The Heat Flow on Metric Random Walk Spaces
aa r X i v : . [ m a t h . A P ] D ec THE HEAT FLOW ON METRIC RANDOM WALK SPACES
JOS´E M. MAZ ´ON, MARCOS SOLERA AND JULI ´AN TOLEDO
Abstract.
In this paper we study the Heat Flow on Metric Random Walk Spaces, which unifies into abroad framework the heat flow on locally finite weighted connected graphs, the heat flow determined byfinite Markov chains and some nonlocal evolution problems. We give different characterizations of theergodicity and prove that a metric random walk space with positive Ollivier-Ricci curvature is ergodic.Furthermore, we prove a Cheeger inequality and, as a consequence, we show that a Poincar´e inequalityholds if, and only if, an isoperimetric inequality holds. We also study the Bakry-´Emery curvature-dimension condition and its relation with functional inequalities like the Poincar´e inequality and thetransport-information inequalities.
Contents
1. Introduction and Preliminaries 11.1. Metric Random Walk Spaces 41.2. Ollivier-Ricci Curvature 82. The Heat Flow on Metric Random Walk Spaces 92.1. The Heat Flow 92.2. Infinite Speed of Propagation and Ergodicity 163. Functional Inequalities 283.1. Spectral Gap and Poincar´e Inequality 283.2. Isoperimetric Inequality 323.3. Cheeger Inequality 333.4. Spectral Gap and Curvature 373.5. Transport Inequalities 41References 47 Introduction and Preliminaries
A metric random walk space is a metric space (
X, d ) together with a family m = ( m x ) x ∈ X of probabilitymeasures that encode the jumps of a Markov chain. Given an initial mass distribution µ on X , the measure µ ∗ m given by µ ∗ m ( A ) := Z X m x ( A ) dµ ( x ) , for all Borel sets A ⊂ X, Date : December 17, 2019.
Key words and phrases.
Random walk, nonlocal operators, Logarithmic-Sobolev inequalities, Cheeger inequality,Ollivier-Ricci curvature, Bakry-´Emery curvature-dimension condition, Concentration of measures, Transport inequalities2010
Mathematics Subject Classification: describes the new mass distribution after a jump. Associated with m , the Laplace operator ∆ m is definedas ∆ m f ( x ) := Z X ( f ( y ) − f ( x )) dm x ( y ) . Assuming that there exists an invariant and reversible measure ν for the random walk, the operator − ∆ m generates in L ( X, ν ) a Markovian semigroup ( e t ∆ m ) t ≥ (Theorem 2.1) called the heat flow on themetric random walk space, which unifies into a broad framework the heat flow on graphs, the heat flowdetermined by finite Markov chains and also some nonlocal heat flows.It is of great importance in many applications to understand the behaviour of the semigroup ( e t ∆ m ) t ≥ as t → ∞ . In this regard, we introduce a new concept, called random walk connectedness or m -connectedness of the metric random walk space, which is related to the geometry of the metric randomwalk space. We then prove that it is equivalent to the infinite speed of propagation of the heat flow(Theorem 2.9) and also to the ergodicity of the Laplacian (Theorem 2.19), that in this context meansthat the only solutions of the equation ∆ m f = 0 are the constant functions, recall further that this is, inturn, equivalent to the ergodicity of the measure ν (see also Theorem 2.21). Moreover, we relate it withgeometric properties of the metric random walk space (Theorem 2.24).In 1969 Jeff Cheeger [17] proved his famous inequality h M ≤ λ (∆ M ) , where λ (∆ M ) is the first non-trivial eigenvalue of the Laplace Beltrami operator ∆ M on L ( M, vol) ofa compact manifold M and the Cheeger constant h M is defined as h M = inf Area( ∂S )min(vol( S ) , vol( M \ S )) , where the infimum runs over all S ⊂ M with sufficiently smooth boundary. This inequality can be tracedback to the paper by Polya and Szego [47]. The first Cheeger estimates on graphs are due to Dodziuk [21]and Alon and Milmann [2]. Since then, these estimates have been improved and various variants havebeen proved. For locally finite weighted connected graphs, the following relation between the Cheegerconstant and the first positive eigenvalue λ ( G ) of the graph Laplacian has been proved in [19] (seealso [10]) h G ≤ λ ( G ) ≤ h G , where h G is the Cheeger constant for graphs. For a general metric random walk space [ X, d, m ] we definethe Cheeger constant h m ( X ) and we obtain the Cheeger inequality (Theorem 3.12) h m ≤ gap( − ∆ m ) ≤ h m , where gap( − ∆ m ) is the spectral gap of the Laplace operator. As a consequence, we show that a Poincar´einequality holds if, and only if, an isoperimetric inequality holds.An important tool in the study of the speed of convergence of the heat flow to the equilibrium is thePoincar´e inequality (see [6]). In the case of Riemannian manifolds and Markov diffusion semigroups, ausual condition required to obtain this functional inequality is the positivity of the corresponding Riccicurvature of the underlying space (see [6], [55]). In [7], Bakry and Emery found a way to define thelower Ricci curvature bound through the heat flow. Moreover, Renesse and Sturm [49] proved that,on a Riemannian manifold M , the Ricci curvature is bounded from below by some constant K ∈ R if, HE HEAT FLOW ON METRIC RANDOM WALK SPACES 3 and only if, the Boltzmann-Shannon entropy is K -convex along geodesics in the 2-Wasserstein space ofprobability measures on M . This was the key observation, used simultaneously by Lott and Villani [34]and Sturm [51], to give a notion of a lower Ricci curvature bound in the general context of length metricmeasure spaces. In these spaces, the relation between the Bakry-´Emery curvature-dimension conditionand the notion of the Ricci curvature bound introduced by Lott-Villani-Sturm, was done by Ambrosio,Gigli and Savar´e in [3], where they proved that these two notions of Ricci curvature coincide under certainassumptions on the metric measure space.When the space under consideration is discrete, for instance, in the case of a graph, the previousconcept of a Ricci curvature bound is not as clearly applicable as in the continuous setting. Indeed, thedefinition by Lott-Sturm-Villani does not apply if the 2-Wasserstein space over the metric measure spacedoes not contain geodesics. Unfortunately, this is the case if the underlying space is discrete. Recently,Erbas and Maas [23], in the framework of Markov chains on discrete spaces, in order to circumvent thenonexistence of 2-Wasserstein geodesics, replace the 2-Wasserstein metric by a different metric, whichwas introduced by Maas in [37]. Here, we do not consider this notion of Ricci curvature bound which, inthe framework of metric random walk spaces, will be the object of the forthcoming paper [40]. Instead,we will use two other concepts of a Ricci curvature bound, the one based on the Bakry-´Emery curvature-dimension condition and the one introduced by Y. Ollivier in [43]. We refer to [42] and the referencestherein for the vibrant research field of discrete curvature.The use of the Bakry-´Emery curvature-dimension condition to obtain a possible definition of a Riccicurvature bound in Markov chains was first considered in 1998 by Schmuckenschlager [50]. Moreover,in 2010, Lin and Yau [33] used this concept for graphs. Subsequently, this concept of curvature in thediscrete setting has been frequently used (see [31] and the references therein). Note that, to deal withthe Bakry-´Emery curvature-dimension condition, one needs a Carr´e du champ
Γ. In the framework ofMarkov diffusion semigroups in order to get good inequalities from this curvature-dimension condition itis essential that the generator A of the semigroup satisfies the chain rule formula A (Φ( f )) = Φ ′ ( f ) A ( f ) + Φ ′′ ( f )Γ( f ) , which characterizes diffusion operators in the continuous setting (see [6]). Unfortunately, this chain ruledoes not hold in the discrete setting and this is one of the main difficulties when working with thiscurvature-dimension condition in metric random walk spaces.In Riemannian geometry, positive Ricci curvature is characterized by the fact that “small balls arecloser, in the 1-Wasserstein distance, than their centers are” (see [49]). In the framework of metricrandom walk spaces, inspired by this, Y. Ollivier [43] introduced the concept of coarse Ricci curvature ,substituting the balls by the measures m x . Moreover, he proved that positive coarse Ricci curvatureimplies positivity of the spectral gap. In Section 3 we give conditions on the Laplace operator ∆ m whichensure the positivity of the spectral gap and we relate bounds on the spectral gap with bounds on theBakry-´Emery curvature-dimension condition.Following the papers by Marton and Talagrand ([36], [53]) about transport inequalities that relateWasserstein distances with entropy and information, this research topic has had a great development (seethe survey [27]). One of the keystones of this theory was the discovery in 1986 by Marton [35] of the linkbetween transport inequalities and the concentration of measure. Concentration of measure inequalitiescan be obtained by means of other functional inequalities such as isoperimetric and logarithmic Sobolevinequalities, see the textbook by Ledoux [32] for an excellent account on the subject. We show thatunder the positivity of the Bakry-´Emery curvature-dimension condition or the Ollivier-Ricci curvature a J. M. MAZ ´ON, M. SOLERA, J. TOLEDO transport-information inequality holds (Theorems 3.22 and 3.28). Moreover, we prove that if a transport-information inequality holds then a transport-entropy inequality is also satisfied (Theorem 3.25) and that,in general, the converse implication does not hold.1.1.
Metric Random Walk Spaces.
Let (
X, d ) be a Polish metric space equipped with its Borel σ -algebra. Definition 1.1. A random walk m on X is a family of probability measures m x on X , x ∈ X , satisfyingthe following two technical conditions:(i) the measures m x depend measurably on the point x ∈ X , i.e., for any Borel subset A of X and anyBorel subset B of R , the set { x ∈ X : m x ( A ) ∈ B } is Borel,(ii) each measure m x has finite first moment, i.e. for some (hence any) z ∈ X , and for any x ∈ X onehas R X d ( z, y ) dm x ( y ) < + ∞ (see [43]).A metric random walk space [ X, d, m ] is a Polish metric space (
X, d ) equipped with a random walk m .Let [ X, d, m ] be a metric random walk space. A Radon measure ν on X is invariant for the randomwalk m = ( m x ) if dν ( x ) = Z X dν ( y ) dm y ( x ) , that is, for any ν -measurable set A , it holds that A is m x -measurable for ν -almost all x ∈ X , x m x ( A )is ν -measurable and ν ( A ) = Z X m x ( A ) dν ( x ) . Hence, for any f ∈ L ( X, ν ), it holds that f ∈ L ( X, m x ) for ν -a.e. x ∈ X , x Z X f ( y ) dm x ( y ) is ν -measurable and Z X f ( x ) dν ( x ) = Z X (cid:18)Z X f ( y ) dm x ( y ) (cid:19) dν ( x ) . Note that, following the notation in the introduction, ν is invariant if ν ∗ m = ν .The measure ν is said to be reversible if, moreover, the detailed balance condition dm x ( y ) dν ( x ) = dm y ( x ) dν ( y ) (1.1)holds. Under suitable assumptions on the metric random walk space [ X, d, m ], such an invariant andreversible measure ν exists and is unique, as we will see below. Note that the reversibility conditionimplies the invariance condition.We will assume that the measure space ( X, ν ) is σ -finite. Example 1.2. (1) Let ( R N , d, L N ), with d the Euclidean distance and L N the Lebesgue measure.Let J : R N → [0 , + ∞ [ be a measurable, nonnegative and radially symmetric function verifying R R N J ( z ) dz = 1. In ( R N , d, L N ) we have the following random walk, m Jx ( A ) := Z A J ( x − y ) d L N ( y ) for every Borel set A ⊂ R N and x ∈ R N . Applying Fubini’s Theorem it easy to see that the Lebesgue measure L N is an invariant andreversible measure for this random walk. HE HEAT FLOW ON METRIC RANDOM WALK SPACES 5 (2) Let K : X × X → R be a Markov kernel on a countable space X , i.e., K ( x, y ) ≥ ∀ x, y ∈ X, X y ∈ X K ( x, y ) = 1 ∀ x ∈ X. Then, for m Kx ( A ) := X y ∈ A K ( x, y ) , [ X, d, m K ] is a metric random walk space for any metric d on X . For irreducible and positive re-current Markov chains (see for example [29]) there exists a unique stationary probability measure(also called steady state) on X , that is, a measure π on X satisfying X x ∈ X π ( x ) = 1 and π ( y ) = X x ∈ X π ( x ) K ( x, y ) ∀ y ∈ X. This stationary probability measure π is said to be reversible for K if the following detailedbalance equation K ( x, y ) π ( x ) = K ( y, x ) π ( y )holds for x, y ∈ X . By Tonelli’s Theorem for series, this balance condition is equivalent to theone given in (1.1) for ν = π : dm Kx ( y ) dπ ( x ) = dm Ky ( x ) dπ ( y ) . (3) A weighted discrete graph G = ( V ( G ) , E ( G )) is a graph with vertex set V ( G ) and edge set E ( G )such that to each edge ( x, y ) ∈ E ( G ) (we will write x ∼ y if ( x, y ) ∈ E ( G )) we assign a positiveweight w xy = w yx . We consider that w xy = 0 if ( x, y ) E ( G ). We say that a vertex x ∈ V ( G )is simple if it has no loops, so that w xx = 0. A graph is said to be simple if all the vertices aresimple.A finite sequence { x k } nk =0 of vertices on a graph is called a path if x k ∼ x k +1 for all k =0 , , ..., n −
1. The length of a path is defined as the number, n , of edges in the path.A graph G = ( V ( G ) , E ( G )) is called connected if, for any two vertices x, y ∈ V , there is a pathconnecting x and y , that is, a sequence of vertices { x k } nk =0 such that x = x and x n = y . If G = ( V ( G ) , E ( G )) is connected then define the graph distance d G ( x, y ) between any two distinctvertices x, y as the minimum of the lengths of the paths connecting x and y .For each x ∈ V ( G ) we define d x := X y ∼ x w xy . When w xy = 1 for every ( x, y ) ∈ E ( G ) with x ∼ y , d x coincides with the degree of the vertex x in the graph, that is, the number of edges containing x . A graph G = ( V ( G ) , E ( G )) is called locally finite if each vertex belongs to a finite number of edges.For each x ∈ V ( G ) we define the following probability measure m Gx = 1 d x X y ∼ x w xy δ y . If G = ( V ( G ) , E ( G )) is a locally finite weighted connected graph, we have that [ V ( G ) , d G , ( m Gx )]is a metric random walk space. Furthermore, it is not difficult to see that the measure ν G definedas ν G ( A ) := X x ∈ A d x , A ⊂ V ( G ) , is an invariant and reversible measure for this random walk. J. M. MAZ ´ON, M. SOLERA, J. TOLEDO (4) From a metric measure space (
X, d, µ ) we can obtain a metric random walk space, the so called ǫ -step random walk associated to µ , as follows. Assume that balls in X have finite measure andthat Supp( µ ) = X . Given ǫ >
0, the ǫ -step random walk on X , starting at point x , consists inrandomly jumping in the ball of radius ǫ around x , with probability proportional to µ ; namely m µ,ǫx := µ B ( x, ǫ ) µ ( B ( x, ǫ )) . Note that µ is an invariant and reversible measure for the metric random walk space [ X, d, m µ,ǫ ].(5) Given a metric random walk space [
X, d, m ] with invariant and reversible measure ν for m , andgiven a ν -measurable set Ω ⊂ X with ν (Ω) >
0, if we define, for x ∈ Ω, m Ω x ( A ) := Z A dm x ( y ) + Z X \ Ω dm x ( y ) ! δ x ( A ) for every Borel set A ⊂ Ω , we have that [Ω , d, m Ω ] is a metric random walk space and it easy to see that ν Ω is reversiblefor m Ω .Given a metric random walk space [ X, d, m ], geometrically we may think of m x as a replacement forthe notion of balls around x , while in probabilistic terms we can rather think of these data as defining aMarkov chain whose transition probability from x to y in n steps is dm ∗ nx ( y ) := Z z ∈ X dm z ( y ) dm ∗ ( n − x ( z ) (1.2)where m ∗ x = m x . Note that m ∗ nx = m ∗ ( n − x ∗ m x for any x ∈ X .Observe that Z y ∈ X f ( y ) dm ∗ nx ( y ) = Z z ∈ X (cid:18)Z y ∈ X f ( y ) dm z ( y ) (cid:19) dm ∗ ( n − x ( z ) . Thus, inductively, Z y ∈ X dm ∗ nx ( y ) = Z z ∈ X (cid:18)Z y ∈ X dm z ( y ) (cid:19) dm ∗ ( n − x ( z ) = Z z ∈ X dm ∗ ( n − x ( z ) = 1 . Hence, [
X, d, m ∗ n ] is also a metric random walk space. Moreover, if ν is invariant and reversible for m ,then ν is also invariant and reversible for m ∗ n . Definition 1.3.
