Ricci curvature and geometric analysis on Graphs

Transcription

1 Ricci curvature and geometric analysis on Graphs Yong Lin Renmin University of China July 9, 2014

2 Ricci curvature on graphs 1

3 Let G = (V, E) be a graph, where V is a vertices set and E is the set of edges. For x, y V, x y means that x is adjacent to y. Let d x denote the degree of the vertex x. If d x < + for all x V, we say that G is a locally finite graph. If d x is same for every x, we say that the graph is a regular graph. For two vertices x and y, the distance between x and y is the number of edges in the shortest path joining x and y. The diameter of a graph G is the maximum distance between any two vertices of G. We always assume that G is connected, which means that any two vertices of G can be connected by a path in G. 2

4 The first definition of Ricci curvature was introduced by Fan Chung Graham and S. T. Yau in In the course of obtaining a good log-sobolev inequality, they found the following definition of Ricci curvature to be useful: We say that a regular graph G has a local k-frame at a vertex x if there exists injective mappings η 1,..., η k from a neighborhood of x into V so that 1. x is adjacent to η i x for 1 i k ; 2. η i x η j x if i j. The graph G is said to be Ricci-flat at x if there is a local k-frame in a neighborhood of x so that for all i, ( ) ( ) ηi η j x = ηj η i x. j j 3

5 For a more general definition of Ricci curvature, we need the following. We first define the Laplace operator on graph without loops and multiple edges. The description in the following can be used for weighted graphs. But for simplicity, we set all weights here equal to 1. Let V R = {f f : V R}. The Laplace operator of a graph G is, for all f V R f (x) = 1 [f (y) f (x)]. d x y x For graphs, we have f (x) 2 = 1 [f (y) f (x)] 2. d x y x 4

6 We first introduce a bilinear operator Γ : V R V R V R by Γ(f, g)(x) = 1 { (f (x) g(x)) f (x) g(x) g(x) f (x)}. 2 The Ricci curvature operator Γ 2 is defined by iterating the Γ: Γ 2 (f, g)(x) = 1 { Γ(f, g)(x) Γ(f, g)(x) Γ(g, f )(x)}. 2 Definition The Laplace operator on graphs satisfies the curvaturedimension type inequality CD(m, K ) (m (1, + ]) if Γ 2 (f, f )(x) 1 m ( (f (x)))2 + k(x) Γ(f, f )(x) We call m the dimension of the operator and k(x) the lower bound of the Ricci curvature of the operator. 5

7 It is easy to see that for m < m, the operator satisfies CD(m, k) if it satisfies CD(m, k). Remark We find that: ( f (x) 2 ) = 1 1 [f (x) 2f (y) + f (z)] 2 d x d x y x y x z y 2 1 [f (x) 2f (y) + f (z)][f (x) f (y)] d x d y z y and Γ(f, f )(x) = 1 1 [f (y) f (x)][ f (y) f (x)]. 2 d y y x 6

8 From the definition of the Ricci curvature operator, we obtain: Γ 2 (f, f )(x) = [f (x) 2f (y) + f (z)] 2 4 d x d y y x z y 1 1 [f (y) f (x)] 2 2 d x y x [ 1 (f (y) f (x))] 2. d x y x We see that the lower bound of the Ricci curvature k(x) only depends on those vertices with distant at most 2 from vertex x. So we can choose the function f (x) to be supported only on those vertices. 7

9 Suppose is the Laplace-Beltrami operator on a m-dimension complete connected Riemannian manifold M. Bochner s formula indicates that Γ 2 (f, f )(x) = Ric( f (x), f (x)) + Hessf (x) 2 2, where Ric is the Ricci tensor on M and Hessf 2 is the Hilbert-Schmidt norm of the tensor of the second derivative of f. Since Hessf m ( f )2, so we say that satisfies the CD(m, k) if and only if the Ricci curvature at x on M is bounded below by k. 8

10 In 1985, Bakry and Emery considered these notions on metric measure space where the operator is the so-called diffusion operator, that is satisfies the following chain-rule formula: for every C function Φ on R and every function f, φ(f ) = φ (f ) f + φ (f )Γ(f, f ). The Laplace operator on graphs does not satisfy the chain-rule formula. But by the following Lemma, it is the correct operator for defining the Ricci curvature operator on graphs. 9

