## Spectral Gap, Logarithmic Sobolev Constant, and Geometric Bunds (2004)

Venue: | Surveys in Diff. Geom., Vol. IX, 219–240, Int |

Citations: | 14 - 0 self |

### BibTeX

@INPROCEEDINGS{Ledoux04spectralgap,,

author = {M. Ledoux},

title = {Spectral Gap, Logarithmic Sobolev Constant, and Geometric Bunds},

booktitle = {Surveys in Diff. Geom., Vol. IX, 219–240, Int},

year = {2004},

pages = {2195409},

publisher = {Press}

}

### OpenURL

### Abstract

We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We examine similarly dimension free isoperimetric bounds using these parameters.

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Citation Context ...perties of distance functions, it will be convenient to use the language of metric measure spaces and of measure concentration (cf. [Le5]). Let thus (X, d, ) be a metric measure space in the sense of =-=[Grom2]-=-, that is (X, d) is a metric space andsis a finite non-negative Borel measure on (X, d), normalized to be a probability measure ((X) = 1). Define, for # # R, the Laplace functional ofson (X, d) as E (... |

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Citation Context ...centration and exponential integrability properties of distance functions under Poincare or logarithmic Sobolev inequalities going back to the work of M. Gromov and V. Milman [G-M] and I. Herbst (cf. =-=[Le5]-=-). In particular, the approach avoids any type of purely geometric arguments and delicate heat kernel bounds, and produces bounds of the correct 1 order of magnitude in the dimension. The investigatio... |

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Citation Context ... = dv V the normalized Riemannian volume element on (X, g). Let furthermore # 1 = # 1 (X) be the first non-trivial eigenvalue of the Laplacian # g on X. By the Raleigh-Ritz variational principle (cf. =-=[Cha1]-=-, [G-H-L]...), # 1 is characterized by the spectral gap, or Poincare, inequality # 1 # X f 2 d # # X f(-# g f)d = # X |#f | 2 d (2.1) for all smooth real-valued functions f on (X, g) such that # X fd ... |

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Citation Context ...compact. Furthermore, if D is the diameter of X, there exists a numerical constant C > 0 such that D # C # n max ## K # 0 , 1 # # 0 # . It is known from the theory of hypercontractive semigroups (cf. =-=[De-S]-=-) that conversely there exists C(n, K, #) such that # 0 # C(n, K, #) D 15 whenever # 1 # # > 0. Proof. By Theorem 4.1 and Corollary 1.2, 1 - (A r ) # 2 e -#0 r 2 /4 (4.5) for every r > 0 and A # X suc... |

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Citation Context ...2 produces equivalently a lower bound on the Cheeger constant defined as the largest h such thatss (#A) # h min # (A), 1 - (A) # (5.6) for all open subsets A of X with smooth boundary #A. Recall from =-=[Chee]-=- that h # 2 # # 1 . In this form, Theorem 5.2 goes back to the work by P. Buser [Bu]. It is a remarkable fact however that Theorem 5.2 yields constants independent of the dimension of the manifold. Pr... |

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Citation Context ... )) # 1 2 . By noncompactness (and completeness), for every r > 0, we can take z at distance r 0 + 2r from x. In particular, B(x, r 0 ) # B(z, 2(r 0 + r)). By the Riemannian volume comparison theorem =-=[C-E]-=-, [Cha2], for every y # X and 0st, (B(y, t)) (B(y, s)) # # t s # n e t # (n-1)K (3.4) 10 where -K, K # 0, is the lower bound on the Ricci curvature of (X, g). Therefore, # B(z, r) # # # r 2(r 0 + r) #... |

