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

In the setting of reversible continuous-time Markov chains, the $CD_\Upsilon$ condition has been shown recently to be a consistent analogue to the Bakry-\'Emery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the $CD_\Upsilon(\kappa,F)$ condition, where the dimension term is expressed by a so called $CD$-function $F$. We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropy-information inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.

1. Introduction 1.1. The curvature-dimension condition of Bakry-Émery. The origins of the Γ-calculus of Bakry andÉmery date back to the seminal work [2]. Meanwhile, this theory, for which the monograph [3] is an excellent source, has been proven itself as a beautiful link between probability theory, geometry and analysis.
For motivational purposes we briefly survey the setting of the Bakry-Émery theory in the sequel. Denoting by L the infinitesimal generator of a Markov semigroup, the carré-du-champ operator Γ and the iterated carré-du-champ operator Γ 2 are defined as Ent µ pf 2 q ď n 2 log´1`4 κn Epf q¯, for f being in a sufficiently large class of functions with ş X f 2 dµ " 1 (where Ent µ denotes the Boltzmann entropy and E the Dirichlet energy on L 2 pµq associated with L and µ). The functional inequality (3) is an important instance of what is called an entropy-energy inequality, that is (4) Ent µ pf 2 q ď Φ`Epf q˘, where Φ : p0, 8q Ñ R is a strictly increasing and concave C 1 -function. We refer to [3,Chapter 7], where applications of entropy-energy inequalities, such as ultracontractivity or diameter bounds, have been discussed.

1.2.
Existing approaches for finding substitutes for curvature and dimension in the discrete setting. The issue of finding suitable substitutes of Ricci curvature lower bounds in the discrete setting has been a very vibrant topic of research in the last decade and a half, see e.g. the recent book [24]. Based on the powerful approach of optimal transport, for which we refer to the seminal works [20,29,30,32], Erbar and Maas successfully developed the theory of entropic Ricci curvature in the context of finite Markov chains in [13] and [21]. Another important notion of discrete curvature that relies on ideas from optimal transport is due to Ollivier (see [26]). The latter curvature notion has been studied in a variety of articles concerning the case that the underlying state space is given by a locally finite graph, see e.g. [15,23].
With regard to the Bakry-Émery approach, it is apparently still possible to define the operators Γ and Γ 2 in the discrete setting, where L now denotes the generator of a Markov chain. However, even though some positive results such as eigenvalue estimates in [17] or diameter bounds in [18] can be deduced, the Bakry-Émery condition is not as applicable as in the continuous setting, in particular with regard to Li-Yau inequalities and (modified) logarithmic Sobolev inequalities. This is caused especially by the major difficulty that the diffusion property (2) does not hold in the discrete setting. There are several modified versions of curvature-dimension inequalities in the discrete setting which are based on the approach of identifying certain discrete substitutes for the chain rule, e.g. in the context of Li-Yau inequalities we refer to [4,11,22]. In particular, in [11] the identity (5) Lplog f q " Lf f´Ψ Υ plog f q has been used as the appropriate replacement for the case of H " log in (2). Here the operator Ψ Υ is defined as in (11) below, with H " Υ, where Υ : R Ñ R is given by Υprq " e r´r´1 , r P R. We will comment on regularity assumptions for f ensuring the validity of (5) in the next subsection. One of the key ideas of [11] is to make use of so called CD-functions in order to express the dimension term in their CD condition. This in fact leads to significantly improved estimates regarding the corresponding Li-Yau inequalities, which are even sharp in some instances. We will follow the approach of using CD-functions in this article as well.
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. The main motivation of this paper is to identify the appropriate discrete counterpart to the logarithmic entropy-energy inequality (3) with regard to the CDpκ, nq condition in the diffusive setting, or in other words, to investigate the case where both is satisfied, a positive curvature bound and a finite dimension term.
1.3. Setting and main results. We consider a time-homogeneous, continuous-time Markov chain`Z t˘t ě0 defined on a probability space`Ω, F , P˘and with (finite or infinite) countable state space X. The generator L of the Markov chain is defined on a suitable class of functions f : X Ñ R by (6) Lf pxq " ÿ yPX kpx, yqf pyq " ÿ yPX kpx, yq`f pyq´f pxq˘.
Here, we assume ř yPX kpx, yq " 0, where kpx, yq ě 0 denotes the transition rate for jumping from x to y if x ‰ y. We remark that L determines naturally a graph structure with vertex set X and edge weights given by kpx, yq for x, y P X, x ‰ y, to which we will refer as the underlying graph to L. If kpx, yq P t0, 1u for any x, y P X with x ‰ y, then the underlying graph to L is an unweighted graph. 3 We denote the associated (sub-)Markov semigroup on the space of bounded functions bỳ P t˘t ě0 , which is given by (7) P t f pxq " Epf pZ t q|Z 0 " x˘.
Further, we suppose that the Markov chain is irreducible and that a unique invariant measure µ exists such that the detailed balance condition (8) µptxuqkpx, yq " µptyuqkpy, xq is valid for any x, y P X. Let π : X Ñ p0, 8q denote the density for µ with respect to the counting measure on X, i.e. dµ " πd#. It is a basic consequence of irreducibility and reversibility that πpxq ą 0 for any x P X.
It is well known that the Markov chain is positive recurrent if and only if µ is finite (and hence can be assumed to be a probability measure) and the Markov chain is non-explosive (see e.g. [25]). In particular, provided that the Markov chain is positive recurrent, stochastic completeness (that is P t 1 " 1) and ergodicity (by which we mean what is sometimes called ordinary ergodicity, see e.g. [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. For more details on the general theory of continuous-time Markov chains we refer the reader to [1] and [25].
We denote by R X the space of real-valued functions on X and by ℓ p pµq, 1 ď p ă 8, the elements of R X that are p-summable with respect to µ. Further, ℓ 8 pXq denotes the space of bounded real-valued functions on X. Throughout this article the mapping }¨} p : ℓ p pµq Ñ r0, 8q, 1 ď p ă 8, denotes the ℓ p pµq-norm.
Moreover, we denote by PpXq the set of probability densities with respect to µ, by P˚pXq the set of elements in PpXq that are strictly positive at any x P X and P`pXq :" P˚pXq X ℓ 8,`p Xq, where ℓ 8,`p Xq " tf P ℓ 8 pXq : Dc ą 0 such that f pxq ě c, @x P Xu.
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, 4 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. This guarantees that for f P ℓ 8 pXq and x P X the operator s well defined. In the case of Hprq " 1 2 r 2 , Ψ H pf q coincides with Γpf q and Ψ 2,H pf q with Γ 2 pf q. For our purposes, the mapping Υprq " e r´r´1 , r P R, will play a key role. Indeed, the choice of Hprq " Υprq, which is motivated by the identity (5), leads to the operators Ψ Υ pf q and Ψ 2,Υ pf q, which are the central objects of investigation in the recent article [31]. Let us remark that in our setting the identity (5) holds true for any f P R X such that f, log f P ℓ 1 pkpx,¨qq for any x P X (cf. [31, Lemma 2.2]), which is for instance the case when f P ℓ 8,`p Xq.
