Let S(v) be a function defined on the vertices v of the infinite binary tree. One algorithm to seek large positive values of S is the Metropolis-type Markov chain (Xn ) defined by P (Xn+1 = wjXn = v) = 1 3 e b(S(w)\GammaS(v)) 1 + e b(S(w)\GammaS(v)) for each neighbor w of v, where b is a parameter ("1=temperature") which the user can choose. We introduce and motivate study of this algorithm under a probability model for the objective function S, in which S is "tree-indexed simple random walk", that is the increments ¸(e) = S(w) \Gamma S(v) along parent-child edges e = (v; w) are independent and P (¸ = 1) = p; P (¸ = \Gamma1) = 1 \Gamma p. This algorithm has a "speed" r(p; b) = lim n n \Gamma1 ES(Xn ). We study the speed via a mixture of rigorous arguments, non-rigorous arguments and Monte Carlo simulations, and compare with a deterministic greedy algorithm which permits rigorous analysis. Formalizing the non-rigorous arguments presents a challenging problem. Mathematically, th...

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We study the limit of functionals of stochastic processes for which an homogenization result holds. All these functionals involve stochastic integrals. Among them, we consider more particularly the Lévy area and those giving the solutions of some SDEs. The main question is to know whether or not the limit of the stochastic integrals is equal to the stochastic integral of the limit of each of its terms. In fact, the answer may be negative, especially in presence of a highly oscillating first-order differential term. This provides us with some counterexamples to the theory of good sequence of semimartingales.

We review old and new results about the limiting behavior of a tagged particle in different interacting particle systems: (a) independent particles with no mass in one dimension with continuous paths like Brownian motions and ideal gases. (b) Reversible processes on R d or Z d like interacting Brownian motions and Kawasaki dynamics. (c) Simple exclusion processes. (d) Zero range processes (e) Conservative nearest particle processes including the Hammersley process. In each case we consider as initial distribution the invariant distribution for the process as seen from the tagged particle. We review two kinds of limiting behavior: the law of large numbers and the invariance principle. Denoting by X(t) the position of the tagged particle, the law of large numbers says that as t ! 1, X(t)=t converges almost surely to a constant. The invariance principle means that when conveniently centered and rescaled, the process converges to Brownian motion. When the initial distribution of th...

Abstract. In this paper we rigorously establish the existence of the mobility coefficient for a tagged particle in a simple symmetric exclusion process with adsorption/desorption of particles, in a presence of an external force field interacting with the particle. The proof is obtained using a perturbative argument. In addition, we show that, for a constant external field, the mobility of a particle equals to the self-diffusivity coefficient of the medium, the so-called Einstein relation. The method can be applied to any system where the environment has a Markovian evolution with a fast convergence to equilibrium (spectral gap property). In this context we find a necessary relation between forward and backward velocity for the validity of the Einstein relation. This relation is always satisfied by reversible systems. We provide an example of a non-reversible system, where the Einstein relation is correct. 1.

This paper contains a study of the long time behavior of a diffusion process in a periodic potential. The first goal is to determine a suitable rescaling of time and space so that the diffusion process converges to some homogeneous limit. The issue of interest is to characterize the effective evolution equation. The main result is that in some cases large drifts must be removed in order to get a diffusive asymptotic behavior. This is applied to the homogenization of parabolic differential equations.

Abstract: Consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction. We investigate the behavior of a tagged particle under the action of an external constant driving force. We prove that the diffusively rescaled position of the test particle ɛX(ɛ −2 t), t>0, converges in probability, as ɛ → 0, to a deterministic function v(t). The function v(·) depends on the initial distribution of the random environment through a non-linear parabolic equation. This law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin. An Einstein relation is satisfied asymptotically when the external force is small.

Abstract. We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of “linear response function ” in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov process X(s) but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is, if the initial measure ν is invariant for the given Markov semi-group, then for any pair of times s < t and nice functions f, g, the dissipation, that is, the derivative in s of the covariance of g(X(t)) and f(X(s)) equals the infinitesimal response at time t and direction g to any Markovian perturbation that alters the invariant measure of X(·) in the direction of f at time s. The same applies in the so called FDT regime near equilibrium, i.e. in the limit s → ∞ with t − s fixed, provided X(s) converges in law to an invariant measure for its dynamics. We provide the response function of two generic Markovian perturbations which we then compare and contrast for pure jump processes on a discrete space, for finite dimensional diffusion processes, and for stochastic spin systems. 1. Introduction and