Results 1  10
of
17
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré
 J. Func. Anal
, 1996
"... Abstract. 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 Poincaré type). We show that they can be linked through new ..."
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Cited by 27 (15 self)
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Abstract. 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 Poincaré type). We show that they can be linked through new inequalities (LyapunovPoincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic FokkerPlanck equation recently studied by HérauNier, HelfferNier and Villani is in particular discussed in the final section.
Isoperimetry between exponential and Gaussian
 Electronic J. Prob
"... We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1 ..."
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Cited by 16 (7 self)
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We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1
Transportationinformation inequalities for Markov processes (II): Relations . . .
, 2009
"... We continue our investigation on the transportationinformation inequalities WpI for a symmetric markov process, introduced and studied in [13]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure µ. We giv ..."
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Cited by 16 (2 self)
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We continue our investigation on the transportationinformation inequalities WpI for a symmetric markov process, introduced and studied in [13]. We prove that WpI implies the usual transportation inequalities WpH, then the corresponding concentration inequalities for the invariant measure µ. We give also a direct proof that the spectral gap in the space of Lipschitz functions for a diffusion process implies W1I (a result due to [13]) and a Cheeger type’s isoperimetric inequality. Finally we exhibit relations between transportationinformation inequalities and a family of functional inequalities (such as Φlog Sobolev or ΦSobolev).
Functional Inequalities for Uniformly Integrable Semigroups and Applications
 Forum Math
"... Let (E; F ; ) be a probability space, (E ; D(E)) a (not necessarily symmetric) Dirichlet form on L 2 (), and P t the associated subMarkov semigroup. The equivalence of the following eight properties is studied: (i) the L 2 uniform integrability of the unit ball in the Sobolev space; (ii) the s ..."
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Cited by 8 (2 self)
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Let (E; F ; ) be a probability space, (E ; D(E)) a (not necessarily symmetric) Dirichlet form on L 2 (), and P t the associated subMarkov semigroup. The equivalence of the following eight properties is studied: (i) the L 2 uniform integrability of the unit ball in the Sobolev space; (ii) the superPoincar'e inequality (1.2); (iii) the FSobolev inequality (1.3); (iv) the L 2 uniform integrability of P t ; (v) the L 2 uniform integrability of the associated resolvents; (vi) the compactness of P t ; (vii) the compactness of the associated resolvents; (viii) empty essential spectrum of the associated generator. The main results can be summarized as follows. In general, (i), (ii) and (iii) are equivalent to each other, and they imply (iv) which is equivalent to (v). If P t has transition density and F is separable, then the first seven properties from (i) to (vii) are equivalent. If in addition (E ; D(E)) is symmetric, then all the above eight properties are equivalent. Moreo...
THE SPECTRAL BOUND AND PRINCIPAL EIGENVALUES OF SCHRÖDINGER OPERATORS ON RIEMANNIAN MANIFOLDS
, 2001
"... Given a complete Riemannian manifold M and a Schrödinger operator − � + m acting on L p (M), we study two related problems on the spectrum of −�+m. The first one concerns the positivity of the L2spectral lower bound s(− � + m). We prove that if M satisfies L2Poincaré inequalities and a local doubl ..."
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Cited by 2 (0 self)
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Given a complete Riemannian manifold M and a Schrödinger operator − � + m acting on L p (M), we study two related problems on the spectrum of −�+m. The first one concerns the positivity of the L2spectral lower bound s(− � + m). We prove that if M satisfies L2Poincaré inequalities and a local doubling property, then s(− � + m)> 0, provided that m satisfies the mean condition 1 inf p∈M B(p, r) m(x) dx> 0
Perturbations of Functional Inequalities Using Growth Conditions
, 2006
"... Perturbations of functional inequalities are studied by using merely growth conditions in terms of a distancelike reference function. As a result, optimal sufficient conditions are obtained for perturbations to reach a class of functional inequalities interpolating between the Poincaré inequality a ..."
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Cited by 1 (0 self)
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Perturbations of functional inequalities are studied by using merely growth conditions in terms of a distancelike reference function. As a result, optimal sufficient conditions are obtained for perturbations to reach a class of functional inequalities interpolating between the Poincaré inequality and the logarithmic Sobolev inequality.
Intrinsic Ultracontractivity on Riemannian Manifolds with Infinite Volume Measures
, 2008
"... By establishing the intrinsic superPoincaré inequality, some explicit conditions are presented for diffusion semigroups on a noncompact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, ..."
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Cited by 1 (1 self)
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By establishing the intrinsic superPoincaré inequality, some explicit conditions are presented for diffusion semigroups on a noncompact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, are sharp for some concrete examples. AMS subject Classification: 58G32, 60J60
Ultracontractivity And Supercontractivity Of Markov Semigroups
"... By using perturbation arguments, a sufficient condition is presented for the ultracontractivity of symmetric diffusion semigroups. As a consequence, a result suggested by D. Stroock is proved: let P t be generated by L = + rV with V = r ( > 0; > 2) on a complete connected Riemannian manifold M , whe ..."
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By using perturbation arguments, a sufficient condition is presented for the ultracontractivity of symmetric diffusion semigroups. As a consequence, a result suggested by D. Stroock is proved: let P t be generated by L = + rV with V = r ( > 0; > 2) on a complete connected Riemannian manifold M , where is the Riemannian distance function from a fixed point, then P t is ultracontractive provided the Ricci curvature is bounded below. Furthermore, kP t k1!1 exp[ 1 + 2 t =( 2) ] for some 1 ; 2 > 0: Next, it is shown that, for a di usion semigroup (not necessarily symmetric) with an invariant probability measure, if the curvature of its generator is bounded from below, then the ultracontractivity is equivalent to kP t exp[ 2 ]k 1 0: Especially, the above estimate of kP t k1!1 holds if L 2 c1 c2 for some c1 ; c2 > 0 and > 2: This estimate is sharp as is shown by examples given at the end of the paper. Corresponding results are proven for supe...
AND
, 2006
"... Abstract. In this paper we derive non asymptotic deviation bounds for ∣∣ ∣ t ..."
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Abstract. In this paper we derive non asymptotic deviation bounds for ∣∣ ∣ t