We recall a representation formula for the operator Ψ 2,Υ , which has been used frequently in order to study a large class of examples in [31], and reads as The detailed balance condition (8) ensures that the generator of the Dirichlet form given by (14) Epf, gq " 1 2 ÿ xPX ÿ yPX kpx, yq`f pyq´f pxq˘`gpyq´gpxq˘πpxq for f, g being suitable functions, coincides with L given by (6) on bounded functions which are contained in the domain of the form generator, see [16]. We will also denote the ℓ 2 pµq operator by L in the sequel. Further, as the corresponding ℓ 2 pµq-semigroup generated by L is an extension of the semigroup given by (7) restricted to ℓ 8 pXq X ℓ 2 pµq, we will also use the notation pP t q tě0 for the corresponding ℓ 2 pµq-semigroup.
An eminent role will be played by the entropy, being given as and the Fisher information Ipf q " Epf, log f q. We refer to the beginning of Section 3 for more details on some elementary properties of the entropy resp. the Fisher information and on corresponding admissible functions. We say that L satisfies CD Υ pκ, F q if olds on X for any f P ℓ 8 pXq, where κ P R and F 0 : R Ñ r0, 8q is the trivial extension of a CD-function F : r0, 8q Ñ r0, 8q (see Definition 2.1 below), i.e. F 0 prq " 0 if r ă 0. Note that the notation of the condition CD Υ pκ, 8q, which states that Ψ 2,Υ pf q ě κΨ Υ pf q holds on X for any f P ℓ 8 pXq, is a bit missleading since it really means that the dimension term vanishes. This terminology is clearly motivated from the case of the quadratic CD-functions F prq " r 2 n , n P r1, 8q, to which we will refer as the CD Υ pκ, nq condition (motivated by the classical Bakry-Emery notation). Note that CD Υ pκ, nq has already been mentioned in slightly different form in [31,Remark 2.9] and also in [22] in a rather implicit form (cf. [31,Section 9]).
We now describe our main results. Assuming that the CD-function is convex, continuously differentiable and such that the mapping r Þ Ñ F prq r 1`δ is increasing on p0, 8q for some δ ą 0, we will be able to show in Theorem 3.3 that CD Υ pκ, F q (with κ ą 0) implies the bound (15) Ent µ pf q ď t for any f P P˚pXq with Ent µ pf q ă 8 and Ipf q P p0, 8q, where G : p0, 8q Ñ p0, 8q denotes the inverse function of the mapping r Þ Ñ F prq r , r ą 0. In particular, in the case of CD Υ pκ, nq with κ ą 0 and n ă 8, (15) reads as (16) Ent µ pf q ď n 2 log´1`I pf q κn¯, see Corollary 3.7. Consequently, (16) with regard to CD Υ pκ, nq is the natural discrete analogue to (3) with regard to CDpκ, nq in the diffusive setting. Note that in the diffusive setting the chain rule implies Ipf 2 q " 4Epf q for suitable functions, which yields that in the classical situation the inequalities (3) and (16) coincide. In particular, (16) is an important example of what will be called an entropy-information inequality, i.e. a functional inequality of the form Ent µ pf q ď ΦpIpf qq for a strictly increasing and concave C 1`p 0, 8q˘-function Φ, to which we will refer as the growth function.
As the modified logarithmic Sobolev inequality differs from the logarithmic Sobolev inequality in the discrete setting, hypercontractivity of the semigroup, which is equivalent to the latter also in the discrete setting (see [9]), does not characterize the modified logarithmic Sobolev inequality. In [5], Bobkov and Tetali established a hypercontractivity formulation for e Ptf being suitable for the modified logarithmic Sobolev inequality. In this spirit, we show in Theorem 4.2 that certain ultracontractivity bounds for e Ptf hold under corresponding entropy-information inequalities. In particular, in case of the CD Υ pκ, nq condition with κ ą 0 and n ă 8, we will be able to derive that holds for any f P ℓ 8 pXq and any t ą 0. In Theorem 5.4 we prove that µ-integrable 1-Lipschitz functions, by which we mean that }Γpf q} 8 ď 1, are exponentially integrable. Moreover, provided that the growth function satisfies ş 8 0 Φps 2 q s 2 ds ă 8, we show that }f´ż X f dµ} 8 ď ż 8 0 Φps 2 q s 2 ds holds for any 1-Lipschitz function f . This in turn implies a finite diameter bound, which in the special case of CD Υ pκ, nq with κ ą 0 and n ă 8 reads as (17) diam ̺ ď π c n κ , see Corollary 5.5. Interestingly, (17) coincides with the diameter bound that has been obtained by Liu, Münch and Peyerimhoff in [18] for the weaker CDpκ, nq condition but under assumptions on the underlying graph to L which can be expected to be more restrictive, see Remark 5.6. Finally, we also show that a new modified version of the celebrated Nash inequality holds under an entropy-information inequality with logarithmic growth function. In the particular case of the CD Υ pκ, nq condition with κ ą 0 and n ă 8, this modified Nash inequality says that }f } n`2 2 ď´}f } 2 2`I pf 2 q κn¯n 2 }f } 2 1 6 holds for any non-vanishing f P ℓ 2 pµq, see Theorem 6.1 and Corollary 6.2. The article is organized as follows. In Section 2 we introduce the curvature-dimension condition CD Υ pκ, F q and discuss several examples. Next, we define the notion of entropy-information inequalities and investigate their relation to the CD Υ pκ, F q condition in the case of power-type CD-functions. In the remaining part of the paper, we discuss applications of entropy-information inequalities and hence also of the CD Υ pκ, F q condition. We derive ultracontractive bounds for e Ptf in Section 4, exponential integrability of Lipschitz functions and diameter bounds in Section 5 and, finally, a modified version of the Nash inequality in Section 6.

The CD Υ condition with finite dimension and some examples
In this section we generalize the CD Υ pκ, 8q condition from the very recent work [31] by adding a general dimension term involving a CD-function. We first recall the notion of a CD-function that originates from the work of [11] and also has been mentioned in [31,Remark 2.9].
is strictly increasing on p0, 8q and 1 F is integrable at 8. For a given CD-function F , we will call the function F 0 : R Ñ r0, 8q given by F 0 prq " F prq if r ě 0 and F 0 prq " 0 otherwise the trivial extension of F . Remark 2.2. If a function F : r0, 8q Ñ r0, 8q with F p0q " 0 is strictly convex on p0, 8q, the mapping r Þ Ñ F prq r is strictly increasing on p0, 8q, cf. [11,Remark 3.3]. However, it can not be deduced in general that F is a CD-function as, for instance, the function r Þ Ñ Υp´rq, r P r0, 8q, serves as a counterexample.
If we impose instead that F p0q " 0 and r Þ Ñ F prq r 1`δ is increasing on p0, 8q for some δ ą 0, as it will be done in Theorem 3.3 (cf. Remark 3.4), then it follows that F is a CD-function. Indeed, it is obvious that r Þ Ñ F prq r is strictly increasing on p0, 8q and further we have where c ą 0.
