Stochastic completeness of graphs: bounded Laplacians, intrinsic metrics, volume growth and curvature
aa r X i v : . [ m a t h . M G ] S e p STOCHASTIC COMPLETENESS OF GRAPHS: BOUNDEDLAPLACIANS, INTRINSIC METRICS, VOLUME GROWTH ANDCURVATURE
RADOS LAW K. WOJCIECHOWSKI
Abstract.
The goal of this article is to survey various results concerningstochastic completeness of graphs. In particular, we present a variety of for-mulations of stochastic completeness and discuss how a discrepancy betweenuniqueness class and volume growth criteria in the continuous and discretesettings was ultimately resolved via the use of intrinsic metrics. Along theway, we discuss some equivalent notions of boundedness in the sense of geom-etry and of analysis. We also discuss various curvature criteria for stochasticcompleteness and discuss how weakly spherically symmetric graphs establishthe sharpness of results.
Contents
1. Introduction 21.1. Stochastic completeness, uniqueness class and volume growth 21.2. Curvature and stochastic completeness 51.3. The structure of this paper 52. The heat equation on graphs 62.1. Weighted graphs 62.2. Laplacians and forms 72.3. A word about essential self-adjointness and ℓ theory 82.4. The heat equation: existence of solutions 93. Formulations of stochastic completeness 123.1. Stochastic completeness and uniqueness of solutions 123.2. A Khas ′ minskii criterion 144. Boundedness of geometry and of Laplacians 144.1. Boundedness of the Laplacian 144.2. Boundedness and stochastic completeness 155. Weakly spherically symmetric graphs, trees and anti-trees 165.1. Weakly spherically symmetric graphs 165.2. Trees and anti-trees 186. Intrinsic metrics 206.1. A brief historical overview 206.2. Intrinsic metrics, combinatorial graph distance and boundedness 206.3. A word about essential self-adjointness and metric completeness 23 Date : October 6, 2020.2000
Mathematics Subject Classification.
Primary 39A12; Secondary 05C63, 58J35.The author gratefully acknowledges financial support from PSC-CUNY Awards, jointly fundedby the Professional Staff Congress and the City University of New York, and the CollaborationGrant for Mathematicians, funded by the Simons Foundation.
7. Uniqueness class, stochastic completeness and volume growth 237.1. Uniqueness class 247.2. Stochastic completeness and volume growth in intrinsic metrics 257.3. Stochastic completeness and volume growth in the combinatorialgraph metric 278. Stochastic completeness and curvature 298.1. Bakry–´Emery curvature and stochastic completeness 298.2. Ollivier Ricci curvature and stochastic completeness 32Acknowledgements 36References 361.
Introduction
The goal of this survey paper is to give an overview of results for the uniquenessof bounded solutions of the heat equation with continuous time parameter, aka sto-chastic completeness, on infinite weighted graphs. We first discuss some equivalentformulations of stochastic completeness and how we have to go beyond the realmof bounded operators on graphs in order for this property to be of interest. Fur-thermore, we discuss how the combinatorial graph distance is not an appropriatechoice of metric for the purpose of finding results analogous to those found in thesetting of Riemannian manifolds. This leads to the use of so-called intrinsic metricswhich give an appropriate notion of volume growth. Finally, we discuss how somerecently developed versions of curvature on graphs can be used to give conditionsfor stochastic completeness.1.1.
Stochastic completeness, uniqueness class and volume growth.
In1986, Alexander Grigor ′ yan published an optimal volume growth condition for thestochastic completeness of a geodesically complete Riemannian manifold. Morespecifically, if a geodesically complete manifold M satisfies Z ∞ r log V ( r ) dr = ∞ where V ( r ) denotes the volume of a ball of radius r with an arbitrary center,then M is stochastically complete [27]. In particular, we note that any Riemannianmanifold with volume growth satisfying V ( r ) ≤ Ce r for C > e r ǫ for any ǫ >
0. In particular, the class of modelmanifolds already provides such examples, see the survey article of Grigor ′ yan forthis and many other results [28].Grigor ′ yan’s volume growth criterion was proven via a uniqueness class resultfor solutions of the heat equation. More specifically, if u is a solution of the heatequation on M × (0 , T ) with initial condition 0 and for all r large u satisfies Z T Z B r u ( x, t ) dµ dt ≤ e f ( r )TOCHASTIC COMPLETENESS 3 where f is a monotone increasing function on (0 , ∞ ) which satisfies Z ∞ rf ( r ) dr = ∞ , then u = 0 on M × (0 , T ). It follows that bounded solutions of the heat equation areunique by taking the difference of two solutions and letting f ( r ) = log( C T V ( r ))where C is a bound on the solutions. See [12, 24, 47, 78] for other techniques forproving volume growth criteria for stochastic completeness in the manifold setting.In the setting of graphs, an explicit study of geometric conditions for the unique-ness of bounded solutions of the heat equation with continuous time parameter canbe found in work of J´ozef Dodziuk and Varghese Mathai [17] as well as that ofDodziuk [15] and subsequently taken up in the author’s Ph.D. thesis [83] and in-dependently in work of Andreas Weber [81]. In particular, Dodziuk/Mathai showthat whenever the Laplacian on a graph with standard weights is a bounded oper-ator, then the graph is stochastically complete. Dodziuk then extends this resultto allow weights on edges. The technique used to establish these results is that ofa minimum principle for the heat equation.In the thesis [83] and follow-up paper [84] we rather use an equivalent formula-tion of stochastic completeness in terms of bounded λ -harmonic functions to derivecriteria for stochastic completeness which allow for unbounded operators. Further-more, we give a full characterization of stochastic completeness in the case of treeswhich enjoy a certain symmetry. This characterization already shows a disparitybetween the graph and manifold settings in that there exist stochastically incom-plete graphs with factorial volume growth, that is, if V ( r ) denotes the number ofvertices within r steps of a center vertex, then the tree is stochastically incompleteand V ( r ) grows factorially with r . However, a more striking disparity appearedin a subsequent paper which introduced a class of graphs called anti-trees. Thesegraphs can be stochastically incomplete and have polynomial volume growth [85].In particular, there exist stochastically incomplete anti-trees with volume growthlike V ( r ) ∼ r ǫ for any ǫ >
0. This provides a very strong contrast with the borderline for themanifold case given by Grigor ′ yan’s result.The volume growth in these examples involves taking balls via the usual combi-natorial graph metric, that is, taking the least number of edges in a path connectingtwo vertices. This notion reflects only the global connectedness properties of thegraph. However, it is natural to expect that a metric should also reflect the localgeometry of a graph, i.e., the valence or degree of vertices. Furthermore, if thegraph has weights on both edges and a measure on vertices, then the metric shouldinteract with both the edge weights and the vertex measure. For Riemannian man-ifolds, there exists the notion of an intrinsic metric which naturally arises from theenergy form as well as from the geometry of the manifold. This notion of an intrin-sic metric for the energy form was then extended to strongly local Dirichlet formsby Karl-Theodor Sturm [77]. Now, graphs which have both a weight on edges anda measure on vertices can be put into a one-to-one correspondence with regularDirichlet forms on discrete measure spaces as discussed in work of Matthias Kellerand Daniel Lenz [50]. However, the Dirichlet forms that arise from graphs are notstrongly local. Thus, the notion of an intrinsic metric from strongly local Dirichlet RADOS LAW K. WOJCIECHOWSKI froms has to be extended to the non-local setting. This was done systematically ina paper by Rupert Frank, Daniel Lenz and Daniel Wingert [22].The notion of intrinsic metrics for non-local Dirichlet forms was quickly putto use by the graph theory community. A first example of a concrete intrinsicmetric for weighted graphs already appers in the Ph.D. thesis of Xueping Huang[39] and can also be found in the work of Matthew Folz on heat kernel estimatesaround the same time [20]. However, even given the tool of intrinsic metrics, thereare still difficulties in proving an analogue to Grigor ′ yan’s criterion for graphs. Inparticular, Huang gives an example of a graph for which there exists a non-zerobounded solution of the heat equation u with zero initial condition which satisfies Z T Z B r u ( x, t ) dµ dt ≤ e f ( r ) for f ( r ) = Cr log r for some constant C , see [39, 40]. Hence, as f in this case clearlysatisfies R ∞ r/f ( r ) dr = ∞ , we see that even when using intrinsic metrics, a directanalogue to Grigor ′ yan’s proof is not possible for all graphs.A recent breakthrough in resolving this issue can be found in the work of XuepingHuang, Matthias Keller and Marcel Schmidt [44]. In this paper, the authors firstprove a uniqueness class result which is valid for a certain class of graphs calledglobally local. They can then reduce the study of stochastic completeness of generalgraphs to that of globally local graphs using the technique of refinements first foundin [45]. With these two results, they are able to establish an exact analogue to thevolume growth criterion of Grigor ′ yan which is valid for all graphs. That is, letting V ̺ ( r ) denote the measure of a ball with respect to an intrinsic metric and lettinglog ( x ) = max { log( x ) , } if Z ∞ r log V ̺ ( r ) dr = ∞ , then the graph is stochastically complete. We note that taking the minimum with1 is only necessary to cover the case of when the entire graph has small measure;the actual value of the constant 1 is not relevant.Let us mention that the volume growth criterion for stochastic completeness ofgraphs involving intrinsic metrics was first proven under some additional assump-tions by Folz [21]. The proof technique of Folz, however, is different from thatof Grigor ′ yan. More specifically, Folz bypasses Grigor ′ yan’s uniqueness class tech-nique via a probabilistic approach involving synchronizing the random walk on thegraph with a random walk on an associated quantum graph and then applying ageneralization of Grigor ′ yan’s result for manifolds to strongly local Dirichlet formsfound in work of Sturm [77]. A similar proof involving quantum graphs but usinganalytic techniques can also be found in a paper by Huang [42].We would also like to highlight earlier work focused on a volume growth criterionby Alexander Grigor ′ yan, Xueping Huang and Jun Masamune [30] using a techniquefrom [12]. While this did not yield the optimal volume growth condition whenusing intrinsic metrics, it did yield an optimal volume growth condition for thecombinatorial graph metric in that V ( r ) ≤ Cr implies stochastic completeness where V ( r ) is the volume defined with respect tothe combinatorial graph metric. Thus, we see that the anti-tree examples found in TOCHASTIC COMPLETENESS 5 [85] are the smallest stochastically incomplete graphs in the combinatorial graphdistance.1.2.
Curvature and stochastic completeness.
Let us now turn to curvature.For Riemannian manifolds, in a paper from 1974, Robert Azencott gave both acurvature criterion for stochastic completeness and the first examples of stochas-tically incomplete manifolds [2]. In Azencott’s example, the curvature decays tonegative infinity rapidly, thus it is natural to expect that lower curvature boundsare necessary for stochastic completeness. An optimal result in this direction in-volving Ricci curvature was established by Nicholas Varopoulos [79] and Pei Hsu[34]. It can be formulated as follows: let M be a geodesically complete Riemannianmanifold and suppose that κ is a positive increasing continuous function on (0 , ∞ )such that for all points away from the cut locus on the sphere of radius r we haveRic( x ) ≥ − Cκ ( r ) for all r large and C >
0. If Z ∞ κ ( r ) dr = ∞ , then M is stochastically complete. This improved the previously known resultswhich gave that Ricci curvature uniformly bounded from below implied stochasticcompleteness as proven by Shing-Tung Yau [88], see also the work of Dodziuk [14].However, due to the connection between Ricci curvature and volume growth, thisresult is already implied by Grigor ′ yan’s volume growth result. There is also anumber of comparison results for stochastic completeness involving curvature, see[46] or Section 15 in the survey of Grigor ′ yan [28].In recent years, there has been a tremendous interest in notions of curvature ongraphs. We focus here on two formulation. One definition of curvature originatesin work of Dominique Bakry and Michele ´Emery on hypercontractive semigroups[3]. Thus, we refer to it as Bakry–´Emery curvature. A second formulation comesfrom the work of Yann Ollivier on Markov chains on metric spaces in [66, 67]. Thiswas later modified to give an infinitesimal version by Yong Lin, Linyuan Lu andShing-Tung Yau [57] and then extended to the case of possibly unbounded operatorson graphs by Florentin M¨unch and the author [65]. In any case, we refer to thisas Ollivier Ricci curvature. For Bakry–´Emery curvature, Bobo Hua and Yong Linproved that a uniform lower bound implies stochastic completeness in [36]. On theother hand, in [65] we prove that for Ollivier Ricci curvature, if κ ( r ) ≥ − C log r for C > r where κ ( r ) denotes the spherical curvature on a sphereof radius r , then the graph is stochastically complete. This is optimal in the sensethat for any ǫ > κ ( r ) decayinglike − (log r ) ǫ . Thus, there is still a disparity in this condition for graphs andfor the Ricci curvature condition for manifolds as presented above. However, thisdisparity cannot be resolved by using the notion of intrinsic metrics as was the casefor stochastic completeness and volume growth.1.3. The structure of this paper.
