Volume doubling, Poincaré inequality and Guassian heat kernel estimate for nonnegative curvature graphs
aa r X i v : . [ m a t h . DG ] D ec Volume doubling, Poincar´e inequality and Gaussianheat kernel estimate for non-negatively curved graphs
Paul Horn, Yong Lin ∗ , Shuang Liu, Shing-Tung Yau † Abstract
By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positivesolutions of the heat equation on graphs, under the assumption of the curvature-dimensioninequality
CDE ′ ( n, The Li-Yau inequality is a very powerful tool for studying positive solutions to the heatequation on manifolds. In its simplest case, it states that a positive solution u (that is apositive u satisfying ∂ t u = ∆ u ) on a compact n -dimensional manifold with non-negativecurvature satisfies |∇ u | u − ∂ t uu ≤ n t . (1.1)Beyond its utility in the study of Riemannian manifolds, variants of the Li-Yau inequalityhave proven to be an important tool in non-Riemannian settings as well. Recently, in[BHLLMY13], the authors have proved a discrete version of Li-Yau inequality on graphs.The discrete setting provided myriad challenges, with many of these stemming from the lackof a chain rule for the Laplacian in a graph setting. Overcoming this, involved introducing anew notion of curvature for graphs, and exploited crucially the fact that a chain rule formulafor the Laplacian does hold in a few isolated case, along with a discrete version of maximum ∗ Supported by the National Natural Science Foundation of China(Grant No11271011). † Supported by the NSF DMS-0804454. u ( x, s ) ≤ C ( x, y, s, t ) u ( y, t ) , (1.2)where C ( x, y, s, t ) depends only on the distance of ( x, s ) and ( y, t ) in space-time. The Li-Yauinequality, and more generally of parabolic Harnack inequalities like (1.2) can also be used toderive further heat kernel estimates. In this direction, one of the most important estimatesto achieve are the following Gaussian type bounds: c l m ( y ) V ( x, √ t ) e − C l d ( x,y )2 t ≤ p ( t, x, y ) ≤ C r m ( y ) V ( x, √ t ) e − c r d ( x,y )2 t , (1.3)where p ( t, x, y ) is a fundamental solution of the heat equation (heat kernel). The Li-Yauinequality can be used to prove exactly such bounds for the heat kernel on non-negativelycurved manifolds. Thus, via the Li-Yau inequality it can be shown that non-negativelycurved manifolds satisfy a strong form of the Harnack inequality (1.2), along with a Gaussianestimate (1.3). It also is known, by combining the Bishop-Gromov comparison theorem [Bi63]and the work of Buser [Bu82] that non-negatively curved manifolds also satisfy the volumegrowth condition known as volume doubling and the Poincar´e inequality (see also the paperof Grigor’yan, [G92]).In manifold setting, Grigor’yan [G92] and Saloff-Coste [SC95] independently gave a com-plete characterization of manifolds satisfying (1.2). They showed that satisfying a volumedoubling property along with Poincar´e inequalities is actually equivalent to satisfying theHarnack inequality (1.2), and is also equivalent to satisfying the Gaussian estimate (1.3).Thus, in the manifold setting the three conditions discussed above that are implied bynon-negative curvature are actually all equivalent. Curvature still plays an important rolehowever, as a local property that certifies that a manifold satisfies the three (equivalent)global properties.In the case of graphs, Delmotte [D99] proved a characterization analogous to that ofSaloff-Coste for both continuous and discrete time. Until now, however, no known notion ofcurvature on graphs has been sufficient to imply that a graph satisfies these three conditions.This is not to say that the question of whether some sort of curvature lower bound impliesstrong geometric properties in a non-Riemannian setting. On metric measure spaces, forinstance, under some curvature lower bound assumptions, Sturm [S06], Rajala [R12], Erbar,Kuwada and Sturm [EKS13] and Jiang, Li and Zhang [JLZ14] studied the volume doublingproperty along with Poincar´e inequalities and Gaussian heat kernel estimates.Despite the successes of [BHLLMY13] in establishing a discrete analogue of the Li-Yauinequality, their ultimate result also had some limitations. Most notably, the results of2BHLLMY13] were unable insufficient to derive the equivalent conditions of volume doublingand Poincar´e inequalities, along with Gaussian heat kernel bounds, and the strongest formof a Harnack inequality. This failure arose from the generalization of (1.1) achieved whenconsidering only a positive solution inside a ball of radius R : in the classical case an extraterm of the form ‘ R ’ occurred, but in the graph case in general the authors were only able toprove a result with an extra term of the factor ‘ R ’. This difference resulted in only being ableto establish weaker bounds on the heat kernel, and polynomial volume growth as opposed tothe stronger condition of volume doubling. Ultimately one of the reasons for these weakerimplications was the methods used: [BHLLMY13] used maximum principle arguments, andultimately ran into problems when cutoff functions were needed.In this paper, we develop a way to apply semigroup techniques in the discrete setting inorder to study the heat kernel of graphs with non-negative Ricci curvature. From here, weobtain a family of global gradient estimates for bounded and positive solutions to the heatequation on an infinite graph. The curvature notion used, as in [BHLLMY13], is a mod-ification of the so-called curvature dimension inequality. Satisfying a curvature dimensioninequality has proven to be an important generalization of having a Ricci curvature lowerbound in the non-Riemannian setting (see, eg. [BE83, BL06]). The utility of satisfying thestandard such inequality is much lower when the Laplace operator does not satisfy the chainrule, such as in the graph case. This led to the modification used in this paper (and in[BHLLMY13]) the so-called exponential curvature dimension inequalities . A more detaileddescription of the curvature notion used in this paper, and the motivation behind it, is givenin Section 2.2. It is, however, important to note that in the Riemannian case (and more gen-erally when the Laplacian generates a diffusive semigroup) the classical curvature dimensioninequality, and the exponential curvature dimension inequalities are equivalent.From our new methods, we show that non-negatively curved graphs (in the sense ofthe exponential curvature dimension inequalities) satisfy volume doubling. This improvesthe results of [BHLLMY13], in which only polynomial volume growth is derived. Thisimprovement is the key point in proving the discrete-time Gaussian lower and upper estimatesof heat kernel, and from this the Poincar´e inequality and Harnack inequality on graphs. Asan important technical point, we do not simply establish volume doubling and the Poincar´einequality and then apply the results of Delmotte [D99] to establish the other (equivalentconditions). Instead, after proving volume doubling we attack the Gaussian bounds directly– using volume doubling along with additional information from our methods to establish theGaussian bounds. Once Gaussian bounds are established, however, we can use the resultsof Delmotte to ‘complete the circle,’ and establish the remaining desired properties. Weemphasize that although a number of notions of curvature for graphs have been introduced(see, eg,[LY10, BHLLMY13]) no previous notion has been shown to imply these conditions –and in fact, [BHLLMY13] was the first paper to show that a non-negative curvature conditionfor graphs implied polynomial volume growth.We further derive continuous-time Gaussian lower estimate of heat kernel. It proves tobe impossible, however, to prove continuous-time Gaussian upper bounds on the heat kernel,however, as from the paper of Davies [DB93] and Pang [P93] the continuous-time Gaussianupper estimate is not true on graphs. 3hile we prove this for any non-negatively curved graph, it is important to note thatGaussian estimates for the heat kernel for Cayley graphs of a finite generated group ofpolynomial growth were proved by Hebisch and Saloff-Coste in [HS93]. For non-uniformtransition case, Strook and Zheng proved related Gaussian estimates on lattices in [SZ97].Establishing that a graph satisfies both volume doubling and the Poincar´e inequalityhas important consequences. For example, under these assumptions on graphs, Delmotte in[D97] proved that the dimension of the space of Harmonic function on graphs with polyno-mial growth is finite. This extends the similar result on Riemannian manifolds by Coldingand Minicozzi in [CM96], see also Li in [Li97]. The original problem came from a conjectureof Yau ([Yau86]) which stated that these space should have finite dimension in Rieman-nian manifolds with non-negative Ricci curvature. Thus, our result answers the analogueconjecture of Yau for graphs in the affirmative.Finally, under the assumption of a graph being positively curved (again, with respect tothe exponential curvature dimension inequality), we derive a Bonnet-Myers type theoremthat the diameter of graphs in terms of the canonical distance is finite. We accomplish thisby proving some logarithmic Sobolev inequalities. Here we establish that certain diameterbounds of Bakry still hold, even though the Laplacian on graphs does not satisfy the diffusionproperty that Bakry used. Under the same assumption, we also can prove that the diameterof graphs in terms of graph distance is finite by proving the finiteness of measure, plus thedoubling property of volume.The paper is organized as follows: We introduce our notation and formally state our mainresults in Section 2. In Section 3, we prove our main variational inequality. This inequalityleads to a different proof of the Li-Yau gradient estimates on graphs from the one given in[BHLLMY13]. From this main inequality we establish an additional exponential integrabilityresult, and ultimately, volume doubling in Section 4. From volume doubling, we can provethe Gaussian heat kernel estimate, parabolic Harnack inequality and Poincar´e inequality inSection 5. Finally, in Section 6, we prove a Bonnet-Myers type theorem on graphs. Acknowledgments
We thank Bobo Hua, Matthias Keller and Gabor Lippner for usefuldiscussion. We also thank Daniel Lenz for many nice comments on the paper. Part of thework of this paper was done when P. Horn and Y. Lin visited S.-T. Yau in The NationalCenter for Theoretical Sciences in Taiwan University in May 2014 and when P. Horn visitedY. Lin in Renmin University of China in June 2014. We acknowledge the support fromNCTS and Renmin University.
In this section we develop the preliminaries needed to state our main results. Through thepaper, we let G = ( V, E ) be a finite or infinite connected graph. We allow the edges on thegraph to be weighted. Weights are given by a function ω : V × V → [0 , ∞ ), the edge xy from x to y has weight ω xy >
0. In this paper, we assume this weight function is symmetric(that is, ω xy = ω yx ). Furthermore, we assume that ω min = inf e ∈ E : ω e > ω e > .
4e furthermore allow loops, so it is permissible for x ∼ x (and hence ω xx > m ( x ) := X y ∼ x ω xy < ∞ , ∀ x ∈ V. For our work, especially in the context of deriving Gaussian heat kernel bounds, oneadditional technical assumption is needed. This is essentially needed to compare the con-tinuous time and discrete time heat kernels. In order for this to go smoothly two thingsneed to happen: no edge can be too ‘small’ (this is essentially the content of our assumption ω min > x ∼ x –this prevents ‘parity problems’ of bipartiteness that would make the continuous and discretetime kernels incomparable. This condition is neatly captured in the following ∆( α ) used byDelmotte in [D99], but has also been used previously by other authors. Definition 2.1.
Let α > G satisfies ∆( α ) if,1) x ∼ x for every x ∈ V , and2) If x, y ∈ V , and x ∼ y , ω xy ≥ αm ( x ) . As a remark, if a loop is on every edge and sup x m ( x ) < ∞ , then the condition ω min > ω min / sup x m ( x )) . In general, this is a rather mildcondition. It is easy to check, for instance, that adding loops does not decrease the curvaturefor our curvature condition (see Section 2.2 below) nor change many the geometric quantitieswe seek to understand (eg. volume growth, and diameter). Thus even graphs without loopsmay safely be altered to satisfy this condition.
