Gia Bao Nguyen, Daniel Remenik

We show that the squared maximal height of the top path among N non-intersecting Brownian bridges starting and ending at the origin is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. This result can be thought of as a discrete version of K. Johansson’s result that the supremum of the Airy_2 process minus a parabola has the Tracy-Widom GOE distribution. The result can also be recast in terms of the probability that the top curve of the stationary Dyson Brownian motion hits an hyperbolic cosine barrier.

Jean-Dominique Deuschel, Xiaoqin Guo, Alejandro F. Ramirez

We prove a quenched central limit theorem for balanced random walks in time dependent ergodic random environments. We assume that the environment satisfies appropriate ergodicity and ellipticity conditions. The proof is based on the use of a maximum principle for parabolic difference operators.

Julien Berestycki, Maria Clara Fittipaldi, Joaquin Fontbona

We introduce the notion of flows of branching processes with competition to describe the evolution of a continuous state branching population in which interactions between individuals give rise to a negative density dependence term. A classical example is the logistic branching processes studied by Lambert. Following the approach developed by Dawson and Li for populations without interactions, such processes are first constructed as the solutions of certain system of stochastic differential equations. We then propose a novel construction of a flow of branching processes with competition based on the interactive pruning of L\’evy-trees, establishing in particular a Ray-Knight representation result for it in terms of the local times of the pruned forest.

Saverio E. Spagnolie, Gregorio R. Moreno-Flores, Denis Bartolo, Eric Lauga

Motivated by recent experiments, we consider the hydrodynamic capture of a microswimmer near a stationary spherical obstacle. Simulations of model equations show that a swimmer approaching a small spherical colloid is simply scattered. In contrast, when the colloid is larger than a critical size it acts as a passive trap: the swimmer is hydrodynamically captured along closed trajectories and endlessly orbits around the colloidal sphere. In order to gain physical insight into this hydrodynamic scattering problem, we address it analytically. We provide expressions for the critical trapping radius, the depth of the «basin of attraction,» and the scattering angle, which show excellent agreement with our numerical findings. We also demonstrate and rationalize the strong impact of swimming-flow symmetries on the trapping efficiency. Finally, we give the swimmer an opportunity to escape the colloidal traps by considering the effects of Brownian, or active, diffusion. We show that in some cases the trapping time is governed by an Ornstein-Uhlenbeck process, which results in a trapping time distribution that is well-approximated as inverse-Gaussian. The predictions again compare very favorably with the numerical simulations. We envision applications of the theory to bioremediation, microorganism sorting techniques, and the study of bacterial populations in heterogeneous or porous environments.

J. Quastel, D. Remenik.

We review the Airy processes; their formulation and how they are conjectured to govern the large time, large distance spatial fluctuations of one dimensional random growth models. We also describe formulas which express the probabilities that they lie below a given curve as Fredholm determinants of certain boundary value operators, and the several applications of these formulas to variational problems involving Airy processes that arise in physical problems, as well as to their local behaviour.

J. Ortmann, J. Quastel, D. Remenik.

We obtain exact formulas for moments and generating functions of the height function of the asymmetric simple exclusion process at one spatial point, starting from special initial data in which every positive even site is initially occupied. These complement earlier formulas of E. Lee [Lee10] but, unlike those formulas, ours are suitable in principle for asymptotics. We also explain how our formulas are related to divergent series formulas for half-flat KPZ of Le Doussal and Calabrese [LDC12], which we also recover using the methods of this paper. In the long time limit, formal asymptotics show that the fluctuations are given by the Airy2→1 marginals.

I. Corwin, J. Quastel, D. Remenik.

The one dimensional Kardar-Parisi-Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a natural renormalization/rescaling on the space of such evolving interfaces. We introduce and describe the renormalization fixed point of the Kardar-Parisi-Zhang universality class in terms of a random nonlinear semigroup with stationary independent increments, and via a variational formula. Furthermore, we compute a plausible formula the exact transition probabilities using replica Bethe ansatz. The semigroup is constructed from the Airy sheet, a four parameter space-time field which is the Airy2 process in each of its two spatial coordinates. Minimizing paths through this field describe the renormalization group fixed point of directed polymers in a random potential. At present, the results we provide do not have mathematically rigorous proofs, and they should at most be considered proposals.