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Citation Context ... - 1)/(p - 1)] 1/2 . The next lemma is a reversed Poincare inequality for heat kernel measures (cf. [Le4]). We use it below as a weak, dimension free, form of the Li-Yau parabolic gradient inequality =-=[L-Y2]-=-. Lemma 5.1. Assume that Ric g # -K, K # 0. Then, for every t # 0 and every smooth function f on (X, g), at every point, c(t) |#P t f | 2 # P t (f 2 ) - (P t f) 2 where c(t) = 1 - e -2Kt K (= 2t if K ... |

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Citation Context ... 2 . By noncompactness (and completeness), for every r > 0, we can take z at distance r 0 + 2r from x. In particular, B(x, r 0 ) # B(z, 2(r 0 + r)). By the Riemannian volume comparison theorem [C-E], =-=[Cha2]-=-, for every y # X and 0st, (B(y, t)) (B(y, s)) # # t s # n e t # (n-1)K (3.4) 10 where -K, K # 0, is the lower bound on the Ricci curvature of (X, g). Therefore, # B(z, r) # # # r 2(r 0 + r) # n e -2(... |

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Citation Context ...The result improve upon [Le1] (see also [Ba-L]). Let (X, g) be a smooth complete connected Riemanian manifold, and let # g be the Laplace operator on (X, g). Let (P t ) t#0 be the heat semigroup (cf. =-=[Da]-=-). It is worthwhile mentioning that, whenever (X, g) is of finite volume V , both the spectral gap # 1 and logarithmic Sobolev constants admit equivalent description in terms of smoothing properties o... |

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Citation Context ...logarithmic Sobolev constants turns out to be of crucial interest in the study of rates of convergence to equilibrium, especially for Markov chains as developed by P. Diaconis and L. Salo#-Coste (cf. =-=[D-SC]-=-, [SC3]). To describe the connection between spectral and logarithmic Sobolev inequalities, and integrability properties of distance functions, it will be convenient to use the language of metric meas... |

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Citation Context ...R n = R 11 n , with equality if and only if X is a sphere (Obata's theorem). This lower bound has been shown to hold similarly for the logarithmic Sobolev constant by D. Bakry and M. Emery [B-E] (cf. =-=[Ba]-=-) so that # 1 # # 0 # R n . (4.3) The case of equality for # 0 is a consequence of Obata's theorem due to an improvement of the preceding by O. Rothaus [Ro2] who showed that when (X, g) is compact and... |

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Citation Context ...claim follows. 3 In these notes, we survey recent developments on spectral and logarithmic Sobolev bounds by the preceding measure concentration tools, mainly taken from the references [SC2], [D-SC], =-=[Le3]-=-. In Section 2, we show how the existence of a spectral gap, or Poincare inequality, implies exponential concentration. This observation turns out to have rather useful consequences to bounds on the d... |

74 |
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Citation Context ...d through the measure concentration and exponential integrability properties of distance functions under Poincare or logarithmic Sobolev inequalities going back to the work of M. Gromov and V. Milman =-=[G-M]-=- and I. Herbst (cf. [Le5]). In particular, the approach avoids any type of purely geometric arguments and delicate heat kernel bounds, and produces bounds of the correct 1 order of magnitude in the di... |

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A Treatise on the Theory of Bessel Functions (Cambridge Univ
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Citation Context ...rst positive zero # n of the Bessel function J n/2 of order n/2 (cf. [Cha1] e.g.). (On a sphere of radius r, there will be a factor r -2 by homogeneity.) In particular, standard methods or references =-=[Wat]-=- show that # n # n as n is large. Denoting by # 0 the logarithmic 17 Sobolev constant on radial functions on B, a simple adaption of the proof of Theorem 4.2 shows that # 0 # Cn for some numerical con... |

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Citation Context ...has non-negative Ricci curvature and if D is the diameter of X, then # 1 # C n D 2 (3.2) where C n > 0 only depends on the dimension n of X. The upper bound (3.2) goes back to the work by S.-Y. Cheng =-=[Chen]-=- in Riemannian geometry (see also [Cha1], [L-Y1] and below). In the opposite direction, it has been shown by P. Li [Li] and H. C. Yang and J. Q. Zhong [Y-Z] that when (X, g) has non-negative Ricci cur... |

