Spectral Gap Estimates Research Articles

Exponential convergence for the linear homogeneous Boltzmann equation for hard potentials

In this paper, we consider the asymptotic behavior of solutions to the linear spatially homogeneous Boltzmann equation for hard potentials without angular cutoff. We obtain an optimal rate of exponential convergence towards equilibrium in a L1-space with a polynomial weight. Our strategy is taking advantage of a spectral gap estimate in the Hilbert space L2(μ−12) and a quantitative spectral mapping theorem developed by Gualdani et al. (2017).

Spectral gap and exponential convergence to equilibrium for a multi-species Landau system

In this paper we prove new constructive coercivity estimates and convergence to equilibrium for a spatially non-homogeneous system of Landau equations with moderately soft potentials. We show that the nonlinear collision operator conserves each species' mass, total momentum, total energy and that the Boltzmann entropy is nonincreasing along solutions of the system. The entropy decay vanishes if and only if the Boltzmann distributions of the single species are Maxwellians with the same momentum and energy. A linearization of the collision operator is computed, which has the same conservation properties as its nonlinear counterpart. We show that the linearized system dissipates a quadratic entropy, and prove existence of spectral gap and exponential decay of the solution towards the global equilibrium. As a consequence, convergence of smooth solutions of the nonlinear problem toward the unique global equilibrium is shown, provided the initial data are sufficiently close to the equilibrium. Our proof is based on new spectral gap estimates and uses a strategy similar to [11] based on an hypocoercivity method developed by Mouhot and Neumann in [27].

A stochastic algorithm finding $p$-means on the circle

A stochastic algorithm is proposed, finding some elements from the set of intrinsic $p$-mean(s) associated to a probability measure $\nu$ on a compact Riemannian manifold and to $p\in[1,\infty)$. It is fed sequentially with independent random variables $(Y_n)_{n\in \mathbb{N}}$ distributed according to $\nu$, which is often the only available knowledge of $\nu$. Furthermore, the algorithm is easy to implement, because it evolves like a Brownian motion between the random times when it jumps in direction of one of the $Y_n$, $n\in\mathbb{N}$. Its principle is based on simulated annealing and homogenization, so that temperature and approximations schemes must be tuned up (plus a regularizing scheme if $\nu$ does not admit a H\"{o}lderian density). The analysis of the convergence is restricted to the case where the state space is a circle. In its principle, the proof relies on the investigation of the evolution of a time-inhomogeneous $\mathbb{L}^2$ functional and on the corresponding spectral gap estimates due to Holley, Kusuoka and Stroock. But it requires new estimates on the discrepancies between the unknown instantaneous invariant measures and some convenient Gibbs measures.

Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation

The aim of the present paper is twofold: 1. We carry on with developing an abstract method for deriving decay estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [10]. We extend the method so as to consider the shrinkage of the functional space. Roughly speaking, we consider a class of operators written as a dissipative part plus a mild perturbation, and we prove that if the associated semigroup satisfies a decay estimate in some reference space then it satisfies the same decay estimate in another—smaller or larger—Banach space under the condition that a certain iterate of the “mild perturbation” part of the operator combined with the dissipative part of the semigroup maps the larger space to the smaller space in a bounded way. The cornerstone of our approach is a factorization argument, reminiscent of the Dyson series. 2. We apply this method to the kinetic Fokker-Planck equation when the spatial domain is either the torus with periodic boundary conditions, or the whole space with a confinement potential. We then obtain spectral gap estimates for the associated semigroup for various metrics, including Lebesgue norms, negative Sobolev norms, and the Monge-Kantorovich-Wasserstein distance W1.

Hypocoercivity for a Linearized Multispecies Boltzmann System

A new coercivity estimate on the spectral gap of the linearized Boltzmann collision operator for multiple species is proved. The assumptions on the collision kernels include hard and Maxwellian potentials under Grad's angular cutoff condition. Two proofs are given: a non-constructive one, based on the decomposition of the collision operator into a compact and a coercive part, and a constructive one, which exploits the “cross effects” coming from collisions between different species and which yields explicit constants. Furthermore, the essential spectra of the linearized collision operator and the linearized Boltzmann operator are calculated. Based on the spectral-gap estimate, the exponential convergence towards global equilibrium with explicit rate is shown for solutions to the linearized multispecies Boltzmann system on the torus. The convergence is achieved by the interplay between the dissipative collision operator and the conservative transport operator and is proved by using the hypocoercivity method of Mouhot and Neumann.

