Lyapunov conditions for Super Poincaré inequalities

@article{Cattiaux2009LyapunovCF,
  title={Lyapunov conditions for Super Poincar{\'e} inequalities},
  author={Patrick Cattiaux and Arnaud Guillin and Feng-Yu Wang and Liming Wu},
  journal={Journal of Functional Analysis},
  year={2009},
  volume={256},
  pages={1821-1841}
}

Weak Poincaré inequalities for convolution probabilities measures

In this paper, weak Poincaré inequalities are obtained for convolution probabilities with explicit rate functions by constructing suitable Lyapunov functions. Here, one of these Lyapunov functions is

A Link Between the Log-Sobolev Inequality and Lyapunov Condition

We give an alternative look at the log-Sobolev inequality (LSI in short) for log-concave measures by semigroup tools. The similar idea yields a heat flow proof of LSI under some quadratic Lyapunov

Weighted Poincar\'{e} Inequalities for Nonlocal Dirichlet Forms

on L(μV ). Taking ρ(r) = e r with 0 < α < 2 and δ > 0, we get some conclusions for general fractional Dirichlet forms, which can be regarded as a complement of our recent work [13], and an

E INEQUALITIES AND HITTING TIMES

Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions are well known. We give here the correspondance (with quantitative results) for reversible

Poincare inequality and exponential integrability of the hitting times of a Markov process

Extending the approach of the paper [Mathieu, P. (1997) Hitting times and spectral gap inequalities, Ann. Inst. Henri Poincare 33, 4, 437 -- 465], we prove that the Poincare inequality for a

Poincaré and Logarithmic Sobolev Inequalities for Nearly Radial Measures

If Poincare inequality has been studied by Bobkov for radial measures, few is known about the logarithmic Sobolev inequalty in the radial case. We try to fill this gap here using different methods:

Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry

This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as
...

References

SHOWING 1-10 OF 38 REFERENCES

Weak Poincaré Inequalities and L2-Convergence Rates of Markov Semigroups

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

Mass Transport and Variants of the Logarithmic Sobolev Inequality

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are

Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general F-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide

Modified logarithmic Sobolev inequalities and transportation inequalities

Abstract.We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type exp (−|x|α) or exp

Modified log-Sobolev inequalities and isoperimetry

We find sufficient conditions for a probability measure μ to satisfy an inequality of the type ∫ Rd fF ( f ∫ Rd f 2 dμ ) dμ ≤ C ∫ Rd f2c∗ ( |∇f | |f | ) dμ + B ∫ Rd f dμ, where F is concave and c (a

Trends to equilibrium in total variation distance

This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a

Sur les in'egalit'es de Sobolev logarithmiques

On logarithmic Sobolev inequalities). — This book is an overview on logarithmic Sobolev inequalities. These inequalities turned out to be a subject of intense activity during the past years, from

Uniformly Integrable Operators and Large Deviations for Markov Processes

Abstract In this paper we introduce and investigate the notion of uniformly integrable operators on L p ( E , μ ). Its relations to classical compactness and hypercontractivity are exhibited. Several

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

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