11 In the classical case, Γ(f, f ) = 1 2 (f ) 2. On graphs, we also have Lemma Suppose G is a locally finite graph and is the Laplace operator on G, then Γ(f, f ) = 1 2 f 2. 10

12 Proof. (f 2 (x)) = 1 [f 2 (y) f 2 (x)] d x y x = 1 [f (y) f (x)] [f (y) + f (x)] d x y x = 2 [f (y) f (x)] f (x) + 1 [f (y) f (x)] 2 d x d x y x y x = 2f (x) f (x) + f (x) 2. So Γ(f, f ) = 1 2 [ (f )2 2f f ] = 1 2 (f ) 2. 11

13 For the Ricci curvature operator on graphs, we have the following theorem. Theorem (Y. Lin, S. T. Yau) Suppose G is a locally finite graph and d = sup x V d x (where d can be infinite). Then we have Γ 2 (f, f ) 1 2 ( f )2 + ( 1 d 1)Γ(f, f ) i.e. the Laplace operator on G satisfies CD(2, 1 d 1). Remark The Ricci-flat graph in the sense of Chung and Yau satisfies CD(, 0). 12

14 We also can obtain an eigenvalue estimate for the finitely connected graph. Suppose a function f (x) V R satisfies ( )f (x) = λf (x), then f (x) is called the eigenfunction of the Laplace operator on G with eigenvalue λ. 13

15 We prove the following result which is similar to the classical result of Li and Yau on compact manifold with Ricci curvature bounded below. Theorem (Y. Lin, S. T. Yau) Suppose G is a connected graph with diameter D, then the non-zero eigenvalue of Laplace operator on G λ 1 d D(exp(d D + 1) 1). 14

16 Notice that there is a well-known estimate for the eigenvalue of Laplacian on graphs where VolG = x V d x. λ 1 D VolG, The proof of the above Theorem is similar to the proof of Li and Yau on manifold by using the following gradient estimate where β > 1. f (x) 2 (β f (x)) 2 d( 1 β 1 λ + 1)2, 15

17 For the graph with Ricci curvature bounded below by a positive number, There are the following eigenvalue estimate which is similar to the Lichnerowicz theorem in Riemannian manifold. We can also show the estimate is shape for m = and m = 2. Theorem (Fan Chung, Y. Lin, Y. Liu) Suppose a finite graph G satisfies the CD(m, k) with k > 0, then the non-zero eigenvalue of on G λ m m 1 k. 16

18 Harnack inequality and eigenvalue estimate on graphs 17

19 Theorem (Fan Chung, Y. Lin, S. T. Yau) Suppose a finite graph G satisfies the CD(m, k), f V R is an eigenfunction of Laplacian with eigenvalue λ. Then the following inequality holds for all x V. f (x) 2 [(8 2 m )λ 4k] sup f 2 (x). z V As a consequence, we can get an eigenvalue estimate for graphs. 18

20 Theorem (Fan Chung, Y. Lin, S. T. Yau) Suppose a finite connected graph G satisfies CD(m, k) and λ is a non-zero eigenvalue of Laplace operator on G. Then λ 1 + 4kdD2 d (8 2 m ) D2 where d is the maximum degree and D denotes the diameter of G. 19

21 Remark This means we still have an eigenvalue lower bound for graphs with Ricci curvature bounded below by some negative number, that is when k > 1 4dD 2. 20

22 For graph with non-negative Ricci curvature. We have. Corollary Suppose a finitely connected graph G satisfies CD(m, 0), then the non-zero eigenvalue of on G λ 1 (8 1. m )d D2 Remark Fan Chung and Yau proved a similar result for so-called Ricci-flat graphs. The Ricci-flat graphs satisfy CD(, 0). Chung and Yau gave an example that showed their eigenvalue estimate λ 1 8d D 2 is a sharp one for Ricci-flat graph. If m <, then we have a better estimate. For example, if G is a triangle graph with three vertices, then m = 2. 21

23 This is the key lemma to prove the functional inequalities in this section. Lemma Suppose G is a finite connected graph satisfying CD(m, k), then for x V, we have ( 4 m 2)( f )2 +(2+2k) f d x d y xy E yz E [f (x) 2f (y)+f (z)] 2. 22

