Schedule for: 18w5069 - Entropies, the Geometry of Nonlinear Flows, and their Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

A central problem of microstructure is to develop technologies capable of producing an arrangement, or ordering, of the material, in terms of mesoscopic parameters like geometry and crystallography, appropriate for a given application. Is there such an order in the first place? We describe very briefly the emergence of the grain boundary character distribution (GBCD), a statistic that details texture evolution, and illustrate why it should be considered a material property. Its identification as a gradient flow by our method is tantamount to exhibiting the harvested statistic as the iterates in a mass transport JKO implicit scheme, which we found astonishing. Consequently the GBCD is the solution, in some sense, of a Fokker-Planck Equation. The development exposes the question of how to understand the circumstances under which a harvested empirical statistic is a property of the underlying process. (joint work with P. Bardsley, K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, X.-Y. Lu and S. Ta'asan).

We are interested in evolutionary biology models for sexual populations. The sexual reproductions are modelled through the so-called Infinitesimal Model, which is similar to an inelastic Boltzmann operator. This kinetic operator is then combined to selection and spatial dispersion operators. In this talk, we will show how the Wasserstein estimates that appear naturally for the kinetic operator can be combined to estimates on the other operators to study the qualitative properties of the solutions. In particular, this approach allows us to recover a well-known (in populations genetics) macroscopic model.

Swelling media (e.g. gels, tumors) are usually described by mechanical constitutive laws (e.g. Hooke or Darcy laws). However, constitutive relations of real swelling media are not well-known. Here, we take an opposite route and consider a simple heuristics relying on the following rule: (i) particles are at packing density; (ii) any two particles cannot swap their position; (iii) motion should be as slow as possible. These heuristics determine the medium velocity uniquely. In general, this velocity cannot be retrieved by a simple Darcy law.

We introduce a spatially extended transport kinetic FitzHugh-Nagumo model with forced local interactions and prove that its hydrodynamic limit converges towards the classical nonlocal reaction-diffusion FitzHugh-Nagumo system. Our approach is based on a relative entropy method, where the macroscopic quantities of the kinetic model are compared with the solution to the nonlocal reaction-diffusion system. This approach allows to make the rigourous link between kinetic and reaction-diffusion models

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

An exemplary drift-reaction system with mass conservation is studied w.r.t. pattern formation. The occurrence of rippling patterns in this system relates to an aggregation equation, whose qualitative behavior will also be discussed. If time permits, aggregation equations with local interactions will be presented, respectively Chemotaxis-models with a non-diffusive chemical. All models have in common, that their qualitative features are more of hyperbolic type. Thus pattern formation and the analysis of these systems is different from the one of their parabolic counterparts.

In this talk, we consider a system of equations arising from reproduction processes in biology, where two densities evolve under diffusion, absorbing reaction and chemotaxis. We prove that chemotaxis plays a crucial role to ensure the efficiency of reaction: Namely, the reaction between the two densities is very slow in the pure diffusion case, while adding a chemotaxis term greatly enhances reaction. While proving our main results we also obtain a weighted Poincare's inequality for the Fokker-Planck equation, which might be of independent interest.

For a range of physical and biological processes—from dynamics of granular media to biological swarming—the evolution of a large number of interacting agents is modeled according to the competing effects of pairwise attraction and (possibly degenerate) diffusion. We prove that, in the slow diffusion limit, the degenerate diffusion becomes a hard height constraint on the density of the population, as arises in models of pedestrian crown motion. We then apply this to develop numerical insight for open conjectures in geometric optimization.

