Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré

By D. Bakry, Patrick Cattiaux, Arnaud Guillin. 2007

296 citations recorded by the source corpus.

We study the relationship between two classical approaches for quantitative ergodic properties : the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincare type). We show that they can be linked through new inequalities (Lyapunov-Poincare inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by Herau-Nier, Helffer-Nier and Villani is in particular discussed in the final section.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/589f49c164af47050fa4bee11ff2eed5dbf11ee0

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Citations in the corpus

Semi Log-Concave Markov Diffusions (2013, methodology): “The long time behavior study of such a system has been considered for a long time and have been tackled by different techniques, see for example: hypocoercivity by Villani [42] or Lyapunov function technique by Bakry&al [3].”
Poincaré inequalities and hitting times (2010, methodology): “Note also that the implications (H1) to (H5) holds also , with additional assumptions ( local Poincaré inequality and ( slight ) conditions on the constants involved in (H1)) using Lyapunov–Poincaré inequalities as in [3].”
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation (2014, background): “Let us mention that there is an important literature in the probability community, see for instance [25, 3], that deals with exponential relaxation to equilibrium for stochastic processes whose law follows kinetic Fokker-Planck equations.”
Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry (2010, methodology): “In [4], connections are given between Lyapunov functions and functional inequalities of weak Poincaré type, improving some existing criteria discussed in [49; 9].”
Mass Transport and Variants of the Logarithmic Sobolev Inequality (2007, background): “[ 8 ]). We estimate the convexity defect by the divergence of a special vector field (usually x or ∇V ) and apply integration by parts.”
Couplings and quantitative contraction rates for Langevin dynamics (2017, methodology): “The choice of the Lyapunov function is motivated by Mattingly, Stuart and Higham [32], see also [39, 46, 1].”
Bernstein type's concentration inequalities for symmetric Markov processes (2010, unknown): “(1.1)”
A simple proof of the Poincaré inequality for a large class of probability measures (2008, methodology): “In the recent paper [2], the authors developed a general strategy to get tractable conditions for the Poincaré inequality (1.”
Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations (2009, methodology): “Subexponential convergence rates via Lyapunov functions has been explored for Harris chains in [MT93, Ver99, BCG08 , DFG09, Hai09].”
Quantitative Estimates for the Long-Time Behavior of an Ergodic Variant of the Telegraph Process (2010, background): “A related and popular approach to the longtime behavior of Markov processes is the Foster–Lyapounov–Meyn–Tweedie theory (see [2], [14], [15], and [19]), which allows one to prove exponential ergodicity under conditions that are”
Derivative formula and applications for degenerate diffusion semigroups (2011, background): “[3, 15] where some long time behaviors of the system have been investigated.”
Sampling from a strongly log-concave distribution with the Unadjusted Langevin Algorithm (2016, background): “Under the stated assumptions on U , π satisfies a log-Sobolev inequalities (see [2, 6, 3]) and the Markov semi-group associated with the Langevin diffusion (Yt)t≥0 converges exponentially fast to π with a rate independent of the dimension of the state space.”
Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes (2016, background): “While the principal result from [28, Section 4] is already contained in [2, 16], the method of proof shows new and interesting aspects avoiding discrete-time approximations.”
The Markov chain Monte Carlo revolution (2008, background): “This interplay between the proof techniques of PDE and Markov chains has evolved into the small but healthy field of functional inequalities [5, 6] which contributes to both subjects.”
Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm (2015, methodology): “More explicit rates of convergence for the semigroup can be obtained using Poincaré inequality; see [3], [9] and [4, Chapter 4] and the references therein.”