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.

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.