Key words. heat equation, gradient estimate, curvature, curvature dimension inequality, continuous

Transcription

1 A SPACIAL GADIENT ESTIMATE FO SOLUTIONS TO THE HEAT EQUATION ON GAPHS PAUL HON Abstract The stdy of positive soltions of the heat eqation =, on both manifolds t and graphs, gives an analytic way of extracting geometric information abot the object In the manifold case, one of the most effective ways of stdying how soltions to the heat eqation evolve is to derive a local gradient estimate of heat change, sing crvatre lower bonds ecently, notions of crvatre for graphs have been developed which enable proving similar estimates for graphs In this article, we derive a gradient estimate for positive heat soltions that considers only how heat varies in space and the time derivative This reslt, de in the manifold case to Hamilton, applies to both finite graphs and infinite graphs of bonded degree As a corollary, a heat comparison theorem is also developed This in trn yields reslts abot the mixing of the continos time random walk on graphs Key words heat eqation, gradient estimate, crvatre, crvatre dimension ineqality, continos time random walk AMS sbject classifications 05C8, 53C, 35K05 The stdy of positive soltions to the heat eqation, t =, is of fndamental importance in iemannian geometry and geometric analysis This is, in large part, becase the evoltion of soltions on manifolds is closely related to the nderlying geometric properties of the manifolds Indeed, as shown independently by Grigor yan [8] and Saloff-Coste [7], Gassian decay of the heat kernel is eqivalent to satisfying a Poincaré ineqality an eigenvale ineqality and volme dobling a statement abot the growth of balls, and in trn is also eqivalent to soltions satisfying a strong Harnack ineqality, allowing comparison of heat at different points at different times In a graph setting, stdying soltions to the heat eqation makes sense for a broad class of Laplace operators For the prposes of this discssion, consider = D A I which is an nsymmetrized version of the normalized Laplacian poplarized by Chng see [3] Here, soltions of the heat eqation have a particlarly nice probabilistic interpretation: they reflect the evoltion of the so-called continos time random walk where a random walker stays on a vertex for an exponentially distribted amont of time In the graph setting, eqivalence of the three properties Gassian decay of the heat kernel, Poincaré pls volme dobling, and strong Harnack ineqality is also known, de to work of Delmotte in [7] These reflect the fact that the evoltion of the continos time random walk is closely related to geometric properties of the graph eg volme dobling, diameter bonds, etc Stdying this evoltion gives an analytic way of nderstanding geometric properties of a graph One qestion remains: how to show that a graph satisfies one, and hence all, of these three eqivalent conditions in an easy fashion In the iemannian setting, a crvatre lower bond provides an easy way to do exactly that throgh the Li-Ya ineqality, derived in [] This ineqality bonds the local change of heat in space and time Crvatre lower bonds with an appropriate notion of crvatre are an appealing way to establish properties of graphs as crvatre is a local property and many of the implications of a crvatre bond are global Ths, sing crvatre to Sbmitted to the editors DATE Fnding: This work was partially fnded by NSA Yong Investigator Grant H Department of Mathematics, University of Denver, palhorn@ded