Let [ X, d, m ] be a metric random walk space. We say that [ X, d, m ] has the strong-Fellerproperty if m x ( A ) = lim n → + ∞ m x n ( A ) for every Borel set A ⊂ X whenever x n → x as n → + ∞ in ( X, d ) . Note that the examples of metric random walk spaces given in Example 1.2 have the strong-Fellerproperty.In [39] we study the concepts of m -perimeter and m -mean curvature associated with a metric randomwalk space [ X, d, m ] with invariant and reversible measure ν with respect to m . For this aim, we introducethe notion of nonlocal interaction between two ν -measurable subsets A and B of X as L m ( A, B ) := Z A Z B dm x ( y ) dν ( x ) . HE HEAT FLOW ON METRIC RANDOM WALK SPACES 7
For L m ( A, B ) < + ∞ , by the reversibility assumption on ν , we have that L m ( A, B ) = L m ( B, A ) . We then define the concept of m -perimeter of a ν -measurable subset E ⊂ X as P m ( E ) = L m ( E, X \ E ) = Z E Z X \ E dm x ( y ) dν ( x ) . If ν ( E ) < + ∞ , we have P m ( E ) = ν ( E ) − Z E Z E dm x ( y ) dν ( x ) . (1.3)It is easy to see that, on account of the reversibility of ν , P m ( E ) = 12 Z X Z X | χ E ( y ) − χ E ( x ) | dm x ( y ) dν ( x ) . In the particular case of a graph [ V ( G ) , d G , m G ], the definition of perimeter of a set E ⊂ V ( G ) is givenby | ∂E | := X x ∈ E,y ∈ V \ E w xy . Then, we have that | ∂E | = P m G ( E ) for all E ⊂ V ( G ) . (1.4)In [39], we also introduce the m -total variation of a function u : X → R as T V m ( u ) := 12 Z X Z X | u ( y ) − u ( x ) | dm x ( y ) dν ( x )and we prove the following Coarea formula . Note that P m ( E ) = T V m ( χ E ) . Theorem 1.4. ( [39, Theorem 2.7] ) For any u ∈ L ( X, ν ) , let E t ( u ) := { x ∈ X : u ( x ) > t } . Then T V m ( u ) = Z + ∞−∞ P m ( E t ( u )) dt. Let E ⊂ X be ν -measurable. For a point x ∈ X we define the m -mean curvature of ∂E at x as H m∂E ( x ) := Z X (cid:16) χ X \ E ( y ) − χ E ( y ) (cid:17) dm x ( y ) = 1 − Z E dm x ( y ) . Note that H m∂E ( x ) can be computed for every x ∈ X , not only for points in ∂E . Furthermore, for a ν -integrable set E , Z E H m∂E ( x ) dν ( x ) = Z E (cid:18) − Z E dm x ( y ) (cid:19) dν ( x ) = ν ( E ) − Z E Z E dm x ( y ) dν ( x ) , hence, having in mind (1.3), we obtain that Z E H m∂E ( x ) dν ( x ) = 2 P m ( E ) − ν ( E ) . (1.5) J. M. MAZ ´ON, M. SOLERA, J. TOLEDO
Ollivier-Ricci Curvature.
Let (
X, d ) be a Polish metric space and M + ( X ) the set of positiveRadon measures on X . Fix µ, ν ∈ M + ( X ) satisfying the mass balance condition µ ( X ) = ν ( X ) . (1.6)The Monge-Kantorovich problem is the minimization problemmin (cid:26)Z X × X d ( x, y ) dγ ( x, y ) : γ ∈ Π( µ, ν ) (cid:27) , where Π( µ, ν ) := { Radon measures γ in X × X : π γ = µ, π γ = ν } , with π α ( x, y ) := x + α ( y − x )for α ∈ { , } .For 1 ≤ p < ∞ , the p -Wasserstein distance between µ, ν is defined as W dp ( µ, ν ) := (cid:18) min (cid:26)Z X × X d ( x, y ) p dγ ( x, y ) : γ ∈ Π( µ, ν ) (cid:27)(cid:19) p . The Monge-Kantorovich problem has a dual formulation that can be stated in this case as follows (seefor instance [54, Theorem 1.14]).
Kantorovich-Rubinstein’s Theorem.
Let µ, ν ∈ M + ( X ) be two measures satisfying the massbalance condition (1.6) . Then, W d ( µ, ν ) = sup (cid:26)Z X u d ( µ − ν ) : u ∈ K d ( X ) (cid:27) = sup (cid:26)Z X u d ( µ − ν ) : u ∈ K d ( X ) ∩ L ∞ ( X, ν ) (cid:27) where K d ( X ) := { u : X R : | u ( y ) − u ( x ) | ≤ d ( y, x ) } . In [43] Y. Ollivier gives the following definition of coarse Ricci curvature that we will call Ollivier-Riccicurvature.
Definition 1.5 ([43]) . On a given metric random walk space [
X, d, m ], for any two distinct points x, y ∈ X , the Ollivier-Ricci curvature of [ X, d, m ] along ( x, y ) is defined as κ m ( x, y ) := 1 − W d ( m x , m y ) d ( x, y ) . The
Ollivier-Ricci curvature of [ X, d, m ] is defined by κ m := inf x, y ∈ Xx = y κ m ( x, y ) . We will write κ ( x, y ) instead of κ m ( x, y ), and κ = κ m , if the context allows no confusion.In the case that ( X, d, µ ) is a smooth complete Riemannian manifold, if ( m µ,ǫx ) is the ǫ -step randomwalk associated to µ given in Example 1.2 (4), then it is proved in [49] (see also [43]) that κ m µ,ǫ ( x, y )gives back the ordinary Ricci curvature when ǫ →
0, up to scaling by ǫ . HE HEAT FLOW ON METRIC RANDOM WALK SPACES 9
Example 1.6.
Let [ R N , d, m J ] be the metric random walk space given in Example 1.2 (1). Let us seethat κ ( x, y ) = 0. Given x, y ∈ R N , x = y , by Kantorovich-Rubinstein’s Theorem, we have W d ( m Jx , m Jy ) = sup (cid:26)Z R N u ( z )( J ( x − z ) − J ( y − z )) dz : u ∈ K d ( R N ) (cid:27) = sup (cid:26)Z R N ( u ( x + z ) − u ( y + z )) J ( z ) dz : u ∈ K d ( R N ) (cid:27) . Now, for u ∈ K d ( R N ), we have Z R N ( u ( x + z ) − u ( y + z )) J ( z ) dz ≤ k x − y k . Thus, W d ( m Jx , m Jy ) ≤ k x − y k . On the other hand, taking u ( z ) := h z,x − y ik x − y k , we have u ∈ K d ( R N ), hence W d ( m Jx , m Jy ) ≥ Z R N ( u ( x + z ) − u ( y + z )) J ( z ) dz = k x − y k . Therefore, W d ( m Jx , m Jy ) = k x − y k , and, consequently, κ ( x, y ) = 0. Example 1.7.
Let [ V ( G ) , d G , ( m Gx )] be the metric random walk space associated to the locally finiteweighted discrete graph G = ( V ( G ) , E ( G )) given in Example 1.2 (3) and let N G ( x ) := { z ∈ V ( G ) : z ∼ x } for x ∈ V ( G ). Then, the Ollivier-Ricci curvature along ( x, y ) ∈ E ( G ) is κ ( x, y ) = 1 − W d G ( m x , m y ) d G ( x, y ) , where W d G ( m x , m y ) = inf µ ∈A X z ∼ x X z ∼ y µ ( z , z ) d G ( z , z ) , being A the set of all matrices with entries indexed by N G ( x ) × N G ( y ) such that µ ( z , z ) ≥ X z ∼ y µ ( z , z ) = w xz d x , X z ∼ x µ ( z , z ) = w yz d y , for ( z , z ) ∈ N G ( x ) × N G ( y ) . There is an extensive literature about Ollivier-Ricci curvature on discrete graphs (see for instance, [8],[11], [18], [28], [30], [33], [43], [44], [45] and [46]).2.
The Heat Flow on Metric Random Walk Spaces
The Heat Flow.
Let [
X, d, m ] be a metric random walk space with invariant measure ν for m . Fora function u : X → R we define its nonlocal gradient ∇ u : X × X → R as ∇ u ( x, y ) := u ( y ) − u ( x ) ∀ x, y ∈ X, and for a function z : X × X → R , its m -divergence div m z : X → R is defined as(div m z )( x ) := 12 Z X ( z ( x, y ) − z ( y, x )) dm x ( y ) . The averaging operator on [
X, d, m ] (see, for example, [43]) is defined as M m f ( x ) := Z X f ( y ) dm x ( y ) , when this expression has sense, and the Laplace operator as ∆ m := M m − I , i.e.,∆ m f ( x ) = Z X f ( y ) dm x ( y ) − f ( x ) = Z X ( f ( y ) − f ( x )) dm x ( y ) . Note that ∆ m f ( x ) = div m ( ∇ f )( x )and ( M m ) n = M m ∗ n for n ∈ N .Due to the invariance of ν for the random walk m , both operators are well defined from L ( X, ν ) to L ( X, ν ), k M m f k ≤ k f k and k ∆ m f k ≤ k f k . Moreover, they map functions which are pointwisebounded by C > C . Observe that the invariance of ν can berewritten as the following property: Z X ∆ m f ( x ) dν ( x ) = 0 ∀ f ∈ L ( X, ν ) . (2.1)In the case of the weighted discrete graph G with the random walk defined in Example 1.2 (3), theabove operator is the graph Laplacian studied by many authors (see e.g. [10], [11], [22] or [30]).By Jensen’s inequality, we have that, for f ∈ L ( X, ν ) ∩ L ( X, ν ), k M m f k L ( X,ν ) = Z X (cid:18)Z X f ( y ) dm x ( y ) (cid:19) dν ( x ) ≤ Z X Z X f ( y ) dm x ( y ) dν ( x ) = Z X f ( x ) dν ( x ) = k f k L ( X,ν ) . Therefore, M m and ∆ m are linear operators in L ( X, ν ) with domain D ( M m ) = D (∆ m ) = L ( X, ν ) ∩ L ( X, ν ) . Moreover, in the case ν ( X ) < + ∞ , M m and ∆ m are bounded linear operators in L ( X, ν ) satisfying k M m k ≤ k ∆ m k ≤ ν is reversible, the following integration by parts formula is straightforward: Z X f ( x )∆ m g ( x ) dν ( x ) = − Z X × X ( f ( y ) − f ( x ))( g ( y ) − g ( x )) dm x ( y ) dν ( x ) (2.2)for f, g ∈ L ( X, ν ) ∩ L ( X, ν ).In L ( X, ν ) we consider the symmetric form given by E m ( f, g ) = − Z X f ( x )∆ m g ( x ) dν ( x ) = 12 Z X × X ∇ f ( x, y ) ∇ g ( x, y ) dm x ( y ) dν ( x ) , with domain for both variables D ( E m ) = L ( X, ν ) ∩ L ( X, ν ), which is a linear and dense subspace of L ( X, ν ). HE HEAT FLOW ON METRIC RANDOM WALK SPACES 11
Recall the definition of generalized product ν ⊗ m x (see, for instance, [1, Definition 2.2.7]), which isdefined as the measure in X × X such that Z X × X g ( x, y ) d ( ν ⊗ m x )( x, y ) := Z X (cid:18)Z X g ( x, y ) dm x ( y ) (cid:19) dν ( x )for every bounded Borel function g with supp( g ) ⊂ A × B , A, B ⊂⊂ X . In the previous definition weneed to assume that the map x m x ( E ) is ν -measurable for any Borel set E ∈ B ( X ). Note that we canwrite E m ( f, g ) = 12 Z X × X ∇ f ( x, y ) ∇ g ( x, y ) d ( ν ⊗ m x )( x, y ) . Theorem 2.1.
Let [ X, d, m ] be a metric random walk space with invariant and reversible measure ν for m . Then, − ∆ m is a non-negative self-adjoint operator in L ( X, ν ) with associated closed symmetric form E m , which, moreover, is a Markovian form.Proof. For f ∈ D (∆ m ), by the integration by parts formula (2.2), we have Z X f ( x )( − ∆ m f )( x ) dν ( x ) = E m ( f, f ) ≥ . Also, as a consequence of (2.2), we have that − ∆ m is a self-adjoint operator in L ( X, ν ).To prove the closedness of E m , consider f n ∈ D ( E m ) such that E m ( f n − f k , f n − f k ) → , when n, k → + ∞ , and k f n − f k k L ( X,ν ) → , when n, k → + ∞ . Since f n → f in L ( X, ν ), we can assume that there exists a ν -null set N such that f n ( x ) → f ( x )for all x ∈ X \ N . Then, ( f n ( x ) − f n ( y )) → ( f ( x ) − f ( y )) for all ( x, y ) ∈ ( X \ N ) × ( X \ N ) =( X × X ) \ [( N × X ) ∪ ( X × N )]. Now, since ν is invariant, we have ν ⊗ m x ([( N × X ) ∪ ( X × N )]) = Z N (cid:18)Z X dm x ( y ) (cid:19) dν ( x ) + Z X (cid:18)Z X χ N ( y ) dm x ( y ) (cid:19) dν ( x )= ν ( N ) + Z X χ N ( y ) dν ( y ) = 2 ν ( N ) = 0 . Then, by Fatou’s Lemma we havelim n →∞ E m ( f n − f, f n − f ) = lim n → + ∞ Z X × X ( ∇ ( f n − f )( x, y )) d ( ν ⊗ m x )( x, y )= lim n → + ∞ Z X × X lim inf k → + ∞ ( ∇ ( f n − f k )( x, y )) d ( ν ⊗ m x )( x, y ) ≤ lim n → + ∞ lim inf k → + ∞ Z X × X ( ∇ ( f n − f k )( x, y )) d ( ν ⊗ m x )( x, y ) = 0 . Therefore, E m is closed. Moreover, for every 1-Lipschitz map η : R → R with η (0) = 0, we have E m ( η ◦ f, η ◦ f ) ≤ E m ( f, f ) for every f ∈ D ( E m ) , and, hence, E m satisfies the Markov property. (cid:3) By Theorem 2.1, as a consequence of the theory developed in [25, Chapter 1], we have that if ( T mt ) t ≥ is the strongly continuous semigroup associated with E m , then ( T mt ) t ≥ is a positivity preserving (i.e., T mt f ≥ f ≥
0) Markovian semigroup (i.e., 0 ≤ T mt f ≤ ν -a.e. whenever f ∈ L ( X, ν ), 0 ≤ f ≤ ν -a.e.). Moreover, ∆ m is the infinitesimal generator of ( T mt ) t ≥ , that is∆ m f = lim t ↓ T mt f − ft , ∀ f ∈ D (∆ m ) . From now on we denote e t ∆ m := T mt and we call { e t ∆ m : t ≥ } the heat flow on the metricrandom walk space [ X, d, m ] with invariant and reversible measure ν for m . For every u ∈ L ( X, ν ), u ( t ) := e t ∆ m u is the unique solution of the heat equation dudt ( t ) = ∆ m u ( t ) for every t ∈ (0 , + ∞ ) ,u (0) = u , (2.3)in the sense that u ∈ C ([0 , + ∞ ) : L ( X, ν )) ∩ C ((0 , + ∞ ) : L ( X, ν )) and verifies (2.3), or equivalently, dudt ( t, x ) = Z X ( u ( t )( y ) − u ( t )( x )) dm x ( y ) for every t > ν -a.e. x ∈ X,u (0) = u . By the Hille-Yosida exponential formula we have that e t ∆ m u = lim n → + ∞ "(cid:18) I − tn ∆ m (cid:19) − n u . As a consequence of (2.1), if ν ( X ) < + ∞ , we have that the semigroup ( e t ∆ m ) t ≥ conserves the mass.In fact ddt Z X e t ∆ m u ( x ) dν ( x ) = Z X ∆ m u ( x ) dν ( x ) = 0 , and, therefore, Z X e t ∆ m u ( x ) dν ( x ) = Z X u ( x ) dν ( x ) . (2.4)Associated with E m we define the energy functional H m ( f ) := E m ( f, f ) , that is, H m : L ( X, ν ) → [0 , + ∞ ] is defined as H m ( f ) := Z X × X ( f ( x ) − f ( y )) dm x ( y ) dν ( x ) if f ∈ L ( X, ν ) ∩ L ( X, ν ),+ ∞ else . We denote D ( H m ) := L ( X, ν ) ∩ L ( X, ν ) . Note that, for f ∈ D ( H m ), we have H m ( f ) = − Z X f ( x )∆ m f ( x ) dν ( x ) . HE HEAT FLOW ON METRIC RANDOM WALK SPACES 13
Remark 2.2.