Let µ : V → R + be a positive measure on the vertices of the G . We denote by V R the spaceof real functions on V . and we denote by ℓ p ( V, µ ) = { f ∈ V R : P x ∈ V µ ( x ) | f ( x ) | p < ∞} ,for any 1 ≤ p < ∞ , the space of ℓ p integrable functions on V with respect to the measure µ . For p = ∞ , let ℓ ∞ ( V, µ ) = { f ∈ V R : sup x ∈ V | f ( x ) | < ∞} be the set of boundedfunctions. For any f, g ∈ ℓ ( V, µ ), we let h f, g i = P x ∈ V µ ( x ) f ( x ) g ( x ) denote the standardinner product. This makes ℓ ( V, µ ) a Hilbert space. As is usual, we can define the ℓ p normof f ∈ ℓ p ( V, µ ) , ≤ p ≤ ∞ : k f k p = X x ∈ V µ ( x ) | f ( x ) | p ! p , ≤ p < ∞ and k f k ∞ = sup x ∈ V | f ( x ) | . We define the µ − Laplacian ∆ : V R → V R on G by, for any x ∈ V ,∆ f ( x ) = 1 µ ( x ) X y ∼ x ω xy ( f ( y ) − f ( x )) . ] X y ∼ x h ( y ) = 1 µ ( x ) X y ∼ x ω xy h ( y ) ∀ x ∈ V. Under our locally finite assumption, it is clear that for any bounded f , ∆ f is likewisebounded. We treat the case of µ Laplacians quite generally, but the two most natural choicesare the case where µ ( x ) = m ( x ) for all x ∈ V , which is the normalized graph Laplacian,and the case µ ≡ D µ := max x ∈ V m ( x ) µ ( x ) < ∞ . It is easy to check that D µ < ∞ is equivalent to the Laplace operator ∆ being bounded on ℓ ( V, µ ) (see also [HKLW12]). The graph is endowed with its natural graph metric d ( x, y ),i.e. the smallest number of edges of a path between two vertices x and y . We define balls B ( x, r ) = { y ∈ V : d ( x, y ) ≤ r } , and the volume of a subset A of V , V ( A ) = P x ∈ A µ ( x ).We will write V ( x, r ) for V ( B ( x, r )). In order to study curvature of non-Riemannian spaces, it is important to have a good defini-tion that allows one to capture the important consequences. One way to do this is throughthe so-called curvature-dimension inequality or CD-inequality. An immediate consequenceof the well-known Bochner identity is that on any n -dimensional manifold with curvaturebounded below by K , any smooth f : M → R satisfies:12 ∆ |∇ f | ≥ h∇ f, ∇ ∆ f i + 1 n (∆ f ) + K |∇ f | . (2.1)It was an important insight by Bakry and Emery [BE83] that one can use 2.1 as a substitutefor a lower Ricci curvature bound on spaces where a direct generalization of Ricci curvatureis not available. Since all known proofs of the Li-Yau gradient estimate exploit non-negativecurvature condition through the CD -inequality, Bakry and Ledoux [BL06] succeeded to useit to generalize (1.1) to Markov operators on general measure spaces when the operatorsatisfies a chain rule type formula.To formally introduce this notion for graphs, we first introduce some notation. Definition 2.2.
The gradient form Γ, associated with a µ -Laplacian is defined by2Γ( f, g )( x ) = (∆( f · g ) − f · ∆( g ) − ∆( f ) · g )( x )= ] X y ∼ x ( f ( y ) − f ( x ))( g ( y ) − g ( x )) . We write Γ( f ) = Γ( f, f ). 6imilarly, Definition 2.3.
The iterated gradient form Γ is defined by2Γ ( f, g ) = ∆Γ( f, g ) − Γ( f, ∆ g ) − Γ(∆ f, g ) . We write Γ ( f ) = Γ ( f, f ). Definition 2.4.
The graph G satisfies the CD inequality CD ( n, K ) if, for any function f and at every vertex x ∈ V ( G ) Γ ( f ) ≥ n (∆ f ) + K Γ( f ) . (2.2)On graphs – where the Laplace operator fails to satisfy the chain rule – satisfying the CD ( n,
0) inequality seems insufficient to prove a generalization of (1.1). None the less,in [BHLLMY13] the authors prove a discrete analogue of the Li-Yau inequality. The curva-ture notion they use is a modification of the standard curvature notion, which they call theexponential curvature dimension inequality. In reality, the authors of [BHLLMY13] intro-duce two slightly different curvature conditions, which they call
CDE and
CDE ′ , both ofwhich we recall below. Definition 2.5.
We say that a graph G satisfies the exponential curvature dimension in-equality CDE ( x, n, K ) if for any positive function f : V → R + such that ∆ f ( x ) <
0, wehave f Γ ( f )( x ) = Γ ( f )( x ) − Γ (cid:18) f, Γ( f ) f (cid:19) ( x ) ≥ n (∆ f )( x ) + K Γ( f )( x ) . (2.3)We say that CDE ( n, K ) is satisfied if CDE ( x, n, K ) is satisfied for all x ∈ V . Definition 2.6.
We say that a graph G satisfies the CDE ′ ( x, n, K ), if for any positivefunction f : V → R + , we have f Γ ( f )( x ) ≥ n f ( x ) (∆ log f ) ( x ) + K Γ( f )( x ) . (2.4)We say that CDE ′ ( n, K ) is satisfied if CDE ′ ( x, n, K ) is satisfied for all x ∈ V .The reason these are known as the exponential curvature dimension inequalities is illus-trated in Lemma 3.15 in [BHLLMY13], which states the following: Proposition 2.1.
If the semigroup generated by ∆ is a diffusion semigroup (e.g. the Lapla-cian on a manifold), then CD ( n, K ) and CDE ′ ( n, K ) are equivalent. To show that
CDE ′ ( n, K ) ⇒ CD ( n, K ) one takes an arbitrary function f , and applies(2.4) to exp( f ) to verify that (2.2) holds. Likewise, to verify that CD ( n, K ) ⇒ CDE ( n, K )one takes an arbitrary positive function f , and applies (2.2) to log( f ) to verify (2.4). Thisequivalence, however, makes strong use of the chain rule, and hence the fact that ∆ generatesa diffusion semigroup.The relation between CDE ′ ( n, K ) and CDE ( n, K ) is the following:7 emark 1. CDE ′ ( n, K ) implies CDE ( n, K ). Proof.
Let f : V → R + be a positive function for which ∆ f ( x ) <
0. Since log s ≤ s − s , we can write∆ log f ( x ) = ] X y ∼ x (log f ( y ) − log f ( x )) = ] X y ∼ x log f ( y ) f ( x ) ≤ ] X y ∼ x f ( y ) − f ( x ) f ( x ) = ∆ f ( x ) f ( x ) < . Hence squaring everything reverses the above inequality and we get( △ f ( x )) ≤ f ( x ) ( △ log f ( x )) , and thus CDE ( x, n, K ) is satisfied f Γ ( f )( x ) ≥ n f ( x ) ( △ log f ) ( x ) + K Γ( f )( x ) > n (∆ f )( x ) + K Γ( f )( x ) . In [BHLLMY13], the
CDE ( n, K ) inequality is preferred: the ∆ log( f ) term occurring inthe CDE ′ inequality is awkward in the discrete case, the CDE ( n, K ) inequality is weakerin general, and the CDE ( n, K ) inequality sufficed for proving the Li-Yau inequality.None the less, as the results in this paper will show, for the purposes of applying semigrouparguments the CDE ′ ( n, K ) inequality is to be preferred. The primary reason for this is thefact that CDE ′ ( n, K ) implies a non trivial lower bound on e Γ ( f ) for a positive function f at every point on a graph, as opposed to just the points where ∆ f <
0. For maximum principlearguments, restricting to points where ∆ f <
CDE ′ ( n, K ) appears to be more useful.We note that, in general, the conditions CDE ′ and CDE better capture the spirit ofa Ricci curvature lower bound than the classical CD condition. For instance, every graphsatisfies CD (2 , −
1) – that is, there is an absolute lower bound to the curvature of of graphs.On the other hand, a k -regular tree satisfies CDE (2 , − d/
2) and this negative curvature is(asymptotically) sharp. Thus with the exponential curvature condition, negative curvatureis unbounded. This is unique amongst graph curvature notions.Moreover, [BHLLMY13] showed that lattices, and more generally Ricci-flat graphs inthe sense of Chung and Yau [CY96] which include the abelian Cayley graphs, have non-negative curvature
CDE ( n,
0) and
CDE ′ ( n, CDE ′ ( n, K ) hasproduct property(see also similar result for CD ( n, K ) in [LP14]). So we can construct lot ofgraphs satisfy the CDE ′ ( n,
0) assumption with different dimension n by taking the Cartesianproduct of graphs which satisfying the CDE ′ ( n, The first main result, alluded to in the introduction, is that satisfying
CDE ′ ( n,
0) is sufficientto imply that a graph satisfies several important conditions: volume doubling, the Poincar´einequality, Gaussian bounds for the heat kernel, and the continuous-time Harnack inequality.For preciseness, we state these conditions now:8 efinition 2.7. ( DV ) A graph G satisfies the volume doubling property DV ( C ) for constant C > x ∈ V and all r > V ( x, r ) ≤ CV ( x, r ) . ( P ) A graph G satisfies the Poincar´e inequality P ( C ) for a constant C > X x ∈ B ( x ,r ) m ( x ) | f ( x ) − f B | ≤ Cr X x,y ∈ B ( x , r ) ω xy ( f ( y ) − f ( x )) , for all f ∈ V R , for all x ∈ V , and for all r ∈ R + , where f B = 1 V ( x , r ) X x ∈ B ( x ,r ) m ( x ) f ( x ) . ( H ) Fix η ∈ (0 ,
1) and 0 < θ < θ < θ < θ and C > G satisfies the continuous-timeHarnack inequality H ( η, θ , θ , θ , θ , C ), if for all x ∈ V and s, R ∈ R + , and everypositive solution u ( t, x ) to the heat equation on Q = [ s, s + θ R ] × B ( x , R ), we havesup Q − u ( t, x ) ≤ C inf Q + u ( t, x ) , where Q − = [ s + θ R , s + θ R ] × B ( x , ηR ), and Q + = [ s + θ R , s + θ R ] × B ( x , ηR ).( H ) Fix η ∈ (0 ,
1) and 0 < θ < θ < θ < θ and C > G satisfies the discrete-timeHarnack inequality H ( η, θ , θ , θ , θ , C ), if for all x ∈ V and s, R ∈ R + , and everypositive solution u ( x, t ) to the heat equation on Q = ([ s, s + θ R ] ∩ Z ) × B ( x , R ), wehave ( n − , x − ) ∈ Q − , ( n + , x + ) ∈ Q + , d ( x − , x + ) ≤ n + − n − implies u ( n − , x − ) ≤ Cu ( n + , x + ) , where Q − = ([ s + θ R , s + θ R ] ∩ Z ) × B ( x , ηR ), and Q + = ([ s + θ R , s + θ R ] ∩ Z ) × B ( x , ηR ).( G ) Fix positive constants c l , C l , C r , c r >
0. The graph G satisfies the Gaussian estimate G ( c l , C l , C r , c r ) if, whenever d ( x, y ) ≤ n , c l m ( y ) V ( x, √ n ) e − C l d ( x,y )2 n ≤ p n ( x, y ) ≤ C r m ( y ) V ( x, √ n ) e − c r d ( x,y )2 n . The following theorem is the first of the main results of this paper.
Theorem 2.2 (cf. Theorem 5.5) . If the graph satisfies
CDE ′ ( n , and ∆( α ) , we have thefollowing four properties. There exists C , C , α > such that DV ( C ) , P ( C ) , and ∆( α ) are true. There exists c l , C l , C r , c r > such that G ( c l , C l , C r , c r ) is true. There exists C H such that H ( η, θ , θ , θ , θ , C H ) is true. ′ There exists C H such that H ( η, θ , θ , θ , θ , C H ) is true. A function u on G is called harmonic function if ∆ u = 0. A harmonic function u on G has polynomial growth if there is positive number d such that ∃ x ∈ V, ∃ C > , ∀ x ∈ V x , | u ( x ) |≤ Cd ( x , x ) d . Combining Theorem 2.2 and Delmotte’s Theorem 3.2 from [D97], we obtain the followingresult which confirms the analogue of Yau’s conjecture ([Yau86]) on graphs.
Theorem 2.3.
If the graph satisfies
CDE ′ ( n , and ∆( α ) , then the dimension of space ofharmonic functions on G has polynomial growth is finite. Our final main result is the following Bonnet-Myers theorems for graphs. We defer thedefinition of canonical distance of graph until Section 6.