2 P HON nderstand properties of a graph involves nderstanding global geometric properties of large graphs by nderstanding the local behavior ie it stdies how one can nderstand global properties of a graph by looking only at behavior of vertices and neighbors to some bonded distance Crvatre also often give a niform way of establishing reslts on both finite and infinite graphs In the graph setting, only recently have crvatre notions been introdced which are strong enogh to show that a graph satisfies these eqivalent properties In [], Baer, Lin, Lippner, Mangobi, Ya and the athor a version of the Li-Ya ineqality for graphs as proved While it not qite strong enogh to complete the cycle and prove that non-negative crvatre implies the three eqivalent conditions, was able to establish that non-negatively crved graphs had polynomial volme growth among other properties Later work of the athor, Lin, Li and Ya [0] sed a slight variant of the crvatre condition of [] and some different argments to establish not only the three eqivalent conditions, bt also to establish from non-negative or positive crvatre a nmber of geometric properties of graphs, sch as diameter and volme bonds In this note, we frther or stdy of the heat eqation The Li-Ya ineqality of [] in its simplest form, for compact n-dimensional manifolds which are nonnegatively crved states that a positive soltion to the heat eqation satisfies t n t This ineqality relates how the soltion changes in space and also in time t in a way that decays as time increases To derive a Harnack ineqality, which compares the heat of two points at two different times, one takes a path in spacetime between the two points and careflly integrates the ineqality to see how the heat can change One interesting aspect to this is that reqires the t term to make sense: the qantity is not able to be bonded by an absolte constant The net effect of this is that the Li-Ya ineqality cannot be integrated to compare the heat of two points at the same time Nonetheless, Hamilton [9] showed that the heat almost behaves well in time, proving that the if is a soltion to the heat eqation, normalized so that =, for a compact n-dimensional non-negatively crved manifold, then t log/ Note that the normalization = appearing is in a sense reqired as the ineqality is not invariant to normalization nlike the original Li-Ya ineqality and the Li-Ya ineqality for graphs proved in [] Or main prpose of this note is to derive a version of Hamilton s Li-Ya ineqality,, for graphs Like Hamilton s version, or ineqality is dimension independent: this is particlarly interesting as this allows s to apply or work to certain directed graphs, which are in a sense infinite dimensional flat graphs We go somewhat beyond the work of Hamilton by providing a version of or ineqality which applies to infinite graphs as well the original work of Hamilton only considers the compact manifold case which translates to finite graphs In the next section we give some preliminary notation and information on the crvatre notions sed In Section 3 we prove or main reslts Finally, we prove a heat comparison theorem, in the vein of a Harnack ineqality, and give some applications and discssion in Section 4

3 GADIENT ESTIMATE FO SOLUTIONS TO THE HEAT EQUATION 3 Backgrond and Notation Let G = V, E be a locally finite graph with maximm degree D, infinite or finite Or graph theory notation is qite standard throghot: we se x y to denote that vertex x is a neighbor of y, dx, y to denote the distance between x and y in the standard graph metric, and diamg to denote the diameter of G for finite graphs We assme edges are weighted with weights w xy > 0, and make the frther assmption that w min = inf w xy > 0 xy E We don t assme a priori that the edge weights are symmetric As a convention, we assign the edge weight w xy to the edge from x to y Most bt not all of the graphs known to be flat with respect to the crvatre conditions below are ndirected, however there do exist some weighted flat graphs and or methods do apply to them Let µ : V + be a positive measre on the vertices with inf x V µx = µ min > 0 The µ-laplace operator on the graph is the following operator: If f : V is a fx = µx w xy fy fx While µ may be chosen to be an arbitrary measre, the most classical choices of µ are µ which leads to the combinatorial Laplace operator and µx = degx = w xy which leads to the normalized Laplace operator While or methods apply to arbitrary Laplace operators, the reader is qite welcome to think of the reslts in terms of these classical cases In the case of finite graphs, these almost agree with the classical combinatorial Laplacian matrix D A and the normalized Laplacian I D / AD / The sign convention is changed to agree with the Laplace-Beltrami operator on manifolds the operator is non-positive semidefinite as opposed to the matrices which are positive semidefinite and the normalized Laplacian I D / AD / of Chng is a symmetrized version of this operator In general, or reslts are agnostic to the measre µ, and we se the notation to denote the types of averaged sm appearing in That is, for real nmbers a x we let x y := w xy x y µx As a convenience we record here some assmptions and notational convetions, we se regarding the measres involved Assmptions: Or measres µ : V + and edge weights w xy satisfy w min = inf xy E w xy > 0, µ min = inf µx > 0, x V D max = sp x degx µx = sp x µ max = sp µx < x V µx w xy <, As an immediate conseqence of or assmptions, we record