It is easy to see that the functional H m is convex and, moreover, with a proof similar to theproof of closedness in Theorem 2.1, we get that the functional H m is closed and lower semi-continuous in L ( X, ν ). Now, it is not difficult to see that ∂ H m = − ∆ m . Consequently, − ∆ m is a maximal monotoneoperator in L ( X, ν ). We can also consider the heat flow in L ( X, ν ). Indeed, if we define in L ( X, ν ) theoperator A as Au = v ⇐⇒ v ( x ) = − ∆ m u ( x ) for all x ∈ X , then A is a completely accretive operator.In fact, let P := { q ∈ C ∞ ( R ) : 0 ≤ q ′ ≤ , supp( q ′ ) is compact and 0 supp( q ) } . Given f ∈ L ( X, ν ), and q ∈ P , applying (2.2), we have Z X q ( f ( x )) Af ( x ) dν ( x ) = 12 Z X × X ( q ( f ( y )) − q ( f ( x )))( f ( y ) − f ( x )) dm x ( y ) dν ( x ) ≥ . Then, by [12, Proposition 2.2], we have that A is a completely accretive operator. Moreover, ∂ H mL ( X,ν ) = A , thus A is m -completely accretive in L ( X, ν ). Therefore, A generates a C -semigroup ( S ( t )) t ≥ in L ( X, ν ) (see [16]) such that S ( t ) f = e t ∆ m f for all f ∈ L ( X, ν ) ∩ L ( X, ν ), verifying k S ( t ) u k L p ( X,ν ) ≤ k u k L p ( X,ν ) ∀ u ∈ L p ( X, ν ) ∩ L ( X, ν ) , ≤ p ≤ + ∞ . (2.5)In the case that ν ( X ) < ∞ , we have that S ( t ) is an extension to L ( X, ν ) of the heat flow e t ∆ m in L ( X, ν ), that we will denote equally.
Example 2.3. (1) Consider the metric random walk space [
X, d, m K ] associated with the Markovkernel K (see Example 1.2 (2)) and assume that the stationary probability measure π is reversible.Then, the Laplacian ∆ m K is given by∆ m K f ( x ) := Z X f ( y ) dm Kx ( y ) − f ( x ) = X y ∈ X K ( x, y ) f ( y ) − f ( x ) ∀ f ∈ L ( X, π ) . Consequently, given u ∈ L ( X, π ), u ( t ) := e t ∆ mK u is the solution of the equation dudt ( t, x ) = X y ∈ X K ( x, y ) u ( t )( y ) − u ( t )( x ) on (0 , + ∞ ) ,u (0) = u . Therefore, e t ∆ mK = e t ( K − I ) is the heat semigroup on X with respect to the geometry determinedby the Markov kernel K . In the case that X is a finite set, we have e t ∆ mK = e t ( K − I ) = e − t + ∞ X n =0 t n K n n ! . (2) If we consider the metric random walk space [ R N , d, m J ], being m J = ( m Jx ) the random walkdefined in Example 1.2 (1), we have that, for the invariant measure ν = L N , the Laplacian isgiven by ∆ m J f ( x ) := Z R N ( f ( y ) − f ( x )) J ( x − y ) dy. Then, given u ∈ L ( R N , L N ) we have that u ( t ) := e t ∆ mJ u is the solution of the J -nonlocalheat equation dudt ( t, x ) = Z R N ( u ( t )( y ) − u ( t )( x )) J ( x − y ) dy in R N × (0 , + ∞ ) ,u (0) = u . (2.6) In the case that Ω is a closed bounded subset of R N , if we consider the metric random walkspace [Ω , d, m J, Ω ], being m J, Ω = ( m J ) Ω (see Example 1.2 (5)), that is m J, Ω x ( A ) := Z A J ( x − y ) dy + Z R n \ Ω J ( x − z ) dz ! δ x ( A ) for every Borel set A ⊂ Ω , we have that∆ m J, Ω f ( x ) = Z Ω ( f ( y ) − f ( x )) dm J, Ω x ( y ) = Z Ω J ( x − y )( f ( y ) − f ( x )) dy. Then we have that u ( t ) := e t ∆ mJ, Ω u is the solution of the homogeneous Neumann problem forthe J -nonlocal heat equation: dudt ( t, x ) = Z Ω ( u ( t )( y ) − u ( t )( x )) J ( x − y ) dx in (0 , + ∞ ) × Ω ,u (0) = u . (2.7)See [4] for a comprehensive study of problems (2.6) and (2.7).Observe that, in general, for a bounded set Ω ⊂ X , and by using m Ω , we have that u ( t ) := e t ∆ m Ω u is the solution of dudt ( t, x ) = Z Ω ( u ( t )( y ) − u ( t )( x )) dm x ( y ) in (0 , + ∞ ) × Ω ,u (0) = u , that, like (2.7), is an homogeneous Neumann problem for the m -heat equation.In [38], it is shown, by means of the Fourier transform, that if D ⊂ R N has L N -finite measure, then e ∆ mJ χ D ( x ) = e − t ∞ X n =0 Z D ( J ∗ ) n ( x − y ) dy t n n ! . (2.8)In the next result we generalize (2.8) for general metric random walk spaces. We use the notationintroduced in (1.2). Theorem 2.4.
Let [ X, d, m ] be a metric random walk space with invariant and reversible measure ν . Let u ∈ L ( X, ν ) ∩ L ( X, ν ) . Then, e t ∆ m u ( x ) = e − t u ( x ) + + ∞ X n =1 Z X u ( y ) dm ∗ nx ( y ) t n n ! ! = e − t + ∞ X n =0 Z X u ( y ) dm ∗ nx ( y ) t n n ! , (2.9) where Z X u ( y ) dm ∗ x ( y ) = u ( x ) .In particular, for D ⊂ X with ν ( D ) < + ∞ , we have e t ∆ m χ D ( x ) = e − t χ D ( x ) + + ∞ X n =1 Z D dm ∗ nx ( y ) t n n ! ! = e − t + ∞ X n =0 m ∗ nx ( D ) t n n ! , where m ∗ x ( D ) = χ D ( x ) . HE HEAT FLOW ON METRIC RANDOM WALK SPACES 15
Proof.
We define u ( x, t ) = e − t u ( x ) + + ∞ X n =1 Z X u ( y ) dm ∗ nx ( y ) t n n ! ! . Note that, since u ∈ L ( X, ν ), then u ∈ L ( X, m ∗ nx ) for ν -a.e. x ∈ X and every n ∈ N , and Z X k X n =0 (cid:12)(cid:12)(cid:12)(cid:12)Z X u ( y ) dm ∗ nx ( y ) (cid:12)(cid:12)(cid:12)(cid:12) t n n ! dν ( x ) = k X n =0 Z X (cid:12)(cid:12)(cid:12)(cid:12)Z X u ( y ) dm ∗ nx ( y ) (cid:12)(cid:12)(cid:12)(cid:12) t n n ! dν ( x ) ≤ k X n =0 Z X Z X | u ( y ) | dm ∗ nx ( y ) dν ( x ) t n n ! = k X n =0 Z X | u ( x ) | dν ( x ) t n n ! ≤ e t k u k L ( X,ν ) . Let f k ( x ) = k X n =0 (cid:12)(cid:12)(cid:12)(cid:12)Z X u ( y ) dm ∗ nx ( y ) (cid:12)(cid:12)(cid:12)(cid:12) t n n !then 0 ≤ f k ( x ) ≤ f k +1 ( x ) < + ∞ and R f k dν ≤ e t k u k L ( ν ) for every k ∈ N so we may apply monotoneconvergence to get that Z X + ∞ X n =0 (cid:12)(cid:12)(cid:12)(cid:12)Z X u ( y ) dm ∗ nx ( y ) (cid:12)(cid:12)(cid:12)(cid:12) t n n ! dν ( x ) ≤ e t k u k L ( X,ν ) , thus the function x + ∞ X n =0 (cid:12)(cid:12)(cid:12)(cid:12)Z X u ( y ) dm ∗ nx ( y ) (cid:12)(cid:12)(cid:12)(cid:12) t n n !belongs to L ( X, ν ) and, consequently, is finite ν -a.e. Note that the same is true for the function x + ∞ X n =0 Z X | u ( y ) | dm ∗ nx ( y ) t n n ! . From this we get that u ( x, t ) is well defined and also the uniform convergence of the series for t in compactsubsets of [0 , + ∞ ). Hence, dudt ( x, t ) = − u ( x, t ) + e − t + ∞ X n =1 Z X u ( y ) dm ∗ nx ( y ) t n − ( n − . Therefore, to prove (2.9), we only need to show that e − t + ∞ X n =1 Z X u ( y ) dm ∗ nx ( y ) t n − ( n − Z X u ( z, t ) dm x ( z ) . Now, by induction it is easy to see that Z X u ( y ) dm ∗ nx ( y ) = Z X (cid:18)Z X u ( y ) dm ∗ ( n − z ( y ) (cid:19) dm x ( z ) . Thus, e − t + ∞ X n =1 Z X u ( y ) dm ∗ nx ( y ) t n − ( n − e − t + ∞ X n =1 Z X (cid:18)Z X u ( y ) dm ∗ ( n − z ( y ) (cid:19) dm x ( z ) t n − ( n − = Z z ∈ X e − t + ∞ X n =1 Z X u ( y ) dm ∗ ( n − z ( y ) t n − ( n − ! dm x ( z ) = Z X u ( z, t ) dm x ( z ) , where we have interchanged the series and integral applying the dominated convergence Theorem because (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − t k X n =1 Z X u ( y ) dm ∗ ( n − z ( y ) t n − ( n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e − t + ∞ X n =1 Z X | u ( y ) | dm ∗ ( n − z ( y ) t n − ( n − F ( z )and F belongs to L ( X, ν ), thus to L ( X, m x ) for ν -a.e. x ∈ X . (cid:3) Infinite Speed of Propagation and Ergodicity.
In this section we study the infinite speed ofpropagation of the heat flow ( e t ∆ m ) t ≥ , that is, if it holds that e t ∆ m u > t > ≤ u ∈ L ( X, ν ) , u . We will see that this property is equivalent to a connectedness property of the space, to the ergodicity ofthe m -Laplacian ∆ m and to the ergodicity of the measure ν .Let [ X, d, m ] be a metric random walk space with invariant measure ν . For a ν -measurable set D , weset N mD = { x ∈ X : m ∗ nx ( D ) = 0 , ∀ n ∈ N } . For n ∈ N , we also define H mD,n = { x ∈ X : m ∗ nx ( D ) > } , and H mD := [ n ∈ N H mD,n = n x ∈ X : m ∗ nx ( D ) > n ∈ N o . Note that N mD and H mD are disjoint and X = N mD ∪ H mD . Observe also that N mD , H mD,n and H mD are ν -measurable. Proposition 2.5.
Let [ X, d, m ] be a metric random walk space with invariant measure ν . For a ν -measurable set D , if N mD = ∅ then:1. m ∗ nx ( H mD ) = 0 for every x ∈ N mD and n ∈ N ,m ∗ nx ( N mD ) = 1 for every x ∈ N mD and n ∈ N .
2. If ν ( X ) < + ∞ or ν is reversible, then m ∗ nx ( H mD ) = 1 for ν -almost every x ∈ H mD , and for all n ∈ N . m ∗ nx ( N mD ) = 0 for ν -almost every x ∈ H mD , and for all n ∈ N . (2.10) Consequently, for every x ∈ N mD and ν -a.e. y ∈ H mD we have m x ⊥ m y , i.e. m x and m y are mutuallysingular. HE HEAT FLOW ON METRIC RANDOM WALK SPACES 17
Proof. 1 : Suppose that m ∗ kx ( H mD ) > x ∈ N mD and k ∈ N , then, since H mD = ∪ n H mD,n thereexists n ∈ N such that m ∗ kx ( H mD,n ) > m ∗ ( n + k ) x ( D ) = Z z ∈ X m ∗ nz ( D ) dm ∗ kx ( z ) ≥ Z z ∈ H mD,n m ∗ nz ( D ) dm ∗ kx ( z ) > m ∗ nz ( D ) > z ∈ H mD,n , and this contradicts that x ∈ N mD . The second statement in 1. isthen immediate. : Fix n ∈ N . Using statement 1. we have that for any finite ν -measurable set A , ν ( A ∩ H mD ) = Z X m ∗ nx ( A ∩ H mD ) dν ( x ) = Z H mD m ∗ nx ( A ∩ H mD ) dν ( x ) (2.11)because m ∗ nx ( H mD ) = 0 for every x ∈ N mD .If ν ( H mD ) is finite, by (2.11), ν ( H mD ) = Z H mD m ∗ nx ( H mD ) dν ( x ) ∀ n ∈ N , and, therefore, m ∗ nx ( H mD ) = 1 for ν -almost every x ∈ H mD , and for all n ∈ N .On the other hand, if ν ( H mD ) is not finite, since the space is σ -finite, we have that H mD = + ∞ [ j =1 H mD ∩ B j ,with B j open and 0 < ν ( B j ) < + ∞ . Now, by (2.11) and using reversibility of ν , ν ( H mD ∩ B j ) = Z H mD m ∗ nx ( H mD ∩ B j ) dν ( x ) = Z H Dm ∩ B j m ∗ nx ( H mD ) dν ( x );thus m ∗ nx ( H mD ) = 1 for ν -almost every x ∈ H Dm ∩ B j .Consequently, m ∗ nx ( H mD ) = 1 for ν -almost every x ∈ H mD , and for all n ∈ N .The second statement in then follows. (cid:3) Proposition 2.6.
Let [ X, d, m ] be a metric random walk space with invariant measure ν such that ν ( X ) < + ∞ or ν is reversible. For a ν -measurable set D , we have that, for every n ∈ N and for anyfinite ν -measurable set A , ν ( A ∩ H mD ) = Z H mD m ∗ nx ( A ) dν ( x ) , and ν ( A ∩ N mD ) = Z N mD m ∗ nx ( A ) dν ( x ) . Proof. If N mD = ∅ the result follows trivially, so let us suppose that N mD = ∅ . By (2.11), and usingProposition 2.5, we have that, for any finite ν -measurable set A , ν ( A ∩ H mD ) = Z H mD m ∗ nx ( A ∩ H mD ) dν ( x ) = Z H mD m ∗ nx ( A ) dν ( x ) since m ∗ nx ( A ) = m ∗ nx ( A ∩ H mD ) + m ∗ nx ( A ∩ N mD ) = m ∗ nx ( A ∩ H mD )for ν -a.e. x ∈ H mD and every n ∈ N . Similarly, one proves the other statement. (cid:3) We have the following corollary.
Corollary 2.7.
Let [ X, d, m ] be a metric random walk space with invariant measure ν such that ν ( X ) < + ∞ or ν is reversible. For any ν -measurable set D , we have that ν ( N mD ∩ D ) = 0 . Consequently, if ν ( D ) > , then D ⊂ H mD up to a ν -null set; therefore, for ν -a.e. x ∈ D there exists n = n ( x ) ∈ N such that m ∗ nx ( D ) > .Proof. If D is finite, from Proposition 2.6, ν ( N mD ∩ D ) = Z N mD m ∗ nx ( D ) dν ( x ) = 0 . If D is not finite, then D = + ∞ [ j =1 D ∩ B j , with B j open and 0 < ν ( B j ) < + ∞ . Then, ν ( N mD ∩ D ) ≤ + ∞ X j =1 ν ( N mD ∩ D ∩ B j ) . Now, from Proposition 2.6, ν ( N mD ∩ D ∩ B j ) = Z N mD m ∗ nx ( D ∩ B j ) dν ( x ) ≤ Z N mD m ∗ nx ( D ) dν ( x ) = 0, thus ν ( N mD ∩ D ) = 0 . (cid:3) Definition 2.8.
A metric random walk space [
X, d, m ] with invariant measure ν is called random-walk-connected or m -connected if, for any D ⊂ X with 0 < ν ( D ) < + ∞ , we have that ν ( N mD ) = 0 . A metric random walk space [
X, d, m ] with invariant measure ν is called weakly- m -connected if, forany open set D ⊂ X with 0 < ν ( D ) < + ∞ , we have that ν ( N mD ) = 0 . Theorem 2.9.
Let [ X, d, m ] be a metric random walk space with invariant and reversible measure ν .1. The space is m -connected if, and only if, for any non-null ≤ u ∈ L ( X, ν ) , we have e t ∆ m u > ν -a.e. for all t > .2. The space is weakly- m -connected if, and only if, for any non-null ≤ u ∈ L ( X, ν ) ∩ C ( X ) , we have e t ∆ m u > ν -a.e. for all t > .Proof. ( , ⇒ ): Given a non-null 0 ≤ u ∈ L ( X, ν ), there exist D ⊂ X with 0 < ν ( D ) < + ∞ and α > u ≥ αχ D . Therefore, by Theorem 2.4, e t ∆ m u ( x ) ≥ αe t ∆ m χ D ( x ) = αe − t ∞ X n =0 m ∗ nx ( D ) t n n ! > ν -a.e. x ∈ X .
