Duality Theorems in Ergodic Transport

We analyze several problems of Optimal Transport Theory in the setting of Ergodic Theory. In a certain class of problems we consider questions in Ergodic Transport which are generalizations of the ones in Ergodic Optimization. Another class of problems is the following: suppose $\sigma$ is the shift acting on Bernoulli space $X=\{0,1\}^\mathbb{N}$, and, consider a fixed continuous cost function $c:X \times X\to \mathbb{R}$. Denote by $\Pi$ the set of all Borel probabilities $\pi$ on $X\times X$, such that, both its $x$ and $y$ marginal are $\sigma$-invariant probabilities. We are interested in the optimal plan $\pi$ which minimizes $\int c d \pi$ among the probabilities on $\Pi$. We show, among other things, the analogous Kantorovich Duality Theorem. We also analyze uniqueness of the optimal plan under generic assumptions on $c$. We investigate the existence of a dual pair of Lipschitz functions which realizes the present dual Kantorovich problem under the assumption that the cost is Lipschitz continuous. For continuous costs $c$ the corresponding results in the Classical Transport Theory and in Ergodic Transport Theory can be, eventually, different. We also consider the problem of approximating the optimal plan $\pi$ by convex combinations of plans such that the support projects in periodic orbits

Flatness is a Criterion for Selection of Maximizing Measures

For a full shift with Np+1 symbols and for a non-positive potential, locally proportional to the distance to one of N disjoint full shifts with p symbols, we prove that the equilibrium state converges as the temperature goes to 0. The main result is that the limit is a convex combination of the two ergodic measures with maximal entropy among maximizing measures and whose supports are the two shifts where the potential is the flattest. In particular, this is a hint to solve the open problem of selection, and this indicates that flatness is probably a/the criterion for selection as it was conjectured by A.O. Lopes. As a by product we get convergence of the eigenfunction at the log-scale to a unique calibrated subaction

Relative entropy and waiting time for continuous-time Markov processes

For discrete-time stochastic processes, there is a close connection between return (resp. waiting) times and entropy (resp. relative entropy). Such a connection cannot be straightforwardly extended to the continuous-time setting. Contrarily to the discrete-time case one needs a reference measure on path space and so the natural object is relative entropy rather than entropy. In this paper we elaborate on this in the case of continuous-time Markov processes with finite state space. A reference measure of special interest is the one associated to the time-reversed process. In that case relative entropy is interpreted as the entropy production rate. The main results of this paper are: almost-sure convergence to relative entropy of the logarithm of waiting-times ratios suitably normalized, and their fluctuation properties (central limit theorem and large deviation principle

Occurrence of patterns in random fields

we study the distribution of the occurrence of patterns in random fields on the lattice Zd , d >_ 2. The knowledge of such distributions is essential for the analysis of lossy and lossless compression schemes of multidimensional arrays. For 1-dimensional mixing processes a distribution of occurrence time t(An) of a pattern An, properly renormalised, converges to an exponential distribution. We generalize this result to higher dimensions. The main difficulty lies in the fact that mixing properties of random fields (d >_ 2) are very different from those of random processes (d = 1). We show that the mixing properties of Gibbsian (and hence Markov) random fields are sufficient for the convergence to the exponential law. As a corollary, we derive other probabilistic results for the distribution of t(An): the central limit theorem and the large deviation principle. Exponential law is also derived for the ¯rst occurrence of an approximate match for the pattern An

Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space SZd where d≥ 1 and S is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.</p

Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space SZd where d≥ 1 and S is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.Accepted author manuscriptApplied Probabilit

Swell impact on reef sedimentary processes: A case study of the La Reunion fringing reef

Two surface-sediment sampling campaigns were carried out in November and December 2003, before and after a strong swell event, in the back-reef area of a microtidal fringing reef on the western coast of La Reunion, Indian Ocean. The spatial distributions of the mean grain size, sorting and skewness parameters are determined, and grain-size trend analysis is performed to estimate the main sediment transport pathways in the reef. The results of this analysis are compared with hydrodynamic records obtained in the same reef area during fair weather conditions and during swell events. Sediment dynamics inferred from the hydrodynamic records show that significant sediment erosion and transport occur only during swell events and under strongly agitated sea states. Under normal wave conditions, there is a potential for onshore sediment transport from the reef-flat to the back-reef, but this transport is episodic and occurs principally during high-tide stages. Sediment transport trends revealed by the grain-size trend analysis method show onshore and alongshore low-energy transport processes that are in agreement with the hydrodynamic records. The grain-size trend analysis method also provides evidence of an offshore high-energy transport trend that could be interpreted as a real physical process associated with return flow from the shore to the reef. The impact of swell on the reef sediment dynamics is clearly demonstrated by onshore and alongshore transport. Considering different combinations of the vector transport trends computed through the grain-size trend analysis approach, more realistic and pertinent results can be obtained by applying an exclusive OR operation (XOR case) on the vectors. The main results presented here highlight a trend towards the accumulation of carbonate sands in the back-reef area of the fringing reef. These sediments can only be resuspended during extreme events such as storms or tropical cyclones