I introduce and study the class of ``linear transfers'' between probability measures. This class contains all cost minimizing mass transports, including ``equivariant mass transports'' and ``martingale mass transports''. It also contain the ``Schroedinger bridge'' associated to a reversible Markov process, and the ``weak mass transports'' of Talagrand, Marton, Gozlan and others. However, what motivated us to develop the concept are the stochastic mass transports in their various forms. We also introduce the cone of ``convex transfers,'' which in addition to linear transfers, include any p-power of a linear transfer, but also the logarithmic entropy, other energy functionals, as well as the Donsker-Varadhan information. The ultimate goal: Stochastic Weak KAM theory.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Entropic regularization of optimal transport is appealing both from a numerical and theoretical perspective. In this talk we will discuss two applications, one from incompressible fluid dynamics and the other from mean-field games theory.

On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich-Rubinstein-Wasserstein square distance from optimal transportation. These notions also make sense in a (nonsmooth) metric measure setting, where they have found powerful applications. In this talk I describe the development of an analogous theory for lower Ricci curvature bounds in time-like directions on a Lorentzian manifold. In particular, by lifting fractional powers of the Lorentz distance (a.k.a. time separation function) to probability measures on spacetime, I show the strong energy condition of Penrose is equivalent to geodesic concavity of the Boltzmann-Shannon entropy there.

We study stochastic processes on the Wasserstein space, together with their infinitesimal generators. One of these processes plays a central role in our work. Its infinitesimal generator defines a partial Laplacian on the space of Borel probability measures, and we use it to define heat flow on the Wasserstein space. We verify a distinctive smoothing effect of this flow for a particular class of initial conditions. To this end, we will develop a theory of Fourier analysis and conic surfaces in metric spaces. We note that the use of the infinitesimal generators has been instrumental in proving various theorems for Mean Field Games, and we anticipate they will play a key role in future studies of viscosity solutions of PDEs in the Wasserstein space (Joint work with Y. T. Chow).

We explore a dynamic formulation of the optimal transportation problem with the additional freedom to choose the end-time of each trajectory. The dual problem is then posed with a Hamilton-Jacobi variational inequality, which we analyze with the method of viscosity solutions. We find properties that imply the optimal stopping-time is the hitting-time of the free boundary to the variational inequality. Joint work with N. Ghoussoub and Y.H. Kim.

BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local equilibrium, a Gaussian with the same macroscopic parameters. Due to the absence of dissipation w.r.t. the spatial direction, convergence to the global equilibrium is only possible thanks to the transport term that mixes various positions. Hence, such models are hypocoercive. We shall prove exponential convergence towards the equilibrium with explicit rates for several linear, space periodic BGK-models in dimension 1 and 2. Their BGK-operators differ by the number of conserved macroscopic quantities (like mass, momentum, energy), and hence their hypocoercivity index. Our discussion includes also discrete velocity models, and the local exponential stability of a nonlinear BGK-model. The proof is based, first, on a Fourier decomposition in space and Hermite function decomposition in velocity. Then, the crucial step is to construct a problem adapted Lyapunov functional, by introducing equivalent norms for each mode. \begin{thebibliography}{10} \bibitem{R1}{\sc F. Achleitner, A. Arnold, E.A. Carlen}, {\sl On linear hypocoercive BGK models}, in Springer Proceedings in Mathematics \& Statistics, Vol. 126, 1-37 (2016) \bibitem{R2}{\sc F. Achleitner, A. Arnold, E.A. Carlen}, {\sl On multi-dimensional hypocoercive BGK models}, to appear in KRM (2018) \bibitem{R3}{\sc F. Achleitner, A. Arnold, B. Signorello}, {\sl On optimal decay estimates for ODEs and PDEs with modal decomposition}, submitted (2018) \end{thebibliography}

We will present here two different approaches to study the long time behaviour of the kinetic Langevin equation : 1) hypocoercivity technique for entropic convergence via a new weighted logarithmic Sobolev inequality ; 2) Wasserstein convergence via a particular reflection coupling.

Using a nonlinear parabolic flow, in this talk I will explain why the optimal regions of symmetry and symmetry breaking for the extremals of critical and subcritical Caffarelli-Kohn-Nirenberg inequalities are related to the spectral gap of the linearized problem around the asymptotic Barenblatt solutions. This is a surprising result since it means that a global test yields a global result. The use of the parabolic flow also allows to get improved inequalities with explicit remainder terms.