4 4 P HON Lemma {a x } x V G are a positive nmbers, a y D max max a y Crvatre Notions on Graphs There have been, over the past few years, a large nmber of crvatre notions developed for graphs These all have the featre that the crvatre at a vertex or, in some cases across an edge, depend on the strctre on the graph in a bonded iterated neighborhood of the vertex Of these two have perhaps attracted the most attention recently The first of those is based on the notion of transportation distance see eg papers of Lott-Villani [5], Ollivier [6], and Lin, L and Ya [3], where crvatre is measred by how well random walks at different points can be copled The other takes a more analytic tilt, and is based on the idea of a crvatre dimension ineqality The idea of crvatre dimension ineqalities, is to define the notion of crvatre lower bonds where the may not other be defined, by via the Bochner formla The Bochner formla states that in an n-dimensional manifold M with crvatre Kn that for all smooth fnctions, the ineqality 7, + n K holds at every point on M emark: For the reader in graph theory, fnctional ineqalities like 7 are perhaps nfamiliar Throghot the paper, when we assert a pointwise ineqality of fnctions, we omit the argments nless it makes or ineqality clearer Also note that if is a fnction, so are and as are all the terms appearing in 7 Bakry and Emery observed see [] that on spaces where a Laplace operator was defined one cold define a gradient operator Γ, formally setting f, g = Γf, g := fg f g g f For graphs, at a vertex x and for two fnctions f, g : V, the fnction Γf, g : V simplifies to Γf, gx = fy fxgy gx As a convenience, we let Γf := Γf, f which serves as a replacement for f in the graph setting Note Γfx = fy fx, and in particlar Γf 0 Bakry and Emery frther noted that whena diffsive semigrop satisfies an analoge of Bochner formla 7 many properties implied by a crvatre lower bond can be recovered To this end, they introdced the iterated gradient form Γ which is defined by Γ f, g := Γf, g Γf, g Γ f, g Then, a space with some Laplace operator eg a graph, manifold, or diffsive semigrop is said to satisfy the crvatre dimension ineqality CDn, K if at every point and for every smooth in the continos case fnction f, Γ fx n fx K Γfx

5 GADIENT ESTIMATE FO SOLUTIONS TO THE HEAT EQUATION 5 The intition is that a graph satisfying CDn, K shold behave like a n-dimensional manifold with a crvatre lower bond of K While it is indeed tre that some analytic conseqences of non-negative crvatre for instance can be established for graphs satisfying CDn, 0 see eg [4,, 4], the standard crvatre dimension ineqality on graphs has some limitations Many applications of the Bochner formla reqire the Laplace operator to satisfy a chain rle This holds in the diffsive semigrop setting considered by Bakry and Emery in [], bt fails in general for graphs To overcome this difficlty, in [] two alternate versions were introdced, the so called exponential crvatre dimension ineqalities, which in a sense bake in the chain rle Definition A graph is said to satisfy the exponential crvatre dimension ineqality CDEn, K if at every vertex x and at for every fnction f : V 0 so that fx 0 Γ f := Γf Γ f, Γf f n f KΓf Definition 3 A graph is said to satisfy the exponential crvatre dimension ineqality CDE n, K if at every vertex x and at for every fnction f : V 0 Γ f := Γf Γ f, Γf f n fx log f KΓf emark: Observe that if G satsifies CDEn, K or CDE n, K then it satisfies CDEn, K for any n n and K K Hence these ineqalities weaken as the dimension increases or the crvatre lower bond decreases Or reslts actally apply to graphs satisfying CDE, K for some K 0; that is we will discard the f term appearing above These two ineqalities are closely related, and indeed it is easy to see by Jensen s ineqality that if a graph satisfies CDE n, K it also satisfies CDEn, K Every graph satisfies CDE, d max bt a nmber of graphs are known to be non-negatively crved with respect to these crvatre notions For instance, it is known that d-reglar icci-flat graphs, in the sense of Chng and Ya see [5, 6] satisfy CDE 3d, 0 and CDEd, 0 by work of [] bit is also known that certain directed icci-flat graphs called in [] icci-flat graphs with weakly consistent weights, where edges in different directions are weighted differently also satisfy CDE, 0 thogh they do not satisfy CDE n, 0 nor CDEn, 0 for any finite n Qite interestingly, or reslts apply to these as well Both the CDE and CDE ineqality are also closely related to the classical CD ineqality Indeed, for diffsive semigrops essentially, when the Laplace operator satisfies the chain rle CDE n, K and CDn, K are eqivalent On graphs, they are not eqivalent Z d, for instance, satisfies CDd, 0, bt does not satisfy CDE d, 0 and only satisfies the weaker ineqality CDE 453d, 0 While BHMMLY se the CDE ineqality in their work on the Li-Ya ineqality, the CDE n, 0 has proven qite sefl for a nmber of argments Indeed, in the work of Lin, Li, Ya and the athor establishing the strong Harnack ineqality for non-negatively crved graphs CDE n, 0 is sed as it allows more global argments to be sed This is de to the fact that the CDE ineqality only gives an ineqality at a point when f < 0, while the CDE ineqality asserts a non-trivial lower bond on Γ at every point In this work, we will se the CDE ineqality as this yields slightly stronger reslts

