Workshop

• Alexis Arnaudon: Noise and dissipation on coadjoint orbits

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Mechanical systems with symmetries can often be described with equations of motion taking values in a phase space of lower dimensions than expected, and with particular geometrical features, called coadjoint orbits. In addition, this convenient representation is compatible with a certain type noise and dissipation mechanisms. With these three ingredients - coadjoint orbits, noise and dissipation - I will explore some new connections between geometric mechanics, random dynamical systems and statistical physics. In particular, I will show the presence of random attractors for isolated systems [1], and some phase transitions in lattices of these systems [2]. This is a joint work with S. Takao, A. De Castro and D. Holm.

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[1] Arnaudon, A., De Castro, A. L., & Holm, D. D. (2018). Noise and dissipation on coadjoint orbits. Journal of nonlinear science, 28(1), 91-145.
[2] Arnaudon, A., & Takao, S. (2018). Networks of Coadjoint Orbits: from Geometric to Statistical Mechanics. arXiv preprint arXiv:1804.11139.

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• Marc Arnaudon: A duality formula and a particle Gibbs sampler for continuous time Feynman-Kac measures on path spaces

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Continuous time Feynman-Kac measures on path spaces are central in applied probability, partial differential equation theory, as well as in quantum physics. I will present a new duality formula between normalized Feynman-Kac distribution and their mean field particle interpretations. Among others, this formula will allow to design a reversible particle Gibbs-Glauber sampler for continuous time Feynman-Kac integration on path spaces. This result extends the particle Gibbs samplers introduced by Andrieu-Doucet-Holenstein in the context of discrete generation models to continuous time Feynman-Kac models and their interacting jump particle interpretations. I will also provide new propagation of chaos estimates for continuous time genealogical tree based particle models with respect to the time horizon and the size of the systems. These results allow to obtain sharp quantitative estimates of the convergence rate to equilibrium of particle Gibbs-Glauber samplers.

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• Yann Brenier: Solving the initial value problem for Euler and Burgers equations by convex minimization

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We show that it possible to solve the initial value problem: i) for short times, in the case of the Euler equations of both compressible and incompressible fluids (and more generally for systems of conservation law admitting a convex entropy), ii) for arbitrarily large time intervals, in the case of Kruzhkov's entropy solutions to the (non-viscous) Burgers equation. The convex minimization problem is related to the concept of sub-solution in the sense of convex integration theory and can also be interpreted as a kind of generalized variational mean-field game.

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• Giovanni Conforti: Stochastic calculus and Otto calculus

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We illustrate how the calculus on the space of probability measures introduced by Otto allows to prove a Newton's law for the marginal flow of a Schrödinger bridge, which we compare to earlier results obtained by Zambrini. An interesting aspect of Otto calculus is that the acceleration has a Riemannian interpretation, i.e. it can be constructed as a covariant derivative. This allows to perform explicit computations, thus opening the door  for a quantitative analysis of Schrödinger bridges (ergodicity, functional inequalities..). In the second part of the talk, we give a probabilistic interpretation of these quantitative results, and provide some ideas that allow to obtain them using instead of Otto calculus the classical toolbox of probability theory: couplings, martingales and FBSDEs. If time allows, some generalizations to mean-field type systems will be sketched.

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• Shizan Fang: Sobolev spaces and Osgood conditions

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Osgood conditions have been involved in the study of stochastic flows of homeomorphisms more than ten years ago. We will present some examples which make appear these conditions naturally.

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• François Gay-Balmaz: On noisy extensions of nonholonomic constraints

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We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange-d’Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of the noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals. The example of the Routh ball allows us to develop a general geometric setting for nonholonomic systems with symmetries.

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• Ivan Gentil: From the Schrödinger problem to a Newton equation and applications

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To understand the link between the Schrödinger problem and  the Newton equation, we first study the finite dimensional case where we define a cost in $R^n$. It appears that the finite dimensional case is a good toy model to deal with the infinite dimensional case, the Schrödinger problem.