Theorem 2.4 (cf. Theorem 6.8 and Theorem 6.10) . Let G = ( V, E ) be a locally finite,connected graph satisfying CDE ′ ( n, K ) , and K > , then the diameter e D of graph G interms of the canonical distance satisfies the inequality e D ≤ √ π r nK , and in particular is finite. Furthermore the diameter D of graph G in terms of the graphdistance is also finite, and satisfies D ≤ π r D µ nK . In this section we establish our main variational inequality which we develop in order toapply semigroup theoretic arguments in the non-diffusive graph case. This is the content ofSection 3.2. Among the immediate applications of this variation inequality are a family ofLi-Yau type inequalities which we derive in Section 3.3.10 .1 The heat kernel on graphs
A function u : [0 , ∞ ) × V → R is a positive solution to the heat equation on G = ( V, E )if u > u satisfies the differential equation∆ u = ∂ t u, at every x ∈ V .In this paper we are primarily interested in the heat kernel , that is the fundamentalsolutions p t ( x, y ) of the heat equation. These are defined so that for any bounded initialcondition u : V → R , the function u ( t, x ) = X y ∈ V µ ( y ) p t ( x, y ) u ( y ) t > , x ∈ V satisfies the heat equation, and lim t → + u ( t, x ) = u ( x )For any subset U ⊂ V , we denote by ◦ U = { x ∈ U : ∀ y ∼ x, y ∈ U } the interior of U .The boundary of U is ∂U = U \ ◦ U . We introduce the following version of the maximumprinciple. Lemma 3.1.
Let U ⊂ V be finite and T > . Furthermore, assume that u : [0 , T ] × U → R is differentiable with respect to the first component and satisfies the inequality ∂ t u ≤ ∆ u on [0 , T ] × ◦ U . Then, u attains its maximum on the parabolic boundary ∂ P ([0 , T ] × U ) = ( { } × U ) ∪ ([0 , T ] × ∂U ) Proof.
Suppose u attains its maximum at a point ( t , x ) ∈ (0 , T ] × U ◦ such that ∂ t u ( t , x ) < ∆ u ( t , x ) (3.1)Then 0 ≤ ∂ t u ( t , x ) < ∆ u ( t , x ) = ] X y ∼ x ( u ( t , y ) − u ( t , x )) , (3.2)contradicting the maximality of u .Otherwise, if at all ( t , x ) ∈ (0 , T ] × U ◦ which are maximum points at u , there is equalityin (3.1) we are done unless there is also equality in 3.2. But this implies that u is constanton (0 , T ] × U , and hence there is a maximum point on the boundary as desired.11 .1.2 The heat equation an a domain Suppose U ⊂ V is a finite subset of the vertex set of a graph. We consider the Dirichletproblem (DP), ∂ t u ( t, x ) − ∆ U u ( t, x ) = 0 , x ∈ ◦ U , t > u (0 , x ) = u ( x ) , x ∈ ◦ U , u | [0 , ∞ ) × ∂U = 0 . where ∆ U : ℓ ( ◦ U , µ ) → ℓ ( ◦ U , µ ) denotes the Dirichlet Laplacian on ◦ U .Note that − ∆ U is positive and self-adjoint, and n := dim ℓ ( ◦ U , µ ) < ∞ . Thus theoperator − ∆ U has eigenvalues 0 < λ ≤ λ < · · · ≤ λ n , along with an orthonormal set ofeigenvectors φ i . Here the orthonormality is with respect to the inner product with respectto the measure µ , ie. h φ i , φ j i = P x ∈ V µ ( x ) φ i ( x ) φ j ( x ).The operator ∆ U is a generator of the heat semigroup P t,U = e t ∆ U , t >
0. Finite dimen-sionality makes the fact that e t ∆ U φ i = e − tλ i φ i transparent. The heat kernel p U ( t, x, y ) forthe finite subset U is then given by p U ( t, x, y ) = P t,U δ y p µ ( y ) ( x ) , ∀ x, y ∈ ◦ U where δ y ( x ) = P ni =1 h φ i , δ y i φ i ( x ) = P ni =1 φ i ( x ) φ i ( y ) p µ ( y ). The heat kernel satisfies p U ( t, x, y ) = n X i =1 e − λ i t φ i ( x ) φ i ( y ) , ∀ x, y ∈ ◦ U .
We record some useful properties of the heat kernel on a finite domain:
Remark 2.
For t, s > ∀ x, y ∈ ◦ U , we have1. p U ( t, x, y ) = p U ( t, y, x )2. p U ( t, x, y ) ≥ P y ∈ ◦ U µ ( y ) p U ( t, x, y ) ≤ t → + P y ∈ ◦ U µ ( y ) p U ( t, x, y ) = 1,5. ∂ t p U ( t, x, y ) = ∆ ( U,y ) p U ( t, x, y ) = ∆ ( U,x ) p U ( t, x, y )6. P z ∈ ◦ U µ ( z ) p U ( t, x, z ) p U ( s, z, y ) = p U ( t + s, x, y ) Proof. (1) and (5) follow from the above fact about the heat kernel, (2) and (3) are immediateconsequences of the maximum principle. Note that (4) follows from the continuity of thesemigroup e t ∆ at t = 0, if the limit is understood in the ℓ sense. As U is finite all normsare equivalent and pointwise convergence follows also. (6) is easy to calculate in ℓ ,and it iscalled the semigroup property of heat kernel.12 .1.3 Heat equation on a infinite graph The heat kernel for an infinite graph can be constructed and its basic properties can bederived using the above ideas by taking an exhaustion of the graph. An exhaustion of G isa sequence ( U k ) of subsets of V , such that U k ⊂ ◦ U k +1 and ∪ k ∈ N U k = V . For any connected,countable graph G such a sequence exists. One may, for instance, fix a vertex x ∈ V takethe sequence U k = B k ( x ) of metric balls of radius k around x . The connectedness of ourgraph G implies that the union of these U k equals V .Denoting by p k , the heat kernel p U k on U k , we may extend p k to all of (0 , ∞ ) × V × V , p k ( t, x, y ) = ( p U k ( t, x, y ) , x, y ∈ ◦ U k ;0 , o.w.Then for any t > x, y ∈ V, we let p ( t, x, y ) = lim k →∞ p k ( t, x, y ) . The maximum principle implies the monotonicity of the heat kernels, i.e. p k ≤ p k +1 , so theabove limit exists (but could a priori be infinite). Similarly, it is not a priori clear that p isindependent of the exhaustion chosen. None the less, the limit is finite and independent ofthe exhaustion and p is the desired heat kernel. This construction is carried out in [WE10]and [WO09] for unweighted graphs, where the measure µ ≡
1. For the general case, we referto [KL12].For convenience, we record some important properties of the heat kernel p which we willuse in the paper. Remark 3.
For t, s > ∀ x, y ∈ V , we have1. p ( t, x, y ) = p ( t, y, x )2. p ( t, x, y ) ≥ P y ∈ V µ ( y ) p ( t, x, y ) ≤ t → + P y ∈ V µ ( y ) p ( t, x, y ) = 1,5. ∂ t p ( t, x, y ) = ∆ y p ( t, x, y ) = ∆ x p ( t, x, y )6. P z ∈ V µ ( z ) p ( t, x, z ) p ( s, z, y ) = p ( t + s, x, y )From here, the semigroup P t : V R → V R acting on bounded functions f : V → R asfollows. for any bounded function f ∈ V R , P t f ( x ) = lim k →∞ X y ∈ V µ ( y ) p k ( t, x, y ) f ( y ) = X y ∈ V µ ( y ) p ( t, x, y ) f ( y )13here lim t → + P t f ( x ) = f ( x ), and P t f ( x ) is a solution of the heat equation. From theproperties of the heat kernel, and the boundedness of f , that is, there exists a constant C >
0, such that for any x ∈ V , sup x ∈ V | f ( x ) | ≤ C , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X y ∈ V µ ( y ) p ( t, x, y ) f ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C lim k →∞ X y ∈ V µ ( y ) p k ( t, x, y ) ≤ C < ∞ , so the semigroup is well-defined. Besides, the different definitions of the heat semigroupcoincide when ∆ is a bounded operator or in finite graphs, that is P t f ( x ) = e t ∆ f ( x ) = + ∞ X k =0 t k ∆ k k ! f ( x ) = X y ∈ V µ ( y ) p ( t, x, y ) f ( y ) . Again we record, without proof, some well known but useful properties of the semigroup P t . Proposition 3.1.
For any bounded function f, g ∈ V R , and t, s > , for any x ∈ V ,1. If ≤ f ( x ) ≤ , then ≤ P t f ( x ) ≤ ,2. P t ◦ P s f ( x ) = P t + s f ( x ) ,3. ∆ P t f ( x ) = P t ∆ f ( x ) . Finiteness of D µ implies boundedness of the operators ∆ and Γ. We in turn derive thefollowing Lemma: Lemma 3.2.
Suppose G = ( V, E ) is a (finite or infinite) graph satisfying the condition CDE ′ ( n, K ) . Then, for a positive and bounded solution u ( t, x ) to the heat equation on G ,the function ∆ u √ u on G is bounded at all t > .Proof. The statement is obvious for finite graphs G , so we restrict our attention to infinitegraphs.Fix R ∈ N and vertex x ∈ V . We define a cutoff function ϕ by letting ϕ ( x ) = , d ( x, x ) > R R − d ( x,x ) R , R ≤ d ( x, x ) ≤ R , d ( x, x ) < R Let F = ϕ · Γ( √ u ) √ u , It is easy to observe that, as 0 ≤ ϕ ( x ) ≤ x ∈ V , | ∆ ϕ | ≤ D µ . As u is bounded,there exists constants c , c so that 0 ≤ Γ( √ u ) ≤ c , and | Γ(Γ( √ u ) , ϕ ) | ≤ c as well.14ix an arbitrary T >
0, let ( x ∗ , t ∗ ) be a maximum point of F in V × [0 , T ]. Clearlysuch a maximum exists, as F ≥ F is only positive on a bounded region. We mayassume F ( x ∗ , t ∗ ) >
0. In what follows all computations take place at the point ( x ∗ , t ∗ ). Let L = ∆ − ∂ t , we apply Lemma 4.1 in [BHLLMY13] with the choice of g = u . This gives L ( √ uF ) ≤ L ( √ u ) F = − F ϕ , and since for any x ∈ V , we have ∂ t Γ( √ u )( x ) = ∂ t ] X y ∼ x (cid:0) √ u ( y ) − √ u ( x ) (cid:1) = ] X y ∼ x ( √ u ( y ) − √ u ( x ))( ∂ t √ u ( y ) − ∂ t √ u ( x ))= 2Γ( √ u, ∆ u √ u )( x ) , then L ( √ uF ) = L ( ϕ · Γ( √ u )) = ∆ ϕ · Γ( √ u ) + 2Γ(Γ( √ u ) , ϕ ) + 2 ϕ · e Γ ( √ u ) . Applying the
CDE ′ ( n, K ) condition and throwing away the n u (∆ log √ u ) term, we obtain − F ϕ ≥ ∆ ϕ · Γ( √ u ) + 2Γ(Γ( √ u ) , ϕ ) + 2 ϕK Γ( √ u ) . From here, we conclude that F ( x ∗ , t ∗ ) ≤ D µ + | K | ) c + c , Thus there exists some
C > F ( x ∗ , t ∗ ) ≤ C. For x ∈ B ( x , R ), Γ( √ u ) √ u ( T, x ) = F ( x, T ) ≤ F ( x ∗ , t ∗ ) ≤ C. From the equation ∆ u = 2 √ u ∆ √ u + 2Γ( √ u ), we can obtain ∆ u √ u is bounded at any positive T > < f ∈ ℓ ∞ ( V, µ ) on G ( V, E ), the function Γ( p P T − t f ),for any 0 ≤ t < T is likewise bounded.Given a positive bounded f , we introduce the function φ ( t, x ) = P t (Γ( p P T − t f ))( x ) , ≤ t < T, x ∈ V. From here we obtain the following (rather crucial) result.15 emma 3.3.