Corrector Estimates for Elliptic Systems with Random Periodic Coefficients

We consider the corrector equation for the second order elliptic system on the $d$-dimensional torus of size $L$ ($d \ge 2$), associated with random coefficients $A$ that are assumed to be coercive and stationary. Using two different approaches we obtain moment bounds on the gradient of the corrector, independent of the domain size $L$. In the first approach we use Green's function representation. For that we require $A$ to be locally Hölder continuous and the distribution of $A$ to satisfy a logarithmic Sobolev inequality. The second method works for nonsmooth (possibly discontinuous) coefficients, and it requires that the statistic of $A$ satisfies a spectral gap estimate.

Cutoff for the East Process

The East process is a 1d kinetically constrained interacting particle system, introduced in the physics literature in the early 1990s to model liquid-glass transitions. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on L sites has order L. We complement that result and show cutoff with an \({O(\sqrt{L})}\)-window.

Reduction in the Resonance Error in Numerical Homogenization II: Correctors and Extrapolation

This paper is the follow-up of Gloria (Math Models Methods Appl Sci 21(8):1601–1630, 2011). One common drawback among numerical homogenization methods is the presence of the so-called resonance error, which roughly speaking is a function of the ratio $$\frac{\varepsilon }{\rho }$$ , where $$\rho $$ is a typical macroscopic lengthscale and $$\varepsilon $$ is the typical size of the heterogeneities. In the present work, we make a systematic use of regularization and extrapolation to reduce this resonance error at the level of the approximation of homogenized coefficients and correctors for general non-necessarily symmetric stationary ergodic coefficients. We quantify this reduction for the class of periodic coefficients, for the Kozlov subclass of almost-periodic coefficients, and for the subclass of random coefficients that satisfy a spectral gap estimate (e.g., Poisson random inclusions). We also report on a systematic numerical study in dimension 2, which demonstrates the efficiency of the method and the sharpness of the analysis. Last, we combine this approach to numerical homogenization methods, prove the asymptotic consistency in the case of locally stationary ergodic coefficients, and give quantitative estimates in the case of periodic coefficients.

Spectral analysis of semigroups and growth-fragmentation equations

The aim of this paper is twofold:(1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, some (quantified) Weyl's Theorems and the Krein–Rutman Theorem. Motivated by evolution PDE applications, the results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part, without assuming any symmetric structure on the operators nor Hilbert structure on the space, and give some growth estimates and spectral gap estimates for the associated semigroup. The approach relies on some factorization and summation arguments reminiscent of the Dyson–Phillips series in the spirit of those used in [96,87,51,86].(2) On the other hand, we present the semigroup spectral analysis for three important classes of “growth-fragmentation” equations, namely the cell division equation, the self-similar fragmentation equation and the McKendrick–Von Foerster age structured population equation. By showing that these models lie in the class of equations for which our general semigroup analysis theory applies, we prove the exponential rate of convergence of the solutions to the associated first eigenfunction or self-similar profile for a very large and natural class of fragmentation rates. Our results generalize similar estimates obtained in [103,73] for the cell division model with (almost) constant total fragmentation rate and in [19,18] for the self-similar fragmentation equation and the cell division equation restricted to smooth and positive fragmentation rate and total fragmentation rate which does not increase more rapidly than quadratically. It also improves the convergence results without rate obtained in [84,36] which have been established under similar assumptions to those made in the present work.

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

We establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergence form, when the diusion coecients satisfy a spectral gap estimate in probability, and for d > 2. This work is based on (5), which is a complete continuum version of (6, 7) (with in addition optimal results for d = 2). The main dierence with respect to the rst part of (5) is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory.

Spectral gap estimates for some block matrices

We estimate the size of the spectral gap at zero for some Hermitian block matrices. Included are quasi-definite matrices, quasi-semidefinite matrices (the closure of the set of the quasi-definite matrices) and some related block matrices which need not belong to either of these classes. Matrices of such structure arise in quantum models of possibly disordered systems with supersymmetry or graphene like symmetry. Some of the results immediately extend to infinite dimension.

Quantitative results on the corrector equation in stochastic homogenization

We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions d \geq 2 . In previous works we studied the model problem of a discrete elliptic equation on \mathbb Z^d . Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions d > 2 and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages – the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions.

