82B44 ; 82C32
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the Kac version of the Hopfield model and prove a Lebowitz-Penrose theorem for the distribution of the overlap parameters. At the same time, we prove a large deviation principle for the standard Hopfield model with infinitely many patterns.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01202783
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  • 12
    Electronic Resource
    Electronic Resource
    Large deviations for Langevin spin glass dynamics (1995)
    Springer
    Probability theory and related fields 102 (1995), S. 455-509 
    Details
    ISSN: 1432-2064
    Keywords: 60F10 ; 60H10 ; 60K35 ; 82C44
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We study the asymptotic behaviour of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime. We study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureQ which is not markovian. We deduce that the quenched law of the empirical measure converges to δ Q . Extending then the preceeding results to replicated dynamics, we investigate the quenched behavior of a single spin. We get quenched convergence toQ in the case of a symmetric initial law and even potential for the free spin.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01198846
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