Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups

By Michael Röckner, Feng-Yu Wang. 2001

214 citations recorded by the source corpus.

Abstract In order to describe L 2 -convergence rates slower than exponential, the weak Poincare inequality is introduced. It is shown that the convergence rate of a Markov semigroup and the corresponding weak Poincare inequality can be determined by each other. Conditions for the weak Poincare inequality to hold are presented, which are easy to check and which hold in many applications. The weak Poincare inequality is also studied by using isoperimetric inequalities for diffusion and jump processes. Some typical examples are given to illustrate the general results. In particular, our results are applied to the stochastic quantization of field theory in finite volume. Moreover, a sharp criterion of weak Poincare inequalities is presented for Poisson measures on configuration spaces.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/c838cde48cecc59352fb6fed37b56426e6c6caa6

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In Applied Mathematics

Citations in the corpus

Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré (2007, methodology): “3 in [25] tells us that, in the symmetric case, if”
Poincaré inequalities and hitting times (2010, methodology): “Let us finally remark that Rockner-Wang [27] proves (H5) to (H6) without symmetry but assuming that L is normal (i.”
Probability distance inequalities on Riemannian manifolds and path spaces (2004, unknown): “[17]), one has μ((Ptf)) → 1 and hence μt → μ weakly as t → ∞.”
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry (2004, methodology): “For α > 0 the measures μα verify a Weak Spectral Gap property introduced by Aida and Kusuoka as shown in [48].”
Functional Inequalities and Spectrum Estimates: The Infinite Measure Case (2002, methodology): “This observation has been applied in several papers for some different inequalities, for instances, in [2, 12] for the Poincaré inequality in the context of jump and diffusion processes, in [13] for the super-Poincaré inequality and in [10] for the weak Poincaré inequality.”
Semi Log-Concave Markov Diffusions (2013, result): “Once we notice that the transformation f 7→ λf does not change h the result follows from [40] Theorem 2.”
Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry (2010, methodology): “For the generalized Cauchy distribution dμ(x) = cβ (1+ |x|)−(n+α), this result is optimal for n = 1 and was shown in [49] (see also [9, Example 2.”
Subgeometric rates of convergence of f-ergodic strong Markov processes (2006, background): “There are another ways to get subexponential rates of convergence (in total variation norm, in L2 or in entropy) such as those based on spectral techniques (as in [12]), or on functional inequalities (weak Poincaré inequalities [29] or weak logarithmic Sobolev inequalities [1]).”
Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions (2011, background): “Proposition 2.8 (Private communication [Röckner and Wang (2001)]).”
Accelerating diffusions (2005, unknown): “By Lemma 2.2 in Röckner and Wang (2001), for s≤ t and π(f2) = 1, ‖T (s)f‖2 ≤ (‖T (t)f‖2)s/t ≤C(N,f)s/te2ρs.”
Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality (2011, methodology): “We use an argument close to one found in [35].”
Large deviations and isoperimetry over convex probability measures with heavy tails (2007, background): “It is shown in (27) that WSGP is equivalent to (8.”
Lyapunov conditions for Super Poincaré inequalities (2009, background): “The relationship between Lyapunov conditions and functional inequalities of Poincaré type (ordinary or weak Poincaré introduced in [24]) is studied in details in the recent work [3].”
Concentration for independent random variables with heavy tails (2005, methodology): “Under mild conditions, we establish concentration inequalities for finite sequences of n independent copies, with good dependence in n.”