Suppose that G satisfies the condition CDE ′ ( n, K ) . Then, for every ≤ t < T ,any x ∈ V , the function φ satisfies ∂ t φ ( t, x ) = 2 P t ( e Γ ( p P T − t f ))( x ) . Proof.
For any x ∈ V , ∂ t P t (Γ( p P T − t f ))( x ) = ∂ t X y ∈ V µ ( y ) p ( t, x, y )Γ( p P T − t f )( y ) ! = X y ∈ V µ ( y ) (cid:16) ∆ p ( t, x, y )Γ( p P T − t f )( y ) + p ( t, x, y ) ∂ t Γ( p P T − t f )( y ) (cid:17) = X y ∈ V µ ( y ) ∆ p ( t, x, y )Γ( p P T − t f )( y ) − p ( t, x, y )Γ( p P T − t f , ∆ P T − t f p P T − t f )( y ) ! = X y ∈ V µ ( y ) p ( t, x, y ) ∆Γ( p P T − t f )( y ) − p P T − t f , ∆ P T − t f p P T − t f )( y ) ! = 2 P t ( e Γ ( p P T − t f ))( x )For the third equality, we observe that for any x ∈ V , ∂ t Γ( p P T − t f )( x ) = ∂ t ] X y ∼ x (cid:16)p P T − t f ( y ) − p P T − t f ( x ) (cid:17) = ] X y ∼ x ( p P T − t f ( y ) − p P T − t f ( x ))( ∂ t p P T − t f ( y ) − ∂ t p P T − t f ( x ))= 2Γ( p P T − t f , ∂ t p P T − t f )( x ) , and, ∂ t p P T − t f = ∂ t P T − t f p P T − t f = − ∆ P T − t f p P T − t f , where ∂ t P T − t f = − ∆ P T − t f .In the fourth step, note that due to the boundedness of f , the function ∆Γ( p P T − t f ) islikewise bounded. Similarly from Lemma 3.2, Γ( p P T − t f , ∆ P T − t f √ P T − t f ) is bounded as well. Likethe proof of Proposition 3.1, we have X y ∈ V µ ( y ) ∆ p ( t, x, y )Γ( p P T − t f )( y ) − p ( t, x, y )Γ( p P T − t f , ∆ P T − t f p P T − t f )( y ) ! = X y ∈ V µ ( y )∆ p ( t, x, y )Γ( p P T − t f )( y ) − X y ∈ V µ ( y )2 p ( t, x, y )Γ( p P T − t f , ∆ P T − t f p P T − t f )( y )= X y ∈ V µ ( y ) p ( t, x, y )∆Γ( p P T − t f )( y ) − X y ∈ V µ ( y )2 p ( t, x, y )Γ( p P T − t f , ∆ P T − t f p P T − t f )( y )= X y ∈ V µ ( y ) p ( t, x, y ) ∆Γ( p P T − t f )( y ) − p P T − t f , ∆ P T − t f p P T − t f )( y ) ! , P t f ( x ) = e t ∆ f ( x ) = + ∞ X k =0 t k ∆ k k ! f ( x ) = X y ∈ V µ ( y ) p ( t, x, y ) f ( y ) . Then consider another function P t f t ( x ), where f t ( x ) is a positive function with respect to t ∈ [0 , + ∞ ) and x ∈ V (here f t ( x ) = 2 e Γ ( p P T − t f )( x )). These above summations have anice convergency when f t ( x ) is a uniform bounded function. Since there exists a constant C > | f t ( x ) |≤ C for any t ∈ [0 , T ], we have | ∆ f t ( x ) | = | ] X y ∼ x ( f t ( y ) − f t ( x )) | ≤ D µ C, and for iteration, we obtain for any k ∈ N ≥ and x ∈ V , | ∆ k f t ( x )( x ) | ≤ k D kµ C. And + ∞ X k =0 T k k ! 2 k D kµ C = Ce D µ T < ∞ . Therefore, the series P t f t ( x ) = X y ∈ V µ ( y ) p ( t, x, y ) f t ( y ) = + ∞ X k =0 t k ∆ k k ! f t ( x )converges uniformly on [0 , T ].This ends the proof of Lemma 3.3.We now obtain some graph theoretical analogues to theorems of Baudoin and Garofalo[BG09] originating on in the manifold setting. In some sense our main observation is that the CDE ′ ( n, K ) condition can be used in order to overcome the diffusive semigroup assumptionusually needed for arguments involving the heat semigroup. This is one of the primaryplaces where we note that the CDE ( n, K ) condition favored in [BHLLMY13] is seeminglyinsufficient to prove the result. Theorem 3.2.
Let G = ( V, E ) be a locally finite, connected graph satisfying CDE ′ ( n, K ) ,then for every α : [0 , T ] → R + be a smooth and positive function and non-positive smoothfunction γ : [0 , T ] → R , we have for any positive and bounded function f∂ t ( αφ ) ≥ ( α ′ − αγn + 2 αK ) φ + 2 αγn ∆ P T f − αγ n P T f. (3.3)17 roof. For any x ∈ V , we have ∂ t ( αφ )( x ) = α ′ φ ( x ) + 2 αP t ( e Γ ( p P T − t f ))( x ) ≥ α ′ φ ( x ) + 2 αP t (cid:18) n (cid:16)p P T − t f ∆ log p P T − t f (cid:17) + K Γ( p P T − t f ) (cid:19) ( x ) ≥ ( α ′ + 2 αK ) φ ( x ) + 2 α X y ∼ x ∆ √ P T − t f ( y ) < µ ( y ) p ( t, x, y ) 1 n (cid:16) ∆ p P T − t f (cid:17) ( y )+ 2 α X y ∼ x ∆ √ P T − t f ( y ) ≥ µ ( y ) p ( t, x, y ) 1 n (cid:16)p P T − t f ∆ log p P T − t f (cid:17) ( y ) ≥ ( α ′ + 2 αK ) φ ( x ) + 2 αn P t ( γ ∆ P T − t f − γ Γ( p P T − t f ) − γ P T − t f )( x )= ( α ′ + 2 αK ) φ ( x ) + 2 αγn P t (∆ P T − t f )( x ) − αγn P t (Γ( p P T − t f ))( x ) − αγ n P t ( P T − t f )( x )= ( α ′ − αγn + 2 αK ) φ ( x ) + 2 αγn ∆ P T f ( x ) − αγ n P T f ( x ) . The first inequality in the above proof comes from applying the
CDE ′ ( n, K ) inequalityto p P T − t f . The second one comes from Jensen’s inequality, under the assumption that(∆ p P T − t f )( y ) <
0. This is essentially the contents of Remark 1 – really we apply the
CDE ( n, K ) inequality at points so that ∆ p P T − t f ( y ) < γ , one has (∆ p P T − t f )( y ) ≥ γ p P T − t f ( y )∆ p P T − t f ( y ) − γ P T − t f ( y ) . Since γ is non-positive, if ∆ p P T − t f ( y ) ≥ (cid:16)p P T − t f △ log p P T − t f (cid:17) ( y ) ≥ γ p P T − t f ( y ) △ p P T − t f ( y ) − γ P T − t f ( y ) , as the left hand side of this inequality is clearly non-negative.Furthermore, by the identity ∆ u = 2 √ u ∆ √ u + 2Γ( u ),2 p P T − t f ∆ p P T − t f = ∆ P T − t f − p P T − t f ) , Therefore, X y ∼ x ∆ √ P T − t f ( y ) < µ ( y ) p ( t, x, y ) (cid:16) ∆ p P T − t f (cid:17) ( y )+ X y ∼ x ∆ √ P T − t f ( y ) ≥ µ ( y ) p ( t, x, y ) P T − t f ( y ) (cid:16) ∆ log p P T − t f (cid:17) ( y ) ≥ P t ( γ ∆ P T − t f − γ Γ( p P T − t f ) − γ P T − t f )( x ) ,
18s desired.
The precise power of Theorem 3.2 is, perhaps, a bit hard to appreciate at first. As anapplication, it can be used to give an alternative derivation of the Li-Yau inequality. Indeed,it can be used to derive a family of similar differential Harnack inequalities. The key inapplying Theorem 3.2 is to choose γ in a careful way so that the things simplifying carefully.For instance, suppose for some (smooth) function α we choose γ in such a way so that α ′ − αγn + 2 αK = 0 . That is, choose γ = n (cid:18) α ′ α + 2 K (cid:19) . If α is chosen appropriately to make γ non-positive, then we may integrate the inequality(3.3) obtained in Theorem 3.2 from 0 to T , and obtain an estimate. If we denote W = √ α ,we obtain the following result. Theorem 3.3.
Let G = ( V, E ) be a locally finite and connected graph satisfying CDE ′ ( n, K ) ,and W : [0 , T ] → R + be a smooth function such that W (0) = 1 , W ( T ) = 0 , and so that W ′ ( t ) ≤ − KW ( t ) for ≤ t ≤ T. Then for any bounded and positive function f ∈ V R , we have Γ( √ P T f ) P T f ≤ (cid:18) − K Z T W ( s ) ds (cid:19) ∆ P T fP T f + n (cid:18)Z T W ′ ( s ) ds + K Z T W ( s ) ds − K (cid:19) . (3.4)Here, the condition W ′ ≤ − KW amounts to the non-positivity of γ . As observed in[BG09], the family obtained by taking W ( t ) = (cid:18) − tT (cid:19) b , for any b > is quite interesting in the regime where − bT < K .For this family, Z T W ( s ) ds = T b + 1 , Z T W ′ ( s ) ds = b (2 b − T .
Thus for such a choice of W , the estimate (3.4) yieldsΓ( √ P T f ) P T f ≤ (cid:18) − KT b + 1 (cid:19) ∆ P T fP T f + n (cid:18) b (2 b − T + K T b + 1 − K (cid:19) . (3.5)When K = 0 and b = 1, this reduces to the familiar Li-Yau inequality on graphs (as derivedby [BHLLMY13]). Indeed, per the identity ∆ P t f = ∂ t P t f = 2 √ P t f ∂ t √ P t f and switchingthe notion T to t , (3.5) reduces to:Γ( √ P t f ) P t f − ∂ t √ P t f √ P t f ≤ n t , t > . While the Li-Yau inequality is an attractive consequence of Theorem 3.2, a version wasalready known to hold on graphs using the
CDE ( n, K ) curvature dimensional inequality(which is slightly weaker than the CDE ′ ( n, K ) inequality used in Theorem 3.2.).In this section, we begin by exhibiting a further application of the variational inequality,and use it derive volume doubling from non-negative curvature, which was out of reach fromprevious results. Theorem 4.1.
Let G = ( V, E ) be a locally finite and connected graph satisfying CDE ′ ( n, ,there exists an absolute positive constant ρ > , and A > , depending only on n , such that P Ar (cid:0) B ( x,r ) (cid:1) ( x ) ≥ ρ, x ∈ V, r > . (4.1) Proof.
