The entropy method under curvature-dimension conditions in the spirit of Bakry-Émery in the discrete setting of Markov chains

Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

“…[1]) hold true. In the recent work [31], which is strongly related to this article, positive recurrence has been an important assumption. If one allows for the Markov chain being explosive, then the semigroup given by ( 7) is only submarkovian, which ensures still that P t f is bounded provided that f is bounded.…”

“…Regarding positive curvature bounds, based on the identity (5), it has been shown very recently in [31] that the CD Υ pκ, 8q condition serves as a consistent analogue to the classical curvaturedimension condition with regard to the strategy of proving entropy decay of an exponential rate using the entropy method. The resulting functional inequality, the modified logarithmic Sobolev inequality, holds with constant κ ą 0 provided that CD Υ pκ, 8q is satisfied (see [31,Corollary 3.5]).…”

“…Regarding positive curvature bounds, based on the identity (5), it has been shown very recently in [31] that the CD Υ pκ, 8q condition serves as a consistent analogue to the classical curvaturedimension condition with regard to the strategy of proving entropy decay of an exponential rate using the entropy method. The resulting functional inequality, the modified logarithmic Sobolev inequality, holds with constant κ ą 0 provided that CD Υ pκ, 8q is satisfied (see [31,Corollary 3.5]). In the sense of the relation between curvature-dimension inequalities and related functional inequalities, the modified logarithmic Sobolev inequality with regard to the CD Υ pκ, 8q condition in the discrete setting serves as the appropriate counterpart to the logarithmic Sobolev inequality with regard to the CDpκ, 8q condition in the diffusive setting.…”

“…Moreover, we refer to the discussion in [31,Remark 2.9] that shows that the CD Υ condition with finite dimension term is strongly related to the articles [22] and [11] (see also Remark 2.4 below) and in particular implies Li-Yau type inequalities. In this sense, the CD Υ condition serves as a suitable analogue to the Bakry-Émery condition with regard to both, positive curvature in terms of the entropy method and finite dimension in terms of Li-Yau inequalities.…”

“…We assume throughout this paper that at any x P X We recall from [31] the definition of the operators Ψ H and Ψ 2,H , where H : R Ñ R is a continuous mapping. We define (11) Ψ H pf qpxq " ÿ yPX kpx, yqHpf pyq ´f pxqq, x P X, for any f P ℓ 8 pXq and ( 12) B H pf, gqpxq " ÿ yPX kpx, yqHpf pyq ´f pxqqpgpyq ´gpxqq, x P X, for suitable functions f and g. In particular, the conditions ( 9) and ( 10) ensure that we can choose in (12) g " Lf for f P ℓ 8 pXq.…”