Gibbs Measures on Brownian Paths: Theory and Applications (original) (raw)

Gibbs Measures Relative to Brownian Motion

The Annals of Probability, 1999

We consider Brownian motion perturbed by the exponential of an action. The action is the sum of an external, one-body potential and a twobody interaction potential which depends only on the increments. Under suitable conditions on these potentials, we establish existence and uniqueness of the corresponding Gibbs measure. We also provide an example where uniqueness fails because of a slow decay in the interaction potential.

A central limit theorem for Gibbs measures relative to Brownian motion

Probability Theory and Related Fields, 2005

We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.

Dynamical Borel-Cantelli lemmas for gibbs measures

Israel Journal of Mathematics, 2001

Let T : X → X be a deterministic dynamical system preserving a probability measure µ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsets A n ⊂ X and µ-almost every point x ∈ X the inclusion T n x ∈ A n holds for infinitely many n. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma.

Sample Path Properties of Brownian Motion

1976

This is a set of lecture notes based on a graduate course given at the Berlin Mathematical School in September 2011. The course is based on a selection of material from my book with Yuval Peres, entitled Brownian motion, which was published by Cambridge University Press in 2010.

Combinatorial criteria for uniqueness of Gibbs measures

Random Structures & Algorithms, 2005

We generalize previously known conditions for uniqueness of the Gibbs measure in statistical physics models by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the total influence of a site is small. Our proofs are combinatorial in nature and use tools from the analysis of discrete Markov chains, in particular the path coupling method. The implications of our conditions for the mixing time of natural Markov chains associated with the models are discussed as well. We also present some examples of models for which the conditions hold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005

On the variational principle for generalized Gibbs measures

2004

We present a novel approach to establishing the variational principle for Gibbs and generalized (weak and almost) Gibbs states. Limitations of a thermodynamic formalism for generalized Gibbs states will be discussed. A new class of intuitively Gibbs measures is introduced, and a typical example is studied. Finally, we present a new example of a non-Gibbsian measure arising from an industrial application.

Excited Brownian motions as limits of excited random walks

Probability Theory and Related Fields, 2012

We obtain the convergence in law of a sequence of excited (also called cookies) random walks toward an excited Brownian motion. This last process is a continuous semi-martingale whose drift is a function, say ϕ, of its local time. It was introduced by Norris, Rogers and Williams as a simplified version of Brownian polymers, and then recently further studied by the authors. To get our results we need to renormalize together the sequence of cookies, the time and the space in a convenient way. The proof follows a general approach already taken by Tóth and his coauthors in multiple occasions, which goes through Ray-Knight type results. Namely we first prove, when ϕ is bounded and Lipschitz, that the convergence holds at the level of the local time processes. This is done via a careful study of the transition kernel of an auxiliary Markov chain which describes the local time at a given level. Then we prove a tightness result and deduce the convergence at the level of the full processes.

On Gibbsianness of infinite-dimensional diffusions

We analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the lattice In a first part, these processes are characterized as Gibbs states on path spaces of the form C([0, T ], R) Z d . In a second part, we study the Gibbsian character on R Z d of ν t , the law at time t of the infinite-dimensional diffusion X(t), when the initial law ν = ν 0 is Gibbsian.

A characterization ofh-Brownian motion by its exit distributions

Probability Theory and Related Fields, 1992

Let X h be an h-Brownian motion in the unit ball D c R d with h harmonic, such that the representing measure of h is not singular with respect to the surface measure on 0D. If Y is a continuous strong Markov process in D with the same killing distributions as X h, then Yis a time change of X h. Similar results hold in simply connected domains in C provided with either the Martin or the Euclidean boundary.

Variational principle for generalized Gibbs measures

2002

We study the thermodynamic formalism for generalized Gibbs measures, such as renormalization group transformations of Gibbs measures or joint measures of disordered spin systems. We first show existence of the relative entropy density and obtain a familiar expression in terms of entropy and relative energy for ”almost Gibbsian measures” (almost sure continuity of conditional probabilities). We also describe these measures as equilibrium states and establish an extension of the usual variational principle. As a corollary, we obtain a full variational principle for quasilocal measures. For the joint measures of the random field Ising model, we show that the weak Gibbs property holds, with an almost surely rapidly decaying translation invariant potential. For these measures we show that the variational principle fails as soon as the measures loses the almost Gibbs property. These examples suggest that the class of weakly Gibbsian measures is too broad from the perspective of a reasonab...

Sample path properties of the stochastic flows

The Annals of Probability, 2004

We consider a stochastic flow driven by a finite-dimensional Brownian motion. We show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the equilibrium state and the central limit theorem. The proof uses new estimates of the mixing rates of the multi-point motion.

Excursions and path functionals for stochastic processes with asymptotically zero drifts

Stochastic Processes and their Applications, 2013

We study discrete-time stochastic processes (X t ) on [0, ∞) with asymptotically zero mean drifts. Specifically, we consider the critical (Lamperti-type) situation in which the mean drift at x is about c/x. Our focus is the recurrent case (when c is not too large). We give sharp asymptotics for various functionals associated with the process and its excursions, including results on maxima and return times. These results include improvements on existing results in the literature in several respects, and also include new results on excursion sums and additive functionals of the form s≤t X α s , α > 0. We make minimal moments assumptions on the increments of the process. Recently there has been renewed interest in Lamperti-type process in the context of random polymers and interfaces, particularly nearest-neighbour random walks on the integers; some of our results are new even in that setting. We give applications of our results to processes on the whole of R and to a class of multidimensional 'centrally biased' random walks on R d ; we also apply our results to the simple harmonic urn, allowing us to sharpen existing results and to verify a conjecture of Crane et al. We benefitted in the early stages of this project from enjoyable discussions with Iain MacPhee, who sadly passed away on 13th January 2012; we dedicate this paper to Iain, in memory of our valued colleague and in gratitude for his generosity.

A sample path large deviation principle for -martingale measure processes

Bulletin des Sciences Mathématiques, 1999

Brownian motion and the diffusion-limit Fleming-Viot process are examples of such infinite-dimensional Markov processes with continuous paths and L 2-martingale measures we study in this work as regards to their sample path large deviation probabilities and their associated large deviation rate functions in the limit of small perturbations. We present a unified approach based on Girsanov transform techniques. We derive the rate function as a Lagrangian functional and, as an alternative representation, via some generalized derivatives in a 'Cameron-Martin space'.

Regular g-measures are not always Gibbsian

Electronic Communications in Probability, 2011

Regular g-measures are discrete-time processes determined by conditional expectations with respect to the past. One-dimensional Gibbs measures, on the other hand, are fields determined by simultaneous conditioning on past and future. For the Markovian and exponentially continuous cases both theories are known to be equivalent. Its equivalence for more general cases was an open problem. We present a simple example settling this issue in a negative way: there exist g-measures that are continuous and non-null but are not Gibbsian. Our example belongs, in fact, to a well-studied family of processes with rather nice attributes: It is a chain with variable-length memory, characterized by the absence of phase coexistence and the existence of a visible renewal scheme.

A representation of Gibbs measure for the random energy model

The Annals of Applied Probability, 2004

In this work we consider a problem related to the equilibrium statistical mechanics of spin glasses, namely the study of the Gibbs measure of the random energy model. For solving this problem, new results of independent interest on sums of spacings for i.i.d. Gaussian random variables are presented. Then we give a precise description of the support of the Gibbs measure below the critical temperature.