Again, we proceed by carefully choosing a γ to apply Theorem 3.2. Let α ( t ) = τ + T − t,γ ( t ) = − n τ + T − t ) , for τ >
0, and K = 0. For such a choice α ′ − αγn + 2 αK = 0 , αγn = − , αγ n = n τ + T − t ) , again simplifying the main inequality. We integrate the inequality from 0 to T , obtaining τ P T (Γ( p f )) − ( T + τ )Γ( p P T f ) ≥ − T P T f − n (cid:18) Tτ (cid:19) P T f. (4.2)20ow, suppose f is a non-positive c -Lipschitz function (that is, | f ( y ) − f ( x ) | ≤ c if x ∼ y .)Fix λ ≥
0, and consider the function ϕ = e λf . Clearly, f is positive and bounded. Let ψ ( λ, t ) = 12 λ log( P t e λf ) , so that P t ϕ = P t ( e λf ) = e λψ . Applying (4.2) to ϕ , and switching notation from T to t , one obtains that τ P t (Γ( e λf )) − ( t + τ )Γ( e λψ ) ≥ − t P t ϕ − n (cid:18) tτ (cid:19) e λψ . (4.3)Fix x ∈ V . Taking C ( λ, c ) = q D µ ce λc < ∞ , we haveΓ( e λf )( x ) = 12 ] X y ∼ x (cid:0) e λf ( y ) − e λf ( x ) (cid:1) = 12 e λf ( x ) ] X y ∼ x (cid:0) e λ ( f ( y ) − f ( x )) − (cid:1) = 12 e λf ( x ) ^ X ≤ f ( y ) − f ( x ) ≤ c (cid:0) e λ ( f ( y ) − f ( x )) − (cid:1) + ^ X − c ≤ f ( y ) − f ( x ) ≤ (cid:0) e λ ( f ( y ) − f ( x )) − (cid:1) ≤ e λf ( x ) e λc ^ X ≤ f ( y ) − f ( x ) ≤ c (cid:0) − e − λc (cid:1) + ^ X − c ≤ f ( y ) − f ( x ) ≤ (cid:0) e − λc − (cid:1) ≤ e λf ( x ) e λc ] X y ∼ x (cid:0) e − λc − (cid:1) ≤ C ( λ, c ) λ e λf ( x ) . This enables us to upper bound the left hand side of (4.3), obtaining τ P t (Γ( e λf )) − ( t + τ )Γ( e λψ ) ≤ τ P t (Γ( e λf )) ≤ C ( λ, c ) λ τ P t ( e λf ) = C ( λ, c ) λ τ e λψ . Combining this with the fact that∆ P t ϕ = ∂ t e λψ = 2 λe λψ ∂ t ψ, we obtain that ∂ t ψ ≥ − λt (cid:18) C ( λ, c ) τ + n λ log(1 + tτ ) (cid:19) . (4.4)Since (4.4) holds for all τ , we optimize. Setting τ to be the optimal value, τ = t (cid:18)r n λ C ( λ, c ) t − (cid:19) , − ∂ t ψ ≤ λC ( λ, c ) G (cid:18) λ C ( λ, c ) t (cid:19) . (4.5)Here, G ( s ) = 12 (cid:18)r n s − (cid:19) + n s log p n s − ! , s > . Note that G ( s ) → s → + , and that G ( s ) ∼ p ns as s → + ∞ . Integrate theinequality (4.5) between t and t (for t ≤ t ) and we obtain that, ψ ( λ, t ) ≤ ψ ( λ, t ) + λC ( λ, c ) Z t t G (cid:18) λ C ( λ, c ) t (cid:19) dt. Jensen’s inequality in ψ yields that2 λψ ( λ, t ) = ln( P t e λf ) ≥ P t (ln e λf ) = 2 λP t f. This yields that λP t f ≤ λψ ( λ, t ), and combining with the previous inequality we have thatfor all t ≤ t . P t ( λf ) ≤ λψ ( λ, t ) + λ C ( λ, c ) Z t t G (cid:18) λ C ( λ, c ) t (cid:19) dt. Replacing t with t , and letting t → + we obtain λf ≤ λψ ( λ, t ) + λ C ( λ, c ) Z t G (cid:18) λ C ( λ, c ) τ (cid:19) dτ. (4.6)Now fix a vertex x ∈ V . Let B = B ( x, r ), and consider the function f ( y ) = − d ( y, x ).Clearly f is 1-Lipschitz. For such a 1-Lipschitz function, we may use C ( λ, c ) = q D µ e λ inthe proceeding.Clearly, e λf ≤ e − λr B c + B . Thus for every t > e λψ ( λ,t )( x ) = P t ( e λf )( x ) ≤ e − λr + P t ( B )( x )and we obtain the lower bound P t ( B )( x ) ≥ e λψ ( λ,t )( x ) − e − λr . (4.6) allows us to estimate the first term in this lower bound. If φ ( λC ( λ, c ) , t ) = λ C ( λ, c ) Z t G (cid:18) λ C ( λ, c ) τ (cid:19) dτ, e λf ( x ) ≤ e λψ ( λ,t )( x ) e φ ( λC ( λ,c ) ,t ) . Hence P t ( B )( x ) ≥ e − φ ( λC ( λ,c ) ,t ) − e − λr . Choose λC ( λ, c ) = r , t = Ar , and we obtain P Ar ( B )( x ) ≥ e − φ ( r ,Ar ) − e − C ( λ,c ) . To finish, we must choose
A > n , and a ρ >
0, sothat for every x ∈ V and r > e − φ ( r ,Ar ) − e − C ( λ,c ) ≥ ρ. (4.7)(Note that, actually, the point that r > simply implies that the term e − C ( λ,c ) is not one.Replacing this by r > ǫ for any positive ǫ would likewise suffice.)To see such an A exists, consider the function φ ( 1 r , Ar ) = 1 r Z Ar G (cid:18) r τ (cid:19) dτ = Z ∞ A − G ( t ) t dt.φ ( r , Ar ) → A → + , and hence such a sufficiently small A exists to ensure that 4.7holds and this completes the proof.In this section we use the previous result to show that non-negatively curved graphs (withrespect to CDE ′ ) satisfying the volume doubling property. That is, we obtain: Theorem 4.2.
Suppose a locally finite, connected graph G satisfies CDE ′ ( n, , then G satisfies the volume doubling property DV ( C ) . That is, there exists a constant C = C ( n ) > such that for all x ∈ V and all r > : V ( x, r ) ≤ CV ( x, r ) . Actually, with some simple computations we can get some slightly stronger conclusionson volume regularity that we will find useful in the proof of a Gaussian estimate.
Remark 4.
For any r ≥ s , V ( x, r ) ≤ V ( x, [ log( rs )log 2 ]+1 s ) ≤ C log( rs )log 2 V ( x, s )= C (cid:16) rs (cid:17) log C log 2 V ( x, s ) , where [ x ] denotes the integer part of x . 23ne final tool in the proof of Theorem 4.2 is an explicit form of a Harnack inequalityarising from the Li-Yau inequality. Such an inequality was derived in [BHLLMY13]. In the(simplified by our assumption that K = 0) form in which we apply it, it states that Corollary 4.3.
Suppose G is a finite or infinite graph satisfying CDE ′ ( n, D := µ max ω min < ∞ , then for every x ∈ V and ( t, y ) , ( t, z ) ∈ (0 , + ∞ ) × V with t < s one has p ( t, x, y ) ≤ p ( s, x, z ) (cid:16) st (cid:17) n exp (cid:18) Dd ( y, z ) s − t (cid:19) . We now turn to the proof of Theorem 4.2.
Proof.
From the semigroup property and the symmetry of the heat kernel in Remark 3, forany y ∈ V and t >
0, we have p (2 t, y, y ) = X z ∈ V µ ( z ) p ( t, y, z ) . Consider now a cut-off function h ∈ V R such that 0 ≤ h ≤ h ≡ B ( x, √ t ) and h ≡ B ( x, √ t ). We thus have P t h ( y ) = X z ∈ V µ ( z ) p ( t, y, z ) h ( z ) ≤ X z ∈ V µ ( z ) p ( t, y, z ) ! X z ∈ V µ ( z ) h ( z ) ! ≤ ( p (2 t, y, y )) (cid:16) V ( x, √ t ) (cid:17) . (4.8)Taking y = x , and t = r , we obtain (cid:16) P r ( B ( x, r ) )( x ) (cid:17) ≤ ( P r h ( x )) ≤ p (2 r , x, x ) V ( x, r ) . (4.9)At this point we use the crucial inequality (4.1), which gives for some 0 < A <
1, dependingon the dimension n , P Ar (cid:0) B ( x,r ) (cid:1) ( x ) ≥ ρ, x ∈ V, r > . Combining the latter inequality with (4.9) and Corollary 4.3, we obtain an on-diagonallower bound p (2 r , x, x ) ≥ ρ ∗ V ( x, r ) , x ∈ V, r > . (4.10)Applying Corollary 4.3 to p ( t, x, y ), one obtains that for every y ∈ B ( x, √ t ), we find p ( t, x, x ) ≤ C ( n ) p (2 t, x, y ) . (4.11)24ntegrating the above inequality over B ( x, √ t ) with respect to y gives p ( t, x, x ) V ( x, √ t ) ≤ C ( n ) X y ∈ B ( x, √ t ) µ ( y ) p (2 t, x, y ) ≤ C ( n ) . Further letting t = 4 r , we obtain an on-diagonal upper bound p (4 r , x, x ) ≤ C ( n ) V ( x, r ) . (4.12)Combining (4.10),(4.11) with (4.12) we finally obtain for any r > , V ( x, r ) ≤ Cp (4 r , x, x ) ≤ C ∗ p (2 r , x, x ) ≤ C ∗∗ V ( x, r ) . When 0 < r ≤ , V ( x, r ) ≡ V ( x, r ) = µ ( x ), the result is obvious. This completes theproof.As a remark, while the proof is fairly simple, it illustrates the power of inequality (4.9).The failing in the [BHLLMY13] paper to obtain volume doubling lay exactly in this point:we previously were obtained a lower bound on this quantity by directly applying the Harnackinequality Corollary 4.3 in a somewhat unusual fashion, rather strongly using the fact thatthe Harnack inequality obtained by integrating the Li-Yau inequality gives a more explicitestimate than the continuous-time Harnack inequality introduced in Section 2. We did this,instead of using the approach of the current proof, the lack of quality cutoff functions in thegraph case meant that our Li-Yau inequality (and hence the obtained Harnack inequality)was insufficient to derive a strong enough lower bound to imply volume doubling. The lessonhere should be taken that using the heat-semigroup arguments as done above allows us towork around the lack of quality cutoff functions for graphs. In this section we focus on the normalized Laplacian: that is, we take our measure µ to be µ ( x ) = m ( x ). We will prove a discrete-time Gaussian estimate on a infinite, connected andlocally finite graph G = ( V, E ).Let P t ( x, y ) = p ( t, x, y ) m ( y ) be the continuous-time Markov kernel on the graph. It isalso a solution of the heat equation. By symmetry, the heat kernel p ( t, x, y ) satisfies P t ( x, y ) m ( y ) = P t ( y, x ) m ( x ) . Let p n ( x, y ) be the discrete-time kernel on G , which is defined by (cid:26) p ( x, y ) = δ xy ,p k +1 ( x, z ) = P y ∈ V p ( x, y ) p k ( y, z ) . p ( x, y ) := ω xy m ( x ) , and δ xy = 1 only when x = y , otherwise equals to 0. We can knowthe two kernels satisfy e − t + ∞ X k =0 t k k ! p k ( x, y ) = P t ( x, y ) . (5.1)In order to obtain our desired Gaussian estimate, we first establish the continuous-timeGaussian on-diagonal estimate for graphs. As demonstrated in [BHLLMY13], the Harnackinequality obtained in that paper suffices to prove a Gaussian upper bound for boundeddegree graphs satisfying CDE ( n, CDE ′ to imply volume doubling.With the new information gleaned from our modified curvature condition, we are howeverable to derive the Gaussian lower bound, as we now illustrate. Theorem 5.1.
Suppose a graph G satisfies CDE ′ ( n , . Then G satisfies the continuous-time Gaussian estimate. That is, there exists a constant C with respect to n so that, forany x, y ∈ V and for all t > , P t ( x, y ) ≤ Cm ( y ) V ( x, √ t ) . Furthermore, for any t > , there exist constants C ′ and c ′ , so that for all t > t : P t ( x, y ) ≥ C ′ m ( y ) V ( x, √ t ) exp (cid:18) − c ′ d ( x, y ) t (cid:19) . Proof.