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Citation Context ...inequalities in Sections 2 and 3. This will be achieved by the Herbst argument from a logarithmic Sobolev inequality to exponential integrability. It goes back to an unpublished argument by I. Herbst =-=[Da-S]-=-, revived in the past years by S. Aida, T. Masuda and I. Shigekawa [AM -S]. Relevance to measure concentration was emphasized in [Le2], and further developed in [Le3] (cf. [Le5] for the historical dev... |

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Citation Context ... R n where R n = R 11 n , with equality if and only if X is a sphere (Obata's theorem). This lower bound has been shown to hold similarly for the logarithmic Sobolev constant by D. Bakry and M. Emery =-=[B-E]-=- (cf. [Ba]) so that # 1 # # 0 # R n . (4.3) The case of equality for # 0 is a consequence of Obata's theorem due to an improvement of the preceding by O. Rothaus [Ro2] who showed that when (X, g) is c... |

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Citation Context ... diameter of X, then # 1 # C n D 2 (3.2) where C n > 0 only depends on the dimension n of X. The upper bound (3.2) goes back to the work by S.-Y. Cheng [Chen] in Riemannian geometry (see also [Cha1], =-=[L-Y1]-=- and below). In the opposite direction, it has been shown by P. Li [Li] and H. C. Yang and J. Q. Zhong [Y-Z] that when (X, g) has non-negative Ricci curvature, # 1 # # 2 D 2 . (3.3) This lower bound i... |

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Citation Context ...her measure on this example the di#erence between the spectral gap and the logarithmic Sobolev constant as the dimension n is large. (On general functions, # 1 and # 0 are both of the order of n, see =-=[Bo]-=-.) As another application, assume Ric g # R > 0. As we have seen, by the Bakry-Emery inequality [B-E], # 0 # R n where R n = R 11 n . Therefore, by Corollary 4.3, D # C # n - 1 R . Up to the numerical... |

35 | On the estimate of the first eigenvalue of a compact Riemannian manifold - Zhong, Yang - 1984 |

32 |
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Citation Context ...sional bounds are known and classical, we emphasize here dimension free estimates of interest in the study of di#usion operators with drifts (cf. [Ba], [Le4]). The result improve upon [Le1] (see also =-=[Ba-L]-=-). Let (X, g) be a smooth complete connected Riemanian manifold, and let # g be the Laplace operator on (X, g). Let (P t ) t#0 be the heat semigroup (cf. [Da]). It is worthwhile mentioning that, whene... |

32 |
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Citation Context ...n Lemma 1.1, it follows from Theorem 2.1 that, for every sets A, B in X, # 1 # 1 d(A, B) 2 log 2 # C (A)(B) # (2.4) with C = 9. (In the context of Theorem 2.2, replace # 1 by # 1 /2.) As discussed in =-=[Bo-L]-=-, the preceding arguments may easily be improved to reach C = 1 in (2.4). Inequalities such as (2.4) have been considered by F. R. K. Chung, A. Grigory'an and S.-T. Yau [C-G-Y1] who showed (2.4) (with... |

28 |
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Citation Context ...er bounds on the logarithmic Sobolev constant in manifolds with non-negative Ricci curvature, similar to the lower bound (3.3) on the spectral gap, are also available. It has been shown by F.-Y. Wang =-=[Wan]-=- (see also [B-L-Q] and [Le3] for slightly improved quantitative estimates) that, if Ric g # 0, # 0 # # 1 1 + 2D # # 1 . In particular, together with (3.1), # 0 # # 2 (1 + 2#)D 2 . The preceding lower ... |