6 6 P HON The Heat Eqation on Graphs In this paper we stdy positive soltions to the heat eqation; that is : V [0, satisfying t = for t 0 As is well known, these soltions are determined by the initial vales 0 : V For any sch 0, it is easy to see that = e t 0 is a soltion to the heat eqation These soltions are closely related to a type of random walk on graphs, the socalled continos time random walk In a continos time random walk, a simple random walk is performed where the amont of time the walker spends before taking a random step is exponentially distribted, instead of the walker spending one time step before taking a step Indeed, if is defined with µv = degv = v w v that is when we consider the normalized Laplace operator the distribtion, ϕ t, of a continos time random walk at time t starting at initial distribtion ϕ 0 ϕ t T = ϕ 0 T e t For a reglar graph, the normalized Laplace operator is symmetric, and so the ϕ t is a soltion to the heat eqation For irreglar graphs, one has to translate between the two bt this is possible For a finite graph and positive, e t tends to the constant fnction as t while T e t tends towards a vector proportional to the degrees that is to the stationary distribtion of the random walk Main eslts In this section we record and prove or main reslts, which are graph theoretical versions of a gradient estimate for soltions to the heat eqation that is originally de to Hamilton Unlike the original Li-Ya ineqality, which controls both a combination of the gradient and time derivative of positive soltions to the heat eqation, the Hamilton gradient estimate is concerned solely with the gradient how a soltion varies in time For convenience, we define the operator L, L = t As we will se the maximm principle repeatedly, we make a brief observation Sppose f : V [0, is maximized on V 0, T at x, t Then fx, t 0 and t fx, t 0, so that Lf 0 Also, we will freqently se that if f, g : V [0, Lf g = f g t f g = f g + f g + Γf, g f t g f g t = Lf g + f Lg + Γf, g As an initial step, we make the record the following comptation which partially motivated the definition of the definition of Γ in [] Lemma 4 If is a positive soltion to the heat eqation t =, then LΓ = Γ

7 GADIENT ESTIMATE FO SOLUTIONS TO THE HEAT EQUATION 7 Proof We record that LΓ = Γ t Γ Then at a vertex x, and time t t Γ x, t = y, t x, t t = y, t t y, t x, t y, t tx, t x, t = y, t y, t x, t x, t y, t x, t = Γ, x, t, and combining gives s what we desire We are ready to prove or main theorem We prove two versions of the theorem; the first for finite graphs Theorem 5 Sppose G = V, E is a finite graph which satisfying CDE, K for some K 0, and is a positive soltion to the heat eqation with = Then at all x V G and t 0,, Γ log emark: A word of explanation is warranted for the form of this ineqality: the original ineqality of Hamilton bonded t log, while we essentially bond instead This yields a slightly weaker gradient estimate, bt for reasons discssed in [] this is really necessary for small time in graphs That we bond the gradient of the sqare root of instead of the gradient of is also tied to the chain rle and the definition of the CDE ineqality Proof of Theorem 5 As a preliminary comptation, we record that at any vertex