J. Ortmann, J. Quastel, D. Remenik.

We obtain a Fredholm Pfaffian formula for an appropriate generating function of the height function of the asymmetric simple exclusion process starting from flat (periodic) initial data. Formal asymptotics lead to the GOE Tracy-Widom distribution.

A. Borodin, I. Corwin, D. Remenik.

We demonstrate how a known Whittaker function integral identity arises from the t=0 and q to 1 limit of the Macdonald polynomial eigenrelation satisfied by Noumi’s q-integral operator.

Pablo Moisset de Espanés, Ivan Rapaport, Daniel Remenik, Javiera Urrutia

In the broadcast version of the congested clique model, n nodes communicate in synchronous rounds by writing O(log n)-bit messages on a whiteboard, which is visible to all of them. The joint input to the nodes is an undirected n-node graph G, with node i receiving the list of its neighbors in G. Our goal is to design a protocol at the end of which the information contained in the whiteboard is enough for reconstructing G. It has already been shown that there is a one-round protocol for reconstructing graphs with bounded degeneracy. The main drawback of that protocol is that the degeneracy m of the input graph G must be known a priori by the nodes. Moreover, the protocol fails when applied to graphs with degeneracy larger than m. In this paper we address this issue by looking for robust reconstruction protocols, that is, protocols which always give the correct answer and work efficiently when the input is restricted to a certain class. We introduce a very simple, two-round protocol that we call Robust-Reconstruction. We prove that this protocol is robust for reconstructing the class of Barabási-Albert trees with (expected) message size O(log n). Moreover, we present computational evidence suggesting that Robust-Reconstruction also generates logarithmic size messages for arbitrary Barabási-Albert networks. Finally, we stress the importance of the preferential attachment mechanism (used in the construction of Barabási-Albert networks) by proving that Robust-Reconstruction does not generate short messages for random recursive trees.

Joaquin Fontbona, Nathael Gozlan, Jean-Francois Jabir

We relate transport-entropy inequalities to the study of critical points of functionals defined on the space of probability measures.

Joaquin Fontbona, Hélène Guérin, Florent Malrieu

We study the long-time behavior of variants of the telegraph process with position-dependent jump-rates, which result in a monotone gradient-like drift toward the origin. We compute their invariant laws and obtain, via probabilistic couplings arguments, some quantitative estimates of the total variation distance to equilibrium. Our techniques extend ideas previously developed for a simplified piecewise deterministic Markov model of bacterial chemotaxis.

Julien Berestycki, Maria Clara Fittipaldi, Joaquin Fontbona

We introduce the notion of flows of branching processes with competition to describe the evolution of a continuous state branching population in which interactions between individuals give rise to a negative density dependence term. A classical example is the logistic branching processes studied by Lambert. Following the approach developed by Dawson and Li for populations without interactions, such processes are first constructed as the solutions of certain system of stochastic differential equations. We then propose a novel construction of a flow of branching processes with competition based on the interactive pruning of L\’evy-trees, establishing in particular a Ray-Knight representation result for it in terms of the local times of the pruned forest.

J.-B Bardet (LMRS), A Christen, J Fontbona (CMM)

We establish explicit exponential convergence estimates for the renewal theorem, in terms of a uniform component of the inter arrival distribution, of its Laplace transform which is assumed finite on a positive interval, and of the Laplace transform of some related random variable. Our proof is based on a coupling construction relying on discrete-time Markovian structures that underly the renewal processes and on Lyapunov-Doeblin type arguments.

Roberto Cortez, Joaquin Fontbona

We study a class of one dimensional particle systems with true (Bird type) binary interactions, which includes Kac’s model of the Boltzmann equation and nonlinear equations for the evolution of wealth distribution arising in kinetic economic models. We obtain explicit rates of convergence for the Wasserstein distance between the law of the particles and their limiting law, which are linear in time and depend in a mild polynomial manner on the number of particles. The proof is based on a novel coupling between the particle system and a suitable system of non-independent nonlinear processes, as well as on recent sharp estimates for empirical measures.

