On quantitative hypocoercivity estimates based on Harris-type theorems

This review concerns recent results on the quantitative study of convergence towards the stationary state for spatially inhomogeneous kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in the $L^1$ (or total variation) setting for a large class of initial data. The convergence rates can be made explicit (both for geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates. We aim to present Harris-type theorems, providing a guideline on how to apply these techniques to the kinetic equations at hand. We discuss recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, giving some perspectives on potential extensions to nonlinear equations.Comment: 40 pages, typos are corrected, new references are added and structure of the paper has change

A cross-diffusion system obtained via (convex) relaxation in the JKO scheme

In this paper, we start from a very natural system of cross-diffusion equations which, unfortunately, is not well-posed as it is the gradient flow for the Wasserstein distance of a functional which is not lower semi-continuous due to lack of convexity of the integral. We then compute the convexification of the integral and prove existence of a solution in a suitable sense for the gradient flow of the corresponding relaxed functional. This gradient flow has also a cross-diffusion structure but the mixture between two different regimes which are determined by the convexification makes this study non-trivial.Comment: 38 pages, 2 figure

Hypocoercivity of linear kinetic equations via Harris's Theorem

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus (x, v) ∈ T d × R d or on the whole space (x, v) ∈ R d × R d with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively L 1 or weighted L 1 norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem

Hypocoercivity of linear kinetic equations via Harris's Theorem

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole space $(x,v) \in \mathbb{R}^d \times \mathbb{R}^d$ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $L^1$ or weighted $L^1$ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem

Hypocoercivity of linear kinetic equations via Harris's Theorem

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus (x,v)∈Td×Rd or on the whole space (x,v)∈Rd×Rd with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively L1 or weightedL1 norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem

Impact of the COVID-19 Pandemic on Inherited Metabolic Diseases: Evaluation of Enzyme Replacement Treatment Adherence with Telemedicine

Aim:During the coronavirus disease-2019 (COVID-19) pandemic, visiting the hospital and getting regular infusions can be difficult for patients with chronic illnesses. Telemedicine may offer a good option for the management of chronic diseases such as lysosomal storage diseases (LSD).Materials and Methods:LSD patients at the Unit of Metabolic Diseases of Ege University were contacted by phone between April, 2020 and March, 2021 during the COVID-19 pandemic. Telemedicine appointments were performed at intervals every month or three months, depending on the patients’ compliance with their treatment.Results:Ninety-two LSD patients [Mucopolysaccharidosis (MPS) I, MPS II, MPS IVA, MPS VI, MPS VII, Gaucher, Fabry, and Pompe] were included in this study. The total skipped treatment rate within one year was 17.1%. Most of the months of interruption were consonant with the time of social isolation. The treatment interruption in patients under 18 years was lower than in patients over 18 years. A positive correlation was detected between the age of patients and the interruption of treatment.Conclusion:The curfew periods might be one of the causes of missed treatment sessions. Telemedicine is a good method to improve the continuity of treatment. This study showed that the number of interrupted enzyme replacement treatments could be decreased via ongoing telemedicine appointments

Asymptotic behaviour of neuron population models structured by elapsed-time

We study two population models describing the dynamics of interacting neurons, initially proposed by Pakdaman et al (2010 Nonlinearity 23 55–75) and Pakdaman et al (2014 J. Math. Neurosci. 4 1–26). In the first model, the structuring variable s represents the time elapsed since its last discharge, while in the second one neurons exhibit a fatigue property and the structuring variable is a generic 'state'. We prove existence of solutions and steady states in the space of finite, nonnegative measures. Furthermore, we show that solutions converge to the equilibrium exponentially in time in the case of weak nonlinearity (i.e. weak connectivity). The main innovation is the use of Doeblin's theorem from probability in order to show the existence of a spectral gap property in the linear (no-connectivity) setting. Relaxation to the steady state for the nonlinear models is then proved by a constructive perturbation argument.MTM2014-52056-P, MTM2017-85067-P, "la Caixa" Foundatio

Spectral gap for the growth-fragmentation equation via Harris's theorem

The work of the first and third authors was supported by the project MTM2017- 85067-P, funded by the Spanish government and the European Regional Development Fund and they gratefully acknowledge the support of the Hausdorff Research Institute for Mathematics (Bonn), through the Junior Trimester Program on Kinetic Theory. The work of the second author was supported by the ANR project NOLO, grant ANR-20-CE40-0015, funded by the French Ministry of Research. The work of the third author was also supported by the Basque Government through the BERC 2018-2021 program, by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2017-0718, by the ``la Caixa"" Foundation, and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme grant 639638.We study the long-time behavior of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalize those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the nonconservativeness of the equation is overcome by performing an h-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.BERCFrench Ministry of ResearchHausdorff Research Institute for Mathematics ANR-20-CE40-0015Spanish governmentEuropean Commission 639638 ECEuropean Research CouncilMinisterio de Economía y Competitividad ¡SEV-2017-0718 MINECOEuropean Regional Development Fun

On the asymptotic behavior of a run and tumble equation for bacterial chemotaxis

We prove that linear and weakly nonlinear run and tumble equations converge to a unique steady state solution with an exponential rate in a weighted total variation distance. In the linear setting, our result extends the previous results to an arbitary dimension while relaxing the assumptions. The main challenge is that even though the equation is a mass-preserving, Boltzmann-type kinetic-transport equation, the classical hypocoercivity methods, e.g., by J. Dolbeault, C. Mouhot, and C. Schmeiser [Trans. Amer. Math. Soc., 367 (2015), pp. 3807–3828], are not applicable for dimension . We overcome this difficulty by using a probabilistic technique, known as Harris’s theorem. We also introduce a weakly nonlinear model via a nonlocal coupling on the chemoattractant concentration. This toy model serves as an intermediate step between the linear model and the physically more relevant nonlinear models. We build a stationary solution for this equation and provide a hypocoercivity result