8 8 P HON x, and any time t Γ x, t + L logx, t 8 = Γ x, t + L logx, t + Llog x, t + Γlogx, t = Γ x, t + log x, t + Γ, log x, t = Γ x, t x, t + x, t logy, t logx, t + y, t x, t logy, t logx, t = Γ x, t x, t + y, t log = 0 y, t x, t [ y, t x, t + x, t y, t y, t log ] x, t y, t Here, in the last ineqality, we se the fact that for a, s > 0 as we consider a positive soltion to the heat eqation, the fnction fs = s a + s a a logs/a is minimized when s = a, at which point fs = 0 Indeed, a simple calclation shows that f s = s a s + a s = s a s + a s This has a single critical point at s = a, which clearly is the global minimm As each of the smmands in 8 is non-negative so is the averaged sm In fact, the ineqality above is strict nless is constant on x and all of its neighbors Let t H = + Kt Γ log/ t = + Kt Γ + log We wish to show that Hx, t 0 for all x, t V G 0, T ] Let x, t be a vertex where H is maximized on V G 0, T ], if sch a point exists We may assme that x, t is not locally constant in the neighborhood of x, as otherwise Hx, t 0 and we will be done At the maximm point, we compte 0 LH = + Kt Γ t + + Kt Γ + L log + Kt + tk Γ + L log + Kt Γ + L log > 0

9 GADIENT ESTIMATE FO SOLUTIONS TO THE HEAT EQUATION 9 Here, the second ineqality follows from CDE, K The third ineqality follows from the fact that + Kt + tk + + KttK = + Kt + Kt The strict ineqality in the last step follows from the fact that is not locally constant in the neighborhood of x, t and CDE, K implies that Γ KΓ This contradiction implies that there is no maximm On the other hand, for every vertex x lim Hx, t 0 t 0 + These combine to imply that Hx, t 0 everywhere This completes the proof Indeed, in this case, at any point x, t so t + Kt Γ log/ 0, Γ log/ A similar statement holds for infinite graphs In this case, however, we mst be more carefl in applying the maximm principle which complicates the proof somewhat Theorem 6 Sppose G = V, E is a bonded degree infinite graph satisfying CDE, K for some K 0, and is a positive soltion to the heat eqation valid for V G 0, with = Then at all x V G and t 0,, Γ log, emark: Observe that we reqire, in this case, to be a positive soltion on the entire graph It wold also be desirable to have a version of Theorem 6 which is valid for soltions which only satisfy the heat eqation on a ball of radis as opposed to the entire graph as sch restrictions occasionally appear in proofs This proves to be a somewhat more tricky bsiness, even in the continos version Indeed, this necessary seems to reqire error term involving and log / as opposed to jst log/, and we are crrently nable to prove sch a version on graphs This ineqality on manifolds, de to Soplet and Zhang [8], reqires a nmber of identities involving the chain rle beyond those which are already cooked in to or method In particlar, the form of or theorem and CDE ineqality is designed to overcome the fact that the identity log = fails for graphs The Soplet and Zhang reslt in the continos case relies on an identity for log log which fails horribly in the discrete case On the other hand, we give a slightly stronger reslt than Soplet and Zhang s for soltions valid everywhere Proof of Theorem 6 Fix x 0 V G, and T 0, and we prove that x 0, T satisfies the desired ineqality As calclated in the finite case, we record that at any vertex x and at any time t L log x, t > Γ x, t