Definition 2.3. We say that the Markov generator L satisfies CD Υ pκ, F q at x P X for κ P R and a CD-function F : r0, 8q Ñ r0, 8q with trivial extension F 0 if olds for all f P ℓ 8 pXq. If L satisfies CD Υ pκ, F q at any x P X, then we say that L satisfies CD Υ pκ, F q.
In the special case of F prq " 1 n r 2 , r ě 0, for some n P r1, 8q, we say that L satisfies CD Υ pκ, nq. According to the ambiguity in the notation, we emphasize that throughout this article a capital letter F in the condition CD Υ pκ, F q always refers to a CD-function, while a small letter n in the condition CD Υ pκ, nq refers to the constant of a quadratic CD-function.
Remark 2.4. (i) It must be pointed out that the condition CD Υ pκ, nq has been introduced in [31, Remark 2.9] in seemingly stricter form, as we only require the dimension term in Definition 2.3 for functions f P ℓ 8 pXq with´Lf pxq ą 0. Further, the latter condition is the only difference of CD Υ pκ, nq and the condition CDψpn, κq for the specific choice of ψ " log, which has been introduced in the case of finite and unweighted graphs in [22]. We refer to [31, Section 9] for a detailed account on the relation of the operators appearing in (18) and those in [22]. The condition´Lf pxq ą 0 also appears in the formulation of other curvature-dimension conditions, such as for instance in the case of the exponential curvature-dimension condition of [4] and the CDpF ; 0q condition in [11], where F denotes a CD-function. We refer to [31, Remark 2.9(iii)], which shows that the condition CD Υ p0, F q suffices to deduce Li-Yau inequalities using the results of [11].
(ii) Importantly, the CD Υ pκ, nq condition (or more generally CD Υ pκ, F q with F prq " 1 n r 2 as r Ñ 0`) implies the Bakry-Émery condition CDpκ, nq. This fact relies on the identities holding true for any f P ℓ 8 pXq, which has been shown in the proof of [31, Proposition 2.11]. As it has been pointed out in [31, Remark 2.13], the procedure extends easily to a quadratic dimension term. More accurately, in the formulation of Definition 2.3, it first implies CDpκ, nq only for f P ℓ 8 pXq with´Lf pxq ą 0, but then extends to any f P ℓ 8 pXq by linearity of L, bilinearity of Γ and the definition of Γ 2 . The fact that the Bakry-Émery condition is necessary for CD Υ with a quadratic CD-function also motivates to study the former condition. In particular, we refer to [28], where CDp0, nq has been studied for a large class of operators with long range jumps and state space Z.
(iii) The property (19) has an important consequence for CD-functions that behave like a power-type function near the origin. Indeed, if CD Υ pκ, F q holds with κ P R and F prq " r 1`δ as r Ñ 0`for some δ ą 0, then we infer from (19) that for any f P ℓ 8 pXq with´Lf pxq ą 0, Consequently, δ ě 1 must hold, or in other words the best behavior of a CD-function near the origin one can hope for is quadratic.
Several concrete examples have been considered in [31,Section 5] to study the CD Υ pκ, 8q condition. It has turned out that the functions (20) ν c,d prq " cΥ 1 prqr`Υp´rq´dΥprq, r P R, with constants c, d P R, are of eminent importance. We refer the reader to the Appendix of [31], where basic properties of these functions have been collected. As a warm-up, we begin with the basic example of the two-point space, for which we provide another property of the functions given by (20) in the Appendix below.
Example 2.5. We consider the two-point space X " t0, 1u with kp0, 1q " a and kp1, 0q " b, where a, b ą 0. Here, the invariant and reversible probability measure µ is given by dµ " πd# with πp0q " b a`b and πp1q " a a`b . We writex " 1´x for x P X and t " f pxq´f pxq. From [31] we know that CD Υ pκ, 8q holds true for some κ ą 0. In order to show that the CD Υ condition is fulfilled with a positive curvature constant and a finite dimension term it hence suffices to show CD Υ p0, F q, where F is a CD-function. Thus it remains to compare Ψ 2,Υ pf qpxq with F p´Lf pxqq " F p´kpx,xqtq, for F being specified below. We have Note that for c, d P R and η ą 0 we have that ν c`η,d`η prq ě ν c,d prq at any r P R, since Υ 1 prqr ě Υprq, r P R, holds by convexity. Hence, we can estimate Note that ν 1`λ,λ is strictly convex by Lemma A.1. Then, we infer from Remark 2.2 and the asymptotic behavior of ν 1`λ,λ that the mapping F : r0, 8q Ñ r0, 8q defined as is a CD-function and we deduce that CD Υ p0, F q holds true.
We continue with the following basic observation.
Proof. For fixed x P X, we observe by Jensen's inequality from which the first claim follows by (21). Clearly, γ is increasing on r0, 8q, which implies the second claim.
Note that the tempting naive approach to deduce a CD Υ condition with non-negative curvature bound and finite dimension term from CD Υ pκ, 8q (with κ ą 0) alone by using Proposition 2.6 does not work. Indeed, the mapping r Þ Ñ Υp´rq can not play the role of the function γ from Proposition 2.6 since it only grows linearly at 8 and is hence not a CD-function. This is a difference to the Bakry-Émery condition in our setting, where the analogous result to Proposition 2.6 yields at least that CDpκ, 8q(with κ ą 0) at x P X implies CDpλκ, p1´λqnq at x P X, where λ P r0, 1s and n ă 8 depends on κ and x. For our purposes however, we need a refined analysis.
In light of the subsequent sections, power-type CD-functions are of particular importance. We note that for any δ ě 1 there exists some optimal c δ ą 0 such that the estimate (22) Υprq`Υp´rq ě c δ |r| 1`δ holds true for any r P R, which follows from the asymptotic and monotonic behavior of the mapping r Þ Ñ Υprq`Υp´rq, r P R, and the fact that Υprq`Υp´rq " r 2 as r Ñ 0. For instance, it can be easily checked that the optimal constant for δ " 1 in (22) is given by c 1 " 1. We illustrate the practical use of (22) in the following example.
Example 2.7. Let X be an arbitrary countable set with at least two elements and l : X Ñ p0, 8q being integrable on X with respect to the counting measure. Further, we set kpx, yq " lpyq for all x, y P X, x ‰ y. Then µ given by dµ " πd# with πpxq " lpxq, x P X, is an invariant and reversible measure. In [31,Example 5.2] it has been shown that L satisfies CD Υ p a 2|l| 1 l˚, 8q, where l˚" inf xPX lpxq and |l| 1 denotes the ℓ 1 -norm with respect to the counting measure on X. Clearly, the integrability of l implies that l˚" 0 if X is infinite. It has also been shown that CD Υ p0, 8q is best possible (concerning the curvature term) in the infinite state space case. Therefore, we will only consider the case of X being finite in the sequel, i.e. l˚ą 0 holds.