Indeed, if x ∈ X \ N mD we have that x ∈ H mD , so there exists n ∈ N such that m ∗ nx ( D ) >
0. Then, since ν ( N mD ) = 0, we conclude. HE HEAT FLOW ON METRIC RANDOM WALK SPACES 19 ( , ⇐ ): Take D ⊂ X with 0 < ν ( D ) < + ∞ , we have that e t ∆ m χ D ( x ) = e − t ∞ X n =0 m ∗ nx ( D ) t n n ! > ν -a.e. x ∈ X. Moreover, since m ∗ x = δ x and m ∗ nx ( D ) = 0 for every x ∈ N mD and n ≥
1, we get e − t ∞ X n =0 m ∗ nx ( D ) t n n ! = e − t χ D ( x ) for x ∈ N mD , thus χ D ( x ) > ν -a.e. x ∈ N mD . Hence, by Corollary 2.7, ν ( N mD ) = 0.The proof of ( , ⇒ ) is similar. ( , ⇐ ): Take D ⊂ X open with 0 < ν ( D ) < + ∞ , since ν is regularthere exists a compact set K ⊂ D with ν ( K ) >
0. By Urysohn’s lemma we may find a continuousfunction 0 ≤ u ≤ u = 0 on X \ D and u = 1 on K , thus u ≤ χ D . Hence e − t ∞ X n =0 m ∗ nx ( D ) t n n ! ≥ e t ∆ m u ( x ) > ν -a.e. x ∈ X. So we conclude as before. (cid:3)
Remark 2.10.
In the preceding proof, when N mD = ∅ , we obtain that in fact e t ∆ m u ( x ) > x ∈ X and for all t > . We will say that the metric random walk space is strong m -connected when this happens for any non- ν -null0 ≤ u ∈ L ( X, ν ), and weakly strong m -connected if it holds for any non-null 0 ≤ u ∈ L ( X, ν ) ∩ C ( X ).The following result gives a characterization of m -connectedness in terms of the m -interaction of sets. Proposition 2.11.
Let [ X, d, m ] be a metric random walk space with reversible measure ν . The followingstatements are equivalent:1. X is m -connected.2. If A, B ⊂ X are ν -measurable non- ν -null sets such that A ∪ B = X , then L m ( A, B ) > .Proof. 1 ⇒ : Assume that X is m -connected and let A, B be as in statement 2. If0 = L m ( A, B ) = Z A Z B dm x ( y ) dν ( x ) , then m x ( B ) = 0 for all x ∈ A \ N , ν ( N ) = 0 . Now, since ν is invariant for m , 0 = ν ( N ) = Z X m x ( N ) dν ( x ) , and, consequently, there exists N ⊂ X , ν ( N ) = 0, such that m x ( N ) = 0 ∀ x ∈ X \ N . Hence, for x ∈ A \ ( N ∪ N ), m ∗ x ( B ) = Z X χ B ( y ) dm ∗ x ( y ) = Z X (cid:18)Z X χ B ( y ) dm z ( y ) (cid:19) dm x ( z )= Z X m z ( B ) dm x ( z ) = Z A m z ( B ) dm x ( z ) + Z B m z ( B ) dm x ( z ) | {z } =0 , since x ∈ A \ N = Z A \ N m z ( B ) dm x ( z ) | {z } =0 , since z ∈ A \ N + Z N m z ( B ) dm x ( z ) | {z } =0 , since x / ∈ N = 0Working as above, we find N ⊂ X , ν ( N ) = 0, such that m x ( N ∪ N ) = 0 ∀ x ∈ X \ N . Hence, for x ∈ A \ ( N ∪ N ∪ N ), we have that m ∗ x ( B ) = Z X χ B ( y ) dm ∗ x ( y ) = Z X (cid:18)Z X χ B ( y ) dm ∗ z ( y ) (cid:19) dm x ( z )= Z X m ∗ z ( B ) dm x ( z ) ≤ Z A m ∗ z ( B ) dm x ( z ) + Z B m ∗ z ( B ) dm x ( z ) | {z } =0 , since x ∈ A \ ( N ∪ N ) ≤ Z A \ ( N ∪ N ) m ∗ z ( B ) dm x ( z ) | {z } =0 , since z ∈ A \ ( N ∪ N ) + Z N ∪ N m ∗ z ( B ) dm x ( z ) | {z } =0 , since x / ∈ N = 0 . Inductively, we obtain that m ∗ nx ( B ) = 0 for ν -a.e x ∈ A and every n ∈ N . Consequently, A ⊂ N mB up to a ν -null set, which is a contradiction. ⇒ : Assume that statement 2 holds. If X is not m -connected, then there exists a ν -measurableset D ⊂ X , 0 < ν ( D ) < + ∞ , such that ν ( N mD ) >
0. Moreover, by Corollary 2.7, D ⊂ H mD . Hence, wehave that ν ( N mD ) > , ν ( H mD ) > X = N mD ∪ H mD , thus, by the hypothesis, L m ( N mD , H mD ) >
0, which is a contradiction. (cid:3)
Observe that the metric random walk space given in Example 1.2 (1) is m -connected. This space hasOllivier-Ricci curvature equal to zero. In the next result we see that metric random walk spaces withpositive Ollivier-Ricci curvature are m -connected. We will also see that connected graphs are always m -connected in Theorem 2.15. HE HEAT FLOW ON METRIC RANDOM WALK SPACES 21
Theorem 2.12.
Let [ X, d, m ] be a metric random walk space with finite invariant measure ν . Assumethat the Ollivier-Ricci curvature κ satisfies κ > . Then, [ X, d, m ] with ν is m -connected and weaklystrong m -connected.Proof. Under the hypothesis κ > −∞ , recall that κ ≤ W contraction property: Let [ X, d, m ] be a metric random walk space. Then, for any two probability distributions, µ and µ ′ , W d ( µ ∗ m ∗ n , µ ′ ∗ m ∗ n ) ≤ (1 − κ ) n W d ( µ, µ ′ ) . (2.12)Hence, under the hypothesis κ >
0, Y. Ollivier in [43, Corollary 21] proves that the invariant measure ν (exists and) is unique up to a multiplicative constant, and that, for ν such that ν ( X ) = 1, the followinghold: ( i ) W d ( µ ∗ m ∗ n , ν ) ≤ (1 − κ ) n W d ( µ, ν ) ∀ n ∈ N , ∀ µ ∈ P ( X ) , ( ii ) W d ( m ∗ nx , ν ) ≤ (1 − κ ) n W d ( δ x , m x ) κ ∀ n ∈ N , ∀ x ∈ X. (2.13)So we will suppose, without loss of generality, that ν ( X ) = 1. By (2.13) and [55, Theorem 6.9], we havethat µ ∗ m ∗ n ⇀ ν weakly as measures, ∀ µ ∈ P ( X ) ,m ∗ nx ⇀ ν weakly as measures, for every x ∈ X. (2.14)Let us now see that the space is m -connected if κ >
0. Take D ⊂ X with 0 < ν ( D ) < + ∞ and supposethat ν ( N mD ) >
0. By Proposition 2.7, we have ν ( H mD ) >
0. Let µ := 1 ν ( H mD ) ν H mD ∈ P ( X ) , and µ ′ := 1 ν ( N mD ) ν N mD ∈ P ( X ) , Now, by Proposition 2.6, µ ∗ m ∗ n = µ, and µ ′ ∗ m ∗ n = µ ′ , but then, by (2.12), we get W ( µ, µ ′ ) = W ( µ ∗ m ∗ n , µ ′ ∗ m ∗ n ) ≤ (1 − κ ) n W ( µ, µ ′ )which is only possible if W ( µ, µ ′ ) = 0 since 1 − κ <
1. Hence, µ = µ ′ , and this implies 1 = µ ′ ( N mD ) = µ ( N mD ) = 0 which gives a contradiction. Therefore, ν ( N mD ) = 0 so thespace is m -connected.If D is open and ν ( D ) > N mD = ∅ . Indeed, for x ∈ N mD , by (2.14), we have0 < ν ( D ) ≤ lim inf n m ∗ nx ( D ) = 0 . (cid:3) Remark 2.13.
In [29], it is shown that uniqueness of the invariant probability measure implies itsergodicity. Consequently, Theorem 2.12 follows from [43, Corollary 21] (see Theorem 2.19 for the relationbetween ergodicity and m -connectedness). We have presented the result for the sake of completeness andusing the framework of m -connectedness.Note that, in the previous result, a condition involving the random walk and the metric on the ambientspace yields the m -connectedness of the metric random walk space, which is, a priori, unrelated to themetric. Proposition 2.14.
Let [ X, d, m ] be a metric random walk space with invariant measure ν such thatsupp ν = X and either ν ( X ) < + ∞ , or ν is reversible. Suppose further that [ X, d, m ] has the strong-Feller property and ( X, d ) is connected, then [ X, d, m ] with ν is strong m -connected.Proof. Recall that, since [
X, d, m ] has the strong-Feller property, [
X, d, m ∗ k ] also has this property forany k ∈ N . Take D with 0 < ν ( D ) < + ∞ . Let us see first that H mD is open or, equivalently, that N mD isclosed; indeed, if we have ( x n ) ⊂ N mD such that lim n →∞ x n = x ∈ X then m ∗ kx ( D ) = lim n →∞ m ∗ kx n ( D ) = 0for any k ∈ N , thus x ∈ N mD .On the other hand, if m x ( H mD ) < x ∈ H mD , since [ X, d, m ] has the strong-Feller property,there exists r > m y ( H mD ) < y ∈ B r ( x ) ⊂ H mD . Therefore, by (2.10), ν ( B r ( x )) = 0,which is a contradiction since supp ν = X . Hence, m x ( H mD ) = 1 if, and only if, x ∈ H mD . Then, given ( x n ) ⊂ H mD such that lim n →∞ x n = x ∈ X , we have m x ( H mD ) = lim n →∞ m x n ( H mD ) = 1 , so x ∈ H mD . Therefore, H mD is also closed and then, since X is connected, we have that X = H mD , whichimplies that N mD = ∅ . (cid:3) Theorem 2.15.
Let [ V ( G ) , d G , ( m Gx )] be the metric random walk space associated with the locally finiteweighted connected graph G = ( V ( G ) , E ( G )) . Then [ V ( G ) , d G , m G ] with ν G is strong m -connected.Proof. Take D ⊂ V ( G ) with ν G ( D ) >
0, and let us see that N m G D = ∅ . Suppose that there exists y ∈ N m G D ,this implies that ( m G ) ∗ ny ( D ) = 0 ∀ n ∈ N . (2.15)Now, given x ∈ D , there is a path of length m , { x, z , z , . . . , z m − , y } , x ∼ z ∼ z ∼ · · · ∼ z m − ∼ y ,and then ( m G ) ∗ ny ( { x } ) ≥ w yz m − w z m − z m − · · · w z z w z x d y d z m − d z m − · · · d z d z > , which is in contradiction with (2.15). (cid:3) The next examples show that there is no relation between m -connectedness and classical connectedness,i.e., there are connected metric random walk spaces that are not m -connected and, conversely, there are m -connected metric random walk spaces that are not connected. HE HEAT FLOW ON METRIC RANDOM WALK SPACES 23
Example 2.16.
Take ([0 , , d ) with d the Euclidean distance and let C be the Cantor set. Let µ bethe Cantor distribution, that is, the probability measure whose cumulative distribution function F ( x ) = µ ([0 , x )) is the Cantor function. We have that µ is singular with respect to the Lebesgue measure and itssupport is the Cantor set. We denote η := L [0 ,
1] and define the random walk m x := η if x ∈ [0 , \ C,µ if x ∈ C. Then, ν = η + µ is invariant and reversible. Indeed, Z (0 , Z (0 , f ( y ) dm x ( y ) dν ( x ) = Z (0 , \ C Z (0 , f ( y ) dydx + Z C Z C f ( y ) dµ ( y ) dµ ( x ) == Z (0 , f ( y ) dy + Z C f ( y ) dµ ( y ) = Z (0 , f ( y ) dν ( y ) , Similarly, we prove that ν is reversible.On the other hand, m ∗ nx = m x for any x ∈ (0 ,
1) and n ∈ N . In fact, if x ∈ C , we have Z y ∈ X f ( y ) dm ∗ x ( y ) = Z z ∈ (0 , Z y ∈ , f ( y ) dm z ( y ) ! dm x ( z ) = Z z ∈ (0 , Z y ∈ (0 , f ( y ) dm z ( y ) ! dµ ( z )= Z z ∈ C Z y ∈ (0 , f ( y ) dm z ( y ) ! dµ ( z ) = Z z ∈ C (cid:18)Z y ∈ C f ( y ) dµ ( y ) (cid:19) dµ ( z ) = Z X f ( y ) dm x ( y ) , and the proof for x ∈ (0 , \ C is similar.Consequently, m ∗ nx ( C ) = 0 for every x ∈ (0 , \ C and for every n ∈ N , so that ν ( N mC ) ≥ ν ((0 , \ C ) =1 > , , d, m ] is not m -connected.For x, y ∈ (0 , x = y , if x, y ∈ C , or x, y ∈ (0 , \ C , then W ( m x , m y ) = 0 and hence κ ( x, y ) = 1;otherwise, if x ∈ C and y ∈ (0 , \ C then W ( m x , m y ) = W ( µ, η ). Hence κ ( x, y ) = 1 − W ( µ,η ) | x − y | .Consequently, since κ = inf x = y κ ( x, y ) and κ ( x, y ) = 1 if x, y ∈ C or x, y ∈ (0 , \ C , we get κ = inf x ∈ C,y / ∈ C (cid:18) − W ( µ, η ) | x − y | (cid:19) = −∞ . Example 2.17.
Let Ω := (cid:16)(cid:3) − ∞ , (cid:3) ∪ (cid:2) , + ∞ (cid:2)(cid:17) × R N − and consider the metric random walk space[Ω , d, m J, Ω ], with d the Euclidean distance and J ( x ) = | B (0) | χ B (0) (see Example (1.2) (1)). It is easyto see that this space with reversible and invariant measure ν = L Ω is m -connected, but (Ω , d ) is notconnected. Let us see that its Ollivier-Ricci curvature κ is negative. Indeed, take x = ( − , , . . . , ∈ Ω,and y = (2 , , . . . , ∈ Ω. Then, we have that u ( x , x , . . . x N ) = − x is a Kantorovich potential for the transport of m Jx to m Jy , consequently W ( m J, Ω x , m J, Ω y ) ≥ Z Ω u ( z ) (cid:0) dm J, Ω x ( z ) − dm J, Ω y ( z ) (cid:1) = Z Ω u ( z ) (cid:0) dm Jx ( z ) − dm Jy ( z ) (cid:1) + Z R N \ Ω dm Jx ( z ) ! u ( x )= Z R N u ( z ) (cid:0) dm Jx ( z ) − dm Jy ( z ) (cid:1) + Z R N \ Ω ( u ( x ) − u ( z )) dm Jx ( z )= W ( m Jx , m Jy ) + Z R N \ Ω (cid:18)
12 + z (cid:19) dm Jx ( z ) > d ( x, y ) . Therefore, the Ollivier-Ricci curvature of [Ω , d, m J, Ω ] κ ≤ κ ( x, y ) = 1 − W ( m J, Ω x , m J, Ω y ) d ( x, y ) < . For Ω = (cid:16)(cid:3) − ∞ , (cid:3) ∪ (cid:2) , + ∞ (cid:2)(cid:17) × R N − , neither [Ω , d, m J, Ω ] with ν = L Ω is m -connected, nor (Ω , d )is connected. As above, we can prove that its Ollivier-Ricci curvature is negative.In a similar way we have that, for Ω = R N \ (0 , N , [Ω , d, m J, Ω ] with ν = L Ω is m -connected, (Ω , d )is connected and its Ollivier-Ricci curvature is κ < m -connectedness property with other known concepts in the literature. Let usbegin with the concept of ergodicity (see, for example, [29]). Definition 2.18.
Let [
X, d, m ] be a metric random walk space with invariant probability measure ν . ABorel set B ⊂ X is said to be invariant with respect to the random walk m if m x ( B ) = 1 whenever x isin B .The invariant probability measure ν is said to be ergodic if ν ( B ) = 0 or ν ( B ) = 1 for every invariantset B with respect to the random walk m . Theorem 2.19.