10 0 P HON if is not locally constant in the neighborhood of x Fix a positive integer, and let { ϕy = max dx } 0, y, 0, where dx 0, y denotes the graph distance between x 0 and y We let t H = + Kt Γ + + D max t log We claim that 9 Hx 0, T T D max Assming this holds for every positive, taking we have that at x 0, T T + KT Γ + log 0, and rearranging yields the reslt Ths it sffices to prove 9 holds for arbitrary Since ϕx 0 =, it sffices to show that ϕh T for all x, t V 0, T ] Since ϕh 0 otside of Bx 0, and lim t 0 + Hx, t 0 for every vertex x V G, if ϕh T at some x, t, then there exists some x, t maximizing ϕh Moreover, cannot be constant in the neighborhood of x at time t Note that as, Lemma gives Γ x, t = y x D max Since log 0, this means that if ϕhx, t T Dmax, then ϕx At x, t, then ϕyhy, t ϕx Hx, t = ϕhx, t LϕHx, t earranging, we obtain 0 = LϕHx, t + Γϕ, Hx t, t + ϕx LHx, t = ϕhx, t + Γϕ, Hx, t + ϕx LHx, t = ϕy ϕx Hx, t + ϕy ϕx Hy, t Hx, t + ϕx LHx, t = ϕy ϕx Hy, t + ϕx LHx, t 0 ϕx Hx, t Hy, t + ϕx LHx, t

11 GADIENT ESTIMATE FO SOLUTIONS TO THE HEAT EQUATION We frther se that and combine it with 0 to obtain 0 ϕx Hx, t ϕx Hx, t ϕyhy, t, ϕx + ϕlhx, t ϕy = Hx, t ϕy ϕx ϕx ϕy + ϕlhx, t D max Hx, t + ϕlhx, t In the last line, we sed Lemma and the fact that ϕy ϕx ϕx ϕy = dx 0, y dx 0, x dx, x 0 dy, x 0 This, in trn, follows from the fact that if dx 0, y = dx 0, x making the above negative dx, x 0 dy, x 0 = dx, x 0 dx, x 0 + Now we handle the LH term Contining from, we have 0 D max Hx, t + ϕlhx, t = D max Hx, t + ϕx t L + t K Γ x, t + x, t log x, t t + ϕx D max L x, t log x, t > D [ max Hx, t + ϕx t D max L log x, t D ] max log x, t > D max Hx, t + D maxϕx [ t Γ x, t ] Dmax ϕx D max Hx, t 0 This contradiction implies that a maximm x, t of ϕh with ϕhx, t T Dmax cannot occr, and completes the proof of the theorem In this string of ineqalities, the first follows from the fact that L t Γ + log > 0 at all points where is locally non-constant, which is the content of the proof of Theorem 5 The second ineqality follows from or recorded ineqality 8 above; that is that L log Γ The third follows from the definition of H, and the final ineqality follows from the fact that ϕx

12 P HON While we have chosen to state these theorems by normalizing so that =, note that by normalizing one has the immediate corollary: Corollary 7 Sppose G = V, E is a finite or bonded degree infinite graph satisfying CDE, K for some K 0, and is a positive soltion to the heat eqation valid for V G 0, with Then at all x V G and t 0,, Γ log Note also that if, t denotes the maximm heat at time t, t, t 0 so that =, 0 3 A Comparison Ineqality We now show how to se Theorems 5 and 6 in order to derive a type of mltiplicative Harnack ineqality, comparing the heat of two different vertices at the same time As a comparison, it is helpfl tbo recall the Harnack ineqality derived in [] which is very mch of the classical form For convenience, we state it only in the simplest form that is when the graph is nweighted, and the the vertices have measre µv = degv for the definition of the Laplacian For a finite or infinite graph satisfying CDEn, 0, the Harnack ineqality then states that n T 4Ddx, y x, T y, T exp, T T T where D denotes the maximm degree of a vertex in G Note that the right hand side of the ineqality is ndefined if T = T The gradient estimate developed in this paper allows s to sidestep this isse, and derive a bond somewhat reminiscent of the above As a first step, we make a record a observation, in the vein of creflem:triv, which follows immediately from the definition of Γf Lemma 8 Sppose x y, and f : V, fx fy µ max w min Γfx Theorem 9 Sppose G = V, E is a finite or bonded degree infinite graph satisfying CDE, K for some K 0, and is a positive soltion to the heat eqation with = Then for all x, y V G and t 0,, x, t y, t exp dx, y µmax /y, t w min Proof We prove this by indction on dx, y, and frther note that we may assme that y, t exp dx, y µmax /y, t, w min as otherwise the ineqality is trivial