Example 2.8. We choose lpxq " 1 in the setting of Example 2.7 for any x P X with given finite state space X consisting of n elements, n ě 2. Then the underlying graph to L is given by the complete graph K n and L satisfies CD Υ p a 2np1´2αq, n α q for any α P p0, 1 2 q by Example 2.7 in the case of δ " 1 (recall that we have c 1 " 1 in (22)). It is natural to ask whether there also exists a dimension bound which is uniform in n while having non-negative curvature as it is the case for the Bakry-Émery condition (see e.g. [15]). Interestingly, in Example 2.14 we are able to give a negative answer to this question.
The procedure described in Example 2.7 can be seen as a guidance for other examples where the mapping ν c,d plays a similar role, e.g. for weighted 4-cycles, finite birth-death processes and weighted stars as discussed in [31].
The case of (R)-Ricci-flat graphs will be discussed seperatly below. For the reader's convenience we recall the definition of (R)-Ricci flat graphs, which originates from the work of [7].
pxq denotes the closed ball with radius 1 and center x with respect to the combinatorical graph distance) for 1 ď i ď d satisfying the following properties: (i) η i puq P B 1 puqztuu for any u P B 1 pxq and i P t1, ..., du, (R)-Ricci-flat graphs constitute a subclass of Ricci-flat graphs with Bakry-Émery condition CDp2, 8q, see [7]. Important examples are given by complete bipartite graphs and, since (R)-Ricci-flat graphs are invariant under tensorization, by the hypercube, cf. [7]. In [31,Example 5.12] it has been shown that a Markov generator with underlying graph being (R)-Ricci-flat even satisfies CD Υ p2, 8q.
Example 2.10. Let the transition rates be given such that the underlying graph to L is a (dregular) pRq-Ricci-flat graph with vertex set X. Further, let x P X be chosen arbitrary and let η i˘i "1,...,d denote the corresponding mappings from Definition 2.9. In [31, Example 5.12] it has been shown that for f P R X the estimate holds true. Now, one readily checks (see also the proof of [31,Lemma A.3]) that ν 2,5 is strictly convex on R. We infer from Proposition 2.6 that L satisfies CD Υ p2, F q with F prq " d 2 ν 2,5 p´r d q, r ě 0, which is a CD-function due to the asymptotic behavior at 8 and by Remark 2.2. Interestingly, since the curvature constant in CD Υ p2, 8q is optimal in general, which follows, for instance, from the case where the underlying graph to L is given by the hypercube (cf. [31]), we do not need a trade off from the curvature constant in order to achieve the CD Υ p2, F q condition (which is, for instance, in contrast to the procedure described in Example 2.7). Further, note that ν 2,5 behaves only quartic near 0. In fact, there does not exist in general a CD-functionF behaving quadratically near zero such that L satisfies CD Υ p2,F q, by combining Remark 2.4(ii) with the fact that the hypercube does not satisfy CDp2, mq for some m ă 8 (see [8]).
Example 2.11. Here we consider a birth-death process with infinite state space X " N 0 . We use the notation originating from [6], which has also been used in [31], and introduce the functions a, b : X Ñ r0, 8q with apxq " kpx, x`1q, bpxq " kpx, x´1q, bp0q " 0, bpxq ą 0 otherwise, and apxq ą 0 for any x P X. Moreover, we set kpx, yq " 0 whenever |x´y| ą 1. The detailed balance condition now reads as (23) apxqπpxq " bpx`1qπpx`1q for any x P X. Note that the measure µ given by dµ " πd# is a finite measure if and only if We assume monotonicity of the rates in the sense that apxq ď apx`1q and bpx`1q ě bpxq for any x P X and moreover that (24) apxq´apx`1q`bpx`1q´bpxq ě κ holds for any x P X and some κ ą 0. Those assumptions led to modified logarithmic Sobolev inequalities in [6], and in [31] it has been shown that they imply CDp κ 2 , 8q. Apparently they also entail that apxq ď ap0q for any x P X and bpxq Ñ 8 as x Ñ 8.
We specify for x P X a function f x P R X such that f x px`2q " f x px`1q`t and f x px´2q " f x px´1q`s, where we define t " f x px`1q´f x pxq and s " f x px´1q´f x pxq (this is called minimizing Ψ 2,Υ pf q over the second neighborhood throughout [31]) and set t " 0. Then we observe from (13) (see also the representation formula for Ψ 2,Υ pf qpxq that has been established in [31, Example 5.13]) 2Ψ 2,Υ pf x qpxq " bpxq´Υpsq`bpx´1q´bpxq´apxq˘`Υp´sqapx´1q`Υ 1 psqs`apx´1q`bpxq´bpx´1q˘¯.
Assuming that 2Ψ 2,Υ pf x qpxq is greater than or equal to 1 n bpxq 2 s 2 (which equals 1 n p´Lf x pxqq 2 ) for some n P r1, 8q implies that hoosing s ă 0 such that Υ 1 psqs´Υpsq´1 n s 2 ă 0 and sending x Ñ 8 yields a contradiction. Interestingly, in [31] it has been shown that under an assumption which is stronger than (24) the CD Υ pκ 0 , 8q condition holds with some positive constant κ 0 ą 0. This shows that it is possible to have positive curvature bounds while having no finite dimension bound regarding the CD Υ pκ, nq condition for some κ ą 0. Furthermore, note that we will show by means of Corollary 3.10 below that a CD Υ condition with positive curvature bound and a non-quadratic power-type CD-function does not hold either.
Next, we give a quite simple negative criterion for the existence of a dimension term with regard to the quadratic CD-function in the infinite state space case.
Proposition 2.12. If there exists a sequence px m q mPN Ă X such that (25) N px m q pM 1 px m qq 2 Ñ 0 as m Ñ 8, then there does not exist some n ă 8 such that CD Υ p0, nq holds true.
Clearly, Proposition 2.12 also yields a necessary condition for families of Markov generators satisfying a uniform CD Υ p0, nq condition. For the sake of clarity we will state this in the following corollary in the case that the underlying graphs to the corresponding Markov generators are unweighted, in which case the mappings N and M 1 are equal. The following corollary follows from the same arguments as in the proof of Proposition 2.12. Despite its simplicity, these findings lead to a remarkable difference between the CD Υ p0, nq and the CDp0, nq condition, as it will be demonstrated by Example 2.14.
Corollary 2.13. Let I be an arbitrary index set and pL i q iPI a family of Markov generators whose respective underlying graphs are unweighted and with corresponding state space pX i q iPI . Assume that there exist sequences pi m q mPN Ă I and px m q mPN Ă X, where X " Ť iPI X i , such that M pimq 1 px m q Ñ 8 as m Ñ 8. Here the upper index denotes that the function M 1 corresponds to the respective Markov generator. Then there exists no n ă 8 such that L i satisfies CD Υ p0, nq for all i P I.
In particular, if the underlying graph to a Markov generator L is given by an unweighted graph with unbounded vertex degree, then there exists no n ă 8 such that CD Υ p0, nq holds true.
Example 2.14. We consider the index set I " tn P N : n ě 2u (in the sense of Corollary 2.13) and the Markov generator L n whose underlying graph is given by the complete graph K n for any n P I. It is known on the one hand that CDp0, 4q holds for any L n (see [15,Proposition 3]), i.e. a dimension-term exists under non-negative curvature with respect to the Bakry-Émery CD-condition that is uniform with regard to n. On the other hand, due to Corollary 2.13 there does not exist a (uniform) d ă 8 such that CD Υ p0, dq holds for any L n , n P I.