Let [ X, d, m ] be a metric random walk space with invariant probability measure ν . Then,the following assertions are equivalent:(i) [ X, d, m ] with ν is m -connected.(ii) ν is ergodic.Proof. ( i ) ⇒ ( ii ). If ν is not ergodic, there exists an invariant set B with respect to the random walk m such that 0 < ν ( B ) <
1. However, note that B is also invariant with respect to m ∗ . Indeed, m ∗ x ( B ) = Z X m z ( B ) dm x ( z ) ≥ Z B m z ( B ) dm x ( z ) = m x ( B ) = 1 HE HEAT FLOW ON METRIC RANDOM WALK SPACES 25 for every x ∈ B . Inductively, we obtain that, in fact, B is invariant for m ∗ n , for every n ∈ N . Therefore,since ν ( B ) = Z X m ∗ nx ( B ) dν ( x ) ≥ Z B m ∗ nx ( B ) dν ( x ) = ν ( B ) for every n ∈ N , we obtain that m ∗ nx ( B ) = 0 for every n ∈ N and ν -a.e. x ∈ X \ B . Therefore, X \ B ⊂ N mB ν -a.e., thus ν ( N mB ) > X, d, m ] with ν is not m -connected.( ii ) ⇒ ( i ). Let D ⊂ X be a ν -measurable set with ν ( D ) >
0. By Proposition 2.5 we have that N mD is invariant with respect to the random walk m . Then, since ν is ergodic, we have that ν ( N mD ) = 0 or ν ( N mD ) = 1. Now, since ν ( D ) >
0, by Corollary 2.7, we have that ν ( N mD ) = 0 and, consequently, [ X, d, m ]with ν is m -connected. (cid:3) Following Bakry, Gentil and Ledoux [6], we give the following definition.
Definition 2.20.
Let [ X, d, m ] be a metric random walk space with invariant measure ν . We say that ∆ m is ergodic if, for u ∈ Dom(∆ m ) , ∆ m u = 0 implies that u is constant (being this constant if ν is notfinite). Theorem 2.21.
Let [ X, d, m ] be a metric random walk space with finite invariant measure ν . Then, ∆ m is ergodic ⇔ [ X, d, m ] is random walk connected.Proof. ( ⇒ ): Suppose that ν ( X ) < + ∞ and [ X, d, m ] is not m -connected, then there exists D ⊂ X with ν ( D ) > ν ( N mD ) >
0. Recall that ν ( H mD ) >
0. Consider the function u ( x ) = χ H mD ( x ) , and note that u ∈ L ( X, ν ) since ν is finite. Now,∆ m u ( x ) = Z X (cid:0) χ H mD ( y ) − χ H mD ( x ) (cid:1) dm x ( y ) = m x ( H mD ) − χ H mD ( x ) , hence, by Proposition 2.5, ∆ m u = 0 ν -a.e. , but u is not equal to a constant ν -a.e., and, consequently, ∆ m is not ergodic. ( ⇐ ): Suppose now that [ X, d, m ] is m -connected and that there exists u ∈ L ( X, ν ) such that ∆ m u = 0 ν -a.e. but u is not ν -a.e. equal to a constant function. Then, we may find U , V ⊂ X with positive ν -measure such that u ( x ) < u ( y ) for every x ∈ U and y ∈ V . Note that∆ m ∗ n u ( x ) = Z X ( u ( y ) − u ( x )) dm ∗ nx ( y )= Z X Z X ( u ( z ) − u ( x )) dm y ( z ) dm ∗ ( n − x ( y )= Z X Z X ( u ( z ) − u ( y )) dm y ( z ) dm ∗ ( n − x ( y ) + Z X Z X ( u ( y ) − u ( x )) dm y ( z ) dm ∗ ( n − x ( y )= Z X ∆ m u ( y ) dm ∗ ( n − x ( y ) + Z X ( u ( y ) − u ( x )) dm ∗ ( n − x ( y )= Z X ∆ m u ( y ) dm ∗ ( n − x ( y ) + ∆ m ∗ ( n − u ( x ) , thus | ∆ m ∗ n u ( x ) | ≤ Z X | ∆ m u ( y ) | dm ∗ ( n − x ( y ) + | ∆ m ∗ ( n − u ( x ) | . (2.16)Now, using the invariance of ν , Z X Z X | ∆ m u ( y ) | dm ∗ ( n − x ( y ) dν ( x ) = Z X | ∆ m u ( x ) | dν ( x ) = 0so Z X | ∆ m u ( y ) | dm ∗ ( n − x ( y ) = 0 for ν -a.e. x ∈ X, thus, by induction on (2.16), ∆ m ∗ n u ( x ) = 0 for ν -a.e. x ∈ X and every n ∈ N . Since [ X, d, m ] is m -connected we have ν ( N mV ) = ν ( X \ H mV ) = 0, so there exists n ∈ N such that ν ( U ∩ H mV,n ) > H m ∗ n ( u ) = Z X Z X ∇ u ( x, y ) dm ∗ nx ( y ) dν ( x ) ≥ Z U ∩ H mV,n Z V ∇ u ( x, y ) dm ∗ nx ( y ) dν ( x ) > . (cid:3) Let [
X, d, m ] be a metric random walk space with invariant and reversible measure ν . It is easy to seethat ∆ m is ergodic if, and only if, e t ∆ m f = f for all t ≥ f is constant. Moreover, we havethe following result. Proposition 2.22.
Let [ X, d, m ] be a metric random walk space with invariant and reversible measure ν . For every f ∈ L ( X, ν ) , lim t →∞ e t ∆ m f = f ∞ ∈ { u ∈ L ( X, ν ) : ∆ m u = 0 } . Suppose further that ∆ m is ergodic:(i) If ν ( X ) = + ∞ , then f ∞ = 0 . (ii) If ν ( X ) < + ∞ , then f ∞ = 1 ν ( X ) Z X f ( x ) dν ( x ) . HE HEAT FLOW ON METRIC RANDOM WALK SPACES 27
Proof.
The first result follows from [15, Theorem 3.11]. The second part is a consequence of the ergodicityof ∆ m and the conservation of mass (2.4). (cid:3) When the invariant measure is a probability measure, the relation between both concepts of ergodicity,the one for the invariant measure and the one for the Laplacian was known; see, for example, [29]. Letus now give another characterization of the ergodicity in terms of geometric properties.
Lemma 2.23.
Let [ X, d, m ] be a metric random walk space with invariant and reversible measure ν andassume that ν ( X ) < + ∞ . Then for every ν -measurable subset D ⊂ X we have ∆ m χ D ( x ) = 0 ⇔ m x ( D ) = χ D ( x ) , (2.17) and ∆ m χ D = 0 ⇔ P m ( D ) = 0 ⇔ Z D H m∂D ( x ) dν ( x ) = − ν ( D ) . Proof.
Since ∆ m χ D ( x ) = Z X (cid:0) χ D ( y ) − χ D ( x ) (cid:1) dm x ( y ) = m x ( D ) − χ D ( x )we have that ∆ m χ D ( x ) = 0 if, and only if, χ D ( x ) − m x ( D ) = 0, and we get (2.17).Suppose now that ∆ m χ D = 0, then χ D ( x ) = m x ( D ), thus integrating this expression over D withrespect to ν , we get P m ( D ) = ν ( D ) − Z D Z D dm x ( y ) dν ( x ) = 0 . Conversely, if P m ( D ) = 0 , we have ν ( D ) = Z D m x ( D ) dν ( x ) . Then, on one hand, m x ( D ) = χ D ( x ) for ν -a.e. x ∈ D ,and, on the other hand, since ν ( D ) = Z X m x ( D ) dν ( x ) = Z D m x ( D ) dν ( x ) + Z X \ D m x ( D ) dν ( x ) , we get Z X \ D m x ( D ) dν ( x ) = 0 , thus m x ( D ) = χ D ( x ) for ν -a.e. x ∈ X \ D .Therefore, m x ( D ) = χ D ( x ) for ν -a.e. x ∈ X and, by (2.17), we get ∆ m χ D = 0.For the second equivalence, by (1.5), we have that Z D H m∂D ( x ) dν ( x ) = 2 P m ( D ) − ν ( D ) , thus P m ( D ) = 0 if, and only if, Z D H m∂D ( x ) dν ( x ) = − ν ( D ). (cid:3) Theorem 2.24.
Let [ X, d, m ] be a metric random walk space with invariant and reversible measure ν and assume that ν ( X ) < + ∞ . The following facts are equivalent:(1) ∆ m is ergodic;(2) ∆ m χ D = 0 for a ν -measurable set D implies that χ D is constant;(3) P m ( D ) > for every ν -measurable set D such that < ν ( D ) < ν ( X ) ;(4) The ν -mean value of the m -mean curvature of ∂D in D satisfies ν ( D ) Z D H m∂D ( x ) dν ( x ) > − for every ν -measurable set D such that < ν ( D ) < ν ( X ) .Proof. Obviously, (1) implies (2), and this yields (3) since P m ( D ) = 0 implies, by Lemma 2.23, that∆ m χ D = 0. Also, by Lemma 2.23, (3) implies (2). Let us now see that (2) implies (1): Suppose that ∆ m is not ergodic, then the space is not m -connected and, consequently, there exists D ⊂ X with ν ( D ) > < ν ( N mD ) <
1; but, from Proposition 2.5,∆ m χ N mD ( x ) = m x ( N mD ) − χ N mD ( x ) = 0 , and this implies that χ N mD should be constant, which is a contradiction with 0 < ν ( N mD ) < ν ( X ). Theequivalence with (4) is evident by the second equivalence in Lemma 2.23. (cid:3) Functional Inequalities
Let [
X, d, m ] be a metric random walk space with invariant and reversible measure ν such that ν ( X ) < + ∞ . In this section we will further assume that ν ( X ) = 1, i.e. that ν is a probability measure. Notethat we may always work with ν ( X ) ν .3.1. Spectral Gap and Poincar´e Inequality.
We denote the mean value of f ∈ L ( X, ν ) (or theexpected value of f ) with respect to ν by ν ( f ) := E ν ( f ) = Z X f ( x ) dν ( x ) . Moreover, given f ∈ L ( X, ν ), we denote its variance with respect to ν byVar ν ( f ) := Z X ( f ( x ) − ν ( f )) dν ( x ) = 12 Z X × X ( f ( x ) − f ( y )) dν ( y ) dν ( x ) . Definition 3.1.
The spectral gap of − ∆ m is defined asgap( − ∆ m ) := inf (cid:26) H m ( f )Var ν ( f ) : f ∈ D ( H m ) , Var ν ( f ) = 0 (cid:27) = inf (cid:26) H m ( f ) k f k : f ∈ D ( H m ) , k f k = 0 , Z X f dν = 0 (cid:27) . (3.1)Observe that, since ν ( X ) < + ∞ , we have D ( H m ) = L ( X, ν ) . HE HEAT FLOW ON METRIC RANDOM WALK SPACES 29
Definition 3.2.
We say that [
X, d, m, ν ] satisfies a
Poincar´e inequality if there exists λ > λ Var ν ( f ) ≤ H m ( f ) for all f ∈ L ( X, ν ) , or, equivalently, λ k f k L ( X,ν ) ≤ H m ( f ) for all f ∈ L ( X, ν ) with ν ( f ) = 0 . Note that, if gap( − ∆ m ) >
0, then [
X, d, m, ν ] satisfies a Poincar´e inequality with λ = gap( − ∆ m ):gap( − ∆ m )Var ν ( f ) ≤ H m ( f ) for all f ∈ L ( X, ν ) , being the spectral gap the best constant in the Poincar´e inequality.With such an inequality at hand and with a similar proof to the one done in the continuous setting(see, for instance, [6]), we have that if gap( − ∆ m ) > e t ∆ m u converges to ν ( u ) with exponentialrate gap( − ∆ m ). Theorem 3.3.
The following statements are equivalent:(i) There exists λ > such that λ Var ν ( f ) ≤ H m ( f ) for all f ∈ L ( X, ν ) . (ii) For every f ∈ L ( X, ν ) k e t ∆ m f − ν ( f ) k L ( X,ν ) ≤ e − λt k f − ν ( f ) k L ( X,ν ) for all t ≥ or, equivalently, for every f ∈ L ( X, ν ) with ν ( f ) = 0 , k e t ∆ m f k L ( X,ν ) ≤ e − λt k f k L ( X,ν ) for all t ≥ . Remark 3.4.
Let k µ − ν k T V be the total variation distance: k µ − ν k T V := sup {| µ ( A ) − ν ( A ) | : A ⊂ X Borel } . Then, for f ∈ L ( X, ν ) and µ t = e t ∆ m f ν , we have k µ t − ν k T V ≤ k f − k L ( X,ν ) e − gap( − ∆ m ) t . Indeed, by Theorem 3.3, for any Borel set A ⊂ X , (cid:12)(cid:12)(cid:12)(cid:12)Z A e t ∆ m f dν − ν ( A ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z A (cid:12)(cid:12) e t ∆ m f − (cid:12)(cid:12) dν ≤ (cid:18)Z X (cid:12)(cid:12) e t ∆ m f − (cid:12)(cid:12) dν (cid:19) ≤ k f − k L ( X,ν ) e − gap( − ∆ m ) t . Hence, it is of interest to elucidate when the spectral gap of − ∆ m is positive. In this section we willdeal with such a question.Let H ( X, ν ) be the subspace of L ( X, ν ) consisting of the functions which are orthonormal to theconstants, i.e., H ( X, ν ) = (cid:8) f ∈ L ( X, ν ) : ν ( f ) = 0 (cid:9) . Since the operator − ∆ m : H ( X, ν ) → H ( X, ν ) is self-adjoint and non-negative and k ∆ m k ≤ σ ( − ∆ m ) of − ∆ m in H ( X, ν ) satisfies σ ( − ∆ m ) ⊂ [ α, β ] ⊂ [0 , , where α := inf {h− ∆ m u, u i : u ∈ H ( X, ν ) , k u k = 1 } ∈ σ ( − ∆ m ) , and β := sup {h− ∆ m u, u i : u ∈ H ( X, ν ) , k u k = 1 } ∈ σ ( − ∆ m ) . If f ∈ L ( X, ν ) and Var ν ( f ) = 0, then u := f − ν ( f ) = 0 belongs to H ( X, ν ), so α ≤ H m (cid:18) u k u k (cid:19) = H m ( u ) k u k = H m ( f )Var ν ( f ) , and, consequently, gap( − ∆ m ) = α = inf {h− ∆ m u, u i : u ∈ H ( X, ν ) , k u k = 1 } . (3.2)Therefore, gap( − ∆ m ) > ⇐⇒ σ ( − ∆ m ) . If we assume that − ∆ m is the sum of an invertible and a compact operator in H ( X, ν ) (this is true,for example, if the averaging operator M m is compact in H ( X, ν )), then, if 0 ∈ σ ( − ∆ m ), by Fredholm’salternative Theorem, we have that there exists u ∈ H ( X, ν ), u = 0, such that − ∆ m u = ( I − M m ) u = 0.Then, if [ X, d, m ] is m -connected, by Theorem 2.21, ∆ m is ergodic so u is constant, thus u = 0 in H ( X, ν ),and we get a contradiction. Consequently, we have the following result.
Proposition 3.5.
Let [ X, d, m ] be an m -connected metric random walk space with invariant-reversibleprobability measure ν . If − ∆ m is the sum of an invertible operator and a compact operator in H ( X, ν ) ,then gap( − ∆ m ) > . Example 3.6. (i) If G = ( V ( G ) , E ( G )) is a finite weighted connected graph, then obviously M m G iscompact and, consequently, gap( − ∆ Gm ) >
0. In this situation, it is well known that, for ♯ ( V ( G )) = N ,the spectrum of − ∆ m G is 0 < λ ≤ λ ≤ . . . ≤ λ N − and 0 < λ = gap( − ∆ m ).(ii) Another example in which − ∆ m is the sum of an invertible and a compact operator is [Ω , d, m J, Ω ]with Ω a bounded domain and the kernel J satisfying: J ∈ C ( R N , R ) is nonnegative, radially symmetricwith J (0) > R R N J ( x ) dx = 1. Indeed, − ∆ m J, Ω f ( x ) = Z Ω J ( x − y ) dyf ( x ) − Z Ω J ( x − y ) f ( y ) dy, where Z Ω J ( x − y ) dyf ( x ) defines an invertible operator and Z Ω J ( x − y ) f ( y ) dy defines a compact operator.Hence, in this case we have (see also [4]):gap( − ∆ m J, Ω ) = inf Z Ω × Ω J ( x − y )( u ( y ) − u ( x )) dxdy Z Ω u ( x ) dx : u ∈ L (Ω) , k u k L ( X,ν ) > , Z Ω u = 0 > . Let us point out that the condition J (0) > − ∆ m ) > m is ergodic and M m ishyperbounded, that is, if there exists p > M m is bounded from L ( X, ν ) to L p ( X, ν ). If we
HE HEAT FLOW ON METRIC RANDOM WALK SPACES 31 have that m x ≪ ν , i.e., m x = f x ν with f x ∈ L ( X, ν ), and we assume that Z X k f x k pL ( X,ν ) dν ( x ) = K < ∞ , (3.3)then, for u ∈ L ( X, ν ), by the Cauchy-Schwarz inequality, we have that k M m u k pp = Z X | M m u ( x ) | p dν ( x ) = Z X (cid:12)(cid:12)(cid:12)(cid:12)Z X u ( y ) dm x ( y ) (cid:12)(cid:12)(cid:12)(cid:12) p dν ( x ) = Z X (cid:12)(cid:12)(cid:12)(cid:12)Z X u ( y ) f x ( y ) dν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) p dν ( x ) ≤ k u k pL ( X,ν ) Z X k f x k pL ( X,ν ) dν ( x ) , hence k M m u k p ≤ K p k u k L ( X,ν ) . Therefore, M m is hyperbounded and, consequently, we have the following result about the spectral gap. Proposition 3.7. If ∆ m is ergodic and (3.3) holds, then gap( − ∆ m ) > . In the next example we see that there exist metric random walk spaces for which the Poincar´e inequalitydoes not hold.
Example 3.8.
Let V ( G ) = { x , x , x . . . , x n . . . } be a weighted linear graph with w x n ,x n +1 = 1 n , w x n +1 ,x n +2 = 1 n , w x n +2 ,x n +3 = 1 n , for n ≥
1, and let f n ( x ) = n if x = x n +1 , x n +2 . Note that ν ( X ) < + ∞ (we avoid its normalization for simplicity). Now,2 H m ( f n ) = Z X Z X ( f n ( x ) − f n ( y )) dm x ( y ) dν ( x )= d x n Z X ( f n ( x n ) − f n ( y )) dm x n ( y ) + d x n +1 Z X ( f n ( x n +1 ) − f n ( y )) dm x n +1 ( y )+ d x n +2 Z X ( f n ( x n +2 ) − f n ( y )) dm x n +2 ( y ) + d x n +3 Z X ( f n ( x n +3 ) − f n ( y )) dm x n +3 ( y )= d x n n n d x n + d x n +1 n n d x n +1 + d x n +2 n n d x n +2 + d x n +3 n n d x n +3 = 4 n . However, we have Z X f n ( x ) dν ( x ) = n ( d x n +1 + d x n +2 ) = 2 n (cid:18) n + 1 n (cid:19) = 2 n (cid:18) n (cid:19) , thus ν ( f n ) = n (cid:0) n (cid:1) ν ( X ) = e O (cid:18) n (cid:19) , where we use the notation ϕ ( n ) = e O ( ψ ( n )) ⇐⇒ ∃ lim n →∞ ϕ ( n ) ψ ( n ) = C = 0 . Therefore, ( f n ( x ) − ν ( f n )) = e O ( n ) if x = x n +1 , x n +2 , e O (cid:0) n (cid:1) otherwise . Finally,Var ν ( f n ) = Z X ( f n ( x ) − ν ( f n )) dν ( x ) = e O (cid:18) n (cid:19) X x = x n +1 ,x n +2 d x + e O ( n )( d x n +1 + d x n +2 )= e O (cid:18) n (cid:19) + 2 e O ( n ) (cid:18) n + 1 n (cid:19) = e O (1) . Consequently, [ V ( G ) , d G , ( m x ) , ν ] does not satisfy a Poincar´e inequality for any λ > H m ( f ) = − R X f ( x )∆ m f ( x ) dν ( x ), if ∆ m f = 0 then H m ( f ) = 0 and, therefore, if[ X, d, m, ν ] satisfies a Poincar´e inequality, we have that f is constant: f ( x ) = Z X f ( x ) dν ( x ) ν − a.e.Consequently, we get the following result. Proposition 3.9. If [ X, d, m, ν ] satisfies a Poincar´e inequality we have that ∆ m is ergodic. Example 3.8 shows that the reverse implication does not hold in general.3.2.
Isoperimetric Inequality.
Recall that, for a ν -measurable set D ⊂ X , P m ( D ) = Z E Z X \ E dm x ( y ) dν ( x ) = T V m ( χ E ) . The Poincar´e inequality, if given only for characteristic functions, implies that there exists λ > λ ν ( D ) (cid:0) − ν ( D ) (cid:1) ≤ P m ( D ) for every ν − measurable set D, (3.4)(observe that this also implies the ergodicity of ∆ m , as we have seen in Theorem 2.24). Hence, sincemin { x, − x } ≤ x (1 − x ) ≤ { x, − x } for 0 ≤ x ≤ , inequality (3.4) implies the following isoperimetric inequality (see [1, Theorem 3.46]):min (cid:8) ν ( D ) , − ν ( D ) (cid:9) ≤ λ P m ( D ) for every ν − measurable set D ; (3.5)and, conversely, the isoperimetric inequality (3.5) implies λ ν ( D ) (cid:0) − ν ( D ) (cid:1) ≤ P m ( D ) for every ν − measurable set D. Definition 3.10.
If there exists λ > satisfying (3.5) , we say that [ X, d, m, ν ] satisfies an isoperimetricinequality. HE HEAT FLOW ON METRIC RANDOM WALK SPACES 33
Cheeger Inequality.
In a weighted graph G = ( V ( G ) , E ( G )) the Cheeger constant is defined as h G := inf D ⊂ V ( G ) | ∂D | min { ν G ( D ) , ν G ( V ( G ) \ D ) } , where | ∂D | := X x ∈ D,y ∈ V \ D w xy . In [19] (see also [10]), the following relation between the Cheeger constant and the first positive eigenvalue λ ( G ) of the graph Laplacian ∆ m G is proved: h G ≤ λ ( G ) ≤ h G . (3.6)The previous inequality appeared in [17], and can be traced back to the paper by Polya and Szego [47].Let [ X, d, m ] be a metric random walk space with invariant and reversible probability measure ν . Wedefine its Cheeger constant as h m ( X ) := inf (cid:26) P m ( D )min { ν ( D ) , ν ( X \ D ) } : D ⊂ X, < ν ( D ) < (cid:27) , or, equivalently, h m ( X ) = inf (cid:26) P m ( D ) ν ( D ) : D ⊂ X, < ν ( D ) ≤ (cid:27) . Having in mind (1.4), we have that this definition is consistent with the definition on graphs. Note that,if h m ( X ) >
0, then h m ( X ) is the best constant in the isoperimetric inequality (3.5).We will now give a variational characterization of the Cheeger constant which generalizes the oneobtained in [52] for the particular case of finite graphs. Recall that, given a function u : X → R , µ ∈ R is a median of u with respect to a measure ν if ν ( { x ∈ X : u ( x ) < µ } ) ≤ ν ( X ) , ν ( { x ∈ X : u ( x ) > µ } ) ≤ ν ( X ) . We denote by med ν ( u ) the set of all medians of u . It is easy to see that µ ∈ med ν ( u ) ⇐⇒ − ν ( { u = µ } ) ≤ ν ( { x ∈ X : u ( x ) > µ } ) − ν ( { x ∈ X : u ( x ) < µ } ) ≤ ν ( { u = µ } ) , from where it follows that0 ∈ med ν ( u ) ⇐⇒ ∃ ξ ∈ sign( u ) such that Z X ξ ( x ) dν ( x ) = 0 , where sign( u )( x ) := u ( x ) > , − u ( x ) < , [ − ,
1] if u = 0 . Let λ m ( X ) := inf { T V m ( u ) : k u k = 1 , ∈ med ν ( u ) } . (3.7) Theorem 3.11. If [ X, d, m ] is a metric random walk space with invariant and reversible probabilitymeasure ν , then h m ( X ) = λ m ( X ) . Proof. If D ⊂ X, < ν ( D ) ≤ , then 0 ∈ med ν ( χ D ). Thus, λ m ( X ) ≤ T V m (cid:18) ν ( D ) χ D (cid:19) = 1 ν ( D ) P m ( D )and, therefore, λ m ( X ) ≤ h m ( X ) . Now, for the other inequality, let u ∈ L ( X, ν ) such that k u k = 1 and 0 ∈ med ν ( u ). Since 0 ∈ med ν ( u ),by the Coarea formula (Theorem 1.4), and having in mind that the set { t ∈ R : ν ( { u = t } ) > } iscountable, we have T V m ( u ) = Z + ∞−∞ P m ( E t ( u )) dt = Z + ∞ P m ( E t ( u )) dt + Z −∞ P m ( X \ E t ( u )) dt ≥ h m ( X ) Z + ∞ ν ( E t ( u )) dt + h m ( X ) Z −∞ ν ( X \ E t ( u )) dt = h m ( X ) (cid:18)Z X u + ( x ) dν ( x ) + Z X u − ( x ) dν ( x ) (cid:19) = h m ( X ) k u k = h m ( X ) . Therefore, taking the infimum in u , we get λ m ( X ) ≥ h m ( X ) . (cid:3) Following [19] and using Theorem 3.11, in the next result we see that the Cheeger inequality (3.6) alsoholds in our context.
Theorem 3.12.
Let [ X, d, m ] be a metric random walk space with invariant and reversible probabilitymeasure ν . The following Cheeger inequality holds h m ≤ gap( − ∆ m ) ≤ h m . Proof.
Let ( f n ) ⊂ D ( H m ), with ν ( f n ) = 0, such thatlim n →∞ H m ( f n ) k f n k = gap( − ∆ m ) . If we take µ n ∈ med ν ( f n ), we have2 H m ( f n ) = Z X Z X ( f n ( y ) − µ n − ( f n ( x ) − µ n )) dm x ( y ) dν ( x )= Z X Z X (cid:2) ( f n ( y ) − µ n ) + − ( f n ( x ) − µ n ) + − (( f n ( y ) − µ n ) − − ( f n ( x ) − µ n ) − ) (cid:3) dm x ( y ) dν ( x )= Z X Z X (cid:0) ( f n ( y ) − µ n ) + − ( f n ( x ) − µ n ) + (cid:1) dm x ( y ) dν ( x )+ Z X Z X (cid:0) ( f n ( y ) − µ n ) − − ( f n ( x ) − µ n ) − (cid:1) dm x ( y ) dν ( x ) − Z X Z X (cid:0) ( f n ( y ) − µ n ) + − ( f n ( x ) − µ n ) + (cid:1) (cid:0) ( f n ( y ) − µ n ) − − ( f n ( x ) − µ n ) − (cid:1) dm x ( y ) dν ( x ) . HE HEAT FLOW ON METRIC RANDOM WALK SPACES 35
Now, an easy calculation gives − Z X Z X (cid:0) ( f n ( y ) − µ n ) + − ( f n ( x ) − µ n ) + (cid:1) (cid:0) ( f n ( y ) − µ n ) − − ( f n ( x ) − µ n ) − (cid:1) dm x ( y ) dν ( x ) ≥ . On the other hand, since ν ( f n ) = 0, we have Z X f n ( x ) dν ( x ) ≤ Z X ( f n ( x ) − µ n ) dν ( x ) . Therefore, 2 H m ( f n ) k f n k ≥ Z X Z X (cid:0) ( f n ( y ) − µ n ) + − ( f n ( x ) − µ n ) + (cid:1) dm x ( y ) dν ( x ) Z X [( f n ( x ) − µ n ) + ] dν ( x ) + Z X [( f n ( x ) − µ n ) − ] dν ( x ) ++ Z X Z X (cid:0) ( f n ( y ) − µ n ) − − ( f n ( x ) − µ n ) − (cid:1) dm x ( y ) dν ( x ) Z X [( f n ( x ) − µ n ) + ] dν ( x ) + Z X [( f n ( x ) − µ n ) − ] dν ( x ) . Having in mind that a + bc + d ≥ min (cid:26) ac , bd (cid:27) for every a, b, c, d ∈ R + , and Z X [( f n ( x ) − µ n ) + ] dν ( x ) + Z X [( f n ( x ) − µ n ) − ] dν ( x ) > , we can assume, without loss of generality, that Z X [( f n ( x ) − µ n ) + ] dν ( x ) > , and that 2 H m ( f n ) k f n k ≥ Z X Z X (cid:0) ( f n ( y ) − µ n ) + − ( f n ( x ) − µ n ) + (cid:1) dm x ( y ) dν ( x ) Z X [( f n ( x ) − µ n ) + ] dν ( x ) . By the Cauchy-Schwartz inequality, we have Z X Z X (cid:12)(cid:12) [( f n ( y ) − µ n ) + ] − [( f n ( x ) − µ n ) + ] (cid:12)(cid:12) dm x ( y ) dν ( x )= Z X Z X (cid:12)(cid:12) ( f n ( y ) − µ n ) + − ( f n ( x ) − µ n ) + (cid:12)(cid:12) (cid:12)(cid:12) ( f n ( y ) − µ n ) + + ( f n ( x ) − µ n ) + (cid:12)(cid:12) dm x ( y ) dν ( x ) ≤ (cid:18)Z X Z X (cid:0) ( f n ( y ) − µ n ) + − ( f n ( x ) − µ n ) + (cid:1) dm x ( y ) dν ( x ) (cid:19) ×× (cid:18)Z X Z X (cid:0) ( f n ( y ) − µ n ) + + ( f n ( x ) − µ n ) + (cid:1) dm x ( y ) dν ( x ) (cid:19) . Now, by the invariance of ν , Z X Z X (cid:0) ( f n ( y ) − µ n ) + + ( f n ( x ) − µ n ) + (cid:1) dm x ( y ) dν ( x ) ≤ Z X [( f n ( x ) − µ n ) + ] dν ( x ) . Thus 2 H m ( f n ) k f n k ≥ Z X Z X (cid:12)(cid:12) [( f n ( y ) − µ n ) + ] − [( f n ( x ) − µ n ) + ] (cid:12)(cid:12) dm x ( y ) dν ( x ) Z X [( f n ( x ) − µ n ) + ] dν ( x ) . Then, since 0 ∈ med ν ([( f n − µ n ) + ] ), by Theorem 3.11, we get2 H m ( f n ) k f n k ≥ h m ( X ) , and, consequently, taking limits as n → ∞ , we obtain h m ≤ gap( − ∆ m ) . To prove the other inequality we can assume that gap( − ∆ m ) >
0. Now, by (3.5), we havemin (cid:8) ν ( D ) , − ν ( D ) (cid:9) ≤ − ∆ m ) P m ( D ) for all D ⊂ X, < ν ( D ) < , from where it follows that gap( − ∆ m ) ≤ h m ( X ). (cid:3) Let A ⊂ X with ν ( A ) = and u = χ A − χ X \ A . It is easy to see that T V m ( u ) = 2 P m ( A ) and H m ( u ) = 4 P m ( A ). Hence, since k u k = k u k = 1, ν ( u ) = 0 and 0 ∈ med ν ( u ), we obtain the followingresult as a consequence of Theorem 3.11. Corollary 3.13.
Let [ X, d, m ] be a metric random walk space with invariant and reversible probabilitymeasure ν . Let A ⊂ X with ν ( A ) = and u = χ A − χ X \ A . Then,1. h m ( X ) = P m ( A ) ν ( A ) ⇐⇒ u = χ A − χ X \ A is a minimizer of (3.7) .2. u is a minimizer of (3.7) and gap( − ∆ m ) = 2 h m ( X ) ⇐⇒ u is a minimizer of (3.1) . Bringing together all the above results we have:
Theorem 3.14.
Let [ X, d, m ] be a metric random walk space with invariant and reversible probabilitymeasure ν . The following statements are equivalent:(1) [ X, d, m, ν ] satisfies a Poincar´e inequality,(2) gap( − ∆ m ) > ,(3) [ X, d, m, ν ] satisfies an isoperimetric inequality,(4) h m ( X ) > . Example 3.15.
It is well known, (see for instance [19]) that for finite graphs G , h m ( G ) > G is connected. This result is not true for infinite graphs. In fact, the graph of the Example 3.8 isconnected and its Cheeger constant is zero since its spectral gap is zero. HE HEAT FLOW ON METRIC RANDOM WALK SPACES 37
Spectral Gap and Curvature.