Joaquin Fontbona, Fabien Panloup (IMT)

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter H>1/2 and multiplicative noise component σ. When σ is constant and for every H∈(0,1), it was proved by Hairer that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order t−α where α∈(0,1) (depending on H). The aim of this paper is to extend such types of results to some multiplicative noise setting. More precisely, we show that we can recover such convergence rates when H>1/2 and the inverse of the diffusion coefficient σ is a Jacobian matrix. The main novelty of this work is a type of extension of Foster-Lyapunov like techniques to this non-Markovian setting, which allows us to put in place an asymptotic coupling scheme without resorting to deterministic contracting properties.

Julio Backhoff, Joaquín Fontbona

We formulate conditions for the solvability of the problem of robust utility maximization from final wealth in continuous time financial markets, without assuming weak compactness of the densities of the uncertainty set, as customary in the literature. Relevant examples of such a situation typically arise when the uncertainty set is determined through moment constraints. Our approach is based on identifying functional spaces naturally associated with the elements of each problem. For general markets these are modular spaces, through which we can prove a minimax equality and the existence of optimal strategies by exploiting the compactness, which we establish, of the image by the utility function of the set of attainable wealths. In complete markets we obtain additionally the existence of a worst-case measure, and combining our ideas with abstract entropy minimization techniques, we moreover provide in that case a novel methodology for the characterization of such measures.

Joaquin Fontbona, Benjamin Jourdain (INRIA Paris-Rocquencourt, CERMICS)

In this paper, we are interested in the long-time behaviour of stochastic systems of n interacting vortices: the position in R2 of each vortex evolves according to a Brownian motion and a drift summing the influences of the other vortices computed through the Biot and Savart kernel and multiplied by their respective vorticities. For fixed n, we perform the rescalings of time and space used successfully by Gallay and Wayne [5] to study the long-time behaviour of the vorticity formulation of the two dimensional incompressible Navier-Stokes equation, which is the limit as n → ∞ of the weighted empirical measure of the system under mean-field interaction. When all the vorticities share the same sign, the 2n-dimensional process of the rescaled positions of the vortices is shown to converge exponentially fast as time goes to infinity to some invariant measure which turns out to be Gaussian if all the vorticities are equal. In the particular case n = 2 of two vortices, we prove exponential convergence in law of the 4-dimensional process to an explicit random variable, whatever the choice of the two vorticities. We show that this limit law is not Gaussian when the two vorticities are not equal.

Joaquin Fontbona, Sylvie Méléard

The spatial behavior of an animal population of dispersive and competitive species is studied. It may be realized as a result of various kinds of biological effects, as heterogeneity of environmental conditions, mutual attractive or repulsive interactions between individuals, competition between individuals for resources and localization of the offspring during a reproduction event. A stochastic model describing the behavior of each individual of such a population is introduced and a large population limit is studied. As consequence, the global existence of a nonnegative weak solution to a multidimensional parabolic strongly coupled model for competing species is proved. The main feature of the model is the nonlocal nonlinearity appearing in the diffusion and drift term. The diffusion matrix is non-symmetric and generally not positive definite and the cross-diffusions terms are allowed to be large if subspecies sizes are large. We prove existence and uniqueness of the finite measure-valued solution and give assumptions under which the solution takes values in a functional space. Then we make the competition kernel converge to a Dirac measure and obtain solution to a locally competitive version of the previous equation. The proofs are essentially based on the underlying stochastic flow related to the dispersive part of the dynamics.

Joaquin Fontbona, Benjamin Jourdain (INRIA Paris-Rocquencourt, CERMICS)

The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales with respect to the tail filtration. The relative entropy is the expected value of a backward submartingale. In the case of (non necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit the Doob-Meyer decomposition of this submartingale. We deduce a stochastic analogue of the well known entropy dissipation formula, which is valid for general convex entropies, including the total variation distance. Under additional regularity assumptions, and using It\^o’s calculus and ideas of Arnold, Carlen and Ju \cite{Arnoldcarlenju}, we obtain moreover a new Bakry Emery criterion which ensures exponential convergence of the entropy to 0. This criterion is non-intrisic since it depends on the square root of the diffusion matrix, and cannot be written only in terms of the diffusion matrix itself. We provide examples where the classic Bakry Emery criterion fails, but our non-intrisic criterion applies without modifying the law of the diffusion process.

Pontificia Universidad Católica de Chile (PUC-Chile)

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