24 By using the above lemma, from the CD(m, 0) condition, we can obtain ( f (x) 2 ) 2λ f (x) 2 4 m λ2 f (x). Meanwhile, we have f 2 (x) = 2λf 2 (x) f (x) 2. We can then use a maximum principle for function f 2 (x) + αλf 2 (x) as Chung and Yau did. For the proof of the corollary, we shall note that for the eigenfunction f (x), we have x V d xf (x) = 0. 23

25 Li-Yau inequality on graphs 24

26 This is a joined work with: Frank Bauer, Paul Horn, Gabor Lippner, Dan Mangoubi, Shing-Tung Yau 25

27 In their celebrated work, Li and Yau proved an upper bound on the gradient of positive solutions of the heat equation. In its simplest form,for an n-dimensional compact manifold with non-negative curvature the inequality, now known as the Li-Yau gradient estimate or Li-Yau inequality, states that a positive solution of the heat equation ( t)u = 0 satisfies u 2 u 2 u t u n 2t. (1) 26

28 Gradient estimates like (1) proved to be powerful tools in geometric analysis. Li and Yau originally used Gradient estimates like (1) to derive a strong form of the Harnack inequality for general manifolds. The parabolic Harnack inequality relates the minimum and maximum values of a positive solution to the heat equation over time and space. Its strong form gives more general bounds: u(x, t) C(x, y, s, t) u(y, s) (2) where u is any positive solution of the heat equation and C(x, y, s, t) is an explicit constant. 27

29 The main goal of this talk is to prove analogous estimates for positive solutions of the heat equation in the discrete setting. In order to do so, we also introduce a reasonable, new notion of curvature for graphs. Our second goal is to derive Harnack type inequalities and heat kernel bounds for graphs that have curvature bounded from below. 28

30 Grigor yan and Saloff-Coste gave a complete geometric characterization of such manifolds. They showed that satisfying a volume growth property known as volume doubling along with an eigenvalue inequality in balls known as the Poincaré inequality is actually equivalent to satisfying the Harnack inequality. In the case of graphs Delmotte proved a characterization analogous to that of Grigor yan and Saloff-Coste. However, just as for manifolds, his conditions are hard to verify because of their global nature. One virtue of our results is that they give local conditions that imply Harnack type inequalities. 29

31 Besides proving an analogue of (1) for non-negatively curved graphs, we also recover generalizations to graphs with curvature bounded from below as well as analogues to the case of non-compact manifolds (corresponding to infinite graphs). We establish analogues for several discrete Laplace operators including the normalized and ordinary graph Laplacian. We also establish gradient estimates for solutions to more general parabolic operators of the form ( t q)u = 0. 30

32 Bakry and Ledoux even manage to use the curvaturedimension inequality we talked above to generalize (1) to Markov operators on general measure spaces when the operator satisfies a chain rule type formula. Unfortunately such a formula cannot hold in a discrete setting and thus new ideas are required that we will explain in the next section. It turns out that the curvature-dimension inequality as stated in Bochner s identity is not quite suitable for our task. Instead, we have introduce our own version of it. Nevertheless, as we shall see, for manifolds our variant of the curvature-dimension inequality is in fact weaker than the classical one. 31

33 Li and Yau use the maximum principle while Bakry and Ledoux use semigroup methods to prove (1). However, in both proofs, the curvature dimension inequality is used with the choice of f = log u together with a simple but key identity that follows from the chain rule formula: log u = u u u 2 u 2 Unfortunately this is false in the discrete setting. = u u log u 2. (3) 32

34 The starting point and probably most important observation is the fact that while there is no chain rule in the discrete setting. But there is a similar equality hold: u = 2 u u + 2Γ( u). 33

35 First we fix our notation. Let G = (V, E) be a graph. We allow the edges on the graph to be weighted; that is, the edge xy from x to y has weight w xy > 0. We do not require that the edge weights be symmetric, so w xy w yx in general, for the proofs of the main theorems, but our key examples satisfying the curvature condition do have symmetric weights. We do, however, require that inf w e =: w min > 0. e E(G) Moreover we assume in the following that the graph is locally finite, i.e. deg(x) := y w xy < for all x V. 34

36 Given a finite measure µ : V R on V, the µ-laplacian on G is the operator : R V R V defined by f (x) = 1 µ(x) w xy (f (y) f (x)). y x 35