Entropy-Information inequalities
From now on we assume that the unique and reversible invariant measure µ is a probability measure.
We consider the entropy for any positive function f P ℓ 1 pµq. It is well known that Ent µ pf q ě 0. Note that we also allow for the value of Ent µ pf q " 8. The Fisher information is given by kpx, yq`f pyq´f pxq˘`log f pyq´log f pxq˘πpxq.
If we assume that M 1 P ℓ 1 pµq then f P ℓ 8,`p Xq ensures that Ipf q ă 8. Further, in the latter case we have the representation see [31,Section 3], where the formula has been established for f P ℓ 8,`p Xq being a probability densitiy with respect to µ, although the proof extends verbatim to the general case. Since we sum up in the right-hand side of (28) over non-negative entries, we can extend the functional I to positive functions f : X Ñ p0, 8q, where we allow for the value of Ipf q " 8.
Note that the assumption of the Markov chain being irreducible implies that Ipf q=0 if and only if f is constant and positive.
Besides that, one readily verifies that the well known scaling behavior (29) Ent µ pcf q " c Ent µ pf q

and (30)
Ipcf q " c Ipf q holds respectively for any constant c ą 0.
Our main object of investigation in the remaining part of this article will be the following family of functional inequalities. Definition 3.1. We say that L satisfies an entropy-information inequality EIpΦq with respect to a strictly increasing and concave C 1 -function Φ : p0, 8q Ñ p0, 8q, which we refer to as the growth function, if for every f P P˚pXq with Ent µ pf q ă 8 and Ipf q ă 8 (31) Ent µ pf q ď ΦpIpf qq holds, where we set Φp0q :" lim rÑ0`Φ prq.
A well known example for an entropy-information inequality is the modified logarithmic Sobolev inequality (32) Ent µ pf q ď 1 2κ Ipf q with constant κ ą 0. See [5] for an extensive account on modified logarithmic Sobolev inequalities in the discrete setting of Markov chains. Further, the functional inequality (32) was subject of investigation in [6], [12], [13] and [14], as well as in [31] where it has been shown that CD Υ pκ, 8q (with κ ą 0) together with positive recurrence and the integrability conditions M 1 P ℓ 2 pµq and M 2 P ℓ 1 pµq imply (32) with constant κ.

Remark 3.2. (i)
The diffusive counterpart to Definition 3.1, so called entropy-energy inequalities, are defined for growth functions mapping to R instead of p0, 8q, see [3]. This generality in the context of [3] allows to include the quite important special case of the Euclidean logarithmic Sobolev inequality (cf. [3, Proposition 6.2.5]). However, assuming that Φ is non-negative is not a restriction in our setting where we have supposed that µ is a probability measure. Indeed, applying (31) to f " 1 shows that lim rÑ0`Φ prq ă 0 is impossible to hold.
(iii) Let f P ℓ 1 pµq be positive with Ent µ pf q ă 8 and Ipf q ă 8. The entropy-information inequality in the form (33) extends to f by (34) Ent µ pf q ď Φ 1 prqIpf q`Θprq ż X f dµ for any r P p0, 8q. This is a consequence of the scaling behavior (29) and (30), after having applied (33) to f }f }1 . Now, we come to the main theorem of this section, that links the previous section to the notion of entropy-information inequalities. Theorem 3.3. Let M 1 P ℓ 2 pµq, M 2 P ℓ 1 pµq and the Markov chain generated by L be positive recurrent. Further, let L satisfy CD Υ pκ, F q with κ ą 0 and a convex CD-function F : r0, 8q Ñ r0, 8q such that F | p0,8q P C 1`p 0, 8q˘and F 1 prqr F prq ě 1`δ holds for any r ą 0 and some δ ą 0. Let G : p0, 8q Ñ p0, 8q denote the inverse function of r Þ Ñ F prq r , r ą 0. Then t holds for any f P P˚pXq with Ent µ pf q ă 8 and Ipf q P p0, 8q.
Proof. It suffices to deduce the claim for f P P`pXq. The full statement follows then from the same standard truncation argument as presented in [31, Lemma 3.2] and the dominated convergence theorem for approximating the right-hand side of (35). For f P P`pXq we set Λptq " Ent µ pP t f q, t ě 0. It is well known that (36) Λ 1 ptq "´IpP t f q is valid provided that M 1 P ℓ 1 pµq. Further, we infer from [31, Theorem 3.4] that (37) Λ 2 ptq " 2 ż X P t f Ψ 2,Υ plog P t f qdµ holds true given the assumptions M 1 P ℓ 2 pµq and M 2 P ℓ 1 pµq. We apply CD Υ pκ, F q to deduce where the latter follows from convexity of the trivial extension F 0 (which follows from convexity of F ), the fact that P t f is a probability density with respect to µ, which follows from µ being invariant for pP t q tě0 , and Jensen's inequality. Now, by the identity (5) and hence we end up with the differential inequality (38) Λ 2 ptq ě´2κΛ 1 ptq`2F p´Λ 1 ptqq.
Note that in fact Λ 1 ptq ă 0 holds, since we have f " 1 otherwise. Further, we observe that where we have applied the condition´Λ 1 ptqF 1 p´Λ 1 ptqq F p´Λ 1 ptqq ě 1`δ and Λ 2 ptq ě 0 in the last step.
Hence, (38) yields that the mapping t Þ Ñ e´2 δκt`1´κ Λ 1 ptq F p´Λ 1 ptqq˘i s increasing. In particular, this implies that which can be rearranged to Then, using Ipf q "´Λ 1 p0q, we obtaiń Consequently, we conclude The claim follows by sending T Ñ 8. Indeed, ΛpT q Ñ 0 as T Ñ 8 follows from the dominated convergence theorem, the Markov chain being ergodic and pP t q tě0 being a Markov semigroup.

Remark 3.4. (i) The crucial assumption that
(40) F 1 prq ě p1`δqF prq r , holds for some δ ą 0 and any r ą 0 implies by Gronwall's inequality that we have for fixed a ą 0 for any r ą a ą 0, i.e. the mapping r Þ Ñ F prq r 1`δ , r ą 0, is increasing. Conversely, differentiating r Þ Ñ F prq r 1`δ , r ą 0, the property (40) follows provided that r Þ Ñ F prq r 1`δ , r ą 0, is increasing. Hence, both properties are equivalent. In particular, we observe that (40) ensures that F grows at least like r 1`δ as r Ñ 8. On the other hand, recall that we have seen in Remark 2.4(iii) that F can not behave better than quadratic at 0 provided that CD Υ pκ, F q holds for some κ P R.
(ii) Note that the mapping G : p0, 8q Ñ p0, 8q is in fact well defined, since assuming that r Þ Ñ F prq r 1`δ is increasing on p0, 8q for some δ ą 0 implies that F prq r Ñ 0 as r Ñ 0 and F prq r Ñ 8 as r Ñ 8.
