Analysis and Geometry of Markov Diffusion Operators by Dominique Bakry Ivan Gentil & Michel Ledoux

Author:Dominique Bakry, Ivan Gentil & Michel Ledoux
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

where p t (x,y) is symmetric since the measure μ is reversible (Definition 1.2.4, p. 14).

By the Chapman-Kolmogorov equation (1.​3.​2), p. 17, for every t>0 and (x,y)∈E×E,

Then, for c(t)>0 to be specified below, and with d the intrinsic distance on (E,μ,Γ) (Sect. 3.​3.​7, p. 166),

By the Cauchy-Schwarz inequality, and symmetry,

Assume now a uniform bound on the kernel p t (x,z)≤K(t) (such bounds will be discussed later in Chaps. 6 and 7). Furthermore, assume that we are in a setting in which the local logarithmic Sobolev inequalities (5.5.5) of Theorem 5.5.2 hold for the measures p t (⋅,z)dμ(z). Therefore, together with the exponential integrability result for the Lipschitz distance function (Proposition 5.4.1), there exists a c(t)>0 such that

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