13 GADIENT ESTIMATE FO SOLUTIONS TO THE HEAT EQUATION 3 First, assme x y Then log x, t x, t = log y, t y, t x, t y, t y, t x, t y, t y, t µ max Γ y, t w min y, t µmax w min y, t Here, the first ineqality follows from the real nmber ineqality logr r, the third from Lemma 8, and the last from Theorem 5 or Theorem 6 earranging and exponentiating then gives the desired reslt in the case dx, y = Now consider the general case: Sppose dx, y = n, and consider a shortest path connecting them, x = x, x,, x n = y By indction, x, t y, t exp n µmax w min y, t Frthermore x, t x, t exp µmax w min x, t Now consider the fnction fz = z expc log/z for some positive constant C and considered for 0 z If f were an increasing

14 4 P HON fnction of z, then we cold simply combine the ineqalities, obtaining x, t x, t exp y, t exp exp y, t exp exp = y, t exp n n n µmax w min x, t µmax µmax w min y, t w min y, t exp µmax w min y, t µmax w min y, t µmax w min y, t Here in the second ineqality we se the fact that the exponential term appearing in the logarithm is at least one as the exponent is positive As log is increasing, discarding exponential term increases the argment in the log and hence we get the claimed ineqality This is exactly what we claimed; and we have verified it modlo showing that f is increasing Unfortnately, f is not a strictly increasing fnction, bt it is nimodal which will sffice for or prposes Note f z = expc C log/z log/z Ths fz is increasing for z exp 4/C and then decreasing Frthermore f =, and hence the maximm vale of f is larger than one To jstify the last comptation, we need to take C = n µmax w min fx, t f y, t exp n If this fails, then the maximm of fz mst occr for some z y, t exp n Since fz for z z, this means f y, t exp n and show that µmax w min y, t µmax w min y, t µmax > w min y, t

15 GADIENT ESTIMATE FO SOLUTIONS TO THE HEAT EQUATION 5 Bt or comptation showed that f y, t exp n y, t exp µmax w min y, t n µmax, w min y, t which we have assmed is at most one by or hypothesis This contradiction flly jstifies and proves the theorem Normalizing, one obtains Corollary 0 Sppose G = V, E is a finite or bonded degree infinite graph satisfying CDE, K for some K 0, and is a positive soltion to the heat eqation Then for all x, y V G and t 0,, x, t y, t exp dx, y µmax /y, t w min As an application, we observe that the maximm heat of a vertex in a finite graph at time t can be bonded in terms of the average heat which is nchanging This is most interesting when one has a non-negatively crved graph for which one obtains: Corollary Sppose G = V, E is an n vertex graph of diameter diamg which satisfies CDE, 0 Sppose x, t is a positive soltion to the heat eqation on G with = Then for all x V G and t 0,, x, t, t n exp diamg t µmax w min n, t This follows immediately from taking the y in Theorem 9 to have the minimm vale which is at most, t /n and noting, as in the proof of Theorem 9 that if this minimm were too large to apply monotonicity, the bond above is larger than one Also note that this corollary holds verbatim if as opposed to = We close with a final corollary related to mixing of random walks Sppose G is an nweighted d-reglar graph As is well known, a positive soltion to the heat eqation with, t = corresponds to the evoltion of the distribtion of the continos time random walk on a graph where is chosen with the measre µv = degv Note that in the irreglar case, it is possible to translate between the two bt they are not qite identical so, for simplicity, we stick to the reglar case For a reglar graph, the stationary distribtion is the niform distribtion and Corollary gives a bond on how mch the distribtion can exceed the stationary distribtion at time t That is, Corollary Sppose G = V, E is a D-reglar graph on n vertices, satisfying CDE, 0 Let, t denote the distribtion of a continos time random walk at time t Then for all x V G and t 0,, x, t n exp diamg D n t

