Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena

Berestycki, Henri; Hamel, François; Nadirashvili, Nikolaï

doi:10.1007/s00220-004-1201-9

Cited by 119 publications

(207 citation statements)

References 28 publications

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“…Actually, as in the reversible case, (13) (for all initial conditions ψ b 0 ) is equivalent to (4). This is because (6) also holds for ψ b solution to (12).…”

Section: Non-reversible Diffusionmentioning

confidence: 89%

“…Furthermore, the optimal convergence rate was obtained for the linear problem (see also Proposition 1 in the present paper) and some explicit examples were presented, for ordinary differential equations in two and three dimensions. The behavior of the generator of the dynamics (8) under a strong nonreversible drift has also been studied [4,6,13]. It was shown in [13] that the spectral gap attains a finite value in the limit as the strength of the perturbation becomes infinite if and only if the operator b · ∇ has no eigenfunctions in an appropriate Sobolev space of index 1.…”

Section: Bibliographymentioning

confidence: 99%

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Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

2013

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We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.Keywords Non-reversible diffusion · Convergence to equilibrium · Wick calculus 1 Introduction MotivationThe problem of convergence to equilibrium for diffusion processes has attracted considerable attention in recent years. In addition to the relevance of this problem for the convergence to equilibrium of some systems in statistical T. Lelièvre at Université Paris-Est, Cermics, and INRIA, MicMac project team, Ecole des ponts, 6-8 avenue Blaise Pascal, 77455 Marne la Vallée Cedex 2, France. E-mail: lelievre@cermics.enpc.fr · F. Nier at IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042 Rennes, France. E-mail: francis.nier@univ-rennes1.fr · G.A. Pavliotis at Imperial College London, Department of Mathematics, South Kensington Campus, London SW7 2AZ, England. E-mail: g.pavliotis@imperial.ac.uk 2 physics, see for example [29], such questions are also important in statistics, for example in the analysis of Markov Chain Monte Carlo (MCMC) algorithms [9]. Roughly speaking, one measure of efficiency of an MCMC algorithm is its rate of convergence to equilibrium, and increasing this rate is thus the aim of many numerical techniques (see for example [5]).Let us recall the basic approach for a reversible diffusion. Suppose that we are interested in sampling from a probability distribution functionwhere V : R N → R is a given smooth potential such that R N e −V dx < ∞ . A natural dynamics to use is the reversible dynamicswhere W t denotes a standard N -dimensional Brownian motion. Let us denote by ψ t the probability density function of the process X t at time t. It satisfies the Fokker-Planck equationUnder appropriate assumptions on the potential V (e.g. that 1 2 |∇V (x)| 2 − ∆V (x) → +∞ as |x| → +∞ , see [41, A.19]), the density ψ ∞ satisfies a Poincaré inequality: there exists λ > 0 such that for all probability density functions φ,The optimal parameter λ in (4) is the opposite of the smallest (in absolute value) non-zero eigenvalue of the Fokker-Planck operator ∇ · (∇V · +∇·), which is self-adjoint in L 2 (R N , ψ −1 ∞ dx) (see (7) below). Thus, λ is also called the spectral gap of the Fokker-Planck operator.It is then standard to show that (4) is equivalent to the following inequality, which shows exponential convergence to the equilibrium for (2): for all initial conditions ψ 0 ∈ L 2 (R N , ψ −1 ∞ dx), for all times t ≥ 0,where · L 2 (ψ −1 ∞ ) denotes the norm in L 2 (R N , ψ −1 ∞ ), namely f 2∞ (x) dx . This equivalence is a simple consequence of th...

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“…Actually, as in the reversible case, (13) (for all initial conditions ψ b 0 ) is equivalent to (4). This is because (6) also holds for ψ b solution to (12).…”

Section: Non-reversible Diffusionmentioning

confidence: 89%

Section: Bibliographymentioning

confidence: 99%

Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

2013

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“…We observe two distinct behaviors of λ with a sharp transition. If A ≫ L 4 , then the principal eigenvalue stays bounded, and can be read off using the variational principle in [3] in the limit A → ∞. This is the averaging regime, and exactly explains (1.10).…”

Section: Introductionmentioning

confidence: 85%

“…In the averaging regime (Theorem 1.1), the proof of the upper bound in (1.13) follows directly using ideas of [3]. The lower bound, however, is much more intricate.…”

Section: Theorem 12 (The Homogenization Regime)mentioning

confidence: 99%

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From homogenization to averaging in cellular flows

Iyer

Komorowski

Novikov

et al. 2014

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A, in a twodimensional domain with L 2 cells. For fixed A, and L → ∞, the problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A → ∞. In this case the solution equilibrates along stream lines.In this paper, we show that if both A → ∞ and L → ∞, then a transition between the homogenization and averaging regimes occurs at A ≈ L 4 . When A ≫ L 4 , the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A ≪ L 4 , the principal eigenvalue behaves likeσ(A)/L 2 , whereσ(A) ≈ √ AI is the effective diffusion matrix. A similar transition is observed for the solution of the exit time problem. The proof in the homogenization regime involves bounds on the second correctors. Miraculously, if the slow profile is quadratic, these estimates can be obtained using drift independent L p → L ∞ estimates for elliptic equations with an incompressible drift. This provides effective sub and super-solutions for our problem.

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Generalized Transition Waves and Their Properties

Berestycki

Hamel

2012

Comm Pure Appl Math

Self Cite

179

205

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In this paper, we generalize the usual notions of waves, fronts and propagation speeds in a very general setting. These new notions, which cover all usual situations, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. We prove the existence of new such waves for some time-dependent reaction-diffusion equations, as well as general intrinsic properties, some monotonicity properties and some uniqueness results for almost planar fronts. The classification results, which are obtained under some appropriate assumptions, show the robustness of our general definitions.

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Elliptic Eigenvalue Problems with Large Drift and Applications to Nonlinear Propagation Phenomena

Cited by 119 publications

References 28 publications

Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

From homogenization to averaging in cellular flows

Generalized Transition Waves and Their Properties

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