A simple proof of the Poincaré inequality for a large class of probability measures

By D. Bakry, Franck Barthe, Patrick Cattiaux, Arnaud Guillin. 2008

228 citations recorded by the source corpus.

Abstract. We give a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on $\mathbb{R}^n$. The proof is based on arguments introduced in Bakry and al, but for the sake of completeness, all details are provided.

Subfields: Mathematical Physics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/313c9f6ae023a3e3b5d5504e1320719fa60404eb

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Citations in the corpus

Central limit theorems for additive functionals of ergodic Markov diffusions processes (2011, methodology): “These examples are then extended to n-dimensional reversible Langevin stochastic differential equations using Lyapunov conditions and results of Bakry et al. (2008b,a); Cattiaux et al. (2010b) to recover Poincaré inequalities or weak Poincaré inequalities through the use of Lyapunov conditions, and ”
Second order Poincaré inequalities and CLTs on Wiener space (2008, background): “Bakry et al. [ 1 ], Bobkov [2], Cacoullos et al. [3], Chen [5–7], Houdre and Perez-Abreu [10], and the references therein.”
Poincaré inequalities and hitting times (2010, methodology): “…can be rewritten : there exists a Lyapunov function W , i . e . there exist a smooth function W ≥ 1, a constant λ > 0 and an open connected bounded subset U such that This formulation is the one used in [2] yielding another bound for the Poincaré constant , namely The bound we will get below ( Eq .”
Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis (2017, methodology): “The following two propositions give sufficient conditions based on so-called Lyapunov function criteria: Proposition A.1 (Bakry et al. (2008)).”
HYPOCOERCIVITY FOR LINEAR KINETIC EQUATIONS CONSERVING MASS (2010, background): “137] for an elementary proof if lim|x|→∞ ( |∇xV |2 − 2∆V ) = ∞, and [1] for some recent considerations on Poincaré inequalities when e is a probability measure.”
Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm (2015, unknown): “(54) Using this inequality and for all u ∈ [0, 1], (1−u)1/2−1 ≤ −u/2, we have for all x ∈ R, satisfiyng ‖x− x⋆‖ ≥ Rc = max(1, 2dη,Mη), ( ‖x− γ∇U(x)− x⋆‖ + 2γd+ 1 )1/2 − φ(x) ≤ φ(x) {( 1− 2γφ(x)(η ‖x− x‖ − d) )1/2 − 1 } ≤ −γφ(x)(η ‖x− x‖ − d) ≤ −(ηγ/2) ‖x− x‖φ(x) ≤ −2ηγ .”
Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes (2016, background): “In particular, it is shown in [1] that if there exists a set B, constants λ,C ∈ (0,∞), and a positive twice-continuously differentiable function V such that LV ≤ −λV + C1B, and there holds a local Poincaré inequality of the form ∫”
Semi Log-Concave Markov Diffusions (2013, background): “The result is due to Bobkov [7] (a short proof is contained in [2]).”
Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry (2010, methodology): “Recall the following elementary lemma whose proof can be found in [3].”
Factorization for non-symmetric operators and exponential H-theorem (2010, background): “We refer to the recent paper [11] for an introduction to this important subject as well as to the references therein for further developments.”
Rogers–Shephard inequality for log-concave functions (2014, methodology): “Many functional inequalities satisfied by Gaussian functions, like Poincaré and Log-Sobolev inequalities, also hold in a more general subclass of log-concave functions [8,9,12].”
Bernstein type's concentration inequalities for symmetric Markov processes (2010, methodology): “− LU U (n) = κ− 1 κ (an − κbn), where it follows that cP < +∞ [4], [5] and so the desired result holds by Theorem 4.”