37 Given a graph and measure, we define deg(x) D w = max x,y V w xy x y and D µ = max x V deg(x) µ(x). 36

38 So far as is possible, we will treat µ-laplacian operators generally. The special cases of most interest, however, are the cases where µ 1 which is the standard graph Laplacian, and the case where µ(x) = y x w xy = deg(x), which yields the normalized graph Laplacian. In the case where the edges are unweighted, we take w xy 1. Throughout the remainder of the talk, we will simply refer to the µ-laplacian as the Laplacian, except when it is important to emphasize the effect of the measure. 37

39 In this paper, we are interested in functions u : V [0, T ] R that are solutions of the heat equation. Let us introduce the operator L = t. We say that u(x, t) is a positive solution to the heat equation for t T, if u > 0 and Łu = 0. It is not hard to see that such solutions can be written as u(x, t) = P t u 0, whenever t T, where P t = e t is the heat kernel and u 0 = u(, 0). Note that the heat equation of course also depends on the measure µ, through the Laplacian it contains. 38

40 Definition The Γ (or µ Γ) gradient operator is defined by 2Γ(f, g)(x) = 2 f, g (x) = ( (f g) f (g) (f ) g ) (x) = = 1 w xy (f (y) f (x))(g(y) g(x)). µ(x) y x We write Γ(f ) = Γ(f, f ) = f 2 for short. Similarly, the iterated gradient operator Γ 2 is defined pointwise as 2Γ 2 (f, g) = Γ(f, g) Γ(f, g) Γ( f, g), and it will be again convenient to write Γ 2 (f ) = Γ 2 (f, f ) for short. 39

41 Definition We say that a graph G satisfies the µ-curvature-dimension inequality (hereafter referred to as simply the curvature-dimension inequality) CD(n, K ) if, for any function f : Γ 2 (f ) 1 n ( f )2 + K Γ(f ). G satisfies infinite dimension curvature dimension inequality CD(, K ) if Γ 2 (f ) K Γ(f ). 40

42 As we have pointed out in the introduction, in the case of an n-dimensional manifold whose curvature is bounded from below by K this inequality is a simple consequence of Bochner s identity. However, following the work of Bakry and Emery, it has proven to be an important definition of curvature in many other settings. 41

43 The semigroup P t = e t is said to be a diffusion semigroup if the following identities are satisfied for any smooth function Φ: Γ(f, g h) = g Γ(f, h) + h Γ(f, g) (4) Γ(Φ(f ), g) = Φ (f )Γ(f, g) (5) (Φ(f )) = Φ (f ) (f ) + Φ (f )Γ(f ). (6) For example if is the Laplace-Beltrami operator on a manifold, then the generated semigroup is a diffusion semigroup. Bakry and Ledoux show that if the operator satisfying CD(n, 0) generates a diffusion semigroup then (1) holds. 42

44 The Laplacian we are interested in does not generate a diffusion semigroup, however, as we mentioned in the introduction, for the choice of Φ(f ) = f a formula similar to a combination of (5) and (6) holds: u = 2 u u + 2Γ( u). (7) This motivates the following key modification of the curvature-dimension inequality. 43

45 Definition We say that a graph G satisfies the exponential curvature dimension inequality CDE(n, K ) if for any vertex x V and any positive function f : V R such that f (x) < 0 we have ( Γ 2 (f ) Γ f, Γ(f ) ) 1 f n ( f )2 + K Γ(f ), or equivalently, using (7) ( 1 2 Γ(f ) Γ f, (f 2 ) ) 1 2f n ( f )2 + K Γ(f ). 44

46 G satisfies the infinite dimensional exponential curvature dimension inequality CDE(, K ) if ( Γ 2 (f ) Γ f, Γ(f ) ) K Γ(f ). f 45

47 To make it easier to write computations concerning CDE, we introduce ( Γ 2 (f ) = Γ 2 (f ) Γ f, Γ(f ) ) = 12 ( f Γ(f ) Γ f, (f 2 ) ) 2f so that G satisfies CDE(n, K ) if Γ 2 (f ) 1 n ( f )2 + K Γ(f ). 46

48 Of course, one hopes that typical graphs which one might consider to have non-negative curvature satisfy CDE(n,0) for some dimension n. As we will show later, the class of Ricci-flat graphs, first introduced by Chung and Yau, which includes abelian Cayley graphs and most notably the lattices Z d (along with finite tori) do indeed satisfy CDE(2d,0). 47