The upper bound follows from the methods of [BHLLMY13], as the Harnack inequal-ity obtained in that paper still applies for graphs satisfying
CDE ′ ( n , t >
0, choosing s = 2 t and for any z ∈ B ( x, √ t ), we have p ( t, x, y ) ≤ p (2 t, z, y )2 n exp(4 D ) . Integrating the above inequality over B ( x, √ t ) with respect to z , gives p ( t, x, y ) ≤ CV ( x, √ t ) X z ∈ B ( x, √ t ) µ ( z ) p (2 t, z, y ) ≤ CV ( x, √ t ) . We now prove the lower bound estimate. Recall that we only claim the result under theassumption that t > t . The result is most transparent if t > /
2. In this case, then taking t > / r = εt for some 0 < ε <
1, equation (4.10) implies that every x ∈ V satisfies p ( εt, x, x ) ≥ ρ ∗ V ( x, q εt ) ≥ ρ ∗ V ( x, √ t ) . (5.2)Applying Corollary 4.3, taking εt as ‘ t ’, taking t to be ‘ s ’, and choosing y = x, z = y , weobtain p ( εt, x, x ) ≤ p ( t, x, y ) ε n exp (cid:18) Dd ( x, y ) (1 − ε ) t (cid:19) . (5.3)26ombining (5.2) with (5.3), we finally obtain for any t > , p ( t, x, y ) ≥ ε − n ρ ∗ V ( x, √ t ) exp (cid:18) − Dd ( x, y ) (1 − ε ) t (cid:19) = C ′ V ( x, √ t ) exp (cid:18) − c ′ d ( x, y ) t (cid:19) . While we assumed that t > here, if we fix any t >
0, it is easy to rework the proofTheorem 4.1 to work with such an arbitrary t and this completes the proof of the theorem.The remaining difficulty is verifiying that the lower bound holds when t is small enough.This we will defer until after proving discrete-time Gaussian estimate to Remark 5. Together,this will complete the proof of Theorem 5.1.As a special case, note that if t ≥ max (cid:8) d ( x, y ) , (cid:9) , then the lower estimate can be write p ( t, x, y ) ≥ C ′′ V ( x, √ t ) . (5.4)Before we ultimately finish the continuous-time lower bound, we address the discrete-timeestimate. We begin with the on-diagonal estimate: Proposition 5.2.
Assume a graph G satisfies CDE ′ ( n , and ∆( β ) , then there exist c d , C d > , for any x, y ∈ V , such that, p n ( x, y ) ≤ C d m ( y ) V ( x, √ n ) , for all n > ,p n ( x, y ) ≥ c d m ( y ) V ( x, √ n ) , if n ≥ d ( x, y ) . This proposition follows the methods of Delmotte from [D99]. To prove it, we firstintroduce some necessary results. Assume ∆( α ) is true (cf. Definition 2.1), so that we canconsider the positive submarkovian kernel p ( x, y ) = p ( x, y ) − αδ xy . Now, compute P n ( x, y ) and p n ( x, y ) with p ( x, y ), P n ( x, y ) = e ( α − n + ∞ X k =0 n k k ! p k ( x, y ) = + ∞ X k =0 a k p k ( x, y ) ,p n ( x, y ) = n X k =0 C kn α n − k p k ( x, y ) = n X k =0 b k p k ( x, y ) . There is a lemma from [D99] to compare the two sums as follows.
Lemma 5.1.
Let c k = b k a k , for ≤ k ≤ n , and suppose α ≤ . c k ≤ C ( α ) , when ≤ k ≤ n , • c k ≥ C ( a, α ) > , when n ≥ a α and | k − (1 − α ) n | ≤ a √ n . Note that the condition that α ≤ implies that n ≤ k ≤ n in the second assertion. Notethat assuming α ≤ does not inhibit us: it is clear from the definition that if ∆( α ) holds,so does ∆( α ′ ) for any α ′ < α .Now we turn to the proof of Proposition 5.2. Proof of Proposition 5.2 .
The proof comes from Delmotte of [D99].The first assertion in Lemma 5.1 implies, for any n ∈ N , p n ( x, y ) ≤ C ( β ) P n ( x, y ) . The upper bound, then, is an immediate consequence of Theorem 5.1: For any x, y ∈ V , p n ( x, y ) ≤ C ( β ) Cm ( y ) V ( x, √ n ) = C d m ( y ) V ( x, √ n ) . The second assertion is a little more complicated.Suppose, for a minute, that for any ε >
0, there exists an a > X | k − (1 − α ) n | >a √ n a k p k ( x, y ) ≤ εm ( y ) V ( x, √ n ) . (5.5)We return briefly to prove that such an a always exists.Fix such an a for a sufficiently small ε , taking, say,0 < ε < C ′ ≤ P n ( x, y ) · V ( x, √ n ) m ( y ) . We set α = β , and n ≥ N = a α .For such choices we have, by the second assertion of Lemma 5.1, that p n ( x, y ) ≥ X | k − (1 − α ) n |≤ a √ n b k p k ( x, y ) ≥ C ( a, α ) X | k − (1 − α ) n |≤ a √ n a k p k ( x, y ) , and furthermore C ( a, α ) P n ( x, y )= C ( a, α ) X | k − (1 − α ) n |≤ a √ n a k p k ( x, y ) + C ( a, α ) X | k − (1 − α ) n | >a √ n a k p k ( x, y ) ≤ p n ( x, y ) + C ( a, α ) X | k − (1 − α ) n | >a √ n a k p k ( x, y ) ≤ p n ( x, y ) + C ( a, α ) εm ( y ) V ( x, √ n ) . n ≥ d ( x, y ) , applying the second assertion of (5.4), we obtain p n ( x, y ) ≥ C ( a, α ) (cid:18) P n ( x, y ) − εm ( y ) V ( x, √ n ) (cid:19) ≥ C ( a, α ) (cid:18) C ′ m ( y ) V ( x, √ n ) − εm ( y ) V ( x, √ n ) (cid:19) = c d m ( y ) V ( x, √ n ) , as we desired.Thus it remains to prove that (5.5) can be satisfied. First we consider another, slightlymodified, Markov kernel p ′ = p − α . Such a kernel is generated by weights ω ′ xy as follows: ω ′ xx = ω xx − αm ( x )1 − α ≥ αm ( x ) , ∀ x ∈ V,ω ′ xy = ω xy − α , ∀ x = y ∈ V,m ′ ( x ) = m ( x ) . Note that ∆( α ) is true in G with the new weights. The condition CDE ′ ( n ,
0) is also stillholds for the new weights, because if one lets ∆ ′ be the new Laplacian for ω ′ xy , then for any f, g ∈ V R we obtain: ∆ ′ f ( x ) = 11 − α ∆ f ( x ) , Γ ′ ( f, g ) = 11 − α Γ( f, g ) , Γ ′ ( f, g ) = 1(1 − α ) Γ ( f, g ) , e Γ ′ ( f ) = 1(1 − α ) e Γ ( f ) . Furthermore, the process of proving DV ( C ) still works when adding loops to every point ofgraph. Then DV ( C ) is still satisfied for the new weights. This yields p ′ k ( x, y ) ≤ C ′ d m ( y ) V ( x, √ k ) , and hence p k ( x, y ) ≤ C ′ d m ( y )(1 − α ) k V ( x, √ k ) . Next, we have to get the estimate as follows e ( α − n X | k − (1 − α ) n | >a √ n ((1 − α ) n ) k k ! 1 V ( x, √ k ) ≤ ε ′ V ( x, √ n ) . The sum for k > a √ n + (1 − α ) n is easier because we simply use V ( x, √ k ) ≥ V (cid:18) x, r n (cid:19) ≥ V (cid:18) x, √ n (cid:19) ≥ V ( x, √ n ) C ,
29o we have, e ( α − n X k>a √ n +(1 − α ) n ((1 − α ) n ) k k ! 1 V ( x, √ k ) ≤ e ( α − n C V ( x, √ n ) X k>a √ n +(1 − α ) n ((1 − α ) n ) k k ! ≤ e ( α − n C V ( x, √ n ) ((1 − α ) n ) (1 − α ) n + a √ n ( a √ n + (1 − α ) n )! 11 − (1 − α ) na √ n +(1 − α ) n ≤ CC V ( x, √ n ) exp (cid:18) a √ n − ( a √ n + (1 − α ) n ) log (cid:18) a (1 − α ) √ n (cid:19)(cid:19) · p a √ n + (1 − α ) n a √ n + (1 − α ) na √ n ≤ ε ′ V ( x, √ n ) , for our (arbitrary) choice of ε ′ , so long as a is sufficiently large. Note that the second to lastinequality follows from the fact that k ! ≥ k k e − k √ kC . To see that this last line holds, note thatas we assume n ≥ a α , the inequality √ a √ n +(1 − α ) n a √ n +(1 − α ) na √ n ≤ a holds. Finally observe that,by the real number inequality log(1 + u ) ≥ u u + u u ) , then the exponential is negativeand for a sufficiently large a the claim holds.Remark 4 allows us to deal with 1 ≤ k < − a √ n + (1 − α ) n . It gives V ( x, √ k ) ≤ C √ k √ k − ! log C log 2 V ( x, √ k − ≤ C V ( x, √ k − . Thus, for the terms 1 ≤ k ≤ (1 − a ) n C , we have((1 − α ) n ) k − ( k − V ( x, √ k − ≤
12 ((1 − α ) n ) k k ! 1 V ( x, √ k ) , and the estimate is straightforward. For the other term (1 − a ) n C < k < − a √ n + (1 − α ) n , andusing Remark 4 again V ( x, √ k ) ≥ V x, s (1 − a ) n C ≥ C V ( x, √ n ) . This completes the proof.In proving the upper bounds of the discrete-time Gaussian estimate on graphs, it is usefulto introduce the following result from [CG98].30 heorem 5.3.
For a reversible nearest neighborhood random walk on the locally finite graph G = ( V, E ) , the following properties are equivalent:1. The relative Faber-Krahn inequality ( F K ) .2. The discrete-time Gaussian upper estimate in conjunction with the doubling property DV ( C ) .3. The discrete-time on-diagonal upper estimate in conjunction with the doubling property DV ( C ) . Now we show the final theorem of the discrete-time Gaussian estimate.
Theorem 5.4.
