On Nonnegative Solutions of the Equation $AD + DA' =  - C$

Snyders, J.; Zakai, Moshe

doi:10.1137/0118063

Cited by 81 publications

(37 citation statements)

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“…Notice that for any (J , Q) ∈ P opt (S) ,J is completely defined (by (41)) as soon as Q is chosen, so that the set P opt (S) can be indexed by the set of matrices Q ∈ S >0 N (R) with N eigenvalues of multiplicity one, and with eigenvectors ψ k satisfying (39). As it will become clear below, the matrix Q of a pair (J , Q) ∈ P opt (S) appears in the quantitative estimates of Theorem 1 and Theorem 2 through the constants C (1) N and C (2) N .…”

Section: ⊓ ⊔mentioning

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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Section: ⊓ ⊔mentioning

confidence: 99%

“…Definition 1 We will denote by P opt (S) the set of pairs (J, Q), where Q is a real symmetric positive definite matrix with N eigenvalues of multiplicity one and associated eigenvectors satisfying (39), andJ is the associated antisymmetric matrix defined by (41).…”

Section: ⊓ ⊔mentioning

confidence: 99%

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

2013

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“…(см., например, [9]). Решение уравнения (8) с межчастичным взаимодействием и ангармоническим потенциалом выводится из частного случая при J ≡ 0, λ = 0 с использованием теоремы Гирсанова, которая определяет меру ρ для полного процесса в терминах меры µ C , полученной при J ≡ 0, λ = 0.…”

Section: интегральное представлениеunclassified

“…По физическим причинам интересно рассмотреть "условие самосо-гласованости" вида ⟨R j ⟩ = 0 в стационарном состоянии. Для линейной динамики существование стационарного состояния и сходимость к нему при t → ∞ являются давно решенными задачами (см., например, [9]). …”

Section: Introductionunclassified

Аналитический Подход К (Ан)гармоническим Кристаллическим Цепочкам С Самосогласованными Стохастическими Резервуарами

Фалкао¹,

Falcao²,

Нето³

et al. 2008

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“…The matrix exponential plays an important role in linear control systems and ordinary differential equations, see [1,2,8,9,11,13] and the references therein. However, the theoretical analysis as well as the numerical computation of e A are still under investigations, see [1,2,5,10].…”

Section: Introductionmentioning

confidence: 99%