We investigate new functional inequalities for the well-known Kac's Walk, and largely resolve the 'Almost' Cercignani Conjecture on the sphere. A new notion of chaoticity plays an essential role. The results we obtain validate Kac's suggestion that functional inequalities for the Kac walk could be used to quantify the rate of approach to equilibrium for the Kac-Boltzmann equation. This is joint work with E. Carlen and A. Einav.

As well as results in Hilbert spaces, Villani's memoire on hypocoercivity contains convergence to equilibrium results measured in relative entropy. Since then hypocoercivity in more general Phi entropies has been studied by several authors. These works have mainly been for diffusion equations which can be put in a 'Hormander sum of squares form'. The linear relaxation Boltzmann equation is a simple equation not of this form for which hypocoercivity in Phi entropies can still be shown but with extra terms added to the functional which would not be needed for a diffusion.

We prove exponential convergence to equilibrium for renormalised solutions to general complex balanced reaction-diffusion systems without boundary equilibria and even for systems with boundary equilibria provided a finite dimensional inequality holds along solutions trajectories. Our proofs are based on the entropy method and represent the most general results on the convergence to equilibrium for complex balanced RD systems currently available. (joint works with Bao Quoc Tang)

The behavior of solutions to the classical porous medium equation is by now well understood: the support of the solution expands at finite speed, and for large times it behaves as the separate-variable solution. When the Laplacian is replaced by a nonlocal diffusion, completely new and surprising phenomena arise depending on the power of the nonlinearity and the one of the diffusion. The aim of the talk is to give an overview of this theory.

The Kac master equation models the behavior of a large number of randomly colliding particles. Due to its simplicity it allows, without too much pain, to investigate a number of issues. E.g., Mark Kac, who invented this model in 1956, used it to give a simple derivation of the spatially inhomogeneous Boltzmann equation. One important issue is the rate of approach to equilibrium, which can be analyzed in various ways, using, e.g., the gap or the entropy. Explicit entropy estimates will be discussed for a Kac type master equation modeling the interaction of a finite system with a large but finite reservoir. This is joint work with Federico Bonetto, Alissa Geisinger and Tobias Ried.

In this talk I would like to present some recent results on the asymptotic behavior of a very fast diffusion PDE with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures. I prove exponential convergence to equilibrium under minimal assumptions on the data, and I also provide sufficient conditions for W2-stability of solutions. Moreover, I will present a work in progress with Filippo Santambrogio and Francesco Saverio Patacchini where we use the JKO scheme to relax the hypotheses of my previous convergence result.

In this talk I will discuss the construction of approximate solutions for the Non-linear Fokker-Planck equation. We utilize the $L^2$-Wasserstein gradient flow structure of this PDEs to perform a semi discretization in time by means of the variational BDF2 method. Our approach can be considered as the natural second order analogue of the Minimizing Movement or JKO scheme. In comparison to our own recent work on constructing solutions to $\lambda$-contractive gradient flows in abstract metric spaces, the technique presented here exploits the differential structure of the underlying $L^2$-Wasserstein space. We directly prove that the obtained limit curve is a weak solution of the non-linear Fokker-Planck equation without using the abstract theory of curves of maximal slope. Additionally, we provide strong $L^m$ convergence instead of merely weak convergence in the $L^2$-Wasserstein topology of the time-discrete approximations.

We discuss two examples of "dynamical optimal transport problems", whose formulations involve a relative entropy functional. The first case is related to the Hellinger-Kantorovich distance and induces an interesting geometric structure on the space of positive measures with finite (but possibly different) mass. In particular, contraction estimates of nonlinear flows are strongly related to geodesic convexity of the generating entropy functionals. In the second example an entropy functional penalizes the density of the connecting measures with respect to a given reference measure (typically the Lebesgue one) and leads to a first order "mean field planning" problem, which is classicaly formulated by a continuity equation and a Hamilton Jacobi equation with a nonlinear coupling. In this case, the variational approach and the displacement convexity of the entropy functionals (in the usual sense of optimal transport) provide crucial tools to give a precise meaning to the PDE system and to prove the existence of a solution.