Assume a graph G satisfies CDE ′ ( n , and ∆( α ) , then the graph satisfiesthe discrete-time Gaussian estimate G ( c l , C l , C r , c r ) .Proof. We have already observed that the discrete-time on-diagonal upper estimate and thedoubling property DV ( C ) both hold for graphs satisfying CDE ′ ( n ,
0) and ∆( α ). Theo-rem 5.3 immediately implies the discrete-time Gaussian upper estimate.The lower bound follows from the on-diagonal one. The strategy is similar to Delmotte of[D99]. We repeatedly apply the second assertion of Proposition 5.2. Set n = n + n + · · · + n j , x = x , x , · · · , x j = y and B = x , B i = B ( x i , r i ), B j = y , such that j − ≤ C d ( x,y ) n ,r i ≥ c √ n i +1 , so that V ( z, √ n i +1 ) ≤ AV ( B i ), when z ∈ B i ,sup z ∈ B i − ,z ′ ∈ B i d ( z, z ′ ) ≤ n i , so that p n i ( z, z ′ ) ≥ c d m ( z ′ ) V ( z, √ n i ) .Such a decomposition allows us to immediately derive the lower bound. Indeed, p n ( x, y ) ≥ X ( z , ··· ,z j − ) ∈ B ×···× B j − p n ( x, z ) p n ( z , z ) · · · p n j ( z j − , y ) ≥ X ( z , ··· ,z j − ) ∈ B ×···× B j − c d m ( z ) V ( x, √ n ) c d m ( z ) V ( z , √ n ) · · · c d m ( y ) V ( z j − , √ n j ) ≥ c jd A − j X ( z , ··· ,z j − ) ∈ B ×···× B j − m ( z ) V ( x, √ n ) m ( z ) V ( B ) · · · m ( y ) V ( B j )= c d m ( y ) V ( x, √ n ) (cid:16) c d A (cid:17) j − . If we choose C l ≥ C log( Ac d ), and V ( x, √ n ) ≤ V ( x, √ n ), the Gaussian lower bound p n ( x, y ) ≥ c d m ( y ) V ( x, √ n ) e − C l d ( x,y )2 n , and thus the theorem, follows. 31ow from the discrete-time Gaussian estimate, we can get the continuous-time Gaussianlower bound estimate when t is small enough we have mentioned before in Theorem 5.1, asfollows. Remark 5. If t is small enough, under CDE ′ ( n , x, y ∈ V , P t ( x, y ) ≥ C ′ m ( y ) V ( x, √ t ) exp (cid:18) − c ′ d ( x, y ) t (cid:19) holds too. Proof. If d ( x, y ) = 0, P t ( x, y ) = m ( y ) p ( t, x, y ) →
1, when t → + . It naturally satisfies thelower bound.Now we consider d ( x, y ) >
0. When k < d ( x, y ), then p k ( x, y ) = 0. When k ≥ d ( x, y ),form the discrete-time Gaussian lower bound, the relationship between the continuous-timeheat kernel and the discrete-time heat kernel (5.1), and the polynomial volume growth ofCorollary 7.8 in [BHLLMY13], which says under CDE ( n , C > V ( x, √ t ) ≤ C µ ( x ) t n , for CDE ′ ( n , ⇒ CDE ( n , µ ( x ) = m ( x ) in this section, we obtain P t ( x, y ) = e − t + ∞ X k =0 t k k ! p k ( x, y ) ≥ e − t + ∞ X k = d ( x,y ) t k k ! c d m ( y ) V ( x, √ k ) e − C l d ( x,y )2 k ≥ e − t + ∞ X k = d ( x,y ) t k k ! c d m ( y ) C k n m ( x ) e − C l d ( x,y ) ≥ c d e − t C m ( y ) m ( x ) · e − C l d ( x,y ) t d ( x,y ) d ( x, y ) n d ( x, y )! ≥ C ′ m ( y ) V ( x, √ t ) exp (cid:18) − c ′ d ( x, y ) t (cid:19) , ( t → + ) . In the last step, when t is small enough, V ( x, √ t ) = m ( x ), and e − t is bounded. Furthermore,for any x, y ∈ V , m = d ( x, y ) is finite in connected graphs, and it mainly dues to if t → + ,the following function for all m ∈ Z + , f ( t, m ) = − tm ln (cid:18) e − C l m t m m n m ! (cid:19) has positive bounds. Since for all m ∈ Z + , f ( t, m ) = − t ln tm + t (cid:18) C l m + n ln mm + ln m ! m (cid:19) → , when t → + . So we can always find a c ′ > d ( x, y ) and the above inequalityholds. This completes the proof. 32ere we have the following result. Theorem 5.5.
If the graph satisfies
CDE ′ ( n , and ∆( α ) , we have the following fourproperties. There exists C , C , α > such that DV ( C ) , P ( C ) , and ∆( α ) are true. There exists c l , C l , C r , c r > such that G ( c l , C l , C r , c r ) is true. There exists C H such that H ( η, θ , θ , θ , θ , C H ) is true. ′ There exists C H such that H ( η, θ , θ , θ , θ , C H ) is true.Proof. The condition
CDE ′ ( n ,
0) implies DV ( C ) (see Theorem 4.2), and Theorem 5.4states that CDE ′ ( n ,
0) and ∆( α ) implies G ( c l , C l , C r , c r ). According to Delmotte of [D99], P ( C ) is true. Moreover, 3) and 3) ′ hold too. In this section we will show another application of Theorem 3.2. We prove that positivelycurved graphs (that is graphs satisfying
CDE ′ ( n, K ) for some K > G associated with a Laplace operator ∆ : e d ( x, y ) = sup f ∈ ℓ ∞ ( V,µ ) , k Γ( f ) k ∞ ≤ | f ( x ) − f ( y ) | , x, y ∈ V. e D = sup x,y ∈ V e d ( x, y ) . In this section we are concerned with simple, connected and loopless graphs.
In this subsection we derive a global heat kernel bound by proving finite measure underthe assumption of positive curvature on graphs, and use this to establish that the diameteris finite. We accomplish this in several steps the most crucial of them being an estimateproving that the total measure of the graph is finite.In Theorem 3.2, we choose the function γ in a such a way that, α ′ − αγn + 2 αK = 0 , that is γ = n (cid:18) α ′ α + 2 K (cid:19) . T , we obtain α ( T ) P T (Γ( √ f )) P T f − α (0) Γ( √ P T f ) P T f ≥ n (cid:18)Z T αγdt (cid:19) ∆ P T ( f ) P T f − n Z T αγ dt. (6.1)Now we introduce the first result in this subsection. Proposition 6.1.
Let G = ( V, E ) be a locally finite, connected graph satisfying CDE ′ ( n, K ) .Then for all < θ < K and t > , there exists a C > such that for every non-negative f satisfying || f || ∞ ≤ , and every t ≥ t , | p P t f ( x ) − p P t f ( y ) | ≤ C e − θ t e d ( x, y ) , x, y ∈ V. Remark:
Of course it is easy to replace the assumption that || f || ∞ ≤ || f || ∞ ≤ M for any M ≥ Proof.
Fix some 0 < θ < K and some 0 < t ≤ T . We show the inequality holds at time T assuming T is sufficiently large. In (6.1), we take α ( t ) = 2 Ke − θt ( e − θt − e − θT ) K/θ − , so that α (0) = 2 K (1 − e − θT ) K/θ − , and α ( T ) = 0 . With such choice a simple computation gives, γ = n (cid:18) − e − θT K − θe − θt − e − θT (cid:19) , which is non-positive for 0 ≤ t ≤ T .Then, for any T > − Kn (1 − e − θT ) K/θ − Γ( √ P T f ) P T f ≥ (cid:18)Z T αγdt (cid:19) ∆ P T ( f ) P T f − Z T αγ dt. (6.2)Now, we can compute Z T αγdt = − nK − e − θT ) K/θ − ( e − θT ) , and Z T aγ dt = Kn − e − θT ) K/θ − e − θT × (cid:18) θ (2 K/θ − K/θ − (cid:19) . We thus obtain from (6.2), that for any
T > t ≥ ≥ − Kn (1 − e − θT ) K/θ − Γ( √ P T f ) P T f ≥ − nK − e − θT ) K/θ − e − θT ∆ P T fP T f − Kn θ (2 K/θ − K/θ −
2) (1 − e − θT ) K/θ − e − θT . (6.3)34ividing, and switching notation from T to t , we obtain thatΓ( p P t f ) ≤ e − θt ∆ P t f + nθ (2 K/θ − K/θ − − e − θt ) e − θt P t f ≤ C e − θt , (6.4)with C = q D µ + nθ (2 K/θ − K/θ − e θt − .We consider the function u ( x ) = C e θ t √ P t f ( x ) ∈ ℓ ∞ ( V, µ ). By construction, we havenormalized u so that for any t ≥ t , k Γ( u ) k ∞ ≤
1. By the definition of the canonical distance e d ( x, y ), | u ( x ) − u ( y ) | ≤ e d ( x, y ) , In turn, | p P t f ( x ) − p P t f ( y ) | ≤ C e − θ t e d ( x, y ) . as desired. Proposition 6.2.
Let G = ( V, E ) be a locally finite, connected graph satisfying CDE ′ ( n, K ) .Then for all < θ < K and t > , there exists a C > such that for every non-negativefunction f with || f || ∞ ≤ , and for every t ≥ t , | ∂ t P t f | ≤ C e − θ t . (6.5) Proof.
Let P t f = u , we have | ∆ u | ≤ ] X y ∼ x | u ( y ) − u ( x ) | = ] X y ∼ x (cid:0) √ u ( y ) + √ u ( x ) (cid:1) |√ u ( y ) − √ u ( x ) |≤ ] X y ∼ x (cid:16)p u ( y ) + p u ( x ) (cid:17) ! ] X y ∼ x (cid:16)p u ( y ) − p u ( x ) (cid:17) ! ≤ p D µ C q Γ( √ u ) . Combing with (6.4), we let C = 2 p D µ · C . This yields the desired result. Proposition 6.3.
Let G = ( V, E ) be a locally finite, connected graph satisfying CDE ′ ( n, K ) with K > . Then the measure µ is finite, that is, µ ( V ) < ∞ .Proof. According to Proposition 6.2 the limit of p ( t, x, · ) exists and is finite when t → ∞ .Moreover Proposition 6.1 and the property 2 in Remark 3, imply that lim t →∞ p ( t, x, · ) issome non-negative c ( x ) ≥
0. The symmetry of the heat kernel implies that c ( x ) actuallydoes not depend on x . 35o show the finiteness of the measure, it will suffice to prove that this limit is actuallystrictly positive under the assumption of CDE ′ ( n, K ) for some K > t > t to ∞ to obtainlim t →∞ p ( t, x, · ) − p ( t , x, y ) = Z ∞ t ∂ t p ( t, x, · ) dt ≥ − C s e − θ t , Let y = x . Theorem 7 of [BHLLMY13] states that there is some constant C ′ >
0, so that p ( t , x, x ) ≥ C ′ t n , under the condition CDE ( n, CDE ( n, K ) – and hence CDE ′ ( n, K ) – for any K >
0. Thus combining: lim t →∞ p ( t, x, x ) ≥ C ′ t n − C θ e − θ t > . This implies that lim t →∞ p ( t, x, y ) = c >
0, for any x, y ∈ V . This, in turn, (from (3) inRemark 3) implies that the measure µ is finite.Finally we introduce the following result (see [GH14]) which says the properties of infinitemeasure and infinite diameter are equivalent properties for a locally compact separable metricspace M , so long as the volume doubling ( DV ) holds. Lemma 6.1.
Assume that ( M, d ) is connected and satisfies DV . Then µ ( M ) = ∞ ⇔ diam ( M ) = ∞ . On graphs, we have the same assertion, and let the above d is the natural distance ongraphs. Since Theorem 4.2, we have already got DV under the assumption CDE ′ ( n, CDE ′ ( n, K ) ⇒ CDE ′ ( n, K >
0. Combining with Proposition 6.3 and the firstequivalence in Lemma 6.1, we get the finiteness of diameter as follows.
Theorem 6.4.
Every locally finite, connected, simple graph satisfying
CDE ′ ( n, K ) with K > has finite diameter in terms of the natural graph distance. Per Proposition 6.3, we may assume µ is probability measure – renormalizing so thatlim t →∞ p ( t, x, · ) = 1. Proposition 6.5.
Suppose G is a connected, locally finite graph satisfying CDE ′ ( n, K ) with K > . Then for any x, y ∈ V , t > , p ( t, x, y ) ≤ (cid:16) − e − K t (cid:17) n . roof. We apply (6.3) with θ = 2 K/
3. Considering p ( τ, x, y ), we obtain ∂ τ log p ( τ, x, y ) ≥ − nK e − θτ − e − θτ . By integrating from t to ∞ , and as lim t →∞ p ( t, x, y ) = 1, we have p ( t, x, y ) ≤ − e − θt ) n . This ends the proof.
In this subsection we derive an explicit diameter bound for graphs satisfying
CDE ′ ( n, K ).The idea is to prove the operator ∆ satisfies an entropy-energy inequality, as mentionedin the introduction. First we derive, for graphs, an analogue of Davies’ theorem([DB89]) onmanifolds. Note that, obviously, if µ is a finite measure, f ∈ ℓ ∞ ( V, µ ) implies f ∈ ℓ p ( V, µ )for any p > Lemma 6.2.
Suppose G is a locally finite, connected graph with µ ( V ) bounded. Let f ∈ ℓ ∞ ( V, µ ) , satisfying k P t f k ∞ ≤ e M ( t ) k f k , for some continuous and decreasing function M ( t ) .If k f k = 1 , then for any t > , X x ∈ V µ ( x ) f ( x ) ln f ( x ) ≤ t X x ∈ V µ ( x )Γ( f )( x ) + 2 M ( t ) . Proof.
