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Any of these conditions can be waived by permission of the Corresponding Author. ledoux@math.univ-toulouse.fr (Michel Ledoux (Chief Editor)) ejpecp@chafai.net (Djalil Chafaï) Thu, 02 Jan 2014 01:40:36 -0800 OJS http://blogs.law.harvard.edu/tech/rss 60 A compact containment result for nonlinear historical superprocess approximations for population models with trait-dependence http://ejp.ejpecp.org/article/view/3506 We consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth- and death-events happen at exponential times. Traits are hereditarily transmitted unless mutation occurs. The present model is an extension of the model used in [Méléard and Tran, EJP, 2012], where for large populations with small individual biomasses and under additional assumptions, the diffusive limit is shown to converge to a nonlinear historical superprocess. The main goal of the present article is to verify a compact containment condition in the more general setup of Polish trait-spaces and general mutation kernels that allow for a dependence on the parent's trait. As a by-product, a result on the paths of individuals is obtained. An application to evolving genealogies on marked metric measure spaces is mentioned where genealogical distance, counted in terms of the number of births without mutation, can be regarded as a trait. Because of the use of exponential times in the modeling of birth- and death-events the analysis of the modulus of continuity of the trait-history of a particle plays a major role in obtaining appropriate bounds. Sandra Kliem http://ejp.ejpecp.org/article/view/3506 Sat, 18 Oct 2014 23:13:12 -0700 On the heat kernel and the Dirichlet form of Liouville Brownian motion http://ejp.ejpecp.org/article/view/2950 In a previous work, a Feller process called Liouville Brownian motion on $\mathbb{R}^2$ has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field $e^{\gamma\, X}$ and is the right diffusion process to consider regarding $2d$-Liouville quantum gravity.  In this note, we discuss the construction of the associated  Dirichlet form, following essentially Fukushima, Oshima, and Takeda, and the techniques introduced in our previous work. Then we carry  out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of  the Liouville heat kernel via a non-trivial theorem of Fukushima and al. One of the motivations which led to introduce the Liouville Brownian motion in our previous work was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. In particular,  the theory developed for example in Stollmann and Sturm, whose aim is to capture the "geometry" of the underlying space out of the Dirichlet form of a process living on that space, suggests a notion of distance associated to a Dirichlet form. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide a distance in the wide sense, called intrinsic metric, which is interpreted as an extension of Riemannian  geometry applicable to non differential structures. We prove  that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of  the metric aspect  of Dirichlet forms. Rémi Rhodes, Christophe Garban, Vincent Vargas http://ejp.ejpecp.org/article/view/2950 Thu, 16 Oct 2014 01:23:55 -0700 Martingale inequalities and deterministic counterparts http://ejp.ejpecp.org/article/view/3270 We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operator involving a concave envelope. Our results yield an explanation for certain inequalities that arise in mathematical finance in the context of robust hedging. Mathias Beiglböck, Marcel Nutz http://ejp.ejpecp.org/article/view/3270 Thu, 16 Oct 2014 01:00:00 -0700 A lognormal central limit theorem for particle approximations of normalizing constants http://ejp.ejpecp.org/article/view/3428 Feynman-Kac path integration models arise in a large variety of scientic disciplines including physics, chemistry and signal processing. Their mean eld particle interpretations, termed Diusion or Quantum Monte Carlo methods in physics and Sequential Monte Carlo or Particle Filters in statistics and applied probability, have found numerous applications as they allow to sample approximately from sequences of complex probability distributions and estimate their associated normalizing constants.This article focuses on the lognormal fuctuations of these normalizing constant estimates when both the time horizon n  and the number of particles N  go to innity in such a way that n/N tends to some number between 0 and 1. To the best of our knowledge, this is the first result of this type for mean field type interacting particle systems. We also discuss special classes of models, including particle absorption models in time-homogeneous environment and hidden Markov models in ergodic random environment, for which more explicit descriptions of the limiting bias and variance can be obtained. Jean Bérard, Pierre Del Moral, Arnaud Doucet http://ejp.ejpecp.org/article/view/3428 Tue, 07 Oct 2014 00:12:12 -0700 Moment bounds and concentration inequalities for slowly mixing dynamical systems http://ejp.ejpecp.org/article/view/3427 We obtain optimal moment bounds for Birkhoff sums, and optimal concentration inequalities, for a large class of slowly mixing dynamical systems, including those that admit anomalous diffusion in the form of a stable law or a central limit theorem with nonstandard scaling $(n\log n)^{1/2}$. Sébastien Gouëzel, Ian Melbourne http://ejp.ejpecp.org/article/view/3427 Fri, 03 Oct 2014 01:55:51 -0700