Since E m admits a Carr´e du champ
Γ (see [6]) defined byΓ( f, g )( x ) := 12 (cid:16) ∆ m ( f g )( x ) − f ( x )∆ m g ( x ) − g ( x )∆ m f ( x ) (cid:17) for all x ∈ X and f, g ∈ L ( X, ν ) , we can study the Bakry-´Emery curvature-dimension condition in this context. We will study its relationwith the spectral gap.According to Bakry and ´Emery [7], we define the Ricci curvature operator Γ by iterating Γ:Γ ( f, g ) := 12 (cid:16) ∆ m Γ( f, g ) − Γ( f, ∆ m g ) − Γ(∆ m f, g ) (cid:17) , which is well defined for f, g ∈ L ( X, ν ). We will write, for f ∈ L ( X, ν ),Γ( f ) := Γ( f, f ) = 12 ∆ m ( f ) − f ∆ m f and Γ ( f ) := Γ ( f, f ) = 12 ∆ m Γ( f ) − Γ( f, ∆ m f ) . It is easy to see thatΓ( f, g )( x ) = 12 Z X ∇ f ( x, y ) ∇ g ( x, y ) dm x ( y ) and Γ( f )( x ) = 12 Z X |∇ f ( x, y ) | dm x ( y ) . Consequently, Z X Γ( f, g )( x ) dν ( x ) = E m ( f, g ) and Z X Γ( f )( x ) dν ( x ) = H m ( f ) . (3.8)Furthermore, by (2.1) and (3.8), we get Z X Γ ( f ) dν = 12 Z X (∆ m Γ( f ) − f, ∆ m f )) dν = − Z X Γ( f, ∆ m f ) dν = −E m ( f, ∆ m f ) , thus Z X Γ ( f ) dν = Z X (∆ m f ) dν. (3.9) Definition 3.16.
The operator ∆ m satisfies the Bakry- ´Emery curvature-dimension condition BE ( K, n )for n ∈ (1 , + ∞ ) and K ∈ R ifΓ ( f ) ≥ n (∆ m f ) + K Γ( f ) ∀ f ∈ L ( X, ν ) . The constant n is the dimension of the operator ∆ m , and K is the lower bound of the Ricci curvature ofthe operator ∆ m . If there exists K ∈ R such thatΓ ( f ) ≥ K Γ( f ) ∀ f ∈ L ( X, ν ) , then it is said that the operator ∆ m satisfies the Bakry- ´Emery curvature-dimension condition BE ( K, ∞ ).Observe that if ∆ m satisfies the Bakry- ´Emery curvature-dimension condition BE ( K, n ) then it alsosatisfies the
Bakry- ´Emery curvature-dimension condition BE ( K, m ) for m > n .This definition is motivated by the well known fact that on a complete n -dimensional Riemannianmanifold ( M, g ), the Laplace-Beltrami operator ∆ g satisfies BE ( K, n ) if, and only if, the Ricci curvatureof the Riemannian manifold is bounded from below by K (see, for example, [6, Appendix C.6]). The use of the Bakry-´Emery curvature-dimension condition as a possible definition of a Ricci curvaturebound in Markov chains was first considered in 1998 [50]. Now, this concept of Ricci curvature in thediscrete setting has been frequently used since the work by Lin and Yau [33] (see [31] and the referencestherein).Integrating the Bakry-´Emery curvature-dimension condition BE ( K, n ) we have Z X Γ ( f ) dν ≥ n Z X (∆ m f ) dν + K Z X Γ( f ) dν. Now, by (3.8) and (3.9), this inequality can be rewritten as Z X (∆ m f ) dν ≥ n Z X (∆ m f ) dν + K H m ( f ) , or, equivalently, as K nn − H m ( f ) ≤ Z X (∆ m f ) dν. (3.10)Similarly, integrating the Bakry-´Emery curvature-dimension condition BE ( K, ∞ ) we have K H m ( f ) ≤ Z X (∆ m f ) dν. (3.11)We call the inequalities (3.10) and (3.11) the integrated Bakry- ´Emery curvature-dimension conditions ,and denote them by IBE ( K, n ) and
IBE ( K, ∞ ), respectively. Theorem 3.17.
Let [ X, d, m ] be a metric random walk space with invariant-reversible probability mea-sure ν . Assume that ∆ m is ergodic. Then gap( − ∆ m ) = sup n λ ≥ λ H m ( f ) ≤ Z X ( − ∆ m f ) dν ∀ f ∈ L ( X, ν ) o . Proof.
By (3.2) we know that gap( − ∆ m ) = α . Set A := sup n λ ≥ λ H m ( f ) ≤ Z X ( − ∆ m f ) dν ∀ f ∈ L ( X, ν ) o . Let ( P λ ) λ ≥ be the spectral projection of the self-adjoint and positive operator − ∆ m : H ( X, ν ) → H ( X, ν ). By the spectral Theorem [48, Theorem VIII. 6], we have H m ( f ) = h− ∆ m f, f i = Z βα λd h P λ f, f i Z X ( − ∆ m f ) dν = h− ∆ m f, − ∆ m f i = Z βα λ d h P λ f, f i . Hence, since λ ≥ αλ , we have that Z X ( − ∆ m f ) dν ≥ α Z βα λd h P λ f, f i = α H m ( f ) , and we get α ≤ A . Finally, let us see that α ≥ A . Since α ∈ σ (∆ m ), given ǫ >
0, there exists0 = f ∈ Range( P α + ǫ ) and, consequently, P λ f = f for λ ≥ α + ǫ . Then, having in mind the ergodicity of HE HEAT FLOW ON METRIC RANDOM WALK SPACES 39 ∆ m , we have0 < Z X ( − ∆ m f ) dν = Z α + ǫα λ d h P λ f, f i ≤ ( α + ǫ ) Z α + ǫα λd h P λ f, f i = ( α + ǫ ) H m ( f ) < ( α + 2 ǫ ) H m ( f ) . This implies that α + 2 ǫ does not belong to the set n λ ≥ λ H m ( f ) ≤ Z X ( − ∆ m f ) dν ∀ f ∈ L ( X, ν ) o , thus A < α + 2 ǫ. Therefore, since ǫ > A ≤ α. (cid:3) Consequently, on account of Theorem 3.17, we can rewrite the Poincar´e inequality via the integratedBakry-´Emery curvature-dimension condition (see [6, Theorem 4.8.4], see also [9, Theorem 2.1]):
Theorem 3.18.
Let [ X, d, m ] be a metric random walk space with invariant-reversible probability measure ν . Assume that ∆ m is ergodic. Then,(1) ∆ m satisfies an integrated Bakry- ´Emery curvature-dimension condition IBE ( K, n ) with K > if, and only if, a Poincar´e inequality with constant K nn − is satisfied.(2) ∆ m satisfies an integrated Bakry- ´Emery curvature-dimension condition IBE ( K, ∞ ) with K > if, and only if, a Poincar´e inequality with constant K is satisfied.Therefore, if ∆ m satisfies the Bakry- ´Emery curvature-dimension condition BE ( K, n ) with K > , wehave gap( − ∆ m ) ≥ K nn − . (3.12) In the case that ∆ m satisfies the Bakry- ´Emery curvature-dimension condition BE ( K, ∞ ) with K > ,we have gap( − ∆ m ) ≥ K. (3.13)In the next example we will see that, in general, the integrated Bakry-´Emery curvature-dimension con-dition IBE ( K, n ) with
K > BE ( K, n )with
K > Example 3.19.
Consider the weighted linear graph G with vertex set V ( G ) = { a, b, c } and where theonly non-zero weights are w a,b = w b,c = 1, and let ∆ := ∆ m G . A simple calculation givesΓ( f )( a ) = 12 ( f ( b ) − f ( a )) = 12 (∆ f ( a )) , Γ( f )( c ) = 12 ( f ( b ) − f ( c )) = 12 (∆ f ( c )) , Γ( f )( b ) = 14 ( f ( b ) − f ( a )) + 14 ( f ( b ) − f ( c )) = 14 (cid:0) (∆ f ( a )) + (∆ f ( c )) (cid:1) = 12 (Γ( f )( a ) + Γ( f )( c )) . Moreover, Γ ( f )( a ) = 18 (∆ f ( c )) + 58 (∆ f ( a )) + 14 ∆ f ( a )∆ f ( c ) (3.14)and Γ ( f )( c ) = 18 (∆ f ( a )) + 58 (∆ f ( c )) + 14 ∆ f ( a )∆ f ( c ) . (3.15) Having in mind (3.14) and (3.15), the BE ( K, n ) conditionΓ ( f ) ≥ n (∆ f ) + K Γ( f ) ∀ f ∈ L ( X, ν )on a or c holds true if, and only if,14 y + 54 x + 12 xy ≥ n x + Kx ∀ x, y ∈ R . (3.16)Now, since (3.16) is true for x = 0, (3.16) holds if, and only if, K ≤ inf x =0 , y y + x + xy − n x x . Moreover, taking y = λx , we obtain that the following inequality must be satisfied K ≤ inf λ (cid:16) λ + 54 + 12 λ − n (cid:17) = 1 − n . In fact, it is easy to see that (3.16) is true for any K ≤ − n .On the other hand, we have thatΓ ( f )( b ) = 12 (∆ f ( b )) + Γ( f )( b ) , and it is easy to see thatΓ ( f )( b ) ≥ n (∆ f ( b )) + K Γ( f )( b ) for all n > , K ≤ − n . Therefore, we have that this graph Laplacian satisfies the Bakry-´Emery curvature-dimension condition BE (cid:18) − n , n (cid:19) for any n > , being K = 1 − n the best constant for a fixed n > − ∆) = 1 thus, by Theorem 3.18, we have that ∆ satisfies the integratedBakry- ´Emery curvature-dimension condition IBE ( K, n ) with K = 1 − n > − n .Note that ∆ satisfies the Bakry-´Emery curvature-dimension condition BE (1 , ∞ ) and hence, in thisexample, the bound in (3.13) is sharp but there is a gap in the bound (3.12). Remark 3.20.
For a metric random walk space [
X, d, m ] with invariant and reversible probability mea-sure ν , Y. Ollivier in [43, Corollary 31], under the assumption that (3.17) Z Z Z d ( y, z ) dm x ( y ) dm x ( z ) dν ( x ) < + ∞ , (3.17)proves that if the Ollivier-Ricci curvature κ m is positive and the space is ergodic, then [ X, d, m, ν ] satisfiesthe Poincar´e inequality κ m Var ν ( f ) ≤ H m ( f ) for all f ∈ L ( X, ν ) , and, consequently, κ m ≤ gap( − ∆ m ) . Observe that, in fact, the ergodicity follows from the positivity of κ m (Theorem 2.12). HE HEAT FLOW ON METRIC RANDOM WALK SPACES 41
Transport Inequalities.
Given a metric random walk space [
X, d, m ] we define, for x ∈ X ,Θ( x ) := 12 (cid:0) W d ( δ x , m x ) (cid:1) = 12 Z X d ( x, y ) dm x ( y ) , and Θ m := ess sup x ∈ X Θ( x ) . Since Θ( x ) ≤ (diam(supp( m x )) , if diam( X ) is finite then we have Θ m ≤ (diam( X )) . Observe that k Γ( f ) k ∞ = sup x ∈ X Z X ( f ( x ) − f ( y )) dm x ( y ) ≤ Θ m k f k Lip . (3.18)Given a metric measure space ( X, d, µ ) as in Example 1.2 (4), if m µ,ǫ is the ǫ -step random walkassociated to µ , that is m µ,ǫx := µ B ( x, ǫ ) µ ( B ( x, ǫ )) for x ∈ X, then Θ m µ,ǫ ≤ ǫ . Let J m ( x ) be the jump of the random walk at x : J m ( x ) := W d ( δ x , m x ) = Z X d ( x, y ) dm x ( y ) . In the particular case of the metric random walk space associated to a locally finite discrete graph[ V ( G ) , d G , m G ], we have J m G ( x ) = 1 d x X y ∼ x, y = x w xy ≤ , thus Θ( x ) = 12 J m G ( x ) = 12 d x X x ∼ y, x = y w xy ≤ . In the case of the metric random walk space [Ω , k . k , m J ] (see Example 2.3 (2)), with J ( x ) = | B r (0) | χ B r (0) ,a simple computation gives Θ( x ) ≤ N N + 2) r . It is well known that in the case of diffusion semigroups the Bakry-´Emery curvature-dimension condi-tion BE ( K, ∞ ) of its generator is characterized by gradient estimates on the semigroup (see for instance[5] or [6]). The same characterization is also true for weighted discrete graphs (see for instance [31]and [26]). With a similar proof we have that in the general context of metric random walk spaces thischaracterization is also true. Theorem 3.21.
Let [ X, d, m ] be a metric random walk space with invariant-reversible probability mea-sure ν and let ( T t ) t> = ( e t ∆ m ) t> be the heat semigroup. Then, ∆ m satisfies the Bakry- ´Emery curvature-dimension condition BE ( K, ∞ ) with K > if, and only if, Γ( T t f ) ≤ e − Kt T t (Γ( f )) ∀ t ≥ , ∀ f ∈ L ( X, ν ) . (3.19) Proof.
Fix t >
0. For s ∈ [0 , t ), we define the function g ( s, x ) := e − Ks T s (Γ( T t − s f ))( x ) , x ∈ X. The same computations as in [31] show that ∂g∂s ( s, x ) = 2 e − Ks T s (Γ ( T t − s f ) − K Γ( T t − s f )) ( x ) . Then, if ∆ m satisfies the Bakry-´Emery curvature-dimension condition BE ( K, ∞ ) with K >
0, we havethat ∂g∂s ( s, x ) ≥ ∂g∂s (0 , x ) ≥ ( T t f ) − K Γ( T t f ) ≥ . Then, letting t →
0, we get Γ ( f ) − K Γ( f ) ≥ . (cid:3) The
Fisher-Donsker-Varadhan information of a probability measure µ on X with respect to ν is definedby I ν ( µ ) := H m ( √ f ) if µ = f ν, f ≥ , + ∞ , otherwise . Observe that D ( I ν ) = { µ : µ = f ν, f ∈ L ( X, ν ) + } since √ f ∈ L ( X, ν ) = D ( H m ) whenever f ∈ L ( X, ν ) + . Here, we use the notation L p ( X, ν ) + := { f ∈ L p ( X, ν ) : f ≥ ν − a.e. } .In the next result we show that the Bakry-´Emery curvature-dimension condition BE ( K, ∞ ) with K >
Theorem 3.22.
Let [ X, d, m ] be a metric random walk space with invariant-reversible probability mea-sure ν , and assume that Θ m is finite. If ∆ m satisfies the Bakry- ´Emery curvature-dimension condition BE ( K, ∞ ) with K > , then ν satisfies the transport-information inequality W d ( µ, ν ) ≤ √ Θ m K p I ν ( µ ) , for all probability measures µ ≪ ν . (3.20) Proof.
Let µ be a probability measure µ ≪ ν , and set µ = f ν . By the Kantorovich-Rubinstein Theoremwe have that W d ( µ, ν ) = sup (cid:26)Z X g ( x )( f ( x ) − dν ( x ) : k g k Lip ≤ g bounded (cid:27) . Let T t = e t ∆ m be the heat semigroup. Given g ∈ L ∞ ( X, ν ) with k g k Lip ≤
1, having in mind Proposition2.22, we have Z X g ( x )( f ( x ) − dν ( x ) = − Z + ∞ ddt Z X ( T t g )( x ) f ( x ) dν ( x ) dt = − Z + ∞ Z X ∆ m ( T t g )( x ) f ( x ) dν ( x ) dt = Z + ∞ E m ( T t g, f ) dt = Z + ∞ Z X Γ( T t g, f )( x ) dν ( x ) dt. Now, using the Cauchy-Schwartz inequality, the reversibility of the measure ν and that( p f ( y ) + p f ( x )) ≤ f ( x ) + f ( y )) , HE HEAT FLOW ON METRIC RANDOM WALK SPACES 43 we have Z X Γ( T t g, f )( x ) dν ( x ) = 12 Z X × X (( T t g )( y ) − ( T t g )( x ))( f ( y ) − f ( x )) dm x ( y ) dν ( x )= 12 Z X × X (( T t g )( y ) − ( T t g )( x ))( p f ( y ) − p f ( x ))( p f ( y ) + p f ( x )) dm x ( y ) dν ( x ) ≤ (cid:18)Z X × X
14 ( p f ( y ) − p f ( x )) dm x ( y ) dν ( x ) (cid:19) × (cid:18)Z X × X (( T t g )( y ) − ( T t g )( x )) ( p f ( y ) + p f ( x )) dm x ( y ) dν ( x ) (cid:19) ≤ (cid:18) Z X Γ( p f )( x ) dν ( x ) (cid:19) (cid:18) Z X (cid:18)Z X (( T t g )( y ) − ( T t g )( x )) dm x ( y ) (cid:19) f ( x ) dν ( x ) (cid:19) . Then, applying Theorem 3.21, we get Z X g ( x )( f ( x ) − dν ( x ) ≤ (cid:18) H m ( p f ) (cid:19) Z + ∞ (cid:18) Z X Γ(( T t g )( x )) f ( x ) dν ( x ) (cid:19) dt ≤ (cid:16) H m ( p f ) (cid:17) Z + ∞ (cid:18) e − Kt Z X T t (cid:0) Γ( g ) (cid:1) ( x ) f ( x ) dν ( x ) (cid:19) dt. Now, by (3.18) and (2.5), we have | T t (cid:0) Γ( g ) (cid:1) ( x ) | ≤ k T t (cid:0) Γ( g ) (cid:1) k ∞ ≤ k Γ( g ) k ∞ ≤ Θ m . Hence Z X g ( x )( f ( x ) − dν ( x ) ≤ (cid:16) H m ( p f ) (cid:17) Z + ∞ (cid:18) e − Kt Θ m Z X f ( x ) dν ( x ) (cid:19) dt ≤ √ Θ m K (cid:16) H m ( p f ) (cid:17) Finally, taking the supremum over all bounden functions g with k g k Lip ≤ (cid:3) Remark 3.23. If ν satisfies a transport-information inequality W d ( µ, ν ) ≤ λ q H m ( p f ) for all µ = f ν , (3.21)then ν is ergodic. In fact, if ν is not ergodic, then by Theorem 2.24 there exists D ⊂ X with 0 < ν ( D ) < m χ D = 0. Now, if µ := ν ( D ) χ D ν , then µ = ν and, therefore, by (3.21), we get H m ( χ D ) > m χ D = 0.As a consequence of the previous Remark and Theorem 3.22, we have that the positivity of the Bakry-´Emery curvature-dimension condition implies ergodicity of ∆ m , then, by Theorem 3.18, we have thefollowing result. Theorem 3.24.