49 Remark If the semigroup generated by is a diffusion semigroup (e.g. the Laplacian on a manifold) then CD(n, K ) implies CDE(n, K ). 48

50 Definition We can also define a curvature dimension inequality CDE (n, K ) if for all f > 0, Γ 2 (f ) 1 n f 2 ( log f ) 2 + K Γ(f ). CDE (n, K ) implies CDE(n, K ) on graphs. In the case of diffusion semigroups, CD(n, K ) and CDE (n, K ) are equivalent. 49

51 First we prove the gradient estimate in the compact case without boundary: for solutions of the heat equation on finite graphs. We start with the simplest case, when the curvature is non-negative. Theorem Let G be a finite graph satisfying CDE(n, 0), and let u be a positive solution to the heat equation on G. Then for all t > 0 Γ( u) u t ( u) u n 2t. 50

52 We can extend the result to the case of graphs satisfying CDE(n, K ) for some K > 0 as follows. Theorem Let G be a finite graph satisfying CDE(n, K ) for some K > 0 and let u be a positive solution to the heat equation on G. Fix 0 < α < 1. Then for all t > 0 (1 α)γ( u) u t ( u) u n (1 α)2t + Kn α. 51

53 We can also extend the result from solutions to the more general operator (L q) = ( t q)u = 0, where q(x, t) is a potential satisfying q ϑ and Γ(q) η 2 for some ϑ and η 0. 52

54 Theorem Let G be a finite graph and q(x, t) : V R + R be a potential satisfying q ϑ and Γ(q) η 2 for all x V and t 0. Suppose u = u(x, t) satisfies (L q)u = 0 on G. 1. If G satisfies CDE(n, 0), then Γ( u) u t ( u) q u 2 < n 2t + 1 n(ϑ + η 2D µ (D w + 1)) Fix 0 < α < 1. If G satisfies CDE(n, K ), for some K 0, then (1 α) Γ( u) t ( u) q u u 2 < n 2(1 α)t + 1 C(α, K, n, ϑ, η), 2 53

55 where C(α, n, K, ϑ, η) = K 2 n 2 α 2 + n ( [ ( ϑ + η (1 α) 2D µ (D w + 1) + α 2D µ D 3 1 α w + 1 )] 54

56 We can prove somewhat weaker results in the presence of boundary. We do not assume finiteness of the graph anymore, and we only assume the heat equation is satisfied in a finite ball. Our estimates will depend on the radius of this ball. 55

57 Theorem Let G(V, E) be a (finite or infinite) graph and R > 0, and fix x 0 V. Let u : V R R a positive function such that Lu(x, t) = 0 if d(x, x 0 ) 2R. If G satisfies CDE(n, 0) then Γ( u) u in the ball of radius R around x 0. t u < n u 2t + n(1 + D w)d µ R 56

58 Corollary If G(V, E) is an infinite, bounded degree graph satisfying CDE(n, 0) and u is a positive solution to the heat equation on G, then on the whole graph. Γ( u) u t u u n 2t 57

59 By using the above gradient estimate, we can also prove the following Lioville theorem on graphs which extend similar results on manifolds. Corollary If G(V, E) is an infinite, bounded degree graph satisfying CDE(n, 0), then there is no non-constant positive harmonic function on G and there is also no non-constant bounded harmonic function on G. 58

60 In order to state the result in complete generality (in particular, when f is a solution to (L q)f = 0 as opposed to a solution to the heat equation), we need to introduce a discrete analogue of the Agmon distance between two points x, and y which are connected in B(x 0, R). 59

61 For a path p 0 p 1... p k define the length of the path to be l(p) = k. Then in a graph with maximum measure µ max : { 2µ max l 2 (P) ϱ q,x0,r,µ max,w min,α(x, y, T 1, T 2 ) = inf w min (1 α)(t 2 T 1 ) k 1 + i=0 ( ti+1 t i q(x i, t)dt + k (T 2 T 1 ) 2 ti+1 where the infinum is taken over the set of all paths t i (t t i ) 2 (q(x i, t) q(x i+1, t))d P = p 0 p 1 p 2 p 3... p k so that p 0 = x, p k = y and having all p i B(x 0, R), and the times T 1 = t 0, t 1, t 2,..., t k = T 2 evenly divide the interval [T 1, T 2 ]. 60