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• Rémi Lassalle: An intrinsic calculus of variations of functionals of laws of semi-martingales with applications to semi-martingale optimal transportation problems

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A construction is provided which extends classical methods of calculus of variations to functional of laws of specific semi-martingales which depends explicitly on their characteristics. Within this general context we encounter a  technical feature : the regularity of characteristics along a family of curves on a set of laws of semi-martingales. This is conveniently handled by introducing a suitable set of variation processes which, in a precise acceptation, preserve information flows through their induced perturbations of laws. This enable to  characterize the dynamic  of specific laws of semi-martingale satisfying a least action principle. Though a straightforward embedding, we recover the corresponding classical method arising from Hamilton's least action principle as a particular case. As an application we obtain informations on optimum to semi-martingale optimal transportation problems ; under further symmetries assumptions on the cost function, we also obtain informations on those optimum through an extension of Noether's theorem to this specific context : within our approach, we recover classical corresponding results as a particular case. Applications to Shr\"{o}dinger bridges also follow as a particular case.
With the collaboration of A.B. Cruzeiro.

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• Christian Léonard: A probabilistic view on optimal transport

 

A decade before the discovery by Kantorovich in 1942 of an efficient relaxation of Monge's optimal transport problem, Schrödinger designed, in 1931, a thought experiment in statistical physics leading to a (neg-)entropy minimization problem. This Schrödinger problem is a probabilistic analogue of the Monge-Kantorovich problem. The object of the talk is (1) to provide an overview of the connections between these two problems, mainly in terms of large deviations, and (2) to present a quick introduction to several talks, to be given later during this week, about connections between the Schrödinger problem, optimal transport and some functional inequalities.

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• Luca Nenna: From Schrödinger to Lasry-Lions via Brenier

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The minimization of a relative entropy (with respect to the Wiener measure) is a very old problem which dates back to Schrödinger. C. Léonard has established strong connections and analogies between this problem and the Monge-Kantorovich problem with quadratic cost (namely the standard Optimal Transport problem). In particular, the entropic interpolation leads to a system of PDEs which present strong analogies with the MFG system with a quadratic Hamiltonian.
In this talk, we will explain how such systems can indeed be obtained by minimization of a relative entropy at the level of measures on paths with an additional term involving the marginals in time. Connection with generalised solutions for incompressible fluid will also be discussed.

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• Gabriel Peyré: Sinkhorn Entropies and Divergences

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Comparing probability distributions is a fundamental problem in data sciences. Simple norms and divergences, such as the total variation and the relative entropy, only compare densities in a point-wise manner and fail to capture the geometric nature of the problem. Maximum Mean Discrepancies (MMD, Euclidean norms defined through a kernel) and Optimal Transport costs (OT) are the two main classes of distances between measures that provide some geometrical understanding, metrizing the convergence in law. In this talk, I will present the Sinkhorn divergences, which is a new family of geometric divergences that interpolates between MMD and OT. These divergences rely on a new notion of metric entropy, which ensures its positivity and metrization of the convergence in law. On the practical side, these divergences can be computed on large scale problem thanks to Sinkhorn's algorithm, and finds numerous application in imaging and machine learning.

This is a joint work with Jean Feydy, Thibault Sejourné, Francois-Xavier Vialard, Shun-ichi Amari and Alain Trouvé.

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• Nicolas Privault: Functional inequalities for marked point processes

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In this talk we will extend functional inequalities for Poisson random measures to the setting of marked temporal point processes, under mild assumptions on their Papangelou conditional intensity. First, we derive a Poincaré inequality which applies to classes of renewal, Cox and Hawkes processes. Second, we derive transportation cost inequalities for functionals of marked point processes, and for the law of marked temporal point processes with a Papangelou conditional intensity. Finally, we provide a variational representation of the Laplace transform of functionals of marked point processes with a Papangelou conditional intensity. The proofs make use of an extension of the Clark-Ocone formula to marked temporal point processes. Joint work with I. Flint and G.L. Torrisi.

