Results 1  10
of
116
Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
 J. FUNCT. ANAL
, 1999
"... We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities. ..."
Abstract

Cited by 175 (9 self)
 Add to MetaCart
(Show Context)
We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities.
On Talagrand's Deviation Inequalities For Product Measures
, 1996
"... We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. ..."
Abstract

Cited by 113 (0 self)
 Add to MetaCart
We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M.
Heat kernel estimates for Dirichlet fractional Laplacian
 J. European Math. Soc
"... In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally ..."
Abstract

Cited by 74 (26 self)
 Add to MetaCart
(Show Context)
In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp twosided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric stable process killed upon leaving a C 1,1 open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a nonlocal operator on open sets.
Functional inequalities for empty essential spectrum
 J. Funct. Anal
, 2000
"... In terms of the equivalence of Poincare ́ inequality and the existence of spectral gap, the superPoincare ́ inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corre ..."
Abstract

Cited by 65 (14 self)
 Add to MetaCart
(Show Context)
In terms of the equivalence of Poincare ́ inequality and the existence of spectral gap, the superPoincare ́ inequality is suggested in the paper for the study of essential spectrum. It is proved for symmetric diffusions that, such an inequality is equivalent to empty essential spectrum of the corresponding diffusion operator. This inequality recovers known Sobolev and Nash type ones. It is also equivalent to an isoperimetric inequality provided the curvature of the operator is bounded from below. Some results are also proved for a more general setting including symmetric jump processes. Moreover, estimates of inequality constants are also presented, which lead to a proof of a result on ultracontractivity suggested recently by D. Stroock. Finally, concentration of reference measures for superPoincare ́ inequalities is studied, the resulting estimates extend previous ones for Poincare ́ and logSobolev inequalities.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
(Show Context)
Intrinsic ultracontractivity and conditional gauge for symmetric stable processes on rough domains, preprint
"... It is shown in this paper that the conditional gauge theorem holds for symmetric:stable processes on bounded C1, 1 domains in Rn where 0<:<2 and n2. Two of the major tools used to prove this conditional gauge theorem are logarithmic Sobolev inequality and intrinsic ultracontractivity. 1997 A ..."
Abstract

Cited by 55 (20 self)
 Add to MetaCart
(Show Context)
It is shown in this paper that the conditional gauge theorem holds for symmetric:stable processes on bounded C1, 1 domains in Rn where 0<:<2 and n2. Two of the major tools used to prove this conditional gauge theorem are logarithmic Sobolev inequality and intrinsic ultracontractivity. 1997 Academic Press 1.
Potential theory of special subordinators and subordinate killed stable processes
 J. Theoret. Probab
, 2006
"... In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one ..."
Abstract

Cited by 41 (21 self)
 Add to MetaCart
In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a onetoone correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κfat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D.
Optimal Smoothing and Decay Estimates for Viscously Damped Conservation Laws, with Application to the 2D Navier Stokes Equation
, 1994
"... Optimal bounds on the Lp (IR n) smoothing and decay are established for certain viscously damped conservations laws, of which the vorticity formulation of the Navier–Stokes equation on IR 2 is a basic example. From the smoothing bounds, we obtain pointwise bounds that provide optimal control on th ..."
Abstract

Cited by 36 (1 self)
 Add to MetaCart
Optimal bounds on the Lp (IR n) smoothing and decay are established for certain viscously damped conservations laws, of which the vorticity formulation of the Navier–Stokes equation on IR 2 is a basic example. From the smoothing bounds, we obtain pointwise bounds that provide optimal control on the spatial decay of solutions. We apply this in a study of the physically important case in which the integral of the initial data (i.e., the total vorticity in the example) vanishes. We show in this case that as the time t increases, the L 1 (IR n) norm of the solution decays to zero in two stages: for large initial data, there is a slow decay period during which the L 1 (IR n) norm falls off with an inverse power of the logarithm of t. Then, once the norm has fallen below a critical value, it decays away to zero with t −1/2. Again, this is optimal, and all of the constants in these estimates are explicitly computable in terms of the initial data.
Spectral Gap, Logarithmic Sobolev Constant, and Geometric Bunds
 Surveys in Diff. Geom., Vol. IX, 219–240, Int
, 2004
"... 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 ch ..."
Abstract

Cited by 36 (0 self)
 Add to MetaCart
(Show Context)
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.