62 Remark In the special case where q 0 and R =, which will arise when f is a solution to the heat equation on the entire graph, then ϱ simplifies drastically. In particular, ϱ µmax,α,w min (x, y, t 1, t 2 ) = 2µ max d(x, y) 2 (1 α)(t 2 T 1 )w min where d(x, y) denotes the usual graph distance. 61

63 Theorem Let G(V, E) be a graph with measure bound µ max, and suppose that a function f : V R R satisfies (1 α) Γ(f ) f 2 (x, t) ttf (x, t) q(x, t) c 1 + c 2 f t whenever x B(x 0, R) for x 0 V along with some R 0, some 0 α < 1 and positive constants c 1, c 2. Then for T 1 < T 2 and x, y V we have f (x, T 1 ) f (y, T 2 ) ( T2 T 1 ) c1 exp ( c 2 (T 2 T 1 ) + ϱ q,x0,r,µmax,wmin,α(x 1, x 2, T 62

64 In the case of unweighted graphs, and when dealing with positive solutions to the heat equation everywhere, the above theorem simplifies greatly. Corollary Suppose G(V, E) is a infinite unweighted graph satisfying CDE(n, 0), and µ(x) = deg(x) for all vertices x V. If u is a positive solution to the heat equation on G, then u(x, T 1 ) u(y, T 2 ) ( T2 T 1 ) n exp ( 4Dd(x, y) 2 T 2 T 1 ), where D denotes the maximum degree of a vertex in G. 63

65 Here we prove that a graph with maximum degree D satisfies CDE(2, D/2 1). We show that this bound is close to sharp for graphs that are locally trees, in particular the curvature of a D-regular large girth graph goes to linearly as D. Theorem deg(x) Suppose G is any graph with D w = max x y w xy and ( ( )) D µ = max deg(x) µ(x). Then G satisfies CDE 2, D Dw µ

66 Remark For unweighted graphs with the normalized Laplacian, the above Theorem states that all graphs satisfy CDE(2, D 2 1). Such a lower bound on curvature is essentially tight in the case of trees. 65

67 Chung and Yau introduced the notion of Ricci-flat (unweighted) graphs as a generalization of Abelian Cayley graphs. 66

68 Definition A d-regular graph G(V, E) is Ricci-flat at the vertex x V if there exists maps η i : V V ; i = 1,..., d that satisfy the following conditions. 1. xη i (x) E for every x V. 2. η i (x) η j (x) if i j, for every x V. 3. for every i we have j η i (η j (x)) = j η j (η i (x)) In fact to test Ricci-flatness at x it is sufficient for the η i to be defined only on x and the vertices adjacent to x. Finally, the graph G is Ricci-flat if it is Ricci-flat at every vertex. 67

69 Theorem Let G be a d-regular Ricci-flat graph. Suppose that the measure µ defining satisfies µ(x) µ for all vertices x G. 1. If the weighting of G is consistent, then G satisfies CDE(d, 0). 2. If the weighting of G is weakly consistent, then G satisfies CDE(, 0) 68

70 Remark For a d-regular Ricci-flat graph and a weakly consistens weighting the two standard choices of the measure µ 1 and µ(x) = deg(x) satisfy µ(x) µ for all x V. Corollary The usual Cayley graph of Z k satisfies CDE(2k, 0), for the regular or normalized graph Laplacian. 69

71 One of the fundamental applications of the Li-Yau inequality, and more generally parabolic Harnack inequalities, is the derivation of heat kernel estimates. As alluded to in the introduction, Grigor yan and Saloff-Coste (in the manifold setting) and Delmotte (in the graph setting) proved the equivalence of several conditions (including Harnack inequalities, and the combination of volume doubling and the Poincaré inequality) to the heat kernel satisfying the following Gaussian type bounds. Let P t (x, y) denote the fundamental solution to the heat equation starting at x. 70

72 Definition G satisfies the Gaussian heat-kernel property G(c, C) if c ( C vol(b(x, t)) exp ) d(x, y)2 P t (x, y) t C ( c vol(b(x, t)) exp ) d(x, y)2 t In the graph setting, Delmotte proved that G(c, C) is equivalent to two other (sets of) properties. The first is the pair of volume doubling and Poincaré inequality. The second is so called H(θ 1, θ 2, θ 3, θ 4, C 0 ). 71

73 Definition G satisfies the volume doubling property VD(C) if for all x V and all r R + : vol(b(x, 2r)) Cvol(B(x, r)) 72