Exponential Convergence to Equilibrium for the Homogeneous Boltzmann Equation for Hard Potentials Without Cut-Off

This paper deals with the long time behavior of solutions to the spatially homogeneous Boltzmann equation. The interactions considered are the so-called (non cut-off and non mollified) hard potentials. We prove an exponential in time convergence towards the equilibrium, improving results of Villani from \cite{Vill1} where a polynomial decay to equilibrium is proven. The basis of the proof is the study of the linearized equation for which we prove a new spectral gap estimate in a $L^1$ space with a polynomial weight by taking advantage of the theory of enlargement of the functional space for the semigroup decay developed by Gualdani and al in \cite{GMM}. We then get our final result by combining this new spectral gap estimate with bilinear estimates on the collisional operator that we establish.

A stochastic algorithm finding generalized means on compact manifolds

A stochastic algorithm is proposed, finding the set of generalized means associated to a probability measure ν on a compact Riemannian manifold and a continuous cost function κ on M×M. Generalized means include p-means for p∈(0,∞), computed with any continuous distance function, not necessarily the Riemannian distance. They also include means for lengths computed from Finsler metrics, or for divergences.The algorithm is fed sequentially with independent random variables (Yn)n∈N distributed according to ν and this is the only knowledge of ν required. It evolves like a Brownian motion between the times it jumps in the direction of the Yn. Its principle is based on simulated annealing and homogenization, so that temperature and approximation schemes must be tuned up. The proof relies on the investigation of the evolution of a time-inhomogeneous L2 functional and on the corresponding spectral gap estimates due to Holley, Kusuoka and Stroock.

Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics

We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice $$\mathbb Z^d$$ with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions $$d>2$$ ) and prove new optimal estimates for regularized versions of the corrector (in dimensions $$d\ge 2$$ ). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

Asymptotic Estimates for the Parabolic-Elliptic Keller-Segel Model in the Plane

We investigate the large-time behavior of the solutions of the two-dimensional Keller-Segel system in self-similar variables, when the total mass is subcritical, that is less than 8 π after a proper adimensionalization. It was known from previous works that all solutions converge to stationary solutions, with exponential rate when the mass is small. Here we remove this restriction and show that the rate of convergence measured in relative entropy is exponential for any mass in the subcritical range, and independent of the mass. The proof relies on symmetrization techniques, which are adapted from a paper of Diaz et al. and allow us to establish uniform estimates for L p norms of the solution. Exponential convergence is obtained by the mean of a linearization in a space which is defined consistently with relative entropy estimates and in which the linearized evolution operator is self-adjoint. The core of proof relies on several new spectral gap estimates which are of independent interest.

Ergodic properties for $\alpha$-CIR models and a class of generalized Fleming-Viot processes

We discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump mechanisms are governed by certain stable laws. The main result gives a lower spectral gap estimate for the generator. As an application, a certain ergodic property is shown for the generalized Fleming-Viot process obtained as the time-changed ratio process.

Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: Geometric conditions ensuring the compactness of the underlying manifold (Bonnet–Myers type results); Volume estimates of metric balls; Gradient bounds and stochastic completeness for the heat semigroup generated by the sub-Laplacian; Spectral gap estimates.

Invariances in variance estimates

We provide variants and improvements of the Brascamp–Lieb variance inequality which takes into account the invariance properties of the underlying measure. This is applied to spectral gap estimates for log-concave measures with many symmetries and to non-interacting conservative spin systems.

A reduced model for spatially structured predator–prey systems with fast spatial migrations and slow demographic evolutions

We consider a spatially structured predator-prey model where fast migrations occur inside a given spatial domain, while slow predator-prey interactions prescribe the demographic evolution. The unknowns of our model are the numbers of predators and prey at each time t and each site x of the domain. In the idealized limit where migrations are infinitely fast, we show one can approximate the global dynamics using the mere two unknowns corresponding to the total number of preys and predators, irrespective of their respective spatial repartition. Besides, the error term induced by this approximation can be made exponentially small with respect to the natural asymptotic parameter. In doing so, we completely characterize how migrations do modify both the qualitative and quantitative properties of the global demography. Our analysis relies on a convenient version of the central manifold theorem, in conjunction with a spectral gap estimate on the involved migration operator.