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Citation Context ...#(#, #) # -1/2 d c (x, y) only yields DQ # n 2 , up to a multiplicative constant. It might be worthwhile observing that in this example, # 0 is of order 1/n log n while it has been shown by B. Maurey =-=[Ma]-=- that concentration (with respect to the combinatoric metric) is satisfied at a rate of the order of 1/n (see below). Consider a N-regular graph with N fixed. Let #(x, y) = 1/N if they are neighbors a... |

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Citation Context ...# 1 # # 0 . Note that, as for # 1 = # 1 (X), one may show (cf. [Gros], [Le3]) that # 0 (X Y ) = min(# 0 (X), # 0 (Y )) for Riemannian manifolds X and Y . It is a non-trivial result, due to O. Rothaus =-=[Ro1]-=-, that whenever X is compact, # 1 # # 0 > 0. (4.2) When the Ricci curvature of (X, g) is uniformly bounded below by a strictly positive constant R, it goes back to A. Lichnerowicz (cf. [Cha1], [G-H-L]... |

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Citation Context ...egular graphs [Al]. In the study of birth and death Markov chains, and especially Poisson point processes, some modified versions of the logarithmic Sobolev inequalities have been recently considered =-=[B-T]-=-. One of them is the entropic inequality that gives rise to the entropic constants# 1 defined as the largest # such that 2# # f log fd # Q(f, log f) when f runs over all finitely supported functions f... |

15 |
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Citation Context ...result goes back independently to M. Gromov and V. Milman [G-M] (in a geometric context) and A. Borovkov and S. Utev [B-U] (in a probabilistic context). (See also [Br].) It has 4 been investigated in =-=[A-M-S]-=- and [A-S] using moment bounds, and in [Sc] using a di#erential inequality on Laplace transforms (similar to the Herbst argument presented in Section 4). We follow here the approach by S. Aida and D. ... |

15 |
A simple analytic proof of an inequality by
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Citation Context ...le sharper dimensional bounds are known and classical, we emphasize here dimension free estimates of interest in the study of di#usion operators with drifts (cf. [Ba], [Le4]). The result improve upon =-=[Le1]-=- (see also [Ba-L]). Let (X, g) be a smooth complete connected Riemanian manifold, and let # g be the Laplace operator on (X, g). Let (P t ) t#0 be the heat semigroup (cf. [Da]). It is worthwhile menti... |

14 |
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Citation Context ...back independently to M. Gromov and V. Milman [G-M] (in a geometric context) and A. Borovkov and S. Utev [B-U] (in a probabilistic context). (See also [Br].) It has 4 been investigated in [A-M-S] and =-=[A-S]-=- using moment bounds, and in [Sc] using a di#erential inequality on Laplace transforms (similar to the Herbst argument presented in Section 4). We follow here the approach by S. Aida and D. Stroock [A... |

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Citation Context ...d functions f : X # R + such that # X fd = 1. The entropic inequality and constant have been considered in [Wu] for Poisson measure on N, and in the present context in the recent contributions [B-T], =-=[G-Q]-=-, [Go]. It is pointed out there that, in general, # 1 # # 1 # # 0 and that # 1 is also suited to control convergence to equilibrium in the total variation distance. In some typical examples, the entro... |

12 |
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Citation Context ...2 d(x) = |#| 2 for all # # R n . By (5.8), the question is thus reduced to the corresponding one for the easier Poincare constant. Discrete versions of Theorems 5.2 and 5.3 are studied in [B-H-T] and =-=[H-T]-=-. 29 While only of logarithmic type with respect to the (power type) isoperimetric comparison theorems of M. Gromov [Grom1] (cf. [Grom2 ]) and [B-B-G], the isoperimetric bound of Theorem 5.3, on the o... |

11 |
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Citation Context ... of Theorems 5.2 and 5.3 are studied in [B-H-T] and [H-T]. 29 While only of logarithmic type with respect to the (power type) isoperimetric comparison theorems of M. Gromov [Grom1] (cf. [Grom2 ]) and =-=[B-B-G]-=-, the isoperimetric bound of Theorem 5.3, on the other hand, involves # 0 rather than the diameter of the manifold, and is independent of the dimension of the manifold (dimension is actually hidden in... |