Let p ( s ) be a bounded, continuous function so that p ( s ) ≥ p ′ ( s ) bounded. Forany function 0 ≤ f ∈ ℓ ∞ ( V, µ ), consider the function ( P s f ) p ( s ) . Note ( P s f ) p ( s ) ∈ ℓ ( V, µ ).Likewise, so are the functions ( P s f ) p ( s ) ln P s f and ∆ P s f ( P s f ) p ( s ) − ∈ ℓ ( V, µ ). (Note here,that at s = 0, if f = 0, we take ( P s f ) p ( s ) ln P s f to be zero as well.) so we have dds k P s f k p ( s ) p ( s ) = dds X x ∈ V µ ( x )( P s f ( x )) p ( s ) = X x ∈ V µ ( x ) dds ( P s f ( x )) p ( s ) = X x ∈ V µ ( x ) (cid:0) p ′ ( s )( P s f ( x )) p ( s ) ln P s f ( x ) + p ( s )( P s f ( x )) ′ ( P s f ( x )) p ( s ) − (cid:1) = p ′ ( s ) X x ∈ V µ ( x )( P s f ( x )) p ( s ) ln P s f ( x ) + p ( s ) X x ∈ V µ ( x )∆ P s f ( x )( P s f ( x )) p ( s ) − . At s = 0, and specializing to p ( s ) = tt − s (where 0 ≤ s ≤ t − t , with t > t > dds k P s f k p ( s ) p ( s ) | s =0 = 2 t X x ∈ V µ ( x ) f ( x ) ln f ( x ) + 2 X x ∈ V µ ( x ) f ( x )∆ f ( x ) .
37n the other hand, we give a lower bound on this derivative. Combining our assumptionthat k P t f k ∞ ≤ e M ( t ) k f k , for continuous and decreasing M ( t ) and our assumption that k f k = 1, and using the Stein interpolation theorem, we obtain k P s f k p ( s ) ≤ e M ( t ) st . Then we obtain dds k P s f k p ( s ) p ( s ) | s =0 ≤ M ( t ) t . We achieve this using the fact that k P s f k p ( s ) p ( s ) | s =0 = k p ( s ) k = 1, and e M ( t ) sp ( s ) t | s =0 = 1.Directly computing yields1 ≥ lim s → + k P s f k p ( s ) p ( s ) − e M ( t ) sp ( s ) t − dds k P s f k p ( s ) p ( s ) | s =0 t M ( t ) . Noting the identity − P x ∈ V µ ( x ) f ( x )∆ f ( x ) = P x ∈ V µ ( x )Γ( f )( x ) holds for any f ∈ ℓ ∞ ( V, µ ), and combining with the above equality, we obtain X x ∈ V µ ( x ) f ( x ) ln f ( x ) ≤ t X x ∈ V µ ( x )Γ( f )( x ) + 2 M ( t ) , t > t . This completes the proof.
Proposition 6.6.
Let G = ( V, E ) be a locally finite, connected graph satisfying CDE ′ ( n, K ) .Any ≤ f ∈ ℓ ∞ ( V, µ ) such that k f k = 1 satisfies X x ∈ V µ ( x ) f ( x ) ln f ( x ) ≤ Φ X x ∈ V µ ( x )Γ( f )( x ) ! , where Φ( x ) = 2 n (cid:20)(cid:18) θn x (cid:19) ln (cid:18) θn x (cid:19) − θn x ln (cid:18) θn x (cid:19)(cid:21) . Proof.
Fix such an f . Using Proposition 6.5 and the Cauchy-Schwartz inequality, we have k P t f k ∞ ≤ − e − θt ) n k f k , where θ = K . Therefore from Lemma 6.2, we obtain X x ∈ V µ ( x ) f ( x ) ln f ( x ) ≤ t X x ∈ V µ ( x )Γ( f )( x ) − n ln(1 − e − θt ) , t > t > . By minimizing the right-hand side of the above inequality over t , we obtain X y ∈ V µ ( y ) f ( y ) ln f ( y ) ≤ − θ x ln (cid:18) xx + θn (cid:19) + 2 n ln (cid:18) x + θnθn (cid:19) = 2 n (cid:20)(cid:18) θn x (cid:19) ln (cid:18) θn x (cid:19) − θn x ln (cid:18) θn x (cid:19)(cid:21) , where x = P y ∈ V µ ( y )Γ( f )( y ). 38e observe that Φ is a non-negative, monotonically increasing, and concave function, aswe shall use these properties later.In order to finish the result, and bound the diameter we first need to introduce somenotation. For a positive bounded real valued function f on V , let E ( f ) denote the entropyof f with respect to µ defined by E ( f ) = X x ∈ V µ ( x ) f ( x ) ln f ( x ) − X x ∈ V µ ( x ) f ( x ) ln X x ∈ V µ ( x ) f ( x ) ! . To ease the notation, we use R f dµ = P x ∈ V µ ( x ) f ( x ). The Laplace operator ∆ satisfies a logarithmic Sobolev inequality if there exists a ρ > f ∈ ℓ ∞ ( V, µ ), ρE ( f ) ≤ Z Γ( f ) dµ, Equivalently, it suffices to say that a logarithmic Sobolev inequality holds if all f ∈ ℓ ∞ ( V, µ ) with k f k = 1 satisfy E ( f ) ≤ Φ (cid:18)Z Γ( f ) dµ (cid:19) (6.6)where Φ is a concave and non-negative function on [0 , ∞ ). Proposition 6.7.
Suppose ∆ satisfies a general logarithmic Sobolev inequality, and the func-tion Φ from (6.6) is non-negative and monotonically increasing. Then G has diameter e D ≤ √ Z ∞ x Φ( x ) dx. Proof.
Consider any g ∈ ℓ ∞ ( V, µ ), with k Γ( g ) k ∞ ≤
1. Let f λ = e λg for some λ ∈ R + . Wewill apply (6.6) to the family of non-negative functions e f λ = f λ/ k f λ/ k . Let G ( λ ) = k f λ/ k = R e λg dµ and observe that G ′ ( λ ) = R ge λg dµ (cid:16) = λ R f λ/ ln f λ/ dµ (cid:17) .On one hand, it is immediate by the definition that, e f , E ( e f λ ) = 1 G ( λ ) ( λG ′ ( λ ) − G ( λ ) ln G ( λ )) . We also must consider the right hand side of the Sobolev inequality, which contains a39erm of the form R Γ( e f λ ) dµ = k f λ/ k R Γ( e λg ) dµ . Such terms can be bounded as follows: Z Γ( e λg ) dµ = 12 X x ∈ V µ ( x ) X y ∼ x ω xy (cid:16) e λg ( y )2 − e λg ( x )2 (cid:17) = 12 X x ∈ V µ ( x ) X y ∼ xg ( x ) > g ( y ) ω xy (cid:16) e λg ( y )2 − e λg ( x )2 (cid:17) + 12 X x ∈ V µ ( x ) X y ∼ xg ( x ) < g ( y ) ω xy (cid:16) e λg ( y )2 − e λg ( x )2 (cid:17) = X x ∈ V µ ( x ) X y ∼ xg ( x ) > g ( y ) ω xy (cid:16) e λg ( y )2 − e λg ( x )2 (cid:17) ≤ X x ∈ V µ ( x ) X y ∼ xg ( x ) > g ( y ) ω xy (cid:16) e λ ( g ( y ) − g ( x )) − (cid:17) e λg ( x ) ≤ λ X x ∈ V µ ( x ) e λg ( x ) X y ∼ xg ( x ) > g ( y ) ω xy ( g ( y ) − g ( x )) ≤ λ Z e λg Γ( g ) dµ. Since Γ( g ) ≤
1, and the function Φ is monotonically increasingΦ (cid:18)Z Γ( e f λ ) dµ (cid:19) = Φ (cid:18) k f λ/ k Z Γ( e λg ) dµ (cid:19) ≤ Φ (cid:18) λ (cid:19) . By the logarithmic Sobolev inequality λG ′ ( λ ) − G ( λ ) ln G ( λ ) ≤ G ( λ )Φ (cid:18) λ (cid:19) . Let H ( λ ) = λ ln G ( λ ). Then the above inequality reads H ′ ( λ ) ≤ λ Φ (cid:18) λ (cid:19) . Since H (0) = lim λ → λ ln G ( λ ) = R gdµ , it follows that H ( λ ) = H (0) + Z λ H ′ ( u ) du ≤ Z gdµ + Z λ u Φ (cid:18) u (cid:19) du. Therefore for any λ ≥ X x ∈ V µ ( x ) e λ ( g ( x ) − R gdµ ) ≤ exp (cid:26) λ Z λ u Φ (cid:18) u (cid:19) du (cid:27) . (6.7)40et C = R ∞ u Φ (cid:16) u (cid:17) du = √ R ∞ x Φ( x ) dx . Then, for every λ ≥ ǫ > g and − g and apply Chebyshev’s inequality, µ ( { x ∈ V : | g ( x ) − Z gdµ | ≥ C + ε } ) ≤ X x ∈ Vg ( x ) ≥ R gdµ + C + ε µ ( x ) + X x ∈ Vg ( x ) ≤ R gdµ − C − ε µ ( x ) ≤ X x ∈ Vg ( x ) ≥ R gdµ + C + ε e λ ( g ( x ) − R gdµ ) e λ ( C + ε ) µ ( x ) + X x ∈ Vg ( x ) ≤ R gdµ − C − ε e λ ( − g ( x )+ R gdµ ) e λ ( C + ε ) µ ( x ) ≤ e − λ ( C + ε ) e λC = 2 e − λε → λ → ∞ ) . That is, we obtain k g ( x ) − Z gdµ k ∞ ≤ C. The diameter bounds follows immediately by the definition of e D : Since g was arbitrary, e D ≤ √ Z ∞ x Φ( x ) dx. That completes the proof.Finally, we obtain
Theorem 6.8.
Let G = ( V, E ) be a locally finite, connected graph satisfying CDE ′ ( n, K ) ,and K > , then the diameter e D of graph G in terms of canonical distance is finite, and e D ≤ √ π r nK . Proof.
Combining Proposition 6.6 and Proposition 6.7, graphs satisfying
CDE ′ ( n, K ) forsome K >
0, also satisfy e D ≤ √ Z ∞ x Φ( x ) dx, where Φ( x ) = 2 n (cid:2)(cid:0) θn x (cid:1) ln (cid:0) θn x (cid:1) − θn x ln (cid:0) θn x (cid:1)(cid:3) , and θ = K . Since Z ∞ x Φ( x ) dx = 12 Z ∞ x Φ( x ) dx = Z ∞ √ x Φ ′ ( x ) dx = − Z ∞ √ x Φ ′′ ( x ) dx < ∞ then the diameter is finite, and Φ ′′ ( x ) = − nx ( x + θn ) , then − Z ∞ √ x Φ ′′ ( x ) dx = 4 π r nθ , so we have completed the proof. 41hile this bounds the canonical diameter, it is possible to recover a bound for the usualgraph distance. In order to accomplish this, we first introduce the notion of an intrinsicmetric. This gives us a way to relate natural distance with the canonical distance.Intrinsic metrics on graphs was first introduced by R. Frank, D. Lenz and D. Wingert in[FLW14]. A function ρ : V × V → R + is called an intrinsic metric if, at all x ∈ V X y ∼ x ω xy ρ ( x, y ) ≤ µ ( x ) . This induces a metric ρ on a graph via finding shortest paths. One example of such afunction, introduced by Xueping Huang in his thesis ([H11]), is to define for all for all x ∈ V and y ∼ x , e ρ ( x, y ) = min (s µ ( x ) m ( x ) , s µ ( y ) m ( y ) ) , where m ( x ) = P y ∼ x ω xy .As mentioned, these metrics give a way of comparing distances with the intrinsic distancewe have been using. Indeed, part ( a ) of the remark following Definition 1.2 in [KLSW15]gives: Proposition 6.9.
For any x, y ∈ V , it holds that √ e ρ ( x, y ) ≤ e d ( x, y ) . Actually from [KLSW15], the above inequality is true for any intrinsic metric. For themetric ˜ ρ however, under the assumption D µ is finite, then e ρ ( x, y ) ≥ d ( x, y ) p D µ for any x and y (by, again, extending the metric ˜ ρ along shortest paths.)The above inequality and Theorem 6.8 and Proposition 6.9 combine to prove the followinginequality Theorem 6.10.
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