16 6 P HON Ths for non-negatively crved graphs one immediately gets a bond on the mixing rate in terms of the order of the graph, the degree and the diameter For graphs satisfying CDE n, K for some K > 0 which also necessarily satisfy CDE, 0 the diameter can also be bonded in terms of K see eg [0] and hence one obtains a mixing rate on sch graphs Note that the corollary ses the fact that for a distribtion fnction, Indeed if a better a priori bond on is available, this can be sed in its stead EFEENCES [] D Bakry and M Émery, Diffsions hypercontractives, in Séminaire de probabilités, XIX, 983/84, vol 3 of Lectre Notes in Math, Springer, Berlin, 985, pp 77 06, doi:0007/bfb , [] F Baer, P Horn, Y Lin, G Lippner, D Mangobi, and S-T Ya, Li-Ya ineqality on graphs, J Differential Geom, 99 05, pp , jdg/ [3] F Chng, Spectral graph theory, vol 9 of CBMS egional Conference Series in Mathematics, Pblished for the Conference Board of the Mathematical Sciences, Washington, DC, 997 [4] F Chng, Y Lin, and S-T Ya, Harnack ineqalities for graphs with non-negative icci crvatre, J Math Anal Appl, 45 04, pp 5 3, doi:006/jjmaa040044, [5] F K Chng and S-T Ya, Logarithmic Harnack ineqalities, Math es Lett, 3 996, pp 793 8, doi:0430/ml996v3n6a8, a8 [6] F K Chng and S-T Ya, Eigenvale ineqalities for graphs and convex sbgraphs, Comm Anal Geom, 5 997, pp , doi:0430/cag997v5n4a, doiorg/0430/cag997v5n4a [7] T Delmotte, Parabolic Harnack ineqality and estimates of Markov chains on graphs, ev Mat Iberoamericana, 5 999, pp 8 3, doi:047/mi/54, 47/MI/54 [8] A A Grigor yan, The heat eqation on noncompact iemannian manifolds, Mat Sb, 8 99, pp [9] S Hamilton, A matrix Harnack estimate for the heat eqation, Comm Anal Geom, 993, pp 3 6, doi:0430/cag993vna6, 993vna6 [0] P Horn, Y Lin, S Li, and S-T Ya, Volme dobling, poincar ineqality and gassian heat kernel estimate for nonnegative crvatre graphs, preprint; arxiv:45087 [] B Klartag, G Kozma, P alli, and P Tetali, Discrete crvatre and abelian grops, Canad J Math, 68 06, pp , doi:0453/cjm , /CJM [] P Li and S-T Ya, On the parabolic kernel of the Schrödinger operator, Acta Math, , pp 53 0, doi:0007/bf039903, [3] Y Lin, L L, and S-T Ya, icci crvatre of graphs, Tohok Math J, 63 0, pp , doi:0748/tmj/ , [4] Y Lin and S-T Ya, icci crvatre and eigenvale estimate on locally finite graphs, Math es Lett, 7 00, pp , doi:0430/ml00v7na3, /ML00v7na3 [5] J Lott and C Villani, icci crvatre for metric-measre spaces via optimal transport, Ann of Math, , pp , doi:04007/annals , doiorg/04007/annals [6] Y Ollivier, icci crvatre of Markov chains on metric spaces, J Fnct Anal, , pp , doi:006/jjfa00800, [7] L Saloff-Coste, Parabolic Harnack ineqality for divergence-form second-order differential operators, Potential Anal, 4 995, pp , doi:0007/bf , org/0007/bf Potential theory and degenerate partial differential operators Parma [8] P Soplet and Q S Zhang, Sharp gradient estimate and Ya s Lioville theorem for the heat eqation on noncompact manifolds, Bll London Math Soc, , pp , doi:0/s ,