(iii) We interpret the integral on the right-hand side of (35) as 8 in the case that the integral is divergent. In fact, this situation appears even under the assumptions of Theorem 3.3 as the CD-function F prq " r m e´1 r m , r ě 0, for some m ą 1 shows. Indeed, one readily verifies that (40) with δ " m´1 and convexity of F respectively hold true. The problem results from the behavior of F in the origin. More precisely, F prq r converges faster to 0 than e´1 r m as r Ñ 0`, which yields that Gprq tends slower to 0 than plog 1 r q´1 m as r Ñ 0`. Consequently the integrand in the right-hand side of (35) dominates a behavior of t´1 m as t Ñ 8, which yields that the integral does not converge.
For general CD-functions the integral on the right-hand side of (35) can not be calculated explicitly and, moreover, it is not clear whether the mapping r Þ Ñ ş 8 0 G`κ e 2κt p1`κ r F prq q´1˘d t is concave. We will focus in the sequel on the situation where the CD-function is given by some power-type function, in which case the mapping G : p0, 8q Ñ p0, 8q of Theorem 3.3 can be given explicitly. In fact, the following result shows in particular that the functional inequality (35) is compatible with Definition 3.1 for power-type CD-functions.
Proposition 3.5. Let M 1 P ℓ 2 pµq, M 2 P ℓ 1 pµq and the Markov chain generated by L be positive recurrent. Further, let L satisfy CD Υ pκ, F q with κ ą 0 and F prq " r δ`1 n , r ě 0, for some δ ě 1 and n P p0, 8q. Then L satisfies EIpΦq with the growth function Moreover, the growth function Φ satisfies the following assertions: Φps 2 q s 2 ds ă 8. Proof. Clearly, F is convex and F 1 prqr " p1`δqF prq holds for any r ą 0. The mapping G : p0, 8q Ñ p0, 8q from Theorem 3.3 is given by Gprq " δ ? nr, r ą 0. Now, it follows from elementary substitution that By means of Definition 3.1 and Theorem 3.3 it suffices to prove that Φ is concave in order to deduce that L satisfies EIpΦq (note that there is nothing to show for the case of Ipf q " 0). To that aim, we differentiate Φ and observe which implies concavity of Φ.
The growth function Φ is bounded if and only if the integral in the right-hand side of (41) converges as r Ñ 8. The latter property holds true if and only if the integral ş 1 δ ? κn r v δ´2 dv converges as r Ñ 8, which happens to be true if and only if δ ą 1.
Regarding the remaining assertion, we note that there is nothing to show for the behavior at 8 by boundedness of Φ in case of δ ą 1 and by the explicit formula for the growth function in the special case of δ " 1, which will be deduced in Corollary 3.7 below. As to the behavior at 0, we observe for ε ą 0 that κn .
In the case of the CD Υ pκ, nq condition, Theorem 3.3 yields the following important entropyinformation inequality.
(43) Ent µ pf q ď n 2 log´1`I pf q κnh olds for any f P P˚pXq with Ent µ pf q ă 8.
As we already highlighted in the introduction, we emphasize that (43) serves as a natural discrete analogue to the logarithmic entropy-energy inequality (3), which plays an important role in the diffusive setting of [3] (in which case it holds true provided that CDpκ, nq is valid).
In the following example we consider one of the most important instances of a birth-death process from Example 2.11. Example 3.8. As a special case of a birth-death process from Example 2.11 (with the notation taken from there), we consider the Poisson case which is given by the choice apxq " λ, where λ ą 0 is called the intensity rate, and bpxq " x, both for any x P N 0 . The invariant and reversible measure is given by the denisity π λ pxq " λ x x! e´λ, x P N 0 . In [31,Example 5.13] it has been shown that there does not exists some κ ą 0 such that L satisfies CD Υ pκ, 8q. However, it is known that the Poisson case of the birth-death prosess satisfies the modified logarithmic Sobolev inequality EIpΦq with Φprq " r, see e.g. [6]. Here we show, that this is the best possible entropy-information inequality for the Poisson case in the sense that if Φ grows slower than linear as r Ñ 8, then L fails to satisfy EIpΦq.
To that aim we repeat an argument that has been used in [6] to show sharpness of the corresponding modified logarithmic Sobolev inequality. Indeed, we consider f k pxq " e kx e λpe k´1 q , which can be readily checked to be an element of P`pXq for any k P N. We have Further, it can be easily checked that the detailed balance condition (23) yields that From this, we can see that Ent µ pf k q Ipf k q Ñ 1, as k Ñ 8.
Consequently, if Φ grows slower than linear at 8, EIpΦq fails for the Poisson case of a birthdeath process. In particular, the Poisson case of a birth-death process not only does not satisfy the CD Υ pκ, nq condition (cf. Example 2.11), but also fails on the level of the corresponding entropy-information inequality.
In a somewhat similar fashion, the next result shows quite remarkable consequences of boundedness of the growth function and of the mapping M 1 , respectively. Theorem 3.9. Let L satisfy EIpΦq, then the following assertions hold true.
(i) If Φ is bounded, then the state space X is finite and the estimate holds true for any positive f P R X . (ii) If M 1,sup ă 8 and the state space X is infinite, then Φ grows linearly as r Ñ 8.
Proof. We consider for x P X and some ε P p0, 1q the function One readily verifies that f x P P˚pXq for any x P X. We have for any x P X. Moreover, we observe that where we have applied the detailed balance condition in the last step. Hence, we conclude for any x P X that (46) 19 After this preliminary work we now show the first assertion. The estimate (44) follows from the definition of EIpΦq for any f P P˚pXq with Ent µ pf q ă 8 and Ipf q ă 8. We then extend (44) to the more general case of positive f P ℓ 1 pµq with Ent µ pf q ă 8 and Ipf q ă 8 by applying (44) to f }f }1 . It remains to show that X is finite. Assuming for contradiction that X is infinite, we find a sequence px m q mPN Ă X such that πpx m q Ñ 0 as m Ñ 8, since µ is assumed to be a probability measure. We infer from (45) that Ent µ pf xm q Ñ 8 as m Ñ 8, which contradicts what has been shown before.
Let us now turn to the second assertion. Choosing a sequence px m q mPN Ă X as above and assuming w.l.o.g. that πpx m q ă 1 for any m P N, we read from (45) that Ent µ pf xm q ě log 1 πpx m q`l ogp1´εq`ε log ε and from (46) that Thus, we have Ent µ pf xm q Ipf xm q ě log 1 πpxmq`l ogp1´εq`ε log ε M 1,sup p1´εq`log 1 πpxmq´l og ε˘. The right hand side of the latter estimate converges to 1 p1´εqM1,sup as m Ñ 8. This yields that there exists a constant C ą 0 and some M P N such that Ent µ pf xm q Ipf xm q ě C ą 0 for any m ě M . Consequently, it follows from EIpΦq that C Ipf xm q ď ΦpIpf xm qq for all m ě M . Since Ipf xm q Ñ 8 as m Ñ 8 and Φ is concave, we conclude that Φprq must grow linearly as r Ñ 8.
