Partially hyperbolic diffeomorphisms and Lagrangian contact structures

In this paper, we classify the three-dimensional partially hyperbolic diffeomorphisms whose stable, unstable, and central distributions $E^s$ , $E^u$ , and $E^c$ are smooth, such that $E^s\oplus E^u$ is a contact distribution, and whose non-wandering set equals the whole manifold. We prove that up to a finite quotient or a finite power, they are smoothly conjugated either to a time-map of an algebraic contact-Anosov flow, or to an affine partially hyperbolic automorphism of a nil- ${\mathrm {Heis}}{(3)}$ -manifold. The rigid geometric structure induced by the invariant distributions plays a fundamental part in the proof.

Insights from Physics-based Hydrologic Models and Stochastic Storm Transposition into the Underlying Assumptions of Flood Quantile Regionalization Techniques

<p>We use three hydrological models and the stochastic storm transposition (SST) framework to investigate the validity of implicit assumptions in the empirical methodology of regionalization of flood frequencies (RFF) for prediction in ungauged basins. In particular, we investigate the long-standing hypothesis that for a set of catchments physical homogeneity of meteorological and infiltration processes implies statistical homogeneity of flood peak distributions. Our modeling (theoretical) results do not support this hypothesis. We also show that power-law regressions (i.e. log-log linearity) do not seem to be an appropriate model to connect distributions across scales (either quantiles or distribution parameters). Finally, even though our results support the most fundamental hypothesis in RFF that the underlying distribution of peak flows is invariant under translation in the river network, our results do not support the simple-scaling or multi-scaling frameworks. First, we show that some moments of the distribution cannot be inferred from area alone, violating the definition put forward by Gupta et al. (1994). Second, the resulting scale invariant distributions that we identified are different from LP-III and GEV and cannot be rejected by data as valid distributions. Our framework provides a new avenue to test methods for flood data analysis and it opens the door towards a unified physics-informed framework for prediction of flood frequencies in ungauged basins embedded in gauged regions.</p>

Urnings: A new method for tracking dynamically changing parameters in paired comparison systems

We introduce a new rating system for tracking the development of parameters based on a stream of observations that can be viewed as paired comparisons. Rating systems are applied in competitive games, adaptive learning systems, and platforms for product and service reviews. We model each observation as an outcome of a game of chance that depends on the parameters of interest (e.g., the outcome of a chess game depends on the abilities of the two players). Determining the probabilities of the different game outcomes is conceptualized as an urn problem, where a rating is represented by a probability (e.g., proportion of balls in the urn). This setup allows for evaluating the standard errors of the ratings and performing statistical inferences about the development of, and relations between, parameters. Theoretical properties of the system in terms of the invariant distributions of the ratings and their convergence are derived. The properties of the rating system are illustrated with simulated examples and its potential for answering research questions is illustrated using data from competitive chess, a movie review system, and an adaptive learning system for math.

Recursive computation of the invariant distributions of Feller processes: Revisited examples and new applications

In this paper, we show that the abstract framework developed in [G. Pagès and C. Rey, Recursive computation of the invariant distribution of Markov and Feller processes, preprint 2017, https://arxiv.org/abs/1703.04557] and inspired by [D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion, Bernoulli 8 2002, 3, 367–405] can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean-reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth.