Stochastic completeness for graphs with curvature dimension conditions

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Advances in Mathematics

www.elsevier.com/locate/aim

Stochastic completeness for graphs with curvature dimension conditions

Bobo Hua^a,b, Yong Lin^c,∗

aSchoolofMathematicalSciences,LMNS,FudanUniversity,Shanghai200433, China

bShanghaiCenterforMathematicalSciences,FudanUniversity,Shanghai200433, China

cDepartmentofMathematics,InformationSchool,RenminUniversityofChina, Beijing100872,China

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received18June2015

Receivedinrevisedform1June2016 Accepted6October2016

Availableonline27October2016 CommunicatedbyAndreasDress

Keywords:

Graphs

Stochasticcompleteness Curvaturedimensionconditions Discretegeometricanalysis

We prove pointwise gradient bounds for heat semigroups associated to general (possibly unbounded) Laplacians on infinitegraphs satisfyingthecurvaturedimension condition CD(K,∞). Using gradient bounds, we show stochastic completeness for graphs satisfying the curvature dimension condition.

©2016ElsevierInc.Allrightsreserved.

1. Introductionandmain results

LetMbeacomplete,noncompactRiemannianmanifoldwithoutboundary.Itiscalled stochasticallycompleteif

M

pt(x, y)dvol(y) = 1, ∀t >0, x∈M, (1)

* Correspondingauthor.

E-mailaddresses:bobohua@fudan.edu.cn(B. Hua),linyong01@ruc.edu.cn(Y. Lin).

http://dx.doi.org/10.1016/j.aim.2016.10.022 0001-8708/©2016ElsevierInc.Allrightsreserved.

wherept(·,·) isthe(minimal)heatkernelonM.Yau[41]firstprovedthatanycomplete Riemannian manifold with a uniform lower bound of Ricci curvature is stochastically complete.KarpandLi[23]showedthestochasticcompletenessintermsofthefollowing volumegrowth property:

vol(Br(x))≤Ce^cr2, somex∈M, ∀r >0, (2) where vol(Br(x)) is the volume of the geodesic ball of radius r and centered at x.

Varopoulos [35], Li [27] and Hsu [19] extended Yau’s result to Riemannian manifolds with general conditions on Ricci curvature. So far, the optimal volume growth condi- tion for stochastic completeness wasgiven byGrigor’yan [14]. We referto [15] for the literatureonstochasticcompletenessofRiemannianmanifolds.Theseresultshavebeen generalizedto aquitegeneralsetting, namely,regularstrongly local Dirichletforms by Sturm[34].

Compared to local operators, graphs(discrete metric measure spaces) are nonlocal innatureand canbe regardedas regularDirichletforms associated tojump processes.

A generalMarkovsemigroup is called adiffusion semigroupif chainrules hold for the associated infinitesimal generator, see Bakry, Gentiland Ledoux [3, Definition 1.11.1], whichis aproperty relatedtothelocalityofthegenerator.Asacommonpointofview to manygraph analysts, theabsence of chainrules for discrete Laplacians is themain difficultyfortheanalysis ongraphs.Thiscausesmanyproblems andvariousinteresting phenomenaemergeongraphs.Agraphiscalledstochasticallycomplete(orconservative) ifanequationsimilarto(1)holdsforthecontinuoustimeheatkernel,seeDefinition 3.1.

The stochastic completeness of graphs has been thoroughly studied by many authors [7,8,13,21,24–26,32,36–39]. In particular, the volume criterion (2) with respect to the graphdistanceisnolongertrueforunboundedLaplaciansongraphs,see[39].Thiscan be circumventedbyusing intrinsicmetricsintroduced byFrank, Lenzand Wingert[9], see e.g.[11,13,22].

Gradient bounds of heat semigroups can be used to prove stochastic completeness.

Nowadays,theso-calledΓ-calculushasbeenwell developedintheframeworkofgeneral Markov semigroups where Γ is called the “carré du champ” operator, see [3, Defini- tion 1.4.2].Givenasmoothfunctionf onaRiemannianmanifold,Γ(f) standsfor|∇f|², seeSection2forthedefinitionongraphs.Heuristically,onaRiemannianmanifoldM if onecanshowthegradientbound fortheheat semigroup

Γ(P_tf)≤C_tP_t(Γ(f)), ∀f ∈C₀^∞(M), (3) where Pt=e^tΔM istheheat semigroupinducedbytheLaplace–BeltramioperatorΔM, Ct a function on t and C₀^∞(M) the space of compactly supported smooth functions on M,then thestochastic completenessfollows from approximatingthe constantfunc- tion1bycompactlysupportedsmoothfunctions.Thegradientbounds(3)canbeproved under curvature assumptions, e.g.auniform lower bound of Ricci curvature, and then

the function Ct depends on the curvature bound. This approach has been systemat- ically generalized to Markov diffusion semigroups, i.e. local operators, see [3]. In this paper,we closelyfollow thisstrategy andprovethestochastic completenessundercur- vature dimension conditions on graphs, see Section 2 for definitions. This shows that the gradient-bound approach workseven in the nonlocalsetting. Note that on graphs one can also interpret the curvature bounds by the bounds of Laplacians of distance functions, and the stochastic completeness has been obtainedin this curvature notion byDodziuk[8,Theorem 4.2],Weber[36, Theorem 4.5]and Huang[21, Theorem 5.3].

Weintroducethesetting ofgraphsandrefertoSection2fordetails.Let(V,E) bea connected,undirected,(combinatorial) infinitegraphwiththeset ofverticesV andthe setofedgesE.Wesayx,y∈V areneighbors,denotedbyx∼y,if(x,y)∈E.Thegraph iscalled locallyfinite ifeach vertexhasfinitely manyneighbors.Inthis paper, weonly considerlocallyfinitegraphs.Weassignaweightmtoeachvertex,m:V →(0,∞),and aweightμtoeachedge,

μ:E→(0,∞), E (x, y)→μ_xy,

andrefertothequadrupleG= (V,E,m,μ) asaweighted graph.Wedenoteby C0(V) :={f :V →R| {x∈V |f(x)= 0}is of finite cardinality}

theset offinitely supportedfunctionsonV andby^p(V,m),p∈[1,∞],the^p spacesof functionsonV with respecttothemeasure m.

For any weighted graph G = (V,E,m,μ), it associates with a Dirichlet form with respecttotheHilbertspace²(V,m) correspondingtotheDirichletboundarycondition,

Q^(D): D(Q^(D))×D(Q^(D))→R (f, g)→ 1

2

x∼y

μ_xy(f(y)−f(x))(g(y)−g(x)), (4) wheretheform domainD(Q^(D)) isdefinedasthecompletionofC₀(V) under thenorm · Q givenby

f²Q =f²²(V,m)+1 2

x∼y

μxy(f(y)−f(x))², ∀f ∈C0(V),

see Keller and Lenz [25]. For the Dirichlet form Q^(D), its (infinitesimal) generator, denoted by L, is called the (discrete) Laplacian. Here we adopt the sign convention such that −L is a nonnegative operator. The associated C0-semigroup is denoted by P_t=e^tL:²(V,m)→²(V,m).Forlocally finitegraphs,thegeneratorL actsas

Lf(x) = 1 m(x)

y∼x

μxy(f(y)−f(x)), ∀f ∈C0(V),

see [25, Theorems 6 and 9]. Obviously, the measure m plays an essential role in the definitionof theLaplacian.GiventheweightμonE,typical choicesof mofparticular interestare:

• m(x)=

y∼xμxy forany x∈V andtheassociatedLaplacianiscalled thenormal- izedLaplacian.

• m(x) = 1 for any x ∈ V and the Laplacian is called combinatorial (or physical) Laplacian.

NotethatnormalizedLaplaciansareboundedoperators,sothatthesegraphsarealways stochastically complete,see[8]or KellerandLenz[24].Thus,theonlyinterestingcases are combinatorialLaplacians,ormoregeneralunboundedLaplacians.

Following the strategy in [3], to show stochastic completeness for the semigroups associated to unboundedLaplacians ongraphs,itsufficesto provethe gradientbounds asin(3).Forthatpurpose,wefirstintroduceacompletenessconditionforinfinitegraphs:

A graphG= (V,E,m,μ) is called complete ifthere exists anondecreasingsequenceof finitely supportedfunctions{ηk}^∞k=1 suchthat

klim→∞η_k=1 and Γ(η_k)≤ 1

k, (5)

where 1 is the constant function 1 on V. Without loss of generality, we may assume 0≤η_k ≤1 forallk∈Nbytakingthepositivepartofη_k,i.e.max{η_k,0}.Notethatthe measure mplaysaroleinthedefinitionofΓ,seeDefinition 2.3,sothatitisessentialto the completenessof aweightedgraph.This conditionwasdefined forMarkovdiffusion semigroups in [3, Definition 3.3.9]; herewe adapt it to graphs. As is well-known, this conditionisequivalenttothegeodesiccompletenessforRiemannianmanifolds,see[33].

Forthediscretesetting,thisconditionissatisfiedforalargeclassofgraphswhichpossess intrinsic metrics,see Theorem 2.8.

For gradient bounds (3), besides completeness we need curvature dimension condi- tions. ForMarkovdiffusionsemigroups,thecurvature dimensionconditionsaredefined via the Γ operator and the iterated operator denoted by Γ₂, see [3, Eq. 1.16.1]. This approach, using curvature dimension conditions to obtain gradient bounds, was ini- tiated in Bakry and Émery [2]. The curvature dimension condition on graphs, the non-diffusion case, was first introduced by Lin and Yau [31] which serves as a com- bination of a lower bound of Ricci curvature and an upper bound of the dimen- sion, see Definition 2.4 for an infinite dimensional version CD(K,∞). For bounded Laplacians on graphs, Bauer et al. [5] introduced an involved curvature dimension condition, the so-called CDE(K,n) condition, to prove the Li–Yau gradient estimate for heat semigroups. Also restricted to bounded Laplacians, Lin and Liu [29] proved the equivalence between the CD(K,∞) condition and the gradient bounds (3) for heat semigroups, see Liu and Peyerimhoff [30] for finite graphs. In this paper, under

some mild assumptions, we prove the gradient bounds for unbounded Laplacians on graphs.

Theorem 1.1 (see Theorem 4.1). Let G = (V,E,m,μ) be a complete graph and m be non-degenerate,i.e.infx∈V m(x)>0.Thenthefollowingareequivalent:

(a) G satisfiesCD(K,∞).

(b) Foranyf ∈C0(V),

Γ(Ptf)≤e^−2KtPt(Γ(f)).

SinceitisnotclearwhatvolumegrowthisforagraphsatisfyingtheCD(K,∞) condi- tion,ourresultcannotbederivedfromthecriteriainvolvingvolumegrowthconditions.

For unbounded Laplacians, standard techniques for bounded Laplacians as in [29,30]

fail due to essential difficulties in the summability of solutions to heat equations. For instance,wedon’t know whetherΓ(Ptf) liesintheform domain (or, morestrongly,in the domain of thegenerator), see Remark 4.2. In order to overcome these difficulties, we add amild assumption onthe measure m, i.e. thenon-degeneracy of the measure, andcriticallyutilizetechniquesfrompartialdifferentialequations,seeLemma 3.4forthe Caccioppoliinequalityand Theorem 4.5.ForCaccioppoli inequalitiesforgeneralgraph Laplacians,onemay referto [18, Lemma 3.4], [9,Theorem 11.1] or [16, Theorem 1.8].

Theassumptionofthenon-degeneracy ofthemeasuremismildsinceitisautomatically satisfiedforany combinatorialLaplacian.

Adirectconsequenceofthegradientboundsisthestochasticcompletenessforgraphs satisfyingtheCD(K,∞) condition.

Theorem 1.2. Let G = (V,E,μ,m) be a complete graph satisfying the CD(K,∞) con- dition for some K ∈ R. Suppose that the measure m is non-degenerate, then G is stochasticallycomplete.

Wegiveanexample,seeExample 2.5,ofaweightedgraphwithunboundedLaplacian satisfyingtheCD(0,∞) conditionwhere wemayapply thistheorem.

Thepaperisorganizedasfollows:Innextsection,wesetupbasicnotationsofweighted graphs.TheΓ-calculusisintroducedtodefinecurvaturedimensionconditions.Wedefine a new concept on the completeness of a graph and prove the completeness under the assumptions involving intrinsic metrics on graphs. In Section 3, we adopt some PDE techniques to provea(discrete) Caccioppoli inequalityfor Poisson’s equations. InSec- tion 4, we prove our main results: the equivalence of curvature dimension conditions andthegradientboundsforheatsemigroupsoncompletegraphs,Theorem 1.1,andthe stochastic completenessfor graphssatisfyingthecurvature dimensioncondition, Theo- rem 1.2.

2. Graphs

2.1. Weightedgraphs

Let (V,E) bea (finite or infinite) undirected graph with the set of vertices V and theset ofedgesE where EisasymmetricsubsetofV ×V.Twoverticesx,y arecalled neighborsif(x,y)∈E,inthiscasedenotedbyx∼y.Atavertexx,if(x,x)∈E,wesay there isaself-loopatx.Inthispaper, wedoallowself-loopsforgraphs.Agraph(V,E) iscalledconnectedifforanyx,y∈V thereisafinitesequenceofvertices,{xi}ⁿi=0,such that

x=x0∼x1∼ · · · ∼xn=y.

Inthispaper, weonlyconsiderlocally finiteconnectedgraphs.

We assign weights,m and μ, on theset ofvertices V and edges E respectivelyand refer tothe quadrupleG= (V,E,m,μ) asaweightedgraph:Here μ:E →(0,∞),E (x,y)→ μxy is symmetric, i.e.μxy =μyx for any(x,y)∈E, and m:V →(0,∞) is a measure on V of full support. Forconvenience, we extendthe function μ onE to the totalset V ×V,μ:V ×V →[0,∞),suchthatμ_xy= 0 foranyxy.

For functions defined on V, we denote by ^p(V,m) or simply ^p_m, the space of ^p summable functions w.r.t. the measure m and by · ^pm the ^p norm of a function.

Given a weighted graph (V,E,m,μ), there is an associated Dirichlet form w.r.t. ²_m corresponding totheNeumannboundarycondition,see[17],

Q^(N): D(Q^(N))×D(Q^(N))→R (f, g)→Q^(N)(f, g) := 1

2

x,y∈V

μ_xy(f(y)−f(x))(g(y)−g(x)),

where D(Q^(N)) := {f ∈ ²_m|

x,yμxy(f(y)−f(x))² < ∞}. For simplicity, we write Q^(N)(f) := ¹₂

x,yμxy(f(y)−f(x))² for any f : V → R. Let D(Q^(D)) denote the completionofC₀(V) underthenorm· Q definedby

fQ=

f²²_m+Q^(N)(f), ∀f ∈C₀(V).

AnotherDirichletformQ^(D),definedastherestrictionofQ^(N)toD(Q^(D)),corresponds to theDirichletboundarycondition,see(4) inSection1.

FortheDirichletform Q^(N),there isauniqueself-adjointoperatorL^(N) on²_mwith D(Q^(N)) = Domain of definition of (−L^(N))¹²

and

Q^(N)(f, g) =

(−L^(N))¹²f,(−L^(N))¹²g

, f, g∈D(Q^(N))

where ·,· denotes the inner product in ²_m. The operator L^(N) is the infinitesimal generatorassociated to the Dirichletform Q^(N), alsocalled the (Neumann)Laplacian.

TheassociatedC₀-semigroupon²_misdenotedbyP_t^(N)=e^tL(N).FortheDirichletform Q^(D), L^(D) and P_t^(D) are defined in the same way. In case that the Dirichlet forms correspondingto NeumannandDirichletboundaryconditionscoincide,i.e.

Q^(N)=Q^(D), weomitthesuperscriptsandsimplywrite

Q=Q^(N)=Q^(D), L=L^(N)=L^(D) etc.

Thefollowing integration byparts formula is usefulin furtherapplications, see [12, Corollary 1.3.1].

Lemma 2.1 (Green’s formula). Let(V,E,m,μ) be a weighted graph. Then for any f ∈ D(Q^(N))andg∈D(L^(N)),

x∈V

f(x)L^(N)g(x)m(x) =−Q^(N)(f, g). (6) Asimilar consequenceholds forthecaseofDirichlet boundary condition.

Forlocallyfinite graphs,wedefine theformal Laplacian,denotedbyΔ,as Δf(x) = 1

m(x)

y∈X

μxy(f(y)−f(x)) ∀f :V →R.

ThisformalLaplaciancanbe usedtoidentifythegeneratorsdefinedbefore.Aresultof KellerandLenz,[25, Theorem 9],statesthat

L^(D)f = Δf, ∀f ∈D(L^(D)), (7)

andasimilarresultholdsforNeumanncondition,see[17]. Notethat Δf ∈C₀(V), ∀f ∈C₀(V).

Differentchoicesforthemeasure minducedifferent Laplacians.Thetypicalchoicesare normalizedLaplaciansandcombinatorialLaplacians,see Section1. Definetheweighted vertexdegree Deg :V →[0,∞) by

Deg(x) = 1 m(x)

y∈V

μxy, x∈V.

Thenitisknown,seee.g.[25],thattheLaplacianassociatedwiththegraph(V,E,m,μ) is aboundedoperatorfrom²_mto²_m ifandonlyif

sup

x∈V

Deg(x)<∞. Themeasure monV iscallednon-degenerate if

δ:= inf

x∈Vm(x)>0. (8)

Thenon-degeneracy of themeasuremyieldsaveryusefulfactfor^p(V,m) spaces.

Proposition 2.2. Let m be a non-degenerate measure on V as in (8). Then for any f ∈^p(V,m),p∈[1,∞),

|f(x)| ≤δ^−1pf^pm ∀x∈V.

Moreover, forany 1≤p< q≤ ∞,^p(V,m)→^q(V,m).

Proof. Thefirstassertionfollowsfrom|f(x)|^pδ≤ |f(x)|^pm(x)≤ f^pp_m.Thesecondone is aconsequenceoftheinterpolationtheorem. 2

Under assumptions of non-degeneracy of the measure m and local finiteness of the graph,theDirichletformscorrespondingtoNeumannandDirichletboundaryconditions coincide,i.e.

Q^(N)=Q^(D),

see [25, Theorem 6] and[17, Corollary 5.3],and the domains ofgenerators arecharac- terized as

D(L^(N)) =D(L^(N)) ={f ∈²_m|Δf ∈²_m}. 2.2. Gamma calculus

WeintroducetheΓ-calculusandcurvaturedimensionconditionsongraphsfollowing [5,31].

First we define two natural bilinear forms associated to the Laplacian. Given f : V →R andx,y ∈V, wedenote by∇xyf :=f(y)−f(x) thedifference ofthe function f ontheverticesxandy.

Definition2.3.ThegradientformΓ,calledthe“carréduchamp”operator,isdefinedby Γ(f, g)(x) = 1

2(Δ(f g)−fΔg−gΔf)(x)

= 1

2m(x)

y

μ_xy∇xyf∇xyg.

Forsimplicity, wewrite Γ(f):= Γ(f,f). Moreover,the iteratedgradientform,denoted byΓ2,isdefinedas

Γ2(f, g) =1

2(ΔΓ(f, g)−Γ(f,Δg)−Γ(g,Δf)).

WewriteΓ2(f):= Γ2(f,f)=¹₂ΔΓ(f)−Γ(f,Δf).

TheCauchy–Schwarzinequalityimpliesthat Γ(f, g)≤

Γ(f)Γ(g)≤1

2(Γ(f) + Γ(g)). (9)

Inaddition,onecaneasilysee thatQ^(N)(f)=Γ(f)¹_m.

Nowwecanintroducecurvaturedimensionconditionsongraphs.

Definition2.4. Wesayagraph(V,E,m,μ) satisfiestheCD(K,∞) condition,K∈R,if foranyx∈V,

Γ2(f)(x)≥KΓ(f)(x).

In the following, we give an example with unbounded weighted vertex degree, i.e.

sup_x∈V Deg(x)=∞,satisfyingtheCD(0,∞) condition.

Example 2.5. Let V =N, E ={(i,j) :|i−j| = 1,i,j ∈N}, m(i)= 1 and μi,i+1 =i,

∀i∈N.

Proof. Consider general weights m and μ on (V,E). To simplify the notation, we set μi =μi,i+1 andmi =m(i) for alli ∈N. First,consider i≥3. Forany function f,set x=f(i−1)−f(i−2),y=f(i)−f(i−1),z=f(i+ 1)−f(i),w=f(i+ 2)−f(i+ 1) whicharearbitrarysincef is. WecalculatethequantityΓ2(f) atthevertex iwhich is aquadratic form in x,y,z and w. Using basic estimates C₁x²+ 2C₁xy ≥ −C₁y² and C₂w²+ 2C₂wz≥ −C₂z² forC₁,C₂>0 toeliminatethevariablesxandw,wehave

2miΓ2(f)(i)≥ μ_i−1(3μ_i−1−μ_i−2) 2mi−1

+μ_i−1(μ_i−1−μ_i) 2mi

y²−2μ_iμ_i−1 mi

yz + μ_i(3μ_i−μ_i+1)

2mi+1

+μ_i(μ_i−μ_i−1) 2mi

z².

Then pluggingintoittheassumptionsofμandmfortheexample,onecanshowthat Γ₂(f)(i)≥0, ∀i≥3.

Fori= 1,2,itisalso truebydirectcalculation. 2

By our theorem,Theorem 1.2, this graphis stochastically complete. Note thatthis canalsobeenobtainedbyusingthecurvaturenotionofLaplaciansofdistancefunctions, e.g.[36]and [21],orother volumegrowth criteria,e.g.[13].

Onecanalsodefineafinitedimensionalversion,CD(K,n) condition(see[31]),which isstrongerthanCD(K,∞).Infact,thepreviousexamplesatisfiestheCD(0,2) condition.

Aninvolvedcurvature dimensioncondition,calledCDE(K,n),wasintroducedin[5].In this paper,weonlyconsider CD(K,∞) conditions.

2.3. Completenessofgraphs

Yau [41]first provedthatcomplete Riemannianmanifoldswith Riccicurvature uni- formlyboundedfrombelowarestochasticallycomplete.Bakry[1]provedthestochastic completeness for weighted Riemannian manifolds satisfying CD(K,∞) condition for weightedLaplacians,see also Li[28].The completenessof Riemannianmanifoldsplays animportantroleintheseproblems.

Foragraph(V,E,m,μ),wedefinethecompletenessofagraphasin(5),seeSection1.

The following lemma shows the importance of thecompleteness of agraph. Note that wedon’tneedthenon-degeneracy ofthemeasure mhere.Asimilarresultcanbefound in[18,Theorem 1].

Lemma2.6.Let(V,E,m,μ)beacompletegraph.Foranyf ∈²_msuchthatQ^(N)(f)<∞ we have

f η_k−fQ →0, k→ ∞.

Thatis,C₀(V)isadensesubsetoftheHilbertspace(D(Q^(N)),·Q)andQ^(N)=Q^(D). Proof. Since we cantake0≤ηk ≤1 and lim_k→∞ηk =1, thedominated convergence theorem yieldsthatfk:=f ηk →f in²_m.Soitsufficestoshow thatQ^(N)(fk−f)→0, k→ ∞.

Q^(N)(f_k−f) = 1 2

x,y

μ_xy|∇xyf(η_k−1)|²

= 1 2

x,y

μ_xy|∇xyf ·(η_k(y)−1) +f(x)∇xyη_k|²

≤

x,y

μxy(|∇xyf|²|ηk(y)−1|²+f²(x)|∇xyηk|²)

=Ik+IIk.

Bythedominatedconvergencetheorem,Ik→0 ask→ ∞.Forthesecondterm, II_k ≤ 2

k²

x

f²(x)m(x)→0, k→ ∞. Thisprovesthelemma. 2

Henceforacompletegraph,Q^(N)=Q^(D).Intherestofthepaper,givenacomplete graphwesimplywrite Q=Q^(N)=Q^(D), andby(7)

L= Δ, onD(L).

2.4. Intrinsic metrics

Inorder to dealwithunboundedLaplacians, weneed thefollowing intrinsicmetrics ongraphsintroducedin[9].

A pseudometric ρis asymmetricfunction, ρ:V ×V → [0,∞), withzero diagonal whichsatisfiesthetriangleinequality.

Definition2.7(Intrinsic metric).ApseudometricρonV iscalled intrinsicif

y∈V

μxyρ²(x, y)≤m(x), ∀x∈V.

In various situationsthe naturalgraph distance, called the combinatorial distance, provestobeinsufficientfortheinvestigationsofunboundedLaplacians,see[26,38,39].For thisreasontheconceptofintrinsicmetricsreceivedquitesomeattentionasacandidateto overcometheseproblems.Indeed,intrinsicmetricsalreadyhavebeenappliedsuccessfully tovarious problemsongraphs[4,6,10,11,13,16,18].

Fixabase pointo∈V and denotethedistanceballsby Br(o) ={x∈V |ρ(x, o)≤r}, r≥0.

The choice of the base point o will be irrelevant to our results later. We say B_r(o) is finite,ifitisoffinite cardinality,i.e.Br(o)<∞.

Theorem2.8.LetG= (V,E,m,μ)beagraphandρbeanintrinsicmetriconG.Suppose that eachballB_r(o),r >0,isfinite,then Gisacomplete graph.

Proof. For any 0 < r < R, we denote by η_r,R the cut-off function on B_R(o)\B_r(o) definedas

ηr,R(·) = min

max

R−ρ(·, o) R−r ,0

,1

.

Set ηk := ηk,2k.Then {ηk}is anondecreasing sequence offinitely supportedfunctions whichconverges totheconstantfunction1pointwise.Moreover,

Γ(η_k)(x) = 1 2m(x)

y∈V

μ_xy|∇xyη_k|²

≤ 1 2m(x)k²

y∈V

μ_xyρ²(x, y)

≤ 1 2k² < 1

k,

where weusedthedefinitionoftheintrinsicmetricρ.This provesthetheorem. 2 Foranyweightedgraph(V,E,m,μ),intrinsicmetricsalwaysexist.Thereisanatural intrinsic metricintroducedbyHuang[20,Lemma 1.6.4].

Example 2.9.Foranygivenweightedgraphthere isanintrinsicpathmetricdefinedby δ(x, y) = inf

x=x0∼...∼xn=y n−1

i=0

(Deg(xi)∨Deg(xi+1))⁻¹², x, y∈V, where theinfimumis takenover allfinite pathsconnectingxandy.

Forthecompletenessofthegraph,itsufficestofindanintrinsicmetricsatisfyingthe conditionsinTheorem 2.8.Forinstance,onecancheckwhethereachballoffiniteradius underthemetricδisfinite.

3. SemigroupsandCaccioppoliinequality 3.1. Semigroupson graphs

In thissection, westudy thepropertiesof heat semigroupsongraphs,whichwill be used later.

Wedenote byP_t^(D)=e^tL(D) theC0-semigroupassociated totheDirichletformQ^(D) on ²_m. It extrapolates to C0-semigroups on ^p_m for all p ∈ [1,∞], for simplicity still denotedbyP_t^(D),see[25].

Definition 3.1.Aweightedgraph(V,E,m,μ) iscalledstochastically completeif P_t^(D)1=1, ∀t >0,

where 1istheconstantfunction1 on V.

The next proposition is a consequence of standard Dirichlet form theory, see [12]

and[25].

Proposition3.2. Forany f ∈^p_m,p∈[1,∞],wehave P_t^(D)f ∈^p_m and P_t^(D)f^pm ≤ f^pm, ∀t≥0.

Moreover,P_t^(D)f ∈D(L^(D))forany f ∈²_m.

Thenextproperty followsfromthespectral theorem.

Proposition3.3. Forany f ∈D(L^(D)),

L^(D)P_t^(D)f =P_t^(D)L^(D)f.

3.2. Caccioppoli inequality

For elliptic partial differential equations on Riemannian manifolds, the Caccioppoli inequalityis well-knownandyieldstheL^p Liouville theoremforharmonicfunctionsfor p∈(1,∞),seeYau[40].

ByadaptingPDEtechniques on manifoldsto graphs,we obtainthe Caccioppoliin- equalityforsubsolutionstoPoisson’s equations.

Lemma3.4.Let(V,E,m,μ)beaweightedgraphandg,h:V →Rsatisfyingthefollowing Δg≥h.

Thenforany η∈C0(V),

Γ(g)η²¹_m ≤C(Γ(η)g²¹_m+ghη²¹_m). (10) Proof. Multiplyingη²gtobothsidesoftheinequality,Δg≥h,andsummingoverx∈V w.r.t.themeasurem,we get

x

η²gh(x)m(x)≤

x

η²gΔg(x)m(x)

=−1 2

x,y

∇xyg∇xy(η²g)μxy

=−1 2

x,y

∇xyg(∇xygη²(x) +g(y)∇xy(η²))μ_xy

=−1 2

x,y

|∇xyg|²η²(x)μxy−1 2

x,y

∇xygg(y)(|∇xyη|²+ 2η(x)∇xyη)μxy,

where weusedGreen’sformula,seee.g.Lemma 2.1,inthesecondlinesinceη∈C0(V).

Forthesecond terminthelastline,bysymmetryonehas

−1 2

x,y

∇xygg(y)|∇xyη|²μ_xy=−1 4

x,y

|∇xyg|²|∇xyη|²μ_xy≤0.

Hence,bythisobservation,thepreviousestimateleadsto 1

2

x,y

|∇xyg|²η²(x)μ_xy

≤ −

x,y

∇xygg(y)η(x)∇xyημxy−

x

η²gh(x)m(x)

≤1 4

x,y

|∇xyg|²η²(x)μ_xy+

x,y

|∇xyη|²g²(y)μ_xy−

x

η²gh(x)m(x),

where we used basic inequality ab ≤ ¹₄a²+b² for a,b ∈ R. The lemma follows from canceling thefirstterminthelastline withthelefthandsideofthesystemofinequali- ties. 2

UsingthisCaccippoliinequality,wegetauniformupperboundoftheDirichletenergy of Ptf fort>0 andf ∈C0(V).

Lemma3.5.Let(V,E,m,μ)beacompletegraph.Thenforanyf ∈C0(V)andt∈[0,∞), Q(Ptf) =Γ(Ptf)¹_m ≤Cf²_mΔf²_m,

where C isauniformconstant.

Proof. Forf ∈C₀(V),thelocalfinitenessofthegraphimpliesthatΔf ∈C₀(V).Bythe completeness of the graph, letη_k ∈ C₀(V) satisfy (5). Since P_tf satisfies the equation

d

dtPtf = ΔPtf for any t > 0, applying the Caccippoli inequality in Lemma 3.4 with g=Ptf,h= _dt^dPtf andη=ηk,wehave

Γ(Ptf)η²_k¹_m ≤C(Γ(ηk)|Ptf|²¹_m+Ptf· d

dtP_tf·η²_k¹_m)

≤C 1

kPtf²²_m+Ptf²_md dtPtf²_m

.

ByProposition 3.2,

Ptf²_m ≤ f²_m

and byProposition 3.3and theequation (7),

d

dtPtf²_m =ΔPtf²_m =PtΔf²_m≤ Δf²_m. Hence

Γ(P_tf)η²_k¹_m ≤C 1

kf²²_m+f²_mΔf²_m

.

By passing to the limit, k → ∞, the monotone convergence theorem yields the lemma. 2

The following result is an improved estimate of the previous lemma which will be usefulinfurtherapplications.

Lemma 3.6. Let(V,E,m,μ)be a complete graph. Thenfor any f ∈C0(V) andT > 0, wehave max_[0,T]Γ(P_tf)∈¹_m and

max

[0,T]Γ(Ptf) ¹_m

≤C1(T, f), (11)

whereC₁(T,f)isaconstant depending on T andf.Moreover, max

[0,T]|Γ(P_tf, d

dtP_tf)| ∈¹_m and max

[0,T]|Γ(Ptf, d dtPtf)|

¹_m

= max

[0,T]|Γ(Ptf,ΔPtf)| ¹_m

≤C2(T, f). (12) Proof. Thelocalfiniteness yieldsthatΔf ∈C₀(V) andΔ²f ∈C₀(V) forf ∈C₀(V).

Forthefirstassertion,theNewton–Leibnizformulayields,forany t>0,

Γ(P_tf) = Γ(f) + t

0

d

dsΓ(P_sf)ds

= Γ(f) + 2 t

0

Γ(P_sf, d

dsP_sf)ds

= Γ(f) + 2 t

0

Γ(P_sf,ΔP_sf)ds

= Γ(f) + 2 t

0

Γ(P_sf, P_s(Δf))ds,

where the last equality follows from Proposition 3.3. Hence by the equation (9) and Lemma 3.5

max

[0,T]Γ(P_tf) ¹_m

≤ Γ(f)¹_m+ 2 T

0

|Γ(P_sf, P_s(Δf))|ds ¹_m

≤ Γ(f)¹_m+ T

0

(Γ(Psf)¹_m+Γ(Ps(Δf))¹_m)ds

≤ Γ(f)¹_m+CTΔf²_m(f²_m+Δ²f²_m) =:C1(T, f).

Thesecondassertionisadirectconsequenceofthefirstone.ByΔf ∈C₀(V) and(9), max

[0,T]|Γ(Ptf, d dtPtf)|

¹_m

= max

[0,T]|Γ(Ptf,ΔPtf)| ¹_m

= max

[0,T]|Γ(Ptf, PtΔf)| ¹_m

≤ 1 2

max

[0,T]Γ(P_tf) ¹_m

+1 2

max

[0,T]Γ(P_tΔf) ¹_m

≤ 1

2(C₁(T, f) +C₁(T,Δf)) =:C₂(T, f).

This provesthelemma. 2

Now we can show that the Dirichlet energy, t → Q(Ptf), decays in time for the semigroupPtoncompletegraphs.

Proposition 3.7. Let(V,E,m,μ)be acompletegraph. Thenforany f ∈C₀(V), Q(P_tf)≤Q(f), ∀t≥0.

Moreover, forany f ∈D(Q),

Q(Ptf)≤Q(f), ∀t≥0.

Proof. Forthefirst assertion,takingthe formalderivativeof timeinQ(P_tf) fort>0, we get

d

dtQ(P_tf) = 2

x∈V

Γ(P_tf, d

dtP_tf)(x)m(x). (13)

Given afixed T > t,notethatforanyt∈[0,T],

|Γ(Ptf, d

dtPtf)(x)| ≤ max

t∈[0,T]|Γ(Ptf, d

dtPtf)(x)|=:g(x)∈¹_m

which follows from (12) in Lemma 3.6. Hence the absolute value of the summand on the right handside of (13)is uniformly (for t∈ [0,T]) bounded abovebya summable

functiong.ThedifferentiabilitytheoremyieldsthatQ(Ptf) isdifferentiableintimeand whosederivativeisgivenby(13).

SincePtf ∈D(L) andΔPtf =PtΔf ∈D(Q),Green’sformulainLemma 2.1yields d

dtQ(Ptf) = 2

x∈V

Γ(Ptf,ΔPtf)(x)m(x)

=−2

x∈V

|ΔPtf(x)|²m(x)≤0.

Thisprovesthefirstassertion.

For the second assertion, set f_k := f η_k for f ∈D(Q). It follows from the previous resultthat

Q(Ptfk)≤Q(fk).

ByLemma 2.6, fk →f inthenorm · Q. Themonotone convergence theorem yields that

P_tf_k →P_tf pointwise.ByFatou’slemma,

Q(P_tf)≤lim inf

k→∞ Q(P_tf_k)≤lim inf

k→∞ Q(f_k) =Q(f).

Thisprovestheresult.

Analternative proof providedby the referee.Letf ∈D(Q^(D)) and μbe thespectral measure of L^(D) with respect to the function g = (−L^(D))¹²f (which is in ²_m since f ∈D(Q^(D))=D((−L^(D))¹²).Since−L^(D)hasnonnegativespectrum,oneconcludesby thespectraltheorem

Q^(D)(P_t^(D)f) =e^tL(D)g, e^tL(D)g= ∞

0

e^−2txdμ(x)≤ ∞

0

dμ(x) =g, g=Q^(D)(f).

Notethatthisproofdoesn’tusethecompletenessofthegraphatthemoment.Thenone mayapply Q^(D)=Q^(N)bythecompleteness. 2

4. Stochasticcompleteness

4.1. Gradient boundsandcurvature dimensionconditions

The curvature dimension condition implies gradient bounds, see [3] for the case of Markovdiffusionsemigroups.Infact,theyareequivalent onlocallyfinite graphsunder somemild assumptions.

Theorem 4.1. Let G= (V,E,m,μ)be a complete graph with a non-degenerate measure m,i.e.infx∈V m(x)>0.Thenthefollowingareequivalent:

(a) Gsatisfies CD(K,∞).

(b) Forany f ∈C₀(V),

Γ(P_tf)≤e^−2KtP_t(Γ(f)).

(c) Forany f ∈D(Q),

Γ(P_tf)≤e^−2KtP_t(Γ(f)).

Remark 4.2. For the case of finite graphsor bounded Laplacians,this resulthas been provenby[29,30].Toillustratetheirproofstrategy,weconsiderafinitegraph(V,E,m,μ) satisfyingtheCD(0,∞) condition.

(a)⇒(b):Foranyf :V →R,set Λ(s)=Ps(Γ(Pt−sf)).Then Λ(s) = ΔP_s(Γ(P_t−sf))−2P_s(Γ(P_t−sf,ΔP_t−sf))

=P_s(ΔΓ(P_t−sf)−2Γ(P_t−sf,ΔP_t−sf))≥0,

where thelastinequalityfollowsfrom theCD(0,∞) condition.However,forthecaseof infinitegraphs,ΔPs(Γ(Pt−sf))=PsΔ(Γ(Pt−sf)) maynotholdsinceingeneralwedon’t know whetherΓ(Pt−sf)∈D(L).

Inaddition, astrongversionofgradientboundshasbeen provedusingthefollowing strongercurvaturecondition,see[3,equation 3.2.4]

Γ(Γ(g))≤4Γ(g)[Γ2(g)−KΓ(g)], ∀g∈C0(V). (14) However,thisstrongercurvatureconditioncanneverbe fulfilledforgraphs.Infact,the inequality(14)failse.g.forg=δ_x.

4.2. Curvaturedimensionconditions andthepropertiesof heat semigroups

InordertoprovethegradientestimateundertheCD(K,∞) condition,weneedsome lemmata. For graphs satisfying the CD(K,∞) condition, the following lemma states thatΓ(Ptf) isasubsolutiontotheheatequationassociatedtotheSchrödingeroperator Δ−2K,astandarddefinitioninthetheoryofPDEs.

Lemma 4.3. Let (V,E,m,μ) be a complete graph satisfying the CD(K,∞) condition.

Then foranyf ∈C0(V) d

dtΓ(P_tf)≤ΔΓ(P_tf)−2KΓ(P_tf).

Proof. This follows from direct calculation by means of the CD(K,∞) condition and localfinitenessofthegraph. 2

Lemma4.4. Let(V,E,m,μ)be acomplete graph.Thenforany f ∈C0(V)andt≥0, d

dtΓ(P_tf) ¹_m

≤2

Q(f)Q(Δf).

Proof. Thisfollows bythecomputation, d

dtΓ(Ptf) ¹_m

= 2

x

Γ(Ptf, d

dtPtf)(x) m(x)

= 2

x

|Γ(Ptf,ΔPtf)(x)|m(x) = 2

x

|Γ(Ptf, PtΔf)(x)|m(x)

≤2

x

Γ(P_tf)m(x)

x

Γ(P_tΔf)m(x)

≤2

x

Γ(f)m(x)

x

Γ(Δf)m(x)<∞,

where weused Proposition 3.3for f ∈C₀(V) inthe thirdequality andProposition 3.7 forf,Δf ∈C₀(V) inthelastone. 2

Forcomplete graphssatisfyingtheCD(K,∞) condition,wehavehighersummability ofthesolutionsto heatequations.

Theorem 4.5. LetG= (V,E,m,μ) be acomplete graph with anon-degenerate measure m.If GsatisfiestheCD(K,∞)condition,then foranyf ∈C0(V)andt≥0,

Γ(Ptf)∈D(Q).

Proof. From the proof of Proposition 3.7, Γ(P_tf) ∈ ¹(V,m). Hence by the non- degeneracy ofm,Γ(P_tf)∈²(V,m).ItsufficestoprovethatQ(Γ(P_tf))<∞.

Let{ηk}bethesequencein(5)bythecompletenessofthegraph.NotethatLemma 4.3 impliesthatΓ(Ptf) isasubsolutiontotheheatequationassociatedtoΔ−2K.Applying theCaccioppoliinequality(10)withg= Γ(Ptf),h= _dt^dg+ 2Kgandη=ηk,weget

Γ(g)η_k²¹_m ≤C(Γ(ηk)g²¹_m+g(d

dtg+ 2Kg)η_k²¹_m)

≤C 1

kg²²_m+gd

dtg¹_m+ 2|K| · g²²_m

≤C(K)(g²²_m+g d dtg¹_m)

=I+II,

where the constant C(K) only depends on K. By the assumption that m is non- degenerate, Propositions 2.2 and3.7yieldthat

I≤CΓ(Ptf)²¹_m ≤CΓ(f)²¹_m <∞.

Fortheotherterm, notingthatg^∞ ≤Cg¹_m,byLemma 4.4,wehave II ≤Cg^∞d

dtg¹_m

≤Cg¹_md

dtg¹_m<∞.

Thus, Γ(g)η²_k¹_m ≤C <∞where theright handside isindependent ofk.Bypassing to thelimit,k→ ∞,Fatou’slemma yieldsthat

Γ(Γ(P_tf))¹_m ≤lim inf

k→∞ Γ(Γ(P_tf))η_k²¹_m ≤C.

This provesthetheorem. 2 4.3. Theproofsof maintheorems

Theorem4.6.Let(V,E,m,μ)beacompletegraph withanon-degeneratemeasuremand satisfying theCD(K,∞)condition. For any f ∈C₀(V),0≤ζ ∈C₀(V)and t>0, the following function

s→G(s) :=

x∈V

Γ(Pt−sf)(x)Psζ(x)m(x) satisfies

G(s)≥2KG(s), 0< s < t.

Proof. First,weshowthatG(s) isdifferentiableins∈(0,t).Withoutlossofgenerality, we assumethat < s< t− forsome >0.Taking theformalderivativeofG(s) ins, we get

−2

x

Γ(Pt−sf,ΔPt−sf)(x)Psζ(x)m(x) +

x

Γ(Pt−sf)(x)Δ(Psζ)(x)m(x) (15) Thisformalderivativeis,infact,thederivativeofG(s) ifonecanshowthattheabsolute valuesofsummandsareuniformly(ins)controlledbysummablefunctions.Forthefirst term in(15), note thatPsζ^∞ ≤ ζ^∞ <∞.Then the equation (12)inLemma 3.6 yields thatforanys∈( ,t− )