11 |
1, vertex isoperimetry and concentration
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Citation Context ... R n #x, ## 2 d(x) = |#| 2 for all # # R n . By (5.8), the question is thus reduced to the corresponding one for the easier Poincare constant. Discrete versions of Theorems 5.2 and 5.3 are studied in =-=[B-H-T]-=- and [H-T]. 29 While only of logarithmic type with respect to the (power type) isoperimetric comparison theorems of M. Gromov [Grom1] (cf. [Grom2 ]) and [B-B-G], the isoperimetric bound of Theorem 5.3... |

11 |
A note on the isoperimetric
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Citation Context ... such thatss (#A) # h min # (A), 1 - (A) # (5.6) for all open subsets A of X with smooth boundary #A. Recall from [Chee] that h # 2 # # 1 . In this form, Theorem 5.2 goes back to the work by P. Buser =-=[Bu]-=-. It is a remarkable fact however that Theorem 5.2 yields constants independent of the dimension of the manifold. Proof. We apply (5.5) to smooth functions approximating the characteristic function #A... |

11 |
A lower bound for the first eigenvalues of Laplacian on a compact manifold
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Citation Context ...e dimension n of X. The upper bound (3.2) goes back to the work by S.-Y. Cheng [Chen] in Riemannian geometry (see also [Cha1], [L-Y1] and below). In the opposite direction, it has been shown by P. Li =-=[Li]-=- and H. C. Yang and J. Q. Zhong [Y-Z] that when (X, g) has non-negative Ricci curvature, # 1 # # 2 D 2 . (3.3) This lower bound is optimal since achieved on the one-dimensional torus. Proof of Theorem... |

11 | Lectures on finite Markov chains. Ecole d’Eté de Probabilités de St-Flour - Saloff-Coste - 1996 |

9 |
Modified logarithmic Sobolev inequalities for some models of random walk. Stochastic Process
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Citation Context ...ions f : X # R + such that # X fd = 1. The entropic inequality and constant have been considered in [Wu] for Poisson measure on N, and in the present context in the recent contributions [B-T], [G-Q], =-=[Go]-=-. It is pointed out there that, in general, # 1 # # 1 # # 0 and that # 1 is also suited to control convergence to equilibrium in the total variation distance. In some typical examples, the entropic co... |

9 |
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Citation Context ...um for finite Markov chains. While it is classical that the spectral gap # 1 governs the asymptotic exponential rate of convergence to equilibrium, it has been shown by P. Diaconis and L. Salo#-Coste =-=[SC1]-=-, [SC2], [SC3], [D-SC], both in the continuous and discrete cases actually, that the logarithmic Sobolev constant # 0 is more closely related to convergence to stationarity than # 1 is. Let us now sur... |

8 |
On the spectrum of non-compact manifold with finite volume
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Citation Context ...of the gradient of f . The following result goes back independently to M. Gromov and V. Milman [G-M] (in a geometric context) and A. Borovkov and S. Utev [B-U] (in a probabilistic context). (See also =-=[Br]-=-.) It has 4 been investigated in [A-M-S] and [A-S] using moment bounds, and in [Sc] using a di#erential inequality on Laplace transforms (similar to the Herbst argument presented in Section 4). We fol... |

5 |
On an inequality and a related characterization of the normal distribution
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Citation Context ... 0, where |#f | denotes the Riemannian length of the gradient of f . The following result goes back independently to M. Gromov and V. Milman [G-M] (in a geometric context) and A. Borovkov and S. Utev =-=[B-U]-=- (in a probabilistic context). (See also [Br].) It has 4 been investigated in [A-M-S] and [A-S] using moment bounds, and in [Sc] using a di#erential inequality on Laplace transforms (similar to the He... |