We now briefly discuss the structure of thispaper. Although we do not give full proofs of results, we also do not assume anyparticular background of the reader and thus try to make the presentation as self-contained as possible in terms of concepts and definitions. We also give specificreferences for all results that we do not prove completely.
RADOS LAW K. WOJCIECHOWSKI
In Section 2 we introduce the setting of weighted graphs and discuss the heatequation. In particular, we outline an elementary construction of bounded solu-tions of the heat equation using exhaustion techniques. In Section 3 we presentsome equivalent formulations for stochastic completeness. In particular, stochasticcompleteness is equivalent to the uniqueness of this bounded solution of the heatequation. In Section 4 we discuss boundedness of the Laplacian and how bound-edness is related to stochastic completeness. In Section 5 we then introduce theclass of weakly spherically symmetric graphs and present the examples of anti-treeswhich show the disparity between the continuous and discrete settings in the caseof the combinatorial graph metric. In Section 6 we introduce intrinsic metrics anddiscuss how they can differ from the combinatorial graph metric and how this isrelated to stochastic completeness. Finally, in Sections 7 and 8 we present the crite-ria for stochastic completeness in terms of volume growth and curvature mentionedabove.We also mention here a recent survey article by Bobo Hua and Xueping Huangwhich has some contact points with our article but also discusses heat kernel esti-mates, ancient solutions of the heat equation and upper escape rates [35].2.
The heat equation on graphs
Weighted graphs.
We start by introducing our setting following [50]. Wenote that this setting is very general in that we allow for arbitrary weights on bothedges and vertices. We also do not assume local finiteness, i.e., that every vertexhas only finitely many edges coming out of the vertex.
Definition 2.1 (Weighted graphs) . We let X be a countably infinite set whoseelements we refer to as vertices . We then let m : X −→ (0 , ∞ ) denote a measure on the vertex set which can be extended to all subsets by additivity. Finally, we let b : X × X −→ [0 , ∞ ) denote a function called the edge weight which satisfies(b1) b ( x, x ) = 0 for all x ∈ X (b2) b ( x, y ) = b ( y, x ) for all x, y ∈ X (b3) P y ∈ X b ( x, y ) < ∞ for all x ∈ X .Whenever b ( x, y ) >
0, we think of the vertices x and y as being connected by anedge with weight b ( x, y ), call x and y neighbors and write x ∼ y . Thus, (b1) givesthat there are no loops, (b2) that edge weights are symmetric and (b3) that thetotal sum of the edge weights is finite. We call the triple G = ( X, b, m ) a weightedgraph or just graph for short.We note, in particular, that condition (b3) above allows for a vertex to haveinfinitely many neighbors. Whenever, each vertex has only finitely neighbors, wecall the graph locally finite . We call the quantityDeg( x ) = 1 m ( x ) X y ∈ X b ( x, y )the weighted vertex degree of x ∈ X or just degree for short. We will see that thisfunction plays a significant role in what follows. Example 2.2.
We now present some standard choices for b and m to help orientthe reader. In particular, we discuss the case of standard edge weights, countingand degree measures. TOCHASTIC COMPLETENESS 7 (1) Whenever b ( x, y ) ∈ { , } for all x, y ∈ X , we say that the graph has standard edge weights . In this case, it is clear that condition (b3) in thedefinition of the edge weights implies that the graph must be locally finite.(2) One choice of vertex measure is the counting measure, that is, m ( x ) = 1for all x ∈ X . In this case, m ( K ) = K is just the cardinality of any finitesubset K . In the case of standard edge weights and counting measure, wethen obtain Deg( x ) = { y | y ∼ x } so that the weighted vertex degree is just the number of neighbors of x ,that is, the valence or degree of a vertex.(3) Another choice for the vertex measure is m ( x ) = X y ∈ X b ( x, y )for x ∈ X . In the case of standard edge weights, it then follows that m ( x ) = { y | y ∼ x } is the number of neighbors of x . In any case, withthis choice of measure, it is clear thatDeg( x ) = 1for all x ∈ X .Often we will assume that graphs are connected in the usual geometric sense,namely, for any two vertices x, y ∈ X , there exists a sequence of vertices ( x k ) nk =0 with x = x , x n = y and x k ∼ x k +1 for k = 0 , , . . . n −
1. We note that we includethe case of x = y when a vertex can be connected to itself via a path consisting ofa single vertex and thus no edges. Such a sequence is called a path connecting x and y . We then let d : X × X −→ [0 , ∞ )denote the combinatorial graph distance on X , that is, d ( x, y ) equals the leastnumber of edges in a path connecting x and y . We note that this metric onlyconsiders the combinatorial properties of the graph encoded in b but not the actualvalue of b ( x, y ) nor the vertex measure m . We will have more to say about thislater.2.2. Laplacians and forms.
We now denote the set of all functions on X by C ( X ), that is, C ( X ) = { f : X −→ R } and the subset of finitely supported functions by C c ( X ). The Hilbert space thatwe will be interested in at various points is ℓ ( X, m ), the space of square summablefunctions on X with respect to the measure m . That is, ℓ ( X, m ) = { f ∈ C ( X ) | X x ∈ X f ( x ) m ( x ) < ∞} with inner product h f, g i = P x ∈ X f ( x ) g ( x ) m ( x ).In order to introduce a formal Laplacian, we first have to restrict to a certainclass of functions as we do not assume local finiteness so that summability becomesan issue. RADOS LAW K. WOJCIECHOWSKI
Definition 2.3 (Formal Laplacian and energy form) . We let F = { f ∈ C ( X ) | X y ∈ X b ( x, y ) | f ( y ) | < ∞ for all x ∈ X } and for f ∈ F , we let L f ( x ) = 1 m ( x ) X y ∈ X b ( x, y )( f ( x ) − f ( y ))for x ∈ X . The operator L is then called the formal Laplacian associated to G . Wefurthermore let D = { f ∈ C ( X ) | X x,y ∈ X b ( x, y )( f ( x ) − f ( y )) < ∞} denote the space of functions of finite energy . For f, g ∈ D , we let Q ( f, g ) = 12 X x,y ∈ X b ( x, y )( f ( x ) − f ( y ))( g ( x ) − g ( y ))denote the energy form associated to G .We denote the restriction of Q to C c ( X ) × C c ( X ) by Q c . It then follows that aversion of Green’s formula holds for Q c : Q c ( ϕ, ψ ) = X x ∈ X L ϕ ( x ) ψ ( x ) m ( x ) = X x ∈ X ϕ ( x ) L ψ ( x ) m ( x )for all ϕ, ψ ∈ C c ( X ) ⊆ ℓ ( X, m ), see, for example [31]. The form Q c is closable andthus there exists a unique self-adjoint operator L with domain D ( L ) ⊆ ℓ ( X, m )associated to the closure of Q c denoted by Q . For a discussion of the closure of aform and the construction of the associated operator in a general Hilbert space, seeTheorem 5.37 in [82]. We refer to L as the Laplacian associated to the graph G .We note that with our sign convention, we have h Lf, f i = Q ( f, f ) ≥ f ∈ D ( L ) so that L is a positive operator.2.3. A word about essential self-adjointness and ℓ theory. Although not amain concern of this article as we mostly deal with bounded solutions, we want tomention another approach to the construction of the Laplacian L . In this viewpoint,one starts by restricting L to C c ( X ) and denoting the resulting operator by L c , thatis, D ( L c ) = C c ( X ) and L c acts as L .However, due to the lack of local finiteness, L does not necessarily map C c ( X )into ℓ ( X, m ). Thus, whenever we want to consider L c as an operator on ℓ ( X, m ),we have to assume that L maps C c ( X ) into ℓ ( X, m ). Under this additional as-sumption, it is easy to see that L c is a symmetric operator on ℓ ( X, m ), i.e., h L c ϕ, ψ i = h ϕ, L c ψ i for all ϕ, ψ ∈ C c ( X ) and that the Green’s formula reads as Q c ( ϕ, ψ ) = h L c ϕ, ψ i for ϕ, ψ ∈ D ( L c ). In this case, the self-adjoint operator associated to the closureof Q c , which is just the Laplacian L , is called the Friedrichs extension of L c , seeTheorem 5.38 in [82] for further details on the construction of this extension forgeneral Hilbert spaces. TOCHASTIC COMPLETENESS 9
We note that L maps C c ( X ) into ℓ ( X, m ) whenever L x ∈ ℓ ( X, m ) for all x ∈ X where 1 x denotes the characteristic function of the singleton set { x } . It is adirect calculation that L x ∈ ℓ ( X, m ) for all x ∈ X if and only if X y ∈ X b ( x, y ) m ( y ) < ∞ for all x ∈ X . In particular, all locally finite graphs or, more generally, all graphswith inf y ∼ x m ( y ) > L x ∈ ℓ ( X, m ) for all x ∈ X is equivalent to a variety of other conditions,for example, that C c ( X ) ⊆ D ( L ), for more details, see [48].We further note that, in general, L c may have many self-adjoint extensions andthat processes associated to these different extensions may have different stochasticproperties. When L c has a unique self-adjoint extension, L c is called essentiallyself-adjoint . It was first shown as Theorem 1.3.1 in [83] that L c is essentially self-adjoint in the case of standard edge weights and counting measure. This was thenextended to allow for general edge weights and any measure such that the measureof infinite paths is infinite as Theorem 6 in [50]. This criterion was further improvedand generalized in [26, 74] which consider more general operators on graphs. Forfurther discussion of essential self-adjointness, see [32, 43, 74] and reference therein.We will also discuss the connection between essential self-adjointness and metriccompleteness in Subsection 6.3 below.2.4. The heat equation: existence of solutions.
We now introduce a contin-uous time heat equation on G . We let ℓ ∞ ( X ) denote the set of bounded functionson X , that is, ℓ ∞ ( X ) = { f ∈ C ( X ) | sup x ∈ X | f ( x ) | < ∞} . We now make precise the requirements for summability, differentiability and bound-edness of a solution.
Definition 2.4 (Bounded solution of the heat equation) . Let u ∈ ℓ ∞ ( X ) . By a bounded solution of the heat equation with initial condition u we mean a boundedfunction u : X × [0 , ∞ ) −→ R such that u ( x, · ) is continuous for every t ≥
0, differentiable for t > x ∈ X and ( L + ∂ t ) u ( x, t ) = 0for all x ∈ X and t > u ( x,
0) = u ( x ).We note, in particular, that as u ( · , t ) ∈ ℓ ∞ ( X ) for every t ≥
0, we obtain that u ( · , t ) ∈ F . Thus, we may apply the formal Laplacian to u at every time t ≥ L on ℓ ( X, m ) so that wemay apply the spectral theorem and functional calculus to obtain a heat semigroup e − tL for t ≥
0, this semigroup acts on ℓ ( X, m ) and we are actually interested inbounded solutions, i.e., solutions in ℓ ∞ ( X ). There is a number of ways around this. One approach taken in [50] is to extend the heat semigroup on ℓ ( X, m ) toall ℓ p ( X, m ) spaces for p ∈ [1 , ∞ ] via monotone limits. Another approach is via thegeneral theory of Dirichlet forms and interpolation between ℓ p ( X, m ) spaces, see[11, 23]. We highlight a slightly different approach in that we rather exhaust thegraph via finite subgraphs, apply the spectral theorem to each operator on the finitesubgraph in order to get a solution and then take the limit. This rather elementaryapproach has its roots in [14] which gave the first construction of the heat kernel ona general Riemannian manifold without any geodesic completeness assumptions.A basic tool behind the construction is the following minimum principle. Wecall a subset K ⊆ X connected if any two vertices in K can be connected via apath that remains within K . Lemma 2.5 (Minimum principle for the heat equation) . Let G be a connectedweighted graph and let K ⊂ X be a finite connected subset. Let T ≥ and let u : X × [0 , T ] −→ R be such that t u ( x, t ) is continuously differentiable on (0 , T ) for every x ∈ K and u ( · , t ) ∈ F for all t ∈ (0 , T ] . Assume that u satisfies (A1) ( L + ∂ t ) u ≥ on K × (0 , T )(A2) u ≥ on ( X \ K × (0 , T ]) ∪ ( K × { } ) . Then, u ≥ on K × [0 , T ] .Proof. Suppose to the contrary that there exists ( x , t ) ∈ K × [0 , T ] such that u ( x , t ) <
0. By continuity, we can assume that ( x , t ) is a minimum for u on K × [0 , T ]. By assumption (A2), it follows that t > ∂ t u ( x , t ) ≤
0. Furthermore, by the definition of L , at a minimum we have L u ( x , t ) ≤ L + ∂ t ) u ( x , t ) ≤ L + ∂ t ) u ( x , t ) = 0from which L u ( x , t ) = 1 m ( x ) X y ∈ X b ( x , y )( u ( x , t ) − u ( y, t )) = 0follows. Therefore, since we are at a minimum, we now obtain that u ( y, t ) = u ( x , t ) < y ∼ x . Iterating this argument and using the connectedness of K now givesa contradiction to (A2) as K = X and we assume that G is connected. (cid:3) Remark . We note that the finiteness of K is not necessary. It suffices to assumethat there is at least one vertex outside of K and that the negative part of u attainsa minimum on K × [0 , T ]. The minimum principle then follows with basically thesame proof, see, for example, Lemma 3.5 in [51]. However, assuming the finitenessof K is sufficient for our purposes. For a much more elaborate discrete integratedminimum principle for solutions of the heat equation, see Lemma 1.1 in [40].We now sketch the construction of the minimal bounded solution of the heatequation. We note that if G is not connected, we work on each connected componentof G separately. Thus, for the construction, we can assume without loss of generalitythat G is connected. We let ( K n ) ∞ n =0 be an exhaustion sequence of the graph G bywhich we mean that each K n is finite and connected, K n ⊆ K n +1 and X = S n K n .For each n , we let L n denote the restriction of L to C ( K n ) = ℓ ( K n , m ). Moreprecisely, for a function f ∈ C ( K n ), we extend f by 0 to be defined on all of X and TOCHASTIC COMPLETENESS 11 let L n f ( x ) = L f ( x ) for x ∈ K n . Then, L n is an operator on a finite dimensionalHilbert space and we can define e − tL n = ∞ X k =0 ( − t ) k k ! L kn for t ≥
0. We then define the restricted heat kernels p nt ( x, y ) for t ≥ x, y ∈ K n via p nt ( x, y ) = e − tL n ˆ1 y ( x )where ˆ1 y = 1 y /m ( y ). It is immediate that u n ( x, t ) = e − tL n u ( x ) = X y ∈ K n p nt ( x, y ) u ( y ) m ( y )satisfies the heat equation on K n × [0 , ∞ ) with initial condition u .Furthermore, applying Lemma 2.5, gives 0 ≤ p nt ( x, y ) m ( y ) ≤ p nt ( x, y ) ≤ p n +1 t ( x, y ) for all x, y ∈ K n , t ≥ n ∈ N . Thus, we may take the limit p nt ( x, y ) → p t ( x, y )as n → ∞ to define p t ( x, y ) which is called the heat kernel on G . Then, by applyingDini’s theorem and monotone convergence, we can show that u ( x, t ) = X y ∈ X p t ( x, y ) u ( y ) m ( y )is a bounded solution of the heat equation with initial condition u on G . For furtherdetails and proofs, see Section 2 in [83] for the case of standard edge weights andcounting measure. An alternative approach for general graphs involving resolventsis given in Section 2 of [50], in particular, Proposition 2.7. Remark . The approach via resolvents in [50] is equivalent to the heat kernelapproach above via the Laplace transform formulas, that is, e − tL n = lim k →∞ kt (cid:18) L n + kt (cid:19) − ! k for all t > L n + α ) − = Z ∞ e − tα e − tL n dt for all α >
0. Both of these formulas also hold for the Laplacian L defined on theentire ℓ ( X, m ) space. We further note that L n is a positive definite operator on ℓ ( K n , m ) as can be seen by direct calculation which gives h L n f, f i = 12 X x,y ∈ K n b ( x, y )( f ( x ) − f ( y )) + X x ∈ K n f ( x ) X y K n b ( x, y )for all f ∈ ℓ ( K n , m ).We mention two further properties that follow from the construction and Lemma 2.5above. First, the solution u is minimal in the following sense: if u ≥ w ≥ u , then u ≤ w . Secondly,as 0 ≤ P y ∈ K n p nt ( x, y ) m ( y ) ≤ n , we get0 ≤ X y ∈ X p t ( x, y ) m ( y ) ≤ by taking the limit n → ∞ . We will return to the second inequality in the followingsection.We note that the approach above also gives that if f ∈ ℓ ∞ ( X ) with 0 ≤ f ≤ ≤ X y ∈ X p t ( x, y ) f ( y ) m ( y ) ≤ . This property is referred to by saying that the heat semigroup is
Markov . The factthat the semigroup is Markov will be used later in our discussion of curvature ongraphs. 3.
Formulations of stochastic completeness
Stochastic completeness and uniqueness of solutions.
We have seenthat given any bounded function, we can construct a bounded solution of the heatequation with the given function as an initial condition. We now address the unique-ness of this solution. In fact, we will see that the uniqueness is equivalent to thefollowing property.
Definition 3.1 (Stochastic completness) . Let G be a weighted graph. If for all x ∈ X and all t ≥ X y ∈ X p t ( x, y ) m ( y ) = 1 , then G is called stochastically complete. Otherwise, G is called stochastically in-complete . Remark . There is a short way to state the definition above which we will haverecourse to later in our discussion of curvature. Namely, letting P t = e − tL denotethe heat semigroup on ℓ ( X, m ) for t ≥ P t can be extended to ℓ ∞ ( X ), the space of bounded functions, via monotonelimits, see Section 6 in [50] where this is actually shown for all ℓ p ( X, m ) spaces with p ∈ [1 , ∞ ]. In particular, letting 1 ∈ ℓ ∞ ( X ) denote the function which is constantly1 on all vertices, we then have P t x ) = X y ∈ X p t ( x, y ) m ( y )for x ∈ X . Thus, stochastic completeness can also be written as P t t ≥ v ∈ F satisfies L v = λv for λ ∈ R , then v is called a λ - harmonic function. In particular, the theorem below characterizesstochastic completeness in terms of non-existence of λ -harmonic bounded functionsfor λ <
0. This will be used in several places in what follows.
Theorem 3.3 (Characterizations of stochastic completeness) . Let G be a weightedgraph. The following statements are equivalent: (i) G is stochastically complete. (ii) Bounded solutions of the heat equations are uniquely determined by initialconditions.
TOCHASTIC COMPLETENESS 13 (ii ′ ) The only bounded solution of the heat equation with initial condition u = 0 is u = 0 . (iii) The only bounded solution to L v = λv for some/all λ < is v = 0 . (iii ′ ) The only non-negative bounded solution to L v ≤ λv for some/all λ < is v = 0(iv) Every bounded function v with v ∗ = sup v > satisfies sup Ω α L v ≥ forevery α < v ∗ where Ω α = { x ∈ X | v ( x ) > v ∗ − α } .Remark . We give a partial history with references forthe equivalences above in various settings. The equivalence of (i), (ii), and (iii) fordiffusion processes on Euclidean spaces goes back to [18, 33]. For manifolds, seeTheorem 6.2 and Corollary 6.3 in [28] which also gives further historical references.For Markov processes on discrete spaces, these equivalence go back to [19, 72]. Fora proof in the case of graphs with standard edge weights and counting measure, seeTheorem 3.1.3 in [83]. For an extension to weighted graphs see Theorem 1 in [50]which deals also with a more general phenomenon called stochastic completenessat infinity. This allows for a discussion of these properties in the case of operatorsof the type Laplacian plus a positive potential, see also [61] for a discussion of thisproperty in the case of manifolds. Condition (iv) is referred to as a weak Omori-Yaumaximum principle after the original work in [68, 87]. The equivalence of (iv) andstochastic completeness was shown for manifolds in [70] and for graphs as Theorem2.2 in [38].We would also like to mention some of the intuition behind the equivalences.Roughly speaking, as mentioned in the introduction and discussed further below,a large volume growth or curvature decay is required for stochastic completenessto fail. Let us discuss how this large volume growth can cause the failure of theother properties listed in the theorem above. First, failure of (i) means that thetotal probability of the process determined by the Laplacian to remain in the graphwhen starting at a vertex x is less than 1 at some time. Hence, under a largeenough volume growth (or curvature decay) the process can be swept off the graphto infinity in a finite time. Second, failure of (ii) means that there exists a non-zerobounded solution of the heat equation with zero initial condition. In other words, alarge volume growth can create something out of nothing. Third, by looking at theequation L v = λv for v > λ <
0, we see that v must increase at some neighborof each vertex. Hence, stochastic incompleteness means that there is a sufficientamount of space in the graph to accommodate this growth while keeping v bounded.This gives the intuition for the failure of condition (iii). Finally, L v ≤ − C < v is near its supremum means that there is always more roomfor v to grow in the graph. This gives the intuition behind the failure of (iv).To summarize, in order for any of the four conditions above to fail requires alarge amount of space in the graph. Conversely, if there is no large growth, then theprocess remains in the graph and the graph cannot accommodate non-zero boundedsolutions to various equations. Thus, stochastic completeness is also referred to as conservativeness or non-explosion . Sketch of the proof of Theorem 3.3.
We now sketch a proof. For full details, pleasesee the references given in the remark directly above. To show the equivalencebetween (i) and (ii), observe that both the constant function 1 and u ( x, t ) = P y ∈ X p t ( x, y ) m ( y ) are bounded solutions of the heat equation with initial con-dition 1. The equivalence between (ii) and (ii ′ ) is shown by taking the difference of two solutions of the heat equation with initial conditions u . The equivalence be-tween (ii ′ ) and (iii) can be established via the fact that if u is a bounded solution ofthe heat equation with initial condition 0, then v ( x ) = R ∞ e tλ u ( x, t ) dt is a bounded λ -harmonic function for λ <
0. To show the equivalence between (iii) and (iii ′ ) onecan use exhaustion and minimum principle arguments. Finally, if (iii) fails and v isa non-trivial bounded λ -harmonic function for λ <
0, then letting α = sup v/
2, itcan be shown that L v ≤ − C < α so that (iv) fails. Conversely, if (iv) fails,then there exists a bounded function v such that L v ≤ − C for C > α forsome 0 < α < v ∗ . Then, w = ( v + α − v ∗ ) + is a non-negative non-trivial boundedfunction with L w ≤ λw for λ = − C/α . Thus, (iii ′ ) fails. (cid:3) Remark . We note that connectedness of the graph and the semigroup propertycan be used to show that if P y ∈ X p t ( x, y ) m ( y ) = 1 holds for one x ∈ X and one t >
0, then it holds for all x ∈ X and all t >
0. However, we do not requireconnectedness for the equivalence of the properties above as later we will need toconsider a possibly unconnected scenario. Thus, in the definition, we assume thatthe sum is 1 for all x ∈ X and all t ≥ A Khas ′ minskii criterion. We will also need another property which im-plies stochastic completeness. This is sometimes referred to as a Khas ′ minskii-typecriterion after [33]. The formulation below is Theorem 3.3 in [38], the proof giventhere uses the weak Omori-Yau maximum principle, that is, condition (iv) in The-orem 3.3. Theorem 3.6.
Let G be a weighted graph. If there exists v ∈ F which satisfies v ≥ , v ( x n ) → ∞ for all sequences of vertices with Deg( x n ) → ∞ and L v + f ( v ) ≥ on X \ K where K ⊆ X is a set such that Deg is a bounded function on K and f : [0 , ∞ ) −→ (0 , ∞ ) is an increasing continuously differentiable function with Z ∞ f ( r ) dr = ∞ , then G is stochastically complete.Remark . This formulation of a Khas ′ minskii-type criterion is very precise as itinvolves the weighted degree function as well as the function f . A more generalformulation is that the existence of a function v which satisfies L v ≥ λv outside ofa compact set and which goes to infinity in all directions implies stochastic com-pleteness, see Corollary 6.6 in [28] for the manifold case and Proposition 5.5 in [51]for weighted graphs. Furthermore, we note that, in the manifold setting, the equiv-alence of this formulation and stochastic completeness was shown as Theorem 1.2in [60]. 4. Boundedness of geometry and of Laplacians
Boundedness of the Laplacian.
We now discuss the boundedness of theLaplacian which turns out to be equivalent to boundedness of the weighted ver-tex degree. Furthermore, it turns out that boundedness always implies stochasticcompleteness.We start by a simple observation. We recall that L is the self-adjoint operator on ℓ ( X, m ) which is obtained from the closure of the form Q c acting on C c ( X ) × C c ( X ) TOCHASTIC COMPLETENESS 15 and that Deg( x ) = 1 /m ( x ) P y ∈ X b ( x, y ) for x ∈ X is the weighted degree of avertex x . We first characterize the boundedness of this operator in terms of theboundedness of the weighted degree function. This fact is certainly well-known,see, for example Theorem 11 in [49]. Theorem 4.1 (Boundedness of L ) . Let G be a weighted graph. The Laplacian L is a bounded operator on ℓ ( X, m ) if and only if Deg is bounded on X .Proof. A direct calculation gives h L x , x i = Deg( x ) m ( x )where 1 x is the characteristic function of the set { x } for x ∈ X . Now, the re-sult follows from the general theory of self-adjoint operators on Hilbert space, seefor example Theorem 4.4 in [82], by noting that { x / p m ( x ) | x ∈ X } forms anorthonormal basis for ℓ ( X, m ). (cid:3) Weighted graphs which satisfy the condition that Deg is a bounded function aresometimes referred to as having bounded geometry . We now have a look at this inthe cases most commonly appearing in the graph theory literature.
Example 4.2.
Let G be a weighted graph.(1) If m ( x ) = P y ∈ X b ( x, y ) is the sum of the edge weights, then Deg( x ) = 1for all x ∈ X . Thus, in this case, L is always a bounded operator.(2) If G has standard edge weights, i.e., b ( x, y ) ∈ { , } for all x, y ∈ X and m is the counting measure, i.e., m ( x ) = 1 for all x ∈ X , thenDeg( x ) = { y | y ∼ x } for all x ∈ X . Thus, Deg is just the usual vertex degree which counts thenumber of neighbors of x . We see that L is bounded in this case if and onlyif there is a uniform upper bound on this quantity.4.2. Boundedness and stochastic completeness.
We now discuss the connec-tion between boundedness and stochastic completeness. In particular, we show thatif Deg is bounded on X , then the graph is stochastically complete. This followsfrom a more general result which allows for some growth of the weighted vertexdegree which we state below. Theorem 4.3 (Boundedness implies stochastic completeness) . Let G be a weightedgraph. If for every infinite path ( x n ) ∞ n =0 ∞ X n =0 x n ) = ∞ , then G is stochastically complete. In particular, if Deg is bounded on X , then G isstochastically complete.Proof. By Theorem 3.3 (iii), it suffices to show that any non-trivial v ∈ F with L v = λv for λ < v ( x ) > x ∈ X . Theequation L v ( x ) = λv ( x ) can be rewritten as1 m ( x ) X y ∈ X b ( x , y ) v ( y ) = (Deg( x ) − λ ) v ( x ) . Hence, there exists x ∼ x such that v ( x ) ≥ (cid:18) − λ Deg( x ) (cid:19) v ( x ) . Now, we iterate this argument to get a sequence of vertices x ∼ x ∼ x . . . suchthat v ( x n +1 ) ≥ (cid:18) − λ Deg( x n ) (cid:19) v ( x n ) ≥ n Y k =0 (cid:18) − λ Deg( x k ) (cid:19) v ( x ) . As P ∞ k =0 / Deg( x k ) = ∞ if and only if Q ∞ k =0 (1 − λ/ Deg( x k )) = ∞ , it follows that v cannot be bounded. (cid:3) Remark . For an even shorter proof of the boundedness portion using the Omori-Yau maximum principle, see Lemma 2.3 in [38]. This is then extended to a bound-endess of a notion of a global weighted degree in Theorem 2.9 in [38]. For a moreprecise result which only considers the maximal outward degree on spheres in thecase of standard edge weights and counting measure, see Theorem 4.2 in [85] andTheorem 5.5 in [38].When b is the standard edge weight and m is the counting measure, the bound-endess result was first shown via a minimum principle in [17]. This proof was thenextended to the case of arbitrary edge weights and counting measure in [15].A more structural proof of the boundedness portion of Theorem 4.3 can befound as Corollary 27 in [49] and can be described as follows. It turns out thatthe boundedness of L on ℓ ( X, m ) also implies boundedness of L acting on ℓ ∞ ( X ).In fact, the boundendess of L restricted to ℓ p ( X, m ) for one p ∈ [1 , ∞ ] implies theboundedness of the restriction of L to ℓ p ( X, m ) for all p ∈ [1 , ∞ ]. This was shownvia the Riesz-Thorin interpolation theorem as Theorem 9.3 in [32] following earlierwork presented as Theorem 11 in [49]. Now, if L gives a bounded operator on ℓ ∞ ( X ), then it is clear that the equation L v = λv cannot have a non-zero boundedsolution for all λ <
0. Thus, by Theorem 3.3 (iii), G is stochastically complete.We also note that the lower bound for the λ -harmonic function appearing in theproof above can also be used to establish the essential self-adjointness of the restric-tion of L to C c ( X ). See Proposition 2.2 in [26] or, more generally, Theorem 11.5.2in [74].5. Weakly spherically symmetric graphs, trees and anti-trees
Weakly spherically symmetric graphs.
We now discuss a class of graphsfor which we will give a full characterization of stochastic completeness. Theseare weakly spherically symmetric graphs. They are an analogue to model mani-folds which are extensively discussed in [28]. The definition we give here was firstpresented in [51] and later generalized in [5].We start with some definitions. We assume that G is connected and recall that d ( x, y ) denotes the combinatorial graph distance between vertices x and y , that is,the least number of edges in a path connecting x and y . For a vertex x ∈ X and r ∈ N , we let S r ( x ) and B r ( x ) denote the sphere and ball of radius r about x ,that is, S r ( x ) = { x ∈ X | d ( x, x ) = r } TOCHASTIC COMPLETENESS 17 and B r ( x ) = S rk =0 S k ( x ) = { x ∈ X | d ( x, x ) ≤ r } . To ensure that these arefinite sets, we now assume that G is locally finite.We will generally suppress the dependence on x and just write S r and B r . Wecan then define the outer and inner degrees of a vertex x ∈ S r asDeg ± ( x ) = 1 m ( x ) X y ∈ S r ± b ( x, y ) . That is, Deg + ( x ) gives the total edge weight of edges going “away” from x dividedby the vertex measure while Deg − ( x ) of those going “back” towards x . Definition 5.1 (Weakly spherically symmetric graphs) . A locally finite connectedweighted graph G is called weakly spherically symmetric if there exists a vertex x ∈ X such that the functions Deg ± depend only on the distance to x . In thiscase, we will write Deg ± ( r ) for Deg ± ( x ) when x ∈ S r ( x ).Again, although all of the concepts above depend on the choice of x , we willsuppress this dependence in our notation. We note that this notion of symmetry isweak in the sense that we do not assume anything about the edge weights betweenvertices on the same sphere nor do we assume anything about the structure of theconnections between vertices on successive spheres.We will now state a full characterization for the stochastic completeness of suchgraphs. In order to do so, we introduce the notion of boundary growth of a ball as ∂B ( r ) = X x ∈ S r X y ∈ S r +1 b ( x, y ) . We note that ∂B ( r ) reflects the total edge weight of edges leaving the ball B r .Furthermore, for weakly spherically symmetric graphs, this can be written as ∂B ( r ) = Deg + ( r ) m ( S r ) = Deg − ( r + 1) m ( S r +1 )as follows directly from the definitions. In particular, we note that ∂B ( r ) = ∂B ( r − + ( r )Deg − ( r ) . In what follows, we also let V ( r ) = m ( B r ) = r X k =0 m ( S k )denote the measure of a combinatorial ball of radius r . Theorem 5.2 (Stochastic completeness of weakly spherically symmetric graphs) . If G is a weakly spherically symmetric graph, then G is stochastically complete ifand only if ∞ X r =0 V ( r ) ∂B ( r ) = ∞ . We give a sketch of the proof. For further details, see the proof of Theorem 5in [51]. For standard edge weights and counting measure, this was first shown asTheorem 4.8 in [85], see also Theorem 5.10 in [38] for an alternative proof in thiscase using the weak Omori-Yau maximum principle.
Proof.
By Theorem 3.3 (iii) it suffices to show that any bounded solution to L v = λv for λ < P ∞ r =0 V ( r ) ∂B ( r ) = ∞ . By applying the characterizationin terms of non-negative subsolutions in Theorem 3.3 (iii ′ ) and the Khas ′ minskiicriterion from Theorem 3.6, it suffices to consider only non-negative solutions v .Finally, by averaging a solution over spheres, it suffices to consider only solutionsdepending on the distance to x .Thus, we may write v ( r ) for v ( x ) for all x ∈ S r and note that stochastic com-pleteness is equivalent to the triviality of v if v is bounded. Now, by induction on r ∈ N , it can be shown by using the formulas above that L v ( r ) = λv ( r ) if andonly if v ( r + 1) − v ( r ) = − λ∂B ( r ) r X k =0 v ( k ) m ( S k ) . In particular, if v (0) >
0, then v is strictly increasing with respect to r . Therefore,we estimate − λV ( r ) ∂B ( r ) v (0) ≤ v ( r + 1) − v ( r ) ≤ − λV ( r ) ∂B ( r ) v ( r )so that v ( r ) − λV ( r ) ∂B ( r ) v (0) ≤ v ( r + 1) ≤ (cid:18) − λV ( r ) ∂B ( r ) (cid:19) v ( r ) . Iterating this down to r = 0, gives − λ r X k =0 V ( k ) ∂B ( k ) v (0) ≤ v ( r + 1) ≤ r Y k =0 (cid:18) − λV ( k ) ∂B ( k ) (cid:19) v (0) . Hence, if v is bounded, then P ∞ k =0 V ( k ) ∂B ( k ) < ∞ . On the other hand, if v is notbounded, then Q ∞ k =0 (cid:16) − λV ( k ) ∂B ( k ) (cid:17) = ∞ which is equivalent to P ∞ k =0 V ( k ) ∂B ( k ) = ∞ .This completes the proof. (cid:3) Trees and anti-trees.
We now illustrate the theorem above linking stochasticcompleteness of weakly spherically symmetric graphs and the ratio of the growthof the ball and the boundary of the ball with several examples. In particular, weintroduce the class of spherically symmetric trees and anti-trees.We start with spherically symmetric trees. For this, we first take standardedge weights and counting measure. Such a graph G is then called a sphericallysymmetric tree if G contains no cycles and there exists a vertex x ∈ X such for all x ∈ S r Deg + ( x ) = { y | y ∼ x, y ∈ S r +1 } only depends on r . Thus, we may write Deg + ( x ) = Deg + ( r ) for all x ∈ S r . Notethat the lack of cycles implies that Deg − ( r ) = 1 for all r ∈ N so that the numberof edges leading back to x is minimal in order to have a connected graph.We note that for spherically symmetric trees, we have m ( S r ) = r − Y k =0 Deg + ( k )and ∂B ( r ) = m ( S r +1 ) as follows by direct calculations. We now apply our char-acterization of stochastic completeness of weakly spherically symmetric graphs tothe case of such trees. TOCHASTIC COMPLETENESS 19
Corollary 5.3 (Stochastic completeness and spherically symmetric trees) . If G isa spherically symmetric tree, then G is stochastically complete if and only if ∞ X r =0 + ( r ) = ∞ . Proof.
From the remarks directly above we obtain ∞ X r =0 V ( r ) ∂B ( r ) = ∞ X r =0 P rk =1 Q k − j =0 Deg + ( j ) Q rk =0 Deg + ( k ) . By the limit comparison test, it then follows that the divergence of the series aboveis equivalent to divergence of the series P ∞ r =0 / Deg + ( r ). Thus, the conclusionsfollows by Theorem 5.2. (cid:3) The result above was first presented as Theorem 3.2.1 in [83]. It establishesthe sharpness of the condition given for stochastic completeness in terms of theweighted vertex degree on paths presented in Theorem 4.3 in the previous section.We note that the case of spherically symmetric trees already provides a contrastwith the manifold case as if we take V ( r ) = m ( B r ) to be the counterpart of thevolume growth in the Riemannian setting, then there exist stochastically incompletetrees with factorial volume growth. However, a much more striking example is thatof anti-trees which we define next. The basic idea is that we choose an arbitrarysequence of natural numbers for the number of vertices on the sphere and thenconnect all vertices between successive spheres. Thus, these are the antithesis oftrees in the sense that for trees the removal of a single edge between spheres createsa disconnected graphs while for anti-trees one must remove all of the edges betweenspheres. Definition 5.4 (Anti-trees) . Let ( a r ) be a sequence with a r ∈ N for r ∈ N and a = 1. A graph G is called an anti-tree with sphere growth ( a r ) if G has standardedge weights and counting measure and the vertex set X can be written as adisjoint union X = S r A r where m ( A r ) = a r and b ( x, y ) = b ( y, x ) = 1 for all x ∈ A r , y ∈ A r +1 for r ∈ N and zero otherwise.Thus, by the definition of the edge weight, an anti-tree with sphere growth ( a r )satisfies m ( S r ) = a r and is weakly spherically symmetric with Deg ± ( r ) = a r ± .Furthermore, ∂B ( r ) = a r a r +1 as each vertex in the sphere S r is connected to allvertices in the sphere S r +1 . Therefore, we obtain the following characterization ofstochastic completeness in the case of anti-trees. Corollary 5.5 (Stochastic completeness and anti-trees) . If G is an anti-tree withsphere growth ( a r ) , then G is stochastically complete if and only if ∞ X r =0 P rk =0 a k a r a r +1 = ∞ . Proof.
This follows directly from the definition of an anti-tree and Theorem 5.2. (cid:3)
We note that if a r grows like r ǫ for any ǫ >
0, then the corresponding anti-tree is stochastically incomplete. Furthermore, V ( r ) grows like r ǫ in this case.Thus, unlike in the case of manifolds, for the combinatorial graph metric, thereexists stochastically incomplete graphs with polynomial volume growth. We will also see later that these are the smallest such examples. This motivates the moveto different graph metrics which take into account not only the combinatorial graphstructure but also the vertex degree. These are the so-called intrinsic metrics whichwe introduce in the next section.The result on stochastic incompleteness of anti-trees presented above originallyappeared as Example 4.11 in [85]. To the best of our knowledge, the first example ofan anti-tree in the special case of sphere growth a r = r + 1 appears as Example 2.5in [16]. This anti-tree is a transient graph with the bottom of the spectrum at 0.The same graph appears in [81] as an example of a stochastically complete graphwith unbounded vertex degree.6. Intrinsic metrics
A brief historical overview.
As we have seen, in order to hope for a coun-terpart for Grigor ′ yan’s volume growth result for graphs, we must go beyond thecombinatorial graph distance when defining volume growth. In this section we in-troduce the notion of an intrinsic metric for a weighted graph. This concept arisesfrom Dirichlet form theory. Although beyond the scope of this article, we mentionthat the form associated to the Laplacian, which is a restriction of the graph energyform, is a regular Dirichlet form which is not strongly local. For background onDirichlet forms see [23], for the connection between graphs and non-local regularDirichlet forms see [50]. Furthermore, let us caution that the notion of an intrinsicmetric for a Dirichlet form is distinct from the notion of an intrinsic metric in thesense of length spaces as discussed, for example, in [7].The concept of an intrinsic metric for strongly local Dirichlet forms was broughtinto full fruition in [77]. This allowed for the extension of a variety of results forRiemannian manifolds, including Grigor ′ yan’s volume growth result, to the settingof strongly local Dirichlet forms. In particular, this covers the Riemannian settingas the Riemannian geodesic distance is an intrinsic metric for the strongly localDirichlet form arising in the manifold setting. However, as mentioned above, theenergy form of a graph is not strongly local so that the notions of [77] do not coverthe graph setting.For non-local Dirichlet forms, such as particular restrictions of the energy formof a graph, the concept of an intrinsic metric was discussed in full generality in[22], see also [62] as well as [20, 21, 30] for the related notion of an adapted metric.However, as noted in [22], the concept of an intrinsic metric for a non-local form ismore complicated than in the local setting as the maximum of two intrinsic metricsis not necessarily an intrinsic metric. This can already be seen in an easy exampleof a graph with three vertices, see Example 6.3 in [22]. The fact that the maximumof two intrinsic metrics is an intrinsic metric for strongly local Dirichlet forms isessential to establish the existence of a maximal intrinsic metric.Thus, there does not exist a maximal intrinsic metric for graphs. However,for proving statements in graph theory which are analogous to the strongly localsetting, the tool of an intrinsic metric is quite useful, see the survey article [48] foran overview of results in this direction and further historical notes and also [35] forsome further recent applications.6.2. Intrinsic metrics, combinatorial graph distance and boundedness.
After this brief discussion of the history of intrinsic metrics, we now present thedefinition for graphs. We call a function mapping pairs of vertices to non-negative
TOCHASTIC COMPLETENESS 21 real numbers a pseudo metric if the map is symmetric, vanishes on the diagonaland satisfies the triangle inequality. In other words, a pseudo metric is a metricexcept for the fact that it might be zero for pairs of distinct vertices. In general,intrinsic metrics are only assumed to be pseudo metrics. However, we will followconvention and refer to them as metrics in any case.
Definition 6.1 (Intrinsic metrics, jump size) . A pseudo metric ̺ : X × X −→ [0 , ∞ )is called an intrinsic metric if X y ∈ X b ( x, y ) ̺ ( x, y ) ≤ m ( x )for all x ∈ X . The quantity j = sup x ∼ y ̺ ( x, y ) is called the jump size of ̺ . When j < ∞ , we say that ̺ has finite jump size .The use of intrinsic metrics often lies in a scenario when we want to estimatethe energy of a cut-off function defined with respect to an intrinsic metric via themeasure. In the easiest example, let ̺ be an intrinsic metric and let ρ ( x ) = ̺ ( x, x )be the distance with respect to ̺ to a fixed vertex x . If K ⊆ X , then X x,y ∈ K b ( x, y )( ρ ( x ) − ρ ( y )) = X x,y ∈ K b ( x, y )( ̺ ( x, x ) − ̺ ( y, x )) ≤ X x ∈ K X y ∈ K b ( x, y ) ̺ ( x, y ) ≤ X x ∈ K m ( x ) = m ( K ) . In particular, we see that if K is a set with finite measure, then the energy of ρ on K is finite. The jump size becomes relevant whenever we have a cut-off functionwhich is supported on K and we need to control how far outside of the set K thesum above reaches.After this brief discussion, let us mention some examples. We recall that d denotes the combinatorial graph distance, that is, the least number of edges in apath connecting two vertices. The case of when the combinatorial graph distance isequivalent to an intrinsic metric can be characterized in terms of the boundednessof the weighted vertex degree. Proposition 6.2.
Let G be a connected weighted graph. The combinatorial graphdistance d is equivalent to an intrinsic metric if and only if Deg is a boundedfunction on X .Proof. We note that d ( x, y ) = 1 for all x ∼ y . Thus, X y ∈ X b ( x, y ) d ( x, y ) = X y ∈ X b ( x, y ) = Deg( x ) m ( x ) . The conclusion now follows directly. (cid:3)
Recall that by Theorem 4.1 this is the case exactly when the Laplacian is abounded operator and by Theorem 5.2, the graph is stochastically complete in thiscase.Thus, we see that the combinatorial graph distance may or may not be intrinsic.We now look at some further examples. In particular, the first example below givesa case when the combinatorial graph distance is in fact intrinsic and the secondgives a pseudo metric which is intrinsic for any given graph.
Example 6.3 (Intrinsic metrics) . We now give two examples.(1) If m ( x ) = P y ∈ X b ( x, y ), then Deg( x ) = 1 for all x ∈ X and thus the com-binatorial graph distance d is equivalent to an intrinsic metric by Proposi-tion 6.2 directly above.(2) For a pair of neighboring vertices x ∼ y we let σ ( x, y ) = (max { Deg( x ) , Deg( y ) } ) − / denote the length of the edge connecting x and y . Now, we can extend fromthe length of an edge to the length of a path in a natural way, that is, if( x k ) = ( x k ) nk =0 is a path, we let l σ (( x k )) = n − X k =0 σ ( x k , x k +1 )denote the length of the path. Finally, we define a pseudo metric via ̺ σ ( x, y ) = inf { l σ (( x k )) | ( x k ) is a path connecting x and y } . As ̺ σ ( x, y ) ≤ σ ( x, y ) for x ∼ y it is then clear that ̺ σ is an intrinsic metricas X y ∈ X b ( x, y ) ̺ σ ( x, y ) ≤ x ) X y ∈ X b ( x, y ) = m ( x ) . This metric was first introduced in [39], see also [20].We note that if the graph is not locally finite, then this intrinsic metricmay be only a pseudo metric; however, in the locally finite case, path met-rics are metrics and give the discrete topology, see for example Lemma A.3in [43].The metric ̺ σ introduced in the second example above shows that there al-ways exists an intrinsic metric on a weighted graph. This metric, which utilizesthe weighted vertex degree function, makes sense in the context of the processdetermined by the heat kernel. Namely, if the Markov process with transitionprobabilities given by the heat kernel is a vertex x , then at the next jump timeit moves with probability b ( x, y ) / P z b ( x, z ) to a neighbor y of x . Furthermore,the wait time at the vertex x is an exponentially distributed random variable withparameter given by the weighted vertex degree, that is, the probability that therandom walker is still at a vertex x after time t without having jumped is given by e − Deg( x ) t . See Section 7 in [49] for a further discussion of the connection betweenthe heat semigroup and Markov processes. Therefore, at vertices with a large ver-tex degree, the process accelerates and thus will more quickly explore neighboringvertices. Hence, for the process, neighbors of vertices of large vertex degree areclose which is consistent with the values of ̺ σ .We note that given an intrinsic metric ̺ , we can always obtain an intrinsic metricof small jump size merely by cutting from above. That is, if ̺ is an intrinsic metricand C >
0, then ̺ C ( x, y ) = min { ̺ ( x, y ) , C } is also intrinsic with jump size at most C . On the other hand, having a uniformlower bound from below on the distance between neighbors is equivalent to boundedgeometry as we now show. TOCHASTIC COMPLETENESS 23
Proposition 6.4.
Let G be a weighted graph. There exists an intrinsic metric ̺ such that ̺ ( x, y ) ≥ C > for all x ∼ y if and only if Deg is a bounded function on X .Proof. If ̺ is an intrinsic metric with ̺ ( x, y ) ≥ C > x ∼ y , then C Deg( x ) m ( x ) ≤ X y ∈ X b ( x, y ) ̺ ( x, y ) ≤ m ( x )so that Deg is bounded. Conversely, if Deg is bounded, then by Proposition 6.2,it follows that the combinatorial graph distance is equivalent to an intrinsic metric ̺ . In particular, there exists a constant C > Cd ( x, y ) ≤ ̺ ( x, y ) for anintrinsic metric ̺ . As d ( x, y ) = 1 for all x ∼ y , the conclusion follows. (cid:3) Thus, we obtain two conditions involving the existence of intrinsic metrics whichimply stochastic completeness.
Corollary 6.5 (Stochastic completeness and intrinsic metrics) . Let G be a con-nected weighed graph. If either the combinatorial graph distance is equivalent to anintrinsic metric or if there exists an intrinsic metric which is uniformly boundedbelow on neighbors, then G is stochastically complete.Proof. This follows immediately by combining Propositions 6.2 and 6.4 with The-orem 4.3. (cid:3)
A word about essential self-adjointness and metric completeness.
Webriefly mention here some additional facts about metrics, geometry and analysis.In Riemannian geometry, there is the famous Hopf-Rinow theorem, which gives aconnection between metric completeness, geodesic completeness and compactness ofballs defined with respect to the geodesic metric, see [13] for example. For locallyfinite graphs, a counterpart is shown in [43], see also [53] for a recent extensionto a more general class of graphs. More specifically, Theorem A.1 in [43] showsthat for locally finite graphs and path metrics, the notions of metric completeness,geodesic completeness in the sense that all infinite geodesics have infinite length,and finiteness of balls are equivalent. Furthermore, if an intrinsic path metric on alocally finite graph satisfies any of these equivalent conditions, then the restriction ofthe formal Laplacian to the finitely supported functions is essentially self-adjoint,see Theorem 2 in [43]. Thus, metric completeness with respect to an intrinsicmetric implies that there exists a unique Laplacian, at least for locally finite graphs.This corresponds to results known for Riemannian manifolds, see [8, 76]. For amore general and thorough discussion which includes this question for magneticSchr¨odinger operators on graphs see [74].Subsequently, the assumption that there exists an intrinsic metric for which ballsare finite has often been used as a substitute for geodesic completeness in the graphsetting. In particular, we will see this assumption appearing in our criteria forstochastic completeness in the following sections.7.
Uniqueness class, stochastic completeness and volume growth
In this section, we will discuss the connections between uniqueness class resultsfor the heat equation, stochastic completeness and volume growth. As we haveseen, using the combinatorial graph metric gives a very different volume growthborderline for stochastic completeness compared to the manifold setting. We will see that the results in these two settings can ultimately be reconciled via the useof intrinsic metrics.7.1.
Uniqueness class.
We start by recalling Grigor ′ yan’s uniqueness class resulton Riemannian manifolds. Specifically, if u is a solution of the heat equation ona Riemannian manifold with zero initial condition and if there exists a monotoneincreasing function f on (0 , ∞ ) such that R ∞ r/f ( r ) dr = ∞ which dominates thegrowth of u in the sense that for all r large Z T Z B r u ( x, t ) dµ dt ≤ e f ( r ) , then u = 0, see Theorem 9.2 in [28] or Theorem 11.9 in [29] for a proof. By The-orem 3.3 above this immediately implies stochastic completeness under a suitablevolume growth restriction. More precisely, if u is a bounded solution of the heatequation with bound given by C , then letting f ( r ) = log( C T V ( r )) shows that u must be zero whenever Z ∞ r log V ( r ) dr = ∞ . Thus, the only bounded solution of the heat equation with trivial initial conditionis trivial.However, in [39, 40], there is already a counterexample to an analogue of thisuniqueness class result when using intrinsic metrics. Namely, for the graph with X = Z , b ( x, y ) = ( | x − y | = 10 otherwiseand counting measure m = 1, there exists an explicit function u which is non-zero,satisfies the heat equation with initial condition 0 as well as the estimate Z T X x ∈ B r u ( x, t ) dt ≤ e f ( r ) for all large r with f ( r ) = log T + Cr log r for some constant C . In particular, it isclear that Z ∞ rf ( r ) dr = ∞ . Thus, no analogue to Grigor ′ yan’s uniqueness class result can hold for all graphs,even when using intrinsic metrics.We note that for this graph Deg( x ) = 2 for all x ∈ X and thus this graph isstochastically complete by Theorem 4.3. Therefore, this non-zero solution cannotbe bounded by Theorem 3.3. Furthermore, we note by Proposition 6.2 that thecombinatorial graph metric is equivalent to an intrinsic metric in this case andthat the volume growth with respect to this metric is only quadratic. Finally, theconstant C appearing in the definition of f in the example above is crucial as for C < /
2, the uniqueness class result holds for all graphs, see Theorem 0.8 in [40].The difference between these results in the discrete and continuous settings wasultimately resolved in [44] by introducing a class of graphs for which a uniquenessclass result analogous to Grigor ′ yan’s does hold. These are the globally local graphswhich we introduce next. The idea for these graphs in the context of a uniquenessclass result is that the growth of the solution of the heat equation is balanced by TOCHASTIC COMPLETENESS 25 a decay in the jump size of an intrinsic metric as we go further out in the graph.We note that in the counter example to the uniqueness class result above we usethe combinatorial graph distance whose jump size is always one and thus does notdecay.
Definition 7.1 (Globally local graphs) . Let G be a weighted graph with a pseudometric ρ . Let B r denote the ball of radius r with respect to ρ and let j r denote thejump size of ρ outside of B r , that is, j r = sup { ρ ( x, y ) | x ∼ y, x, y B r } .G is said to be globally local in ρ with respect to a monotone increasing function f : (0 , ∞ ) −→ (0 , ∞ ) if G has finite jump size, i.e., j < ∞ and if there exists aconstant A > r →∞ j r f ( Ar ) r < ∞ . Thus, globally local graphs not only have finite jump size but provided that f has a certain growth, the jump size must decay outside of balls. In the borderlinecase for the uniqueness class result, f is of the order r log r so that j r must takecare of the growth of log r . We note, in particular, that the example mentionedabove, that the combinatorial metric used there as an intrinsic metric will not beglobally local with respect to f ( r ) = r log r .In any case, with this notion of globally local graphs, Theorem 1.3 in [44] presentsthe following result. Theorem 7.2 (Uniqueness class for globally local graphs) . Let G be a weightedgraph. Let ̺ be an intrinsic metric with finite balls B ̺r and assume that G is globallylocal in ̺ with respect to a monotone increasing function f : (0 , ∞ ) −→ (0 , ∞ ) suchthat Z ∞ rf ( r ) dr = ∞ . If u : X × [0 , T ] −→ R is a solution of the heat equation with initial condition and Z T X x ∈ B ̺r u ( x, t ) m ( x ) dt ≤ e f ( r ) for all r > , then u = 0 . The proof of Theorem 7.2 can be found in Section 2 of [44]. Though ratherlong and technical, the main idea is to estimate the size of a solution of the heatequation over a small ball at some time via the size of the solution over a larger ballat an earlier time. Then one iterates this estimate down to time zero to show thatthe solution must be trivial. Along the way, the use of cut-off functions involvingintrinsic metrics is crucial. This is, in part, because of the fact that in the discretesetting there is no good substitute for the chain rule which is used throughout theproof of Grigor ′ yan’s uniqueness class result for manifolds.7.2. Stochastic completeness and volume growth in intrinsic metrics.
Wewant to use the uniqueness class result above to establish stochastic completenessunder a volume growth restriction which is valid for all graphs which allow for anintrinsic metric with finite distance balls. However, we note that the uniquenessclass result above involves the additional assumption of being globally local. Thus, some additional considerations are required in order to reduce from general graphsto the class of globally local ones.As a first step, it turns out that one can reduce to the case of an intrinsic metricwith finite jump size via the notion of truncating the edge weights which is alreadycontained in [30], see also [62]. More specifically, if ̺ is an intrinsic metric for G = ( X, b, m ), we define new edge weights on X via b s ( x, y ) = ( b ( x, y ) if ̺ ( x, y ) ≤ s ̺ is also intrinsic for G s = ( X, b s , m ) and ̺ now has finitejump size of at most s on G s . Furthermore, Lemma 3.4 in [44] gives that if G s is stochastically complete, then G is stochastically complete. We note that G s is not necessarily connected even if we start with a connected graph; however,the equivalent notions of stochastic completeness presented in Theorem 3.3 do notrequire connectedness of the graph. As an alternate viewpoint, one may applythem on connected components of the graph. In particular, Lemma 3.4 in [44] usesthe weak Omori-Yau characterization of stochastic completeness, that is, condition(iv) in Theorem 3.3 above to establish the result. See also Theorem 2.2 in [30]for a more general statement involving Dirichlet forms associated to general jumpprocesses.Thus, without loss of generality, we may assume that the intrinsic metric hasfinite jump size. The assumption of finite jump size along with finiteness of ballsis easily seen to imply local finiteness of the graph, see, for example Lemma 3.5 in[48]. Thus, we have reduced to the case of locally finite graphs with finite jumpsize and finite distance balls.Finally to reduce to the case of globally local graphs, the authors of [44] use thenotion of refinements for locally finite graphs found in [45]. The idea is to insertadditional vertices within edges and extend the definitions of the edge weights,vertex measure and the intrinsic metric in such a way that both the finiteness ofballs is preserved and that the measure of balls is only rescaled by a constant.Furthermore, as the inserted vertices are now closer together with respect to thenew intrinsic metric, it follows by Lemma 3.3 in [44] that this can be done insuch a way that the refined graph is globally local with respect to an arbitrarilychosen function. Finally, Theorem 1.5 in [44] shows that stochastic completeness ispreserved during the process of refining the graph via the use of the weak Omori-Yau maximum principle.Putting everything together, we get the following analogue to Grigor ′ yan’s vol-ume growth result which can be found as Theorem 1.1 in [44]. Theorem 7.3 (Volume growth and stochastic completeness) . Let G be a weightedgraph with an intrinsic metric ̺ with finite distance balls B ̺r . Let V ̺ ( r ) = m ( B ̺r ) and let log ( x ) = max { log( x ) , } . If Z ∞ r log V ̺ ( r ) dr = ∞ , then G is stochastically complete.Sketch of proof. From the discussion above, we can reduce to the case of finite jumpsize and finite balls and, thus, to locally finite graphs. In this case, the technique of
TOCHASTIC COMPLETENESS 27 refinements allows us to reduce to the case of graphs which are globally local withrespect to f ( r ) = log V ̺ ( r ). Thus, given a bounded solution of the heat equation u with initial condition 0 and bound C , we obtain Z T X x ∈ B ̺r u ( x, t ) m ( x ) dt ≤ C T e f ( r ) = e f ( r )+ C for some constant C . Therefore, by Theorem 7.2, we obtain that u = 0 and byTheorem 3.3 (ii ′ ) we get that G is stochastically complete. (cid:3) Remark . (1) We note that the use of log instead of just log is to deal withthe case when the measure m is small. In particular, this covers the casewhen the entire vertex set has finite measure, that is, m ( X ) < ∞ . In thiscase, stochastic completeness is actually equivalent to two other properties,namely, to recurrence and to form uniqueness, see Theorem 16 in [73] orTheorem 7.1 in [25] for further details. Thus, we see that in the case offinite measure, the existence of an intrinsic metric which gives finite ballsimplies all three of these properties. A partial converse to this result wasrecently proven in [71]. More specifically, if a graph is recurrent, then thereexists a finite measure and an intrinsic metric which has finite distanceballs. For a precise statement, see Theorem 11.6.15 in [74].(2) In the case of locally finite graphs with counting measure and an intrinsicmetric with finite jump size and finite balls, Theorem 7.3 was first shown asTheorem 1.2 in [21]. However, not only does the formulation have additionalassumptions on the graph, but the proof is quite different from Grigor ′ yan’soriginal proof on manifolds. Namely, the approach in [21] is to synchronizethe random walk on the discrete graph with a random walk on a metricgraph. Metric graphs are graphs where edges are intervals of real numbers.In particular, the energy form on a metric graph is a strongly local Dirichletform. Thus, the extension of Grigor ′ yan’s result to strongly local Dirichletforms shown in [77] implies stochastic completeness given that the volumegrowth of the discrete graph is comparable to the volume growth of themetric graph. We note that the proof in [77] also does not invoke the heatequation.An analytic proof of the result in [21] using the Omori-Yau maximumprinciple and which allows for an arbitrary vertex measure but still usesmetric graphs and assumes local finiteness and finite jump size can be foundin [42]. Thus, while an analogue of Grigor ′ yan’s result was known for someclasses of graphs since [21], the paper [44] contains the first proof whichdoes not assume finite jump size nor local finiteness and does not invokemetric graphs.7.3. Stochastic completeness and volume growth in the combinatorialgraph metric.
We now briefly discuss how in the case of standard edge weightsand counting measure the paper [30] already contains the optimal growth rateresult for stochastic completeness when using the combinatorial graph distance.More specifically, the authors of [30] first use the method of [10] from the stronglylocal setting to establish that the volume growth conditionlim inf r →∞ log V ̺ ( r ) r log r < implies stochastic completeness of general jump processes, see also [63] where itis shown that the 1 / ∞ . Here, thevolume growth is defined with respect to what the authors of [30] call an adapted metric. The idea for an adapted metric in [30] is that the intrinsic condition needsto be only satisfied for pairs of vertices that are close with respect to ̺ , that is,one truncates the metric before imposing the intrinsic condition. More specifically,letting ̺ be a pseudo metric and ̺ = min { ̺, } , then ̺ must satisfy X y ∈ X b ( x, y ) ̺ ( x, y ) ≤ m ( x )for all x ∈ X in order for ̺ to be called adapted.Clearly any intrinsic metric is adapted. On the other hand, we can also modifyExample 6.3 (2) in a natural way to get an adapted metric. More specifically, bydefining the length of an edge now by σ ( x, y ) = (Deg( x ) ∨ Deg( y )) − / ∧ a ∨ b = max { a, b } and a ∧ b = min { a, b } then we can extend to paths to getan adapted metric ̺ σ . Then, for locally finite graphs, Corollary 4.3 in [30] givesthat if inf x ∈ X m ( x ) > V ̺ σ ( r ) ≤ e Cr for C > r , then G is stochastically complete. We note that theadditional assumption that inf x ∈ X m ( x ) > ̺ σ are finite. Thus, theseassumptions are used as a replacement for the finiteness of balls assumption foundin Theorem 7.3 above.Although clearly not optimal when compared with Theorem 7.3, it turns out thatthis volume growth result already gives an optimal volume growth condition in thecase of standard edge weights, counting measure and combinatorial growth distance.More specifically, letting B r denote the ball of radius r defined with respect to thecombinatorial graph metric d and V ( r ) = m ( B r ) which is the number of verticesin the ball in this case, Theorem 1.4 in [30] gives the following result. Theorem 7.5.
Let G be a graph with standard edge weights and counting measure.If V ( r ) ≤ Cr for some constant C > and all large r , then G is stochastically complete.Idea of proof. It can be shown that under the assumption V ( r ) ≤ Cr , there aresufficiently many vertices with small degree so that a ball defined with respect ̺ σ is contained in a ball of larger radius with respect to the combinatorial graphdistance d . In particular, the volume growth restriction on V ( r ) can be used to geta volume growth restriction on V ̺ σ ( r ) and then stochastic completeness followsfrom the volume growth criterion for V ̺ σ ( r ) mentioned above. For more details,see Section 4 in [30]. (cid:3) We note that the characterization of stochastic completeness of anti-trees pro-vided in Corollary 5.5 above gives the sharpness of Theorem 7.5 as for an anti-treewith sphere growth a r of the order r ǫ for any ǫ >
0, we have that V ( r ) grows like r ǫ and that the anti-tree is stochastically incomplete. Furthermore, in this casethe weighted degree of vertices grows like r ǫ so that balls defined with respect to TOCHASTIC COMPLETENESS 29 ̺ σ are not finite and thus the graph is not geodesically complete. Finally, we notethat Theorem 7.5 remains valid in the more general setting where the edge weightsand vertex measure satisfy b ( x, y ) ≤ Cm ( x ) m ( y )for all x, y ∈ X and some C >
0, see Remark 4.4 in [30]. This type of assumptionis sometimes called an ellipticity condition on graphs, for more details and someapplications of this condition in the context of curvature see [54].8.
Stochastic completeness and curvature
In recent years there has been a surge of interest in various notions of curvaturein the discrete setting. We do not even attempt to give a comprehensive overview ofdefinitions nor of results. We rather confine ourselves to two of the most prominentdefinitions of curvature and discuss results which relate curvature and stochasticcompleteness. The two notions of curvature that we will discuss are that of Bakry–`Emery which arises from the Γ-calculus as outlined in [3] and Ollivier Ricci whicharises from optimal transportation theory and is defined for general Markov chainsin [67]. We will briefly introduce each and present the relevant results for stochasticcompleteness.We note that, unlike in the manifold case where the Bishop–Gromov inequalityrelates lower bounds on Ricci curvature to volume growth, there is no analogousconnection between lower curvature bounds and volume growth in full generalityin the discrete setting thus far. Therefore, we are not able to relate the volumegrowth criteria for stochastic completeness presented in the previous section to thecurvature conditions given in this section. However, for Bakry–´Emery curvature, seeTheorem 1.8 in [37] for some recent progress in connecting curvature and volumegrowth for a specific class of graphs and Theorem 4.1 in [64] for a connectionbetween lower curvature bounds and the volume doubling property for finite graphs.Furthermore, [1] establishes comparisons between averaged inner and outer degreesand volume growth and also gives an example of two graphs with equal OlliverRicci curvatures but different volume growths.8.1.
Bakry–´Emery curvature and stochastic completeness.
We start withBakry–´Emery curvature. This notion has origins in work on hypercontractive semi-groups found in [3]. For early manifestations in the graph setting, see [58, 75]. Wecaution the reader at the outset that in the curvature on graphs community, oneusually takes the Laplacian with the opposite sign of ours.We first introduce the Γ-calculus. In order to take care of convergence of sums,we now assume that all graphs are locally finite. In this case, we note that thedomain of the formal Laplacian is the set of all functions on X , that is, F = C ( X ).For f, g ∈ C ( X ) and x ∈ X , we letΓ( f, g )( x ) = −
12 ( L ( f g ) − f L g − g L f ) ( x ) . We will follow convention and write Γ( f ) for Γ( f, f ). By a direct calculation wethen obtain Γ( f )( x ) = 12 m ( x ) X y ∈ X b ( x, y )( f ( x ) − f ( y )) . In particular, if Γ( f ) = 0, then f is constant on any connected component of thegraph. In some sense, Γ can be thought of as an analogue to the norm squared ofthe gradient from the continuous setting.We then define Γ ( f ) = − L Γ( f ) + Γ( f, L f ) . With these notations G is said to satisfy CD ( K, ∞ ) at x ∈ X for K ∈ R ifΓ ( f )( x ) ≥ K Γ( f )( x )for all f ∈ C ( X ). The CD ( K, ∞ ) condition on G is then just the fact that G satisfies the conditions at all x ∈ X . Definition 8.1.
A locally finite weighted graph G is said to satisfy CD ( K, ∞ ) for K ∈ R if Γ ( f ) ≥ K Γ( f )for all f ∈ C ( X ).The idea of the definition is to mimic the inequality obtained via Bochner’sformula and a lower Ricci curvature bound in the Riemannian manifold setting.The number K is then thought to be a lower curvature bound on the graph.We now work towards giving criteria for stochastic completeness involving Bakry–´Emery curvature conditions. We start with the main result found as Theorem 1.2 in[36] which gives stochastic completeness under the condition CD ( K, ∞ ), finitenessof distance balls and a uniform lower bound on the measure. We recall that wedenote the heat semigroup by P t = e − tL for t ≥
0. This semigroup is originally defined on ℓ ( X, m ) via the spectral theoremand can then be extended to all ℓ p ( X, m ) spaces for p ∈ [1 , ∞ ] via monotone limits.The heat semigroup is also strongly continuous, that is, P t f → f as t → + forall f ∈ ℓ ∞ ( X ). Furthermore, P t is Markov, specifically, for 0 ≤ f ≤
1, we have0 ≤ P t f ≤
1. Stochastic completeness is then equivalent to the fact that P t t ≥ Theorem 8.2 (Stochastic completeness and Bakry–´Emery curvature) . Let G bea locally finite graph connected with inf x m ( x ) > and an intrinsic metric ̺ withfinite distance balls. If G satisfies CD ( K, ∞ ) , then G is stochastically complete.Idea of proof. The bulk of the work is in showing that CD ( K, ∞ ) is actually equiv-alent to the following gradient estimate on the heat semigroupΓ( P t ϕ ) ≤ e − Kt P t (Γ( ϕ ))for all ϕ ∈ C c ( X ) and t ≥
0, see Theorem 4.1 in [36]. Furthermore, the finitenessof balls with respect to an intrinsic metric allows for the construction of a sequence ϕ n ∈ C c ( X ) such that 0 ≤ ϕ n ≤ ϕ n ( x ) → n → ∞ for every x ∈ X and suchthat Γ( ϕ n ) ≤ n for all n ∈ N . In particular, one lets ϕ n ( x ) = (cid:18) n − ̺ ( x, x ) n (cid:19) ∨ ∧ TOCHASTIC COMPLETENESS 31 where x ∈ X is an arbitrary vertex, a ∨ b = max { a, b } and a ∧ b = min { a, b } . Itis then clear from the definition that 0 ≤ ϕ n ≤ ϕ n = 1 on B n , ϕ n is supportedon B n , and thus ϕ n ∈ C c ( X ) by the assumption of finite distance balls, and usingthe fact that ̺ is intrinsic, a direct calculation gives Γ( ϕ n ) ≤ n .As the semigroup is Markov on C c ( X ), we then obtain P t (Γ( ϕ n )) ≤ n . Given these ingredients, the proof is now straightforward as P t ϕ n ( x ) → P t x ) forall x ∈ X by the monotone convergence theorem and thusΓ( P t x ) = lim n →∞ Γ( P t ϕ n )( x ) ≤ lim inf n →∞ e − Kt P t (Γ( ϕ n ))( x ) ≤ lim inf n →∞ e − Kt n = 0for all x ∈ X . Thus, P t t ≥
0. From the heat equation weobtain ∂ t P t −L P t t >
0. As P P t t ≥ (cid:3) Remark . We note that Theorem 8.2 includes not only the assumption of finite-ness of balls but also the additional assumption that inf x m ( x ) >
0. Both ofthese assumptions appear in the context of essential self-adjointness. In particular,inf x m ( x ) > L maps C c ( X ) into ℓ ( X, m ) and that the restric-tion of L to C c ( X ) is essentially self-adjoint, see Theorem 6 in [50]. In the contextof the proof of Theorem 8.2 in [36], this assumption is used to establish the con-vergence of sums in the proof of the estimate Γ( P t ϕ ) ≤ e − Kt P t (Γ( ϕ )). However,it is not clear whether this assumption is really necessary for the ultimate result ofstochastic completeness.On the other hand, the result in [36] actually assumes a seemingly weaker as-sumption than finiteness of balls found in the formulation of Theorem 8.2 above.Namely, Theorem 1.2 in [36] only assumes the existence of a sequence of finitelysupported functions ( ϕ n ) such that 0 ≤ ϕ n ≤ , ϕ n ( x ) → ϕ n ) ≤ /n for all n ∈ N . This is sometimes referred to as a completeness assumption on thegraph as, in the manifold setting, the existence of such a sequence is known to beequivalent to geodesic completeness, see [4, 76]. It is clear in the proof above thatthe existence of an intrinsic metric with finite distance balls implies the existence ofsuch a sequence. On the other hand, Marcel Schmidt recently communicated to usthat the converse is also true, that is, that the existence of an intrinsic metric withfinite distance balls is actually equivalent to completeness as defined above. Fora proof of this fact, see Appendix A in the updated version of [56]. Furthermore,for graphs satisfying the ellipticity condition b ( x, y ) ≤ Cm ( x ) m ( y ) for all x, y ∈ X and which are stochastically complete and satisfy an additional condition calledthe Feller property, see [69, 86] for more details, CD (0 , ∞ ) implies that the graphis complete, see Theorem 6.1 in [54].Theorem 8.2 gives stochastic completeness in the case of a uniform lower Bakry–´Emery curvature bound in the spirit of [88]. However, looking at the optimalcurvature results from the Riemannian setting mentioned in the introduction, we would expect to allow for some decay of Ricci curvature as in [34, 79]. One improve-ment of Theorem 8.2 in this direction is contained for a special class of graphs in[37]. More specifically, if G is a graph with X = N , x ∼ y if and only if | x − y | = 1, m is the counting measure and ̺ is an intrinsic metric on G , then letting κ ( x ) = sup { K ∈ R | G satisfies CD ( K, ∞ ) at x } if κ ( x ) decays like − ̺ (0 , x ), then G is stochastically complete, see Theorem 1.6 in[37] for a more precise statement and proof.8.2. Ollivier Ricci curvature and stochastic completeness.