For a natural class of discretisations of a convex domain in $R^n$, we consider the dynamical optimal transport metric for probability measures on the discrete mesh. Although the associated discrete heat flow converges to the continuous heat flow, we show that the transport metric may fail to converge to the 2-Kantorovich metric. Under an additional symmetry condition on the mesh, we show that Gromov-Hausdorff convergence to the 2-Kantorovich metric holds. This is joint work with Peter Gladbach and Eva Kopfer.

We study the main consequences of the existence of a Gradient Flow (GF for short), in the form of Evolution Variational Inequalities (EVI), in the very general framework of an abstract metric space. In particular, no volume measure is needed. The hypotheses on the functional associated with the GF are also very mild: we shall require at most completeness of the sublevels (no compactness assumption is made) and, for some convergence and stability results, approximate $\lambda$-convexity. The main results include: quantitative regularization properties of the flow (in terms e.g. of slope estimates and energy identities), discrete-approximation estimates of a minimizing-movement scheme and a stability theorem for the GF under suitable gamma-convergence-type hypotheses on a sequence of functionals approaching the limit functional. Existence of the GF itself is a quite delicate issue which requires some concavity-type assumptions on the metric, and will be addressed in a future project. This is a joint work with G. Savaré.

Nonlinear convection and nonlocal aggregation equations are known to feature a "formal" gradient flow structure in presence of a "nonlinear mobility", in terms of the generalized Wasserstein distance "à la" Dolbeault-Nazaret-Savaré. Such a structure is inherited by the discrete Lagrangian approximations of those equations in a quite natural way in one space dimension, and this simple remark allows to formulate a discrete-to-continuum "many particle" approximation. I will describe some recent results in this direction, which include the discrete (deterministic) particle approximation for scalar conservation laws and (more recently) a large class of nonlocal aggregation equations as main examples. The results are in collaboration with M. D. Rosini (Ferrara), S. Fagioli and E. Radici (L'Aquila).

One of the archetypical aggregation-diffusion models is the so-called classical parabolic-elliptic Patlak-Keller-Segel (PKS for short) model. This model was classically introduced as the simplest description for chemotatic bacteria movement in which linear diffusion tendency to spread fights the attraction due to the logarithmic kernel interaction in two dimensions. For this model there is a well-defined critical mass. In fact, here a clear dichotomy arises: if the total mass of the system is less than the critical mass, then the long time asymptotics are described by a self-similar solution, while for a mass larger than the critical one, there is finite time blow-up. In this talk we will show some recent results concerning the symmetry of the stationary states for a nonlinear variant of the PKS model, of the form \begin{equation} \label{aggregation} \partial_t \rho = \Delta \rho^m + \nabla \cdot (\rho\nabla(W*\rho)), \end{equation} being $W\in C^{1}(\R^{d}\setminus \{0\})$, $d\geq2$, a suitable aggregation kernel, in the assumptions of dominated diffusion, i.e. when $m>2-2/d$. In particular, if $W$ represents the classical logarithmic kernel in the bidimensional case, we will show that there exists a unique stationary state for the model \eqref{aggregation} and it coincides, according to one of the main results in the work \cite{CCV}, with the global minimizer of the free energy functional associated to \eqref{aggregation}. In the case $d=2$ we will also show how such steady state coincides with the aymptotic profile of \eqref{aggregation}. Finally, we will also discuss some recent results concerning the model \eqref{aggregation} with a Riesz potential aggregation, namely when $W(x)=c_{d,s}|x|^{2s-d}$ for $s\in(0,d/2)$, again in the diffusion dominated regime, \emph{i.e.} for $m>2-(2s)/d$. In particular, all stationary states of the model are shown to be radially symmetric decreasing and that global minimizers of the associated free energy are compactly supported, uniformly bounded, Hˆlder regular, and smooth inside their support. These results are objects of the joint works \cite{CHVY}, \cite{CHMV}. \vspace{12pt} \parindent=0pt \begin{thebibliography}{10} \bibitem{CCV} {\sc J.~A.~Carrillo, D.~Castorina, B.~Volzone}, \emph{Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction,} SIAM J. Math. Anal. 47 (2015), no. 1, 1--25. \bibitem{CHVY} {\sc J.~A.~Carrillo, S. Hittmeir, B. Volzone, Y. Yao}, \emph{Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics,}, arXiv:1603.07767. \bibitem{CHMV} {\sc J.~A.~Carrillo, F. Hoffmann, E. Mainini, B. Volzone}, \emph{Ground States in the Diffusion-Dominated Regime}, arXiv:1705.03519. \end{thebibliography}