While the corresponding growth function in the case of the CD Υ pκ, nq condition (with κ ą 0 and n ă 8) grows logarithmically at 8, we have seen in Proposition 3.5 that for power type CD-functions of higher order the respective growth function is bounded. Combining Proposition 3.5 with Theorem 3.9 leads to the following interesting observation. Corollary 3.10. Let M 1 P ℓ 2 pµq, M 2 P ℓ 1 pµq and the Markov chain generated by L be positive recurrent. Further, let L satisfy CD Υ pκ, F q with κ ą 0 and F prq " r 1`δ n , r ě 0, for some δ ě 1 and n P p0, 8q. Then the state space X is finite if and only if either δ ą 1 or δ " 1 and M 1,sup ă 8.

Ultracontractive Bounds under Entropy-Information inequalities
In the classical diffusive setting, entropy-energy inequalities imply under the condition that r Þ Ñ Φ 1 prq r is integrable at 8 ultracontractivity of the semigroup, cf. [3]. But as the entropyinformation inequality compares to entropy-energy inequalities like the modified logarithmic Sobolev inequality to logarithmic Sobolev inequalities, it is natural to expect that ultracontractive bounds come in the form of the hypercontractivity bounds from [5] for the modified logarithmic Sobolev inequality, i.e. not with respect to the respective norm of the semigroup, but of e Ptf instead. We recall the following auxiliary result, whose proof is contained in the proof of [5,Theorem 7.1], where the authors have considered an even more general setting. Note in fact that the assumptions (1)-(4) of [5,Section 7] are satisfied provided that f P ℓ 8 pXq. Moreover, f P ℓ 8 pXq implies that we have e ηptqPtf P ℓ 8,`p Xq for any fixed t ą 0 (with ηptq P R), which yields that Ipe ηptqPtf q ă 8 if we assume in addition that M 1 P ℓ 1 pµq.
Lemma 4.1. Let M 1 P ℓ 1 pµq, f P ℓ 8 pXq, t ą 0 and q : p0, C 0 q Ñ p0, 8q be some differentiable mapping, where C 0 P p0, 8s. Then we have Proof. We briefly repeat the calculation of [5] for the reader's convenience and refer for more details to [5,Section 7]. Note that the assumption of M 1 P ℓ 1 pµq in fact justifies to interchange integration and differentation in the lines below. We have Theorem 4.2. Let L satisfy EIpΦq and M 1 P ℓ 1 pµq. Then for every 1 ď p ď q ď 8, every f P ℓ 8 pXq and every ̺ ą 0 }e P tp̺q f } q ď }e f } p e mp̺q holds true, where Here the case of q " 8 has to be understood in the limit q Ñ 8 in both formulas in (47) and can be reached only if r Þ Ñ Φ 1 prq r is integrable at 8.
Proof. We define Λptq " }e Ptf } qptq for a strictly increasing and differentiable q : p0, C 0 q Ñ p0, 8q, which, together with C 0 , will be specified below. By Lemma 4.1 we have Applying EIpΦq in the form of (34) to e qptqPtf yields for any r ą 0 For given r " rpqq (which will be made precise below) we choose qptq such that the differential equation q 1 Φ 1 prpqqq " q is satisfied, which in fact can be done by separation of variables. Indeed, let T : pp, 8q Ñ p0, C 0 q be defined as T psq " dq (the value C 0 " 8 is allowed). Then T is bijective and q : p0, C 0 q Ñ pp, 8q, given by qptq " T´1ptq, solves the ODE mentioned above on p0, C 0 q. In particular, qp0q " p extends q continuously onto r0, C 0 q. We conclude that Λ 1 ptq ď q 1 ptq qptq 2 ΘprpqptqqqΛptq 21 holds for any t P p0, C 0 q, which is equivalent to the differential inequality Integrating (48) yields Θprpqqq q 2 dq.
Note that we can write t " Choosing rpqq " ̺q establishes the formula for tp̺q. Moreover, recalling that Θpsq " ΦpsqΦ 1 psqs, s P p0, 8q, we deduce from a simple application of integration by parts that which yields the claim.
We see from Theorem 4.2 that q " 8 can be reached provided that Φ 1 prq r is integrable at 8. In particular, in case of the modified logarithmic Sobolev inequality, Theorem 4.2 does not lead to ultracontractive bounds, which is consistent to the role of the logarithmic Sobolev inequality in the diffusive setting. Otherwise, for a growth function Φ that behaves as r α with 0 ă α ă 1 at 8 we have that Φ 1 prq r is integrable at 8. In this sense, the modified logarithmic Sobolev inequality constitutes an extreme case.
The growth function resulting from CD Υ pκ, F q, with F being a power-type CD-function from Proposition 3.5, satisfies the integrability condition that we have mentioned throughout the previous lines. This fact can be seen from the identity (42). We close this section with an application of Theorem 4.2 in this particular context. Corollary 4.3. Let M 1 P ℓ 2 pµq, M 2 P ℓ 1 pµq and the Markov chain generated by L be positive recurrent. Further, let L satisfy CD Υ pκ, F q with κ ą 0 and F prq " r 1`δ n , r ě 0, for some δ ě 1 and n P p0, 8q. Then we have for any t ą 0 that holds for any f P ℓ 8 pXq, where Φ denotes the growth function given by (41). In particular, in case of the CD Υ pκ, nq condition, (49) reads as Proof. Due to Proposition 3.5, L satisfies EIpΦq with growth function given by (41). We choose p " 1 and q " 8 (in the limit sense) in (47), recall the formula (42) for the derivative of the growth function and observe for ̺ ą 0 tp̺q " n 2 From this we infer by monotonicity of the growth function that mp̺q " Φp̺q ď Φ´δ c n 2δtp̺q¯.
Consequently, by Theorem 4.2 we get that ? n 2δtp̺q˘} e f } 1 holds for any ̺ ą 0. But as the mapping ̺ Þ Ñ tp̺q, ̺ ą 0, is bijective onto p0, 8q since it is decreasing with tp̺q Ñ 0 as ̺ Ñ 8 and tp̺q Ñ 8 as ̺ Ñ 0, (49) follows. The special case of (50) now can be established by the explicit formula for the growth function in the case of δ " 1, see Corollary 3.7.

Exponential Integrability of Lipschitz functions and Diameter bounds
Exponential integrability of Lipschitz functions and diameter bounds (see the definitions below) are both important properties to investigate in the classical theory of [3]. In order to reach finite diameter bounds in the diffusive setting, Poincaré inequalities resp. logarithmic Sobolev inequalities are not sufficient. Instead, Sobolev inequalities resp. certain entropy-energy inequalities ensure the validity of a finite diameter. Speaking on the level of CD-inequalities this means that in the diffusive setting positive curvature and finite dimension suffices to deduce finite diameter bounds, while positive curvature alone does not. In this section we will be able to show that the CD Υ condition behaves consistently in the discrete setting of Markov chains. Now, we recall the definitions of Lipschitz functions and the diameter.