5 |
Convergence to equilibrium and logarithmic Sobolev constant on manifolds with Ricci curvature bounded below
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Citation Context ...2 # 2 /2 . The claim follows. 3 In these notes, we survey recent developments on spectral and logarithmic Sobolev bounds by the preceding measure concentration tools, mainly taken from the references =-=[SC2]-=-, [D-SC], [Le3]. In Section 2, we show how the existence of a spectral gap, or Poincare inequality, implies exponential concentration. This observation turns out to have rather useful consequences to ... |

4 |
Lévy’s isoperimetric inequality
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(Show Context)
Citation Context ...constant. Discrete versions of Theorems 5.2 and 5.3 are studied in [B-H-T] and [H-T]. 29 While only of logarithmic type with respect to the (power type) isoperimetric comparison theorems of M. Gromov =-=[Grom1]-=- (cf. [Grom2 ]) and [B-B-G], the isoperimetric bound of Theorem 5.3, on the other hand, involves # 0 rather than the diameter of the manifold, and is independent of the dimension of the manifold (dime... |

4 |
Hypercontractivity and the Bakry-Emery criterion for compact Lie groups
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(Show Context)
Citation Context ...onstant by D. Bakry and M. Emery [B-E] (cf. [Ba]) so that # 1 # # 0 # R n . (4.3) The case of equality for # 0 is a consequence of Obata's theorem due to an improvement of the preceding by O. Rothaus =-=[Ro2]-=- who showed that when (X, g) is compact and Ric g # R (R # R), # 0 # # n # 1 + (1 - # n )R n (4.4) where # n = 4n/(n + 1) 2 . In particular, # 1 and # 0 are of the same order if (X, g) has non-negativ... |

4 |
Poincar'e type inequalities, and deviation inequalities
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(Show Context)
Citation Context ...d V. Milman [G-M] (in a geometric context) and A. Borovkov and S. Utev [B-U] (in a probabilistic context). (See also [Br].) It has 4 been investigated in [A-M-S] and [A-S] using moment bounds, and in =-=[Sc]-=- using a di#erential inequality on Laplace transforms (similar to the Herbst argument presented in Section 4). We follow here the approach by S. Aida and D. Stroock [A-S]. Theorem 2.1. Let (X, g) be a... |

3 |
Eigenvalues and diameters for manifolds and graphs, Tsing Hua lectures on geometry & analysis
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- 1990
(Show Context)
Citation Context ...). Inequalities such as (2.4) have been considered by F. R. K. Chung, A. Grigory'an and S.-T. Yau [C-G-Y1] who showed (2.4) (with C = 4) using heat kernel expansions, and then using the wave equation =-=[C-G-Y2]-=- (with C = e). They actually establish similar inequalities for the all sequence of eigenvalues, something not considered here. They also establish similar results on graphs. 3. Spectral and diameter ... |

3 |
Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter
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(Show Context)
Citation Context ...egrability. It goes back to an unpublished argument by I. Herbst [Da-S], revived in the past years by S. Aida, T. Masuda and I. Shigekawa [AM -S]. Relevance to measure concentration was emphasized in =-=[Le2]-=-, and further developed in [Le3] (cf. [Le5] for the historical developments) . The principle is similar to the application of spectral properties to concentration presented in Section 2.1, but logarit... |

3 |
The geometry of Markov di#usion generators
- Ledoux
- 2000
(Show Context)
Citation Context ...ithmic Sobolev informations. While sharper dimensional bounds are known and classical, we emphasize here dimension free estimates of interest in the study of di#usion operators with drifts (cf. [Ba], =-=[Le4]-=-). The result improve upon [Le1] (see also [Ba-L]). Let (X, g) be a smooth complete connected Riemanian manifold, and let # g be the Laplace operator on (X, g). Let (P t ) t#0 be the heat semigroup (c... |