We now discuss asecond commonly appearing manifestation of curvature in the discrete setting. Thisformulation comes from optimal transport theory, see [80] for a general background,and was defined for Markov chains in [66, 67]. For graphs, the basic idea is totransport a mass, which is given by the transition probability of a simple randomwalker starting at a vertex, to that of a mass at another vertex with minimal effort.This definition was then modified to give an infinitesimal version for the case ofbounded degree in [57] and extended to graphs with general measure and edgeweights in [65]. We note that, in contrast to the Bakry–´Emery formulation whichis defined at a vertex, this curvature is defined for pairs of vertices.We start with some basic definitions. As in the previous subsection, we assumethat all graphs are locally finite. We will further assume that all graphs are con-nected. For a vertex x ∈ X and ǫ > µ ǫx on X via µ ǫx ( y ) = ( − ǫ Deg( x ) if y = xǫb ( x, y ) /m ( x ) otherwisewhere Deg( x ) is the weighted degree of x . This is a probability distribution providedthat ǫ ≤ / Deg( x ). We note that a connection to the Laplacian is given by µ ǫx ( y ) = 1 y ( x ) − ǫL y ( x )as follows by a direct calculation.Now, for two vertices x , x ∈ X , we define the Wasserstein distance between µ ǫx and µ ǫx via W ( µ ǫx , µ ǫx ) = inf π X x,y ∈ X π ( x, y ) d ( x, y )where the infimum is taken over all π : X × X −→ [0 ,
1] with P y ∈ X π ( x, y ) = µ ǫx ( x )and P x ∈ X π ( x, y ) = µ ǫx ( y ) and d is the combinatorial graph distance. The ideabehind this is that π transports the mass distribution from µ ǫx to µ ǫx and thus W ( µ ǫx , µ ǫx ) minimizes the effort required to carry out this transport.We let Lip (1) denote the set of functions with Lipshitz constant 1 with respectto the combinatorial graph distance, that is,
Lip (1) = { f ∈ C ( X ) | | f ( x ) − f ( y ) | ≤ d ( x, y ) for all x, y ∈ X } and let ℓ ∞ ( X ) denote the set of bounded functions. By Kantorovich duality, seeTheorem 1.14 in [80], we have W ( µ ǫx , µ ǫx ) = sup f ∈ Lip (1) ∩ ℓ ∞ ( X ) X x ∈ X f ( x )( µ ǫx ( x ) − µ ǫx ( x )) . TOCHASTIC COMPLETENESS 33
Finally, following [57, 67], in [65] we define the Ollivier Ricci curvature between twovertices as follows.
Definition 8.4 (Ollivier Ricci curvature) . For vertices x, y ∈ X with x = y , we let κ ǫ ( x, y ) = 1 − W ( µ ǫx , µ ǫy ) d ( x, y )and define the Ollivier Ricci curvature as κ ( x, y ) = lim ǫ → + κ ǫ ( x, y ) ǫ . Remark . We note that [67] often considers the case when m ( x ) = P y ∈ X b ( x, y )and ǫ = 1. In this case, Deg( x ) = 1 so that µ x ( y ) = b ( x, y ) P z ∈ X b ( x, z )is just the one step transition probability at x of the simple random walk on G .More generally, when ǫ <
1, we get that µ ǫx ( y ) = ( − ǫ if y = xǫb ( x, y ) / P z ∈ X b ( x, z ) otherwise . In this case, the simple random walk is given a positive probability to remain atthe vertex x . As such, the constant 1 − ǫ is sometimes referred to as the idlenessparameter in this setting, see [6].The idea of letting ǫ → + in the case when m ( x ) = P y ∈ X b ( x, y ) then appears in[57]. In particular, the concavity of the function κ ǫ is used to establish the existenceof the limit. However, as noted previously, the assumption m ( x ) = P y ∈ X b ( x, y )gives that Deg( x ) = 1 and thus all such graphs are stochastically complete byTheorem 4.3. The contribution of [65] is to allow for this definition in the caseof possibly unbounded vertex degree which makes the question of stochastic com-pleteness interesting. The existence of the limit as ǫ → + follows analogously tothe argument in [57], see also [6].We note that the Ollivier Ricci curvature as defined above can be explicitlycalculated in many cases. For example, if the vertices x ∼ y are not contained inany 3-, 4-, or 5-cycles, then κ ( x, y ) = 2 b ( x, y ) (cid:18) m ( x ) + 1 m ( y ) (cid:19) − Deg( x ) − Deg( y )see Example 2.3 in [65]. As a concrete illustration, in the case of standard edgeweights and counting measure, the curvature of an edge in a k -regular tree is κ ( x, y ) = 4 − k . This confirms the notion that regular trees of degree greaterthan 2 are the analogues of hyperbolic space as they have constant negative curva-ture.As a second example, let X = N and let x ∼ y if and only if | x − y | = 1. Suchgraphs will be referred to as birth-death chains . We note that they automaticallyfall into the framework of weakly spherically symmetric graphs as discussed inSection 5. Let j, k ∈ N with k > j . The Ollivier Ricci curvature on a birth-deathchain can be calculated directly as κ ( j, k ) = 1 k − j (cid:18) b ( j, j + 1) − b ( j, j − m ( j ) − b ( k, k + 1) − b ( k, k − m ( k ) (cid:19) see Theorem 2.10 in [65]. We note that for j ∼ k , this reduces to the formula above.Both of the examples above are easily derived from the following formula whichallows us to compute the curvature in terms of the Laplacian. To state it, we let ∇ xy f = f ( x ) − f ( y ) d ( x, y )for x = y . For a proof of the following formula see Theorem 2.1 in [65]. Theorem 8.6 (Ollivier Ricci curvature and Laplacian) . Let G be a locally finiteconnected weighted graph. For vertices x = y , we have κ ( x, y ) = inf f ∈ Lip (1) ∇ xy f =1 ∇ xy L f = inf f ∈ Lip (1) ∩ C c ( X ) ∇ xy f =1 ∇ xy L f. We now show how Theorem 8.6 allows us to prove a Laplacian comparison result.In this result, we compare the Laplacian applied to a distance function on a generalgraph to the Laplacian applied to a distance function on a birth-death chain. Wefirst define the notion of sphere curvature which will be involved. We recall thatfor a vertex x ∈ X , S r denotes the sphere of radius r around x with respect tothe combinatorial graph distance. We then let κ ( r ) = min y ∈ S r max x ∈ S r − x ∼ y κ ( x, y )for r ∈ N with κ (0) = 0 and call κ the sphere curvature . Theorem 8.7 (Laplacian comparison) . Let G be a locally finite connected weightedgraph. If x ∈ X , ρ ( x ) = d ( x, x ) and κ denotes the sphere curvature, then L ρ ( x ) ≥ ρ ( x ) X j =1 κ ( j ) − Deg( x ) for all x ∈ X .Proof. We note that in general L ρ ( x ) = Deg − ( x ) − Deg + ( x )where Deg ± ( x ) are the outer and inner degrees as defined in Section 5. In particular, L ρ ( x ) = − Deg( x ) so taking the sum to be zero in this case gives the statementfor x = x .The proof is now by induction on r = ρ ( x ) for x ∈ S r . Assume that the statementis true for r −
1. Let y ∈ S r and let x ∈ S r − be such that κ ( x, y ) ≥ κ ( x, z ) for all z ∼ x with z ∈ S r . We note that κ ( r ) ≤ κ ( x, y ), ∇ yx ρ = 1 and ρ ∈ Lip (1) so thatTheorem 8.6 gives κ ( r ) ≤ κ ( x, y ) ≤ ∇ yx L ρ = L ρ ( y ) − L ρ ( x ) . Therefore, by the inductive hypothesis, r X j =1 κ ( j ) − Deg( x ) = r − X j =1 κ ( j ) − Deg( x ) + κ ( r ) ≤ L ρ ( x ) + κ ( r ) ≤ L ρ ( y )which completes the proof. (cid:3) TOCHASTIC COMPLETENESS 35
We note that the Laplacian comparison result is sharp on birth-death chains ascan be seen by a direct calculation. That is, for birth-death chains, we get L ρ ( x ) = ρ ( x ) X j =1 κ ( j ) − Deg( x ) . Thus, Theorem 8.7 compares the Laplacian of a distance function on a generalgraph to that of a birth-death chain.We now put the various pieces together to obtain a stochastic completeness resultfor Ollivier Ricci curvature. This can be found as Theorem 4.11 in [65].
Theorem 8.8.
Let G be a locally finite connected weighted graph. If κ ( r ) ≥ − C log r for some constant C > and all large r , then G is stochastically complete.Proof. Let ρ ( x ) = d ( x, x ). It follows from Theorem 8.7 that L ρ ( x ) ≥ ρ ( x ) X j =1 κ ( j ) − Deg( x ) . Now, from the assumption that κ ( r ) ≥ − C log r we may choose an increasingcontinuously differentiable function f : [0 , ∞ ) −→ (0 , ∞ ) such that L ρ + f ( ρ ) ≥ Z ∞ f ( r ) dr = ∞ . As ρ ( x n ) → ∞ along any sequence of vertices ( x n ) with Deg( x n ) → ∞ , G isstochastically complete by Theorem 3.6. (cid:3) It turns out that Theorem 8.8 is sharp in the sense that for any ǫ >
0, thereexists a stochastically incomplete graph with κ ( r ) ≥ − (log r ) ǫ . This can alreadybe seen from the case of birth-death chains for which stochastic completeness isequivalent to ∞ X r =0 r + 1 b ( r, r + 1) = ∞ by Theorem 5.2 above. For further details, see Theorem 4.11 in [65].We note that the optimal curvature criterion for stochastic completeness in termsof curvature in the manifold setting gives the borderline for stochastic completenessaround the curvature decay of order − r , see [34, 79]. One might be tempted to tryand reconcile the difference between the manifold and graph setting by using intrin-sic metrics as was successfully carried out in the case of volume growth, however,for Ollivier Ricci curvature this approach turns out to not work, see Example 4.13in [65].We note that this is not the only difference between the continuous and discretesettings when it comes to curvature. As another example, there exist infinite graphswhich have uniformly positive Ollivier Ricci curvature, see Example 4.18 in [65].This is known to be impossible in the manifold setting by the Bonnet–Myers theo-rem. Here, anti-trees also prove to be a source of counterexamples as they provideexamples of infinite graphs satisfying and CD ( K, ∞ ) for K > uniformly positive Ollivier Ricci curvature, see [9] in [52] for the Bakry–´Emery andOllivier Ricci curvature of anti-trees. However, as soon as one either imposes up-per bounds on the vertex degree or lower bounds the measure, then a graph withuniformly positive lower curvature bounds must be finite, see [59] for the case ofBakry–´Emery and [65] for the case of Ollivier Ricci.
Acknowledgements.
I would like to thank J´ozef Dodziuk for suggesting sucha fruitful area of study and for sustaining me over the years. I am also happyto acknowledge the inspiration and support offered by Isaac Chavel and LeonKarp. Furthermore, I would like to thank Alexander Grigor ′ yan for encourage-ment and support, early in my career up until the present moment. And also manythanks to my coauthors who contributed to this story and whom I also considerto be good friends. In alphabetical order: Sebastian Haeseler, Bobo Hua, XuepingHuang, Matthias Keller, Daniel Lenz, Jun Masamune, Florentin M¨unch and Mar-cel Schmidt. Finally, I would like to thank Isaac Pesenson for the invitation tocontribute this article. References [1] Andrea Adriani and Alberto G. Setti,
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