An analytic approach to existence and uniqueness for martingale problems in infinite dimensions

We prove existence and uniqueness for a class of martingale problems in a Hilbert space. We solve the associated Kolmogorov equation and prove that the corresponding semigroup is determined by a kernel of measures if a Schauder-type regularity is satisfied. Received: 18 May 1998 / Revised version: 27 September 1999 / Published online: 5 September 2000     

Discrete schemes for processes in random media

We present discrete schemes for processes in random media. We prove two results. The first one is the convergence of Sinai's random walks in random environments to the Brox model. The second one is the convergence of random walks in media with random “gates” to a continuous process in a Poisson potential. The proofs are based on the following idea: we consider the discrete media as random potentials for continuous models. Received: 6 May 1999 / Revised version: 18 October 1999 / Published online: 20 October 2000     

A law of the iterated logarithm for heavy-tailed random vectors

For a random vector belonging to the (generalized) domain of operator semistable attraction of some nonnormal law we prove various variants of Chover's law of the iterated logarithm for the partial sum. Furthermore we also derive some large deviation results necessary for the proof of our main theorems. Received: 30 September 1998 / Revised version: 28 May 1999     

Almost sure Kallianpur–Robbins laws for Brownian motion in the plane

The Kallianpur–Robbins law describes the long term asymptotic behaviour of integrable additive functionals of Brownian motion in the plane. In this paper we prove an almost sure version of this result. It turns out that, differently from many known results, this requires an iterated logarithmic average. A similar result is obtained for the small scales asymptotic by means of an ergodic theorem of Chacon–Ornstein type, which allows an exceptional set of scales. Received: 8 May 1998 / Revised version: 1 December 1999 / Published online: 8 August 2000     

A superatomic Boolean algebra with few automorphisms

Assuming GCH, we prove that for every successor cardinal μ > ω₁, there is a superatomic Boolean algebra B such that |B| = 2^μ and |Aut B| = μ. Under ◊_ω1, the same holds for μ = ω₁. This answers Monk's Question 80 in [Mo]. Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000     

Exponential decay rate of survival probability in a disastrous random environment

Summary. We study the exponential decay rate of the survival probability up to time t>0 of a random walker moving in Zopf; ^d in a temporally and spatially fluctuating random environment. When the random walker has a speed parameter κ>0, we investigate the influence of κ on the exponential decay rate λ(d,κ). In particular we prove that for any fixed d≥1, λ(d,κ) behaves like as logκ as κ↘0. Received: 21 May 1996 / In revised form: 2 February 1997     

Inégalités de concentration pour les processus empiriques de classes de parties

We propose new concentration inequalities for maxima of set-indexed empirical processes. Our approach is based either on entropy inequalities or on martingale methods. The improvements we get concern the rate function which is exactly the large deviations rate function of a binomial law in most of the cases. Received: 11 January 2000 / Revised version: 12 May 2000 / Published online: 14 December 2000     

Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations

We consider adaptive estimating the value of a linear functional from indirect white noise observations. For a flexible approach, the problem is embedded in an abstract Hilbert scale. We develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the Hilbert scale. It is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals. The case of discretized indirect white noise observations is studied, and the adaptive estimator in this setting is developed. Received: 5 May 1999 / Revised version: 25 October 1999 / Published online: 5 September 2000     

Rigorous low-temperature results for the mean field p-spins interaction model

We prove that, just below the critical temperature, the mean field p-spins interaction model, for p suitably large, spontaneously decomposes into different states. The asymptotic overlaps between any two different states are zero. Under a mild (unproven) hypothesis on the weight distribution of these states, we prove that they are pure states. This situation is called in physics “one level of symmetry breaking”. Received: 15 January 1998 / Revised version: 10 November 1999 / Published online: 21 June 2000     

A criterion of density for solutions of Poisson-driven SDEs

We construct a Dirichlet structure related to a Poisson measure on ℝ⁺×M, where M is a general measured space, with compensator dt⊗dv. We obtain a criterion of density for variables in the domain of the Dirichlet form and we apply it to S.D.E. driven by this Poisson measure. Received: 15 May 1999 / Revised version: 23 February 2000 / Published online: 12 October 2000     

Large deviation principles for Euclidean functionals and other nearly additive processes

We prove a large deviation principle for a process indexed by cubes of the multidimensional integer lattice or Euclidean space, under approximate additivity and regularity hypotheses. The rate function is the convex dual of the limiting logarithmic moment generating function. In some applications the rate function can be expressed in terms of relative entropy. The general result applies to processes in Euclidean combinatorial optimization, statistical mechanics, and computational geometry. Examples include the length of the minimal tour (the traveling salesman problem), the length of the minimal matching graph, the length of the minimal spanning tree, the length of the k-nearest neighbors graph, and the free energy of a short-range spin glass model. Received: 3 April 1999 / Revised version: 23 June 1999 / Published online: 8 May 2001     

Elliptic equations for measures on infinite dimensional spaces and applications

We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models. Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001     

Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold

We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on ℤ ^d at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap of the generator of the dyamics in a box of side L centered at the origin scales like L ⁻². Such an estimate is then used to prove a decay to equilibrium for local functions of the form where ε is positive and arbitrarily small and α = ? for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition when β crosses the critical value β _c of the pure system. Received: 10 April 2000 / Revised version: 23 October 2000 / Published online: 5 June 2001     

Stochastic functional differential equations on manifolds

In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters. Received: 6 July 1999 / Revised version: 19 April 2000 /?Published online: 14 June 2001     

Homogenization of a parabolic signorini boundary value problem in a thick plane junction

We consider a parabolic Signorini boundary value problem in a thick plane junction Ω _ε which is the union of a domain Ω₀ and a large number of ε−periodically situated thin rods. The Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is done as ε → 0, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove a convergence theorem and show that the Signorini conditions are transformed (as ε → 0) in differential inequalities in the region that is filled up by the thin rods in the limit passage. Bibliography: 31 titles. Illustrations: 1 figure.     

à propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

Groupwise dense families

We show that the Filter Dichotomy Principle implies that there are exactly four classes of ideals in the set of increasing functions from the natural numbers. We thus answer two open questions on consequences of ? < ?. We show that ? < ? implies that ? = ?, and that Filter Dichotomy together with ? < ? implies ? < ?. The technical means is the investigation of groupwise dense sets, ideals, filters and ultrafilters. With related techniques we prove the new inequality ?≤ cf(?). Received: 9 October 1998 / Revised version: 18 August 1999 / Published online: 21 December 2000     

Martingale convergence and the stopped branching random walk

We discuss the construction of stopping lines in the branching random walk and thus the existence of a class of supermartingales indexed by sequences of stopping lines. Applying a method of Lyons (1997) and Lyons, Pemantle and Peres (1995) concerning size biased branching trees, we establish a relationship between stopping lines and certain stopping times. Consequently we develop conditions under which these supermartingales are also martingales. Further we prove a generalization of Biggins' Martingale Convergence Theorem, Biggins (1977a) within this context. Received: 21 December 1998 / Revised version: 28 June 1999 / Published online: 7 February 2000