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• Tudor Ratiu: Nonholonomic stochastic systems

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• Luigia Ripani: Talagrand inequality for the entropic transportation cost

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In the past few years several links between the Schrödinger problem and functional inequalities have been established. In this talk we focus on one particular example. We introduce a generalization of the classical Talagrand inequality in which the Wasserstein distance is replaced by the entropic transportation cost, that is the minimal value of the Schrödinger problem. We provide equivalent characterizations in terms of reverse hypercontractivity for the heat semigroup, contractivity of the Hamilton-Jacobi-Bellman semigroup and dimension-free concentration of measure. Finally we show the relation to other known functional inequalities such as the log-Sobolev and the classical Talagrand inequalities.

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• Sylvie Roelly: Reciprocal processes: a stochastic analysis approach

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Reciprocal processes  (whose  concept  can  be  traced  back  to  E. Schroedinger) form  a class of stochastic processes constructed as mixture of bridges, that satisfy a temporal Markovian field property.  We discuss an  unified  approach  to  characterize  different  types  of  reciprocal processes via  duality  formulae  on  path  spaces.  Two main  examples  are  treated:  the  case  of reciprocal processes  with  continuous  paths  associated  to  Brownian diffusions and  the  case  of reciprocal processes associated to pure jump processes.
This talk is a review, based on joint works with G. Conforti, P. Dai Pra, C. Léonard, R. Murr, M. Thieullen and J.-C. Zambrini.

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• Esmeralda Sousa Dias: Geometry and dynamics of maps from mutation-periodic quivers

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The maps arising from mutation-periodic quivers are birational maps which preserve a presympletic form defined by the quiver. In certain cases, these maps are also Poisson maps with respect to distinct Poisson structures of quadratic type. The interplay between the presymplectic and Poisson structures will be addressed as well as the respective consequences to the dynamics of such maps.

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• Luca Tamanini: Variational representations of the Schrödinger problem on manifolds and beyond

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Although completely different in the motivation, the Schrödinger problem has a deep connection with the Monge-Kantorovich one, as in very great generality the latter appears as the zero-noise limit of the former. In the recent years this connection has been further investigated and variational representations of the entropic cost have been proved in the Euclidean setting, in line with analogous formulas for the Wasserstein distance. However, these results have been obtained with techniques that strongly rely on probability or stochastic control, hence can not be easily adapted to non-Euclidean settings (e.g. manifolds, Ricci limits).
For this reason, in the first part of the talk we will recall the most famous representation formulas for the Wasserstein distance, namely Benamou-Brenier formula, Hamilton-Jacobi duality and Kantorovich duality. In the second part a similar picture for the Schrödinger problem will be provided, emphasizing analogies and differences with respect to optimal transport. The main novelty is a unifying and purely analytical approach that allows us to extend to non-smooth spaces the variational representations of the entropic cost.
The talk is based on a joint work with Nicola Gigli.

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• François-Xavier Vialard: Generalized flows for the Camassa-Holm equation

 

We study the variational problem of existence of length minimizing geodesics for the Hdiv right-invariant metric on the diffeomorphism group. We first show that the geodesic flow of the Hdiv metric can be understood as particular solution of the incompressible Euler equation. Taking advantage of this formulation, we propose a new convex relaxation à la Brenier, we prove that it recovers smooth geodesics for short time as unique minimizer. Uniqueness of the pressure also holds. Last, we give hints on the tightness of this relaxation in 2D and higher.

This is joint work with Andrea Natale and Thomas Gallouët.

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• Pierre Vuillermot: Bernstein processes, parabolic problems and spectral theory

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Bernstein processes, also named Schrödinger or reciprocal processes in the literature, constitute a generalization of Markov processes and have played an increasingly important role in various areas of Mathematics and Mathematical Physics over the years, particularly in view of the recent advances in the Monge-Kantorovitch formulation of Optimal Transport Theory and Stochastic Geometric Mechanics. In this talk I will show how to construct such processes from an infinite hierarchy of forward-backward systems of linear deterministic parabolic partial differential equations, when the elliptic part of the parabolic operators may be realized as an unbounded Schrödinger operator with compact resolvent in standard L2-space. I will also discuss many important properties of such processes, including those of a natural entropy function associated with them.

This is joint work with J. C. Zambrini.

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