74 Definition G satisfies the Poincaré inequality P(C) if x B(x 0,r) µ(x)(f (x) f B ) 2 Cr 2 x,y B(x 0,2r) w xy (f (y) f (x)) 2, for all f : V R, for all x 0 V and for all r R +, where f B = 1 vol(b(x 0, r)) x B(x 0,r) µ(x)f (x). 73

75 In general, however, we only have for graphs satisfying CDE(n, 0) the Harnack inequality u(x, T 1 ) u(y, T 2 ) ( T2 T 1 ) c1 ( c2 exp R (T d(x, y) 2 ) 2 T 1 ) + c 3. T 2 T 1 This will not suffice for proving H(θ 1, θ 2, θ 3, θ 4, C 0 ). Indeed, if T 2 T 1 = cr 2, then this only implies that sup Q u(x, t) exp(cr + c ) inf u(x, t), Q + where the constant depends now on R. Where 74

76 Nevertheless, we can derive heat kernel upper bounds that are Gaussian, and lower bounds that are not quite Gaussian but still have a similar form. The heat kernel bound then allows us to derive volume growth bounds: we show that if G satisfies CDE(n, 0) then G has polynomial volume growth. We derive here only on-diagonal upper and lower bounds, but it is well known that off-diagonal bounds can be established using the on-diagonal bounds. 75

77 Theorem Suppose G satisfies CDE(n, 0) and has maximum degree D. Then there exist constants so that, for t > 1, C 1 ( t n exp C d 2 ) (x, y) P t (x, y) C µ(y) t 1 vol(b(x, t)). 76

78 By using a method from Davies, we can imply the following Li-Yau heat kernel Gaussian upper bound. Theorem Suppose G satisfies CDE(n, K ) for some K > 0 and has maximum degree D. Then there exist constants so that, for all x, y and ct d for some constant c, ) P t (x, y) C µ(y) (c vol 1 2 (B(x, t))vol 1 2 (B(y, t)) exp d(x, y)2 Kt c. t Where c depends on c. 77

79 If we have the volume doubling, then we can prove the Gaussian lower bound. But we still can t prove the volume doubling now. Nevertheless we can prove the the polynomial volume growth from the heat kernel bound. Corollary Let G be a graph satisfying CDE(n, 0). Then G has a polynomial volume growth. 78

80 As another application of the gradient estimate, we prove a Buser-type estimate for the smallest nontrivial eigenvalue of a finite graph. For now on we assume that the edge weights are symmetric, i.e. w xy = w yx for all x y. 79

81 The Cheeger constant h of a graph is defined as h = inf =U V :vol(u) 1/2 vol(v ) U vol(u), where U = x U,y V \U w xy and vol(u) = x U µ(x). 80

82 Theorem Let G be a finite graph satisfying CDE(n, K ) for some K > 0 and fix 0 < α < 1. Then where the constant λ 1 max{2c K h, 4C 2 h 2 }, ( ) 1 (2 α)n 2 C = 8 3µ max α(1 α) 2 only depends on the dimension n and µ max. 81

83 Remark h The Cheeger inequality states that 2 2D µ λ 1. Thus in particular if K = 0, the above Theorem implies that h 2 2D µ λ 1 4C 2 h 2, i.e. λ 1 is of the order h 2. 82

84 Remark Klartag and Kozma show a similar but stronger result for graphs satisfying the original curvature-dimension inequality. Namely they prove, following the arguments of Ledoux, that if a finite graphs satisfies CD(, K ) then λ 1 8 max{ K h, h 2 }. Note that their condition does not involve dimension, and hence their constant is also dimension independent. 83

85 The Li-Yau gradient estimate has been generalized to many important setting in geometric analysis. The most notable one was made by Hamilton on the Ricci flow. We can also get a similar estimate and derive the Harnack inequality for the scalar curvature of solutions to the Ricci flow on graphs with scalar curvature nonnegative. We can also prove the nonnegative of scalar curvature are preserved under the Ricci flow. 84

86 Doubling property and Gaussian heat kernel estimate 85

87 Let u = u t to be the heat equation on a graph G = (V, E), the goal of this talk is to prove the following results on graphs. 1. Gaussian heat kernel estimate: There exists positive constants c 1, c 2, c 3, c 4 > 0, when d(x, y) n, the heat kernel on graph satisfied c 1 µ(b(x, d(x,y) 2 n)) e c 2 n h(n, x, y) c 3 µ(b(x, d(x,y) 2 n)) e c 4 n. 86