Definition 5.1. A function f P R X is called Lipschitz function if Γpf qpxq exists at any x P X (in the sense that the sum in (11) with Hprq " r 2 2 is finite) and }f } Lip :" a }Γpf q} 8 ă 8. Moreover, we say that f is C-Lipschitz, where C ą 0, when }f } Lip ď C holds true.
These definitions have been used in the diffusive situation of [3], but also in the discrete setting in [18], where diam ̺ has been called the resistance diameter. Definition 5.2 is further closely related to the diameter with respect to the combinatorical graph distance on the underlying graph to L. In fact, in [18] it has been shown that the estimate holds true for any s P R.
which contradicts (56) in the asymptotic behavior of t Ñ 8. Simultaneously, by considering the asymptotic behavior as t Ñ´8, one obtains that f pxq´ş X f dµ ě´C holds for all x P X. This establishes (53) for any 1-Lipschitz function. From that we conclude for any x, y P X and 1-Lipschitz function f . But (57) implies that f must be bounded. Hence we deduce by scaling that any Lipschitz function is bounded. Furthermore, by (57) and the definition of ̺, we deduce (54).
Theorem 5.4 yields finite bounds on diam ̺ if L satisfies CD Υ pκ, F q with κ ą 0 and a powertype CD-function by means of Proposition 3.5(ii). In the special case of the quadratic CDfunction, we get the following bound. The claim follows by combining Corollary 3.7 with Theorem 5.4.
Remark 5.6. By (58) we recover (by different methods) exactly the same diameter bound as in [18], where there it is assumed on the one hand only CDpκ, nq but on the other hand that the underlying graph to L is locally finite and satisfies the completeness assumption and nondegeneracy of the vertex measure. Note that by [19, Theorem 2.2] the latter boils down to the case of finite graphs since κ ą 0. Hence, although the curvature-dimension condition of Corollary 5.5 is more restrictive, the setting where it applies can be expected to be more general compared to the one of [18].
With the following example we aim to emphasize that the property that Lipschitz functions are bounded is quite strong in the sense that it fails for a large class of examples that all satisfy corresponding modified logarithmic Sobolev inequalities. In particular, it turns out that CD Υ pκ, 8q, with κ ą 0, is not sufficient for deducing a finite diameter bound.
Example 5.7. We consider a birth-death process on N 0 as in Example 2.11 and employ the notation that has been used therein. In particular, we assume that the rate functions a and b are monotone as in Example 2.11 and that condition (24) holds for some κ ą 0. We set f p0q " 0 and define the sequence of partial sums (59) f pnq " n ÿ k"1 1 a bpkq , n P N.
We claim that Lipschitz functions are bounded if and only if the partial sums given by (59) converge. 26 First, we assume that the partial sums given by (59) diverge as n Ñ 8. Then, we have 2Γpf qpnq " apnqpf pn`1q´f pnqq 2`b pnqpf pn´1q´f pnqq 2 " apnq bpn`1q`1 for any n P N. From the monotonicity assumption on the rates we infer that Γpf q is bounded, or in other words that f is a Lipschitz function. But apparently, f is unbounded. This yields that the corresponding generator does not satisfy an entropy-information inequality with growth function Φ such that ş 8 0 Φps 2 q s 2 ds ă 8. On the other hand, it is known by [6] that L satisfies a corresponding modified logarithmic Sobolev inequality. Moreover, we emphasize that among those birth-death processes of the present example are also processes that even satisfy the condition CD Υ pκ, 8q, see [31]. This shows that the condition CD Υ pκ, 8q with κ ą 0 is in general not sufficient to obtain a finite diameter. The latter finding is consistent to the Bakry-Émery condition in the diffusive setting. Now, let us assume conversely that the partial sums given by (59) converge and let g be C-Lipschitz for some C ą 0. In particular, 2bpnq`gpn´1q´gpnq˘2 ď C 2 holds for any n P N. Consequently, we have |gpn´1q´gpnq| ď C ? 2bpnq , n P N, and by the triangle inequality we deduce |gpN q´gp0q| ď C ? 2 f pN q ď C ? 2 ÿ nPN 1 a bpnq for any N P N, which yields that g is bounded.

Modified Nash inequalities
In the diffusive setting, logarithmic entropy-energy inequalities are known to be equivalent to Nash inequalities. Regarding the discrete setting of Markov chains we refer to [10], and also [27], for an extensive account on Nash inequalities. Clearly, it can not be expected that logarithmic entropy-information inequalities are linked to the classical Nash inequality in the discrete setting as they are in the diffusive setting by the lack of chain rule. We refer to the natural analogue as the modified Nash inequality, which can be induced from corresponding logarithmic entropyinformation inequality as will be shown subsequently. We say that a function f P R X is nonvanishing if f pxq ‰ 0 for any x P X. Theorem 6.1. If L satisfies EIpΦq with Φprq " α log`A`r β˘a nd α, β ą 0, A ě 1, then the following modified Nash inequality holds for any non-vanishing f P ℓ 2 pµq.
Proof. Clearly, we can assume that Ipf 2 q ă 8. Further, it suffices to prove (60) for bounded non-vanishing functions by a standard truncation argument. Indeed, let pf N q N PN denote the sequence of bounded functions that has been considered in the proof of Theorem 5.4. Then it follows readily by means of the monotone convergence theorem that }f N } 2 Ñ }f } 2 , }f N } 1 Ñ }f } 1 and Ipf 2 N q Ñ Ipf 2 q as N Ñ 8. It is a well known consequence of Hölder's inequality that the mapping r Þ Ñ }f } 1 r , r P p0, 1s is log-convex. Then, for a given non-vanishing f P ℓ 8 pXq with }f } 2 " 1, we consider the convex mapping Λprq " log }f } 1 r , r P p0, 1s, which is well defined since µ is a probability measure. One readily verifies by a similar calculation as in the proof of Lemma 4.1 that where we use that |f | is bounded in order to interchange differentation and integration. In particular, using }f } 2 " 1, we observe that Λ 1`1 2˘"´E nt µ pf 2 q. By convexity, we thus have 2`Λp1q´Λ`1 2˘˘ě Λ 1`1 2˘. Consequently, by the entropy-information inequality EIpΦq and the fact that Λ`1 2˘" 0, we observe log 1 }f } 2 1 ď Ent µ pf 2 q ď log´A`I pf 2 q β¯α , which implies (61) 1 ď`A`I pf 2 q β˘α }f } 2 1 .
Now, for the non-normalized case we apply (61) to f }f }2 . By the scaling behavior of the Fisher information (cf. (30)), we deduce from which the claim follows.
Combining Corollary 3.7 with Theorem 6.1, we observe the following result.

Appendix A. Auxiliary Lemma
In this section we provide an auxiliary result, which has been used to investigate the example of the two-point space in Example 2.5. For further properties of the functions ν c,d : R Ñ R, given by ν c,d prq " cΥ 1 prqr`Υp´rq´dΥprq, c, d P R, which also have been used throughout Section 2, we refer to the Appendix of [31].
In particular, this implies that ν 3 1`λ,λ prq ă 0 and hence we have r˚ąr by monotonicity, which establishes the claim.