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not. This is joint work with J. A. Carrillo and M. G. Delgadino.

We propose a generalization of the Schr\"odinger problem by replacing the usual entropy with a functional $\mathcal F$ which approaches the Wasserstein distance along the gradient of $\mathcal F$. From an heuristic point of view by using Otto calculus, we show that interpolations satisfy a Newton equation, extending the recent result of Giovani Conforti. Various inequalities as Evolutional-Variational-inequalities are also established from a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the usual Schr\"odinger problem under Ricci bound on a smooth and compact Riemannian manifold. This is a joint work with L. Ripani and C. L\'eonard.

Stein kernels are a way of measuring distance between probability measures, defined via integration by parts formulas. I will present a connection between these kernels and optimal transport. The main result is a way of deriving rates of convergence in the classical central limit theorem using regularity estimates for a variant of the Monge-Ampere PDE. As an application, we obtain new rates of convergence for the multi-dimensional CLT, with explicit dependence on the dimension.

I will present a discrete notion of super Ricci flow that applies to time dependent Markov chains or weighted graphs. This notion can be characterized equivalently in terms of a discrete time-dependent Bochner inequality, gradient estimates for the heat propagator on the evolving graph, contraction estimates in discrete transport distances, or dynamic convexity of the entropy. I will also discuss several examples. This is joint work with Eva Kopfer.

We revisit a result in probability known as the Harris theorem and give a simple proof which is well-suited for some applications in PDE. The proof is not far from the ideas of Hairer \& Mattingly (2011) but avoids the use of mass transport metrics and can be readily extended to cases where there is no spectral gap and exponential relaxation to equilibrium does not hold. We will also discuss some contexts where this result can be useful, particularly in a model for neuron populations structured by the elapsed time since the last discharge. This talk is based on joint works with Stéphane Mischler and Havva Yolda.

In the talk, the McKean-Vlasov equation on the flat torus is studied. The model is obtained as the mean field limit of a system of interacting diffusion processes enclosed in a periodic box. The system acts as a model for several real-world phenomena from statistical physics, opinion dynamics, collective behaviour, and stellar dynamics. This work provides a systematic approach to the qualitative and quantitative analysis of the McKean-Vlasov equation. We comment on the longtime behaviour and convergence to equilibrium, for which we introduce a notion of H-stability. The main part of the talk considers the stationary problem. We show that the system exhibits multiple equilibria which arise from the uniform state through continuous bifurcations, under certain assumptions on the interaction potential. Finally, criteria for the classification of continuous and discontinuous transitions of this system are provided. This classification is based on a fine analysis of the free energy. The results are illustrated by proving and extending results for a wide range of models, including the noisy Kuramoto model, Hegselmann-Krause model, and Keller-Segel model. (joint work with José Carrillo, Rishabh Gvalani, and Greg Pavliotis)

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.