88 2. Volume doubling: x V, r > 0, µ(b(x, 2r)) cµ(b(x, r)). Poincaré inequality: µ(x)(f (x) f B ) 2 cr 2 ω xy (f (x) f (y)) 2, x B(x 0,r) x,y B(x 0,2r) for all f V R, all x 0 V and all r R +, where f B = 1 µ(b(x 0, r)) x B(x 0,r) µ(x)f (x). 87

89 3. Parabolic Harnack inequality: Let η (0, 1), 0 < θ 1 < θ 2 < θ 3 < θ 4, for every non-negative solution on [s, s + θ 4 r 2 ] B(x 0, r), we have sup (t,x) [s+θ 1 r 2,s+θ 2 r 2 ] B(x 0,ηr) u(t, x) C inf (t,x) [s+θ 3 r 2,s+θ 4 r 2 ] B(x 0,ηr) u(t, 88

90 In 1986, Li-Yau proved the results (1) and (3), under the non-negatice Ricci curvature assumption on Riemannian manifold for continuous setting heat equation. Grigorgan(1992) and Saloff-Coste (1995) proved that (1) (2) on manifold. 89

91 On graph, in 1997, Delmotte proved that (1) (2) (3) + (α). where (α): x, y V, α > 0, s.t. p(x, y) α, x y p(x, x) α. 90

92 Main tools: Li-Yau gradient estimate, weak Harnack parabolic inequality. We prove the following theorem: Theorem (P.Horn,Y.Lin, S.Liu, S.T.Yau) Suppose a infinite graph satisfies CDE (n, 0), and (α), then we have the Gaussian estimate (1) for discrete-time heat kernel. From the result of Delmotte, we obtain: Corollary Under the same assumption, (2) and (3) are also true. 91

93 Remark From a example of Pang in 1994, the upper Gaussian estimate for continuous-time heat kernel may not be true even on lattice Z 1.(which satisfies CDE (n,0).) 92

94 A graph G = (V, E), V is the vertex set and E is the edge set, the metric between two vertices x and y: d(x, y) = the length of minimal path between x and y. then (G, d) is a metric space. We let µ(x) = degree of x = {y : y x}. µ(b(x, r)) = µ(y). y B(x,r) And define p 1, l p = {f V R : µ(x) f (x) p < + }. x V when p = 2, we can let f, g = x V µ(x)f (x)g(x). 93

95 The Laplace operator on G is f (x) = 1 µ(x) is a bounded operator on l p. (f (y) f (x)), y x Markov heat semigroup operator: P t = e t = k=0 t k k! k. 94

96 Let p n (x, y) be defined by { pn (x, y) = z V p n 1(x, z)p(z, y), p 0 (x, y) = δ x (y). where p(x, y) = { 1 µ(x), y x, 0, o.w. The discrete-time heat kernel is defined by h(n, x, y) = p n(x, y), µ(y) which is the solution of discrete-time heat equation, u(n, x) = u(n + 1, x) u(n, x) 95

97 Let it is easy to know p(t, x, y) = e t k=0 t k k! p k(x, y), p(t, x, y) µ(y) = p(t, y, x). µ(x) The continue-time heat kernel h(t, x, y) is defined by h(t, x, y) = p(t, x, y), µ(y) and it is the solution of continue-time heat equation { u = u t, p 0 (x, y) = δ x (y). 96

98 We need a modify of CD(n, K ), let Definition Γ2 (f ) := Γ 2 (f ) Γ(f, Γ(f ) ). f We say that a graph satisfying CDE (n, 0) at x V for some n R +, K R, if for any positive function f V R, Γ 2 (f )(x) 1 n f (x)2 ( log f (x)) 2 + K Γ(f )(x). 97

99 Remark For diffusion operator, CD(n, K ) CDE (n, K ) Example If G is a Ricci-flat graph in the sense of Chung and Yau, e.g. an Abelian Cayley graph, then it satisfies the CDE (n, 0). 98

100 1. Volume doubling; 2. Gaussian lower bound estimate for continuous-time heat kernel; 3. On-diagonal lower and upper estimate for discrete-time heat kernel; 4. (3)+Volume doubling Guassian estimate. 99

101 Thank you! 100