Let [ X, d, m ] be a metric random walk space with invariant-reversible probability mea-sure ν , and assume that Θ m is finite. Then: If ∆ m satisfies the Bakry- ´Emery curvature-dimension condition BE ( K, n ) with K > , we have gap( − ∆ m ) ≥ K nn − . In the case that ∆ m satisfies the Bakry- ´Emery curvature-dimension condition BE ( K, ∞ ) with K > ,we have gap( − ∆ m ) ≥ K. The relative entropy of 0 ≤ µ ∈ M ( X ) with respect to ν is defined byEnt ν ( µ ) := Z X f log f dν − ν ( f ) log (cid:0) ν ( f ) (cid:1) if µ = f ν, f ≥ , f log f ∈ L ( X, ν ) , + ∞ , otherwise,with the usual convention that f ( x ) log f ( x ) = 0 if f ( x ) = 0.The next result shows that a transport-information inequality implies a transport-entropy inequalityand, therefore, normal concentration (see for example [13, 32]). Theorem 3.25.
Let [ X, d, m ] be a metric random walk space with invariant-reversible probability mea-sure ν and assume that Θ m is finite and that there exists some x ∈ X such that R d ( x, x ) dν ( x ) < ∞ .Then the transport-information inequality W d ( µ, ν ) ≤ K p I ν ( µ ) , for all probability measures µ ≪ ν , (3.22) implies the transport-entropy inequality W d ( µ, ν ) ≤ s √ m K Ent ν ( µ ) , for all probability measures µ ≪ ν . (3.23) Proof.
By [13, Theorem 1.3], inequality (3.23) is equivalent to Z X e λf ( x ) dν ( x ) ≤ e λ √ Θ m √ K , (3.24)for every bounded function f on X with k f k Lip ≤ ν ( f ) = 0, and all λ ∈ R .Given f ∈ L ∞ ( X, ν ) with k f k Lip ≤ ν ( f ) = 0, we define the functionΛ( λ ) := Z X e λf ( x ) dν ( x ) , and the probabilities µ λ := 1Λ( λ ) e λf dν. By the Kantorovich-Rubinstein Theorem and the assumption (3.22), we have ddλ log(Λ( λ )) = 1Λ( λ ) Z X f ( x ) e λf ( x ) dν ( x ) = Z X f ( x )( dµ λ ( x ) − dν ( x )) ≤ W d ( µ λ , ν ) ≤≤ K vuut H m s λ ) e λf ! = √ K vuutZ X Γ s λ ) e λf ! ( x ) dν ( x ) HE HEAT FLOW ON METRIC RANDOM WALK SPACES 45 = √ K sZ X λ ) Γ (cid:16) e λf (cid:17) ( x ) dν ( x ) . Now, since 1 − a ≤ log a for a ≥
1, having in mind the reversibility of ν , we have Z X Γ( g )( x ) dν ( x ) ≤ Z X g ( x )Γ(log g )( x ) dν ( x ) , and, consequently, by (3.18), we get ddt log(Λ( λ )) ≤ √ K sZ X λ ) e λf ( x ) Γ (cid:18) λf (cid:19) ( x ) dν ( x ) = λ √ K sZ X λ ) e λf ( x ) Γ ( f ) ( x ) dν ( x )= λ √ K sZ X Γ ( f ) ( x ) dµ λ ( x ) ≤ √ Θ m √ K λ.
Then, integrating we get (3.24). (cid:3)
In the next example we see that, in general, a transport-entropy inequality does not imply transport-information inequality.
Example 3.26.
Let Ω = [ − , ∪ [2 ,
3] and consider the metric random walk space [Ω , d, m J, Ω ], with d the Euclidean distance in R and J ( x ) = χ [ − , (see Example (1.2) (5)). An invariant and reversibleprobability measure for m J, Ω is ν := L Ω. By the Gaussian integrability criterion [20, Theorem 2.3] ν satisfies a transport-entropy inequality. However, ν does not satisfy a transport-information inequality,since if ν satisfies a transport-information inequality, then ν must be ergodic (see Remark 3.23). Now itis easy to see that [Ω , d, m J, Ω ] is not m -connected and then by Theorem 2.19, ν is not ergodic.By Theorems 2.12 and 2.19, we have that the metric random walk space [Ω , d, m J, Ω ] of the aboveexample has non-positive Ollivier-Ricci curvature. In the next theorem we will see that, under positiveOllivier-Ricci curvature, a transport-information inequality holds. First we need the following result. Lemma 3.27.
Let [ X, d, m ] be a metric random walk space with invariant-reversible probability measure ν .Then, if f ∈ L ( X, ν ) with k f k Lip ≤ , we have k e t ∆ m f k Lip ≤ e − tκ m .Proof. By [43, Proposition 25], we have that κ m ∗ ( n + l ) ≥ κ m ∗ n + κ m ∗ l − κ m ∗ n κ m ∗ l ∀ n, l ∈ N . where κ m ∗ = κ m . Hence, 1 − κ m ∗ n ≤ (1 − κ m ) n ∀ n ∈ N . (3.25)By Theorem 2.4 and (3.25), we have | e t ∆ m f ( x ) − e t ∆ m f ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − t + ∞ X n =0 Z X f ( z )( dm ∗ nx ( z ) − dm ∗ ny ( z )) t n n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e − t + ∞ X n =0 W d ( m ∗ nx , m ∗ ny ) t n n ! ≤ e − t + ∞ X n =0 (1 − κ m ∗ n ) d ( x, y ) t n n ! ≤ e − t + ∞ X n =0 (1 − κ m ) n t n n ! d ( x, y )= e − t e t (1 − κ m ) d ( x, y ) = e − tκ m d ( x, y ) , from where it follows that k e t ∆ m f k Lip ≤ e − tκ m . (cid:3) Theorem 3.28.
Let [ X, d, m ] be a metric random walk space with invariant-reversible probability mea-sure ν , and assume that Θ m is finite. If κ m > then the following transport-information inequalityholds: W d ( µ, ν ) ≤ √ m κ m p I ν ( µ ) , for all probability measures µ ≪ ν .Proof. Let T t = e t ∆ m be the heat semigroup and µ = f ν be a probability measure in X . We use, as inthe proof of Theorem 3.22, the Kantorovich-Rubinstein Theorem. Let g ∈ L ∞ ( X, ν ) with k g k Lip ≤ Z X g ( x )( f ( x ) − dν ( x ) = − Z + ∞ ddt Z X ( T t g )( x ) f ( x ) dν ( x ) dt = − Z + ∞ Z X ∆ m ( T t g )( x ) f ( x ) dν ( x ) dt = Z + ∞ Z X × X (( T t g )( y ) − ( T t g )( x ))( f ( y ) − f ( x )) dm x ( y ) dν ( x ) dt ≤ Z + ∞ k T t g k Lip Z X × X d ( x, y ) | f ( y ) − f ( x ) | dm x ( y ) dν ( x ) dt ≤ Z + ∞ e − tκ m Z X × X d ( x, y ) | f ( y ) − f ( x ) | dm x ( y ) dν ( x ) dt = 12 κ m Z X × X d ( x, y ) | f ( y ) − f ( x ) | dm x ( y ) dν ( x )= 12 κ m Z X × X d ( x, y ) | p f ( y ) − p f ( x ) | (cid:16)p f ( y ) + p f ( x ) (cid:17) dm x ( y ) dν ( x ) ≤ √ κ m q H m ( p f ) sZ X × X d ( x, y ) (cid:16)p f ( y ) + p f ( x ) (cid:17) dm x ( y ) dν ( x ) . Now, using reversibility of ν , Z X × X d ( x, y ) (cid:16)p f ( y ) + p f ( x ) (cid:17) dm x ( y ) dν ( x )= Z X × X d ( x, y ) (cid:18) f ( x ) + 2 f ( y ) − (cid:16)p f ( y ) − p f ( x ) (cid:17) (cid:19) dm x ( y ) dν ( x ) ≤ Z X × X d ( x, y ) ( f ( x ) + f ( y )) dm x ( y ) dν ( x ) ≤ m . Therefore, we get Z X g ( x )( f ( x ) − dν ( x ) ≤ √ Θ m κ m q H m ( p f ) , so, taking the supremum over the functions g , W d ( µ, ν ) ≤ √ m κ m q H m ( p f ) = √ m κ m p I ν ( µ ) . (cid:3) Acknowledgment.
The authors have been partially supported by the Spanish MCIU and FEDER,project PGC2018-094775-B-I00. The second author was also supported by Ministerio de Econom´ıa yCompetitividad under Grant BES-2016-079019.
HE HEAT FLOW ON METRIC RANDOM WALK SPACES 47
References [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. OxfordMathematical Monographs, 2000.[2] N. Alon and V.D. Milman, λ , Isoperimetric inequalities for graphs, and superconcentrators . J. Combin. Theory Ser.B , (1985) 73–88.[3] L. Ambrosio, N. Gigli and G. Savar´e, Bakry- ´Emery curvature-dimension condition and Riemannian Ricci curvaturebounds.
Ann. Probab. (2015), 339-404.[4] F. Andreu-Vaillo, J. M. Maz´on, J. D. Rossi and J. Toledo, Nonlocal Diffusion Problems. Mathematical Surveys andMonographs, vol. 165. AMS, 2010.[5] D. Bakry, Functional inequalities for Markov semigroups.
In: Probability Measures on Groups: Recent Directions andTrends, pp. 91–147. Tata Institute of Fundamental Research, Mumbai (2006)[6] D. Bakry, I. Gentil and M. Ledoux,
Analysis and Geometry of Markov Diffusion Operators . Grundlehren der Mathe-matischen Wissenschafter,
Springer, 2014.[7] D. Bakry and M. ´Emery,
Diffusions hypercontractives . (French) [Hypercontractive diffusions] S´eminaire de probabilit´es,XIX, 1983/84, 177-206, Lecture Notes in Math., 1123, Springer, Berlin, 1985.[8] B. B. Bhattacharya and S. Mukherjee,
Exact and asymptotic results on coarse Ricci curvature of graphs . DiscreteMathematics (2015), 23–42.[9] F. Bauer, F. Chung, Y. Lin and Y. Liu,
Curvature aspect of Graphs . Proc. Amer. Math. Soc. (2017), 2033–2042.[10] F. Bauer and J. Jost,
Bipartite and neighborhood graps and the spectrum of the normalized graph laplace operator .Comm. in Analysis and geometry (2013), 787–845.[11] F. Bauer, J. Jost and S. Liu, Ollivier-Ricci Curvature and the spectrum of the normalized graph Laplace operator .Math. Res. Lett. (2012), 1185–1205.[12] Ph. B´enilan and M. G. Crandall, Completely Accretive Operators . In Semigroups Theory and Evolution Equations(Delft, 1989), Ph. Clement et al. editors, volume 135 of
Lecture Notes in Pure and Appl. Math. , Marcel Dekker, NewYork, 1991, pp. 41–75.[13] S.G. Bobkov and F. G¨otze,
Exponential Integrability and Transportation Cost Related to Logarithmic Inequalities .Journal of Functional Analysis (1999), 1–28.[14] S.G. Bobkov and P. Tetali,
Modified Logarithmic Sobolev Inequalities in Discrete Setting . Journal of TheoreticalProbability (2006), 289–336.[15] H. Brezis, Operateurs Maximaux Monotones. North Holland, Amsterdam, 1973.[16] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential equations. Universitext, Springer, 2011.[17] J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian,
Problems in Analysis, (R. C. Gunning, ed.)Princeton Univ. Press (1970) 195–199.[18] H. J. Cho and S-H. Paeng,
Olivier’s Ricci curvature and coloring . European Journal of Combinatorics (2013),916–922.[19] F. Chung, Spectral Graph Theory (CBMS Regional Conference Series in Mathematics, No. 92), American MathematicalSociety, 1997.[20] H. Djellout, A. Guillin, and L.Wu,
Transportation cost-information inequalities for random dynamical systems anddiffusions.
Annals of Probability, (2004), 2702–2732.[21] J. Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks . Trans. Amer. Math.Soc. (1984), 787–794.[22] J. Dodziuk and L. Karp,
Spectral and Function Theory for Combinatorial Laplacian.
Comptemp. Math. vol. 73.American Mathematical Society, Providence, RI, 2001.[23] M. Erbas and J. Maas,
Ricci Curvature of Finite Markov Chains via Convexity of the Entropy . Arch. Rat. Mech. Anal. (2012), 997–1038.[24] M. Fathi and Y. Shu,
Curvature and transport inequalities for Markov chains in discrete spaces . Bernouilli (2018),672–698.[25] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet forms and symetric Markov processes , Studies in Mathematics , De Gruyter, 2011.[26] Ch. Gong and Y. Lin, Equivalent properties for CD inequalities on graphs with unbounded Laplacians.
Chin. Ann.Math. Ser. B (2017), 1059–1070.[27] N. Gozlan and C. L´eonard, Transport inequalities. A survey. Markov Process.
Related Fields (2010), 635–736.[28] N. Gozlan, C. Roberto, P-M. Samson and P. Tetali, Displacement convexity of entropy and related inequalities ongraphs . Probab. Theory and Related Fields (2014), 47–94.[29] O. Hern´andez-Lerma and J. B. Laserre,
Markov Chains and Invariant Probabilities . Birkh¨auser Verlag, Basel, 2003. [30] J. Jost and S. Liu,
Ollivier’s Ricci Curvature, Local Clustering and Curvature-Dimension Inequalities on Graphs .Discrete Compt. Geom. (2014), 300–322.[31] B. Klartag, G. Kozma, P. Ralli and P. Tetali, Discrete Curvature and Abelian Groups . Canad. J. Math. (2016),655–674.[32] M. Ledoux, The Concentration of Measures Phenomenon . Math Surveys and Monographs. Vol 89, Amer. Math.Soc,2001.[33] Y. Lin and and S-T. Yau,
Ricci Curvature and eigenvalue estimates on locally finite graphs . Math. Res. Lett. (2010),343–356.[34] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport.
Ann. of Math. (2009),903–991.[35] K. Marton,
A simple proof of the blowing-up lemma.
IEEE Trans. Inform. Theory, (1986), 445–446.[36] K. Marton, A measure concentration inequality for contracting Markov chains.
Geom. Funct. Anal., (1996), 556–571.[37] J. Maas, Gradient flows of the entropy for finite Markov chains.
J.Funct.Anal. (2011), 2250–2292.[38] J. M. Maz´on, J. D. Rossi and J. Toledo,
The Heat Content for Nonlocal Diffusion with Non-singular Kernels.
Adv.Nonlinear Stud. (2017), 255–268.[39] J. M. Maz´on, M. Solera and J. Toledo, The Total Variation Flow in Metric Random Walk Spaces . arXiv:1905.01130[40] J. M. Maz´on, M. Solera and J. Toledo,
Gradient Flows and Ricci Curvature on Metric Random Walk Spaces . Forth-coming paper.[41] L. Miclo,
On hyperboundedness and spectrum of Markov operators . Invent. math. (2015), 311–343.[42] L. Najman and P. Romon (ed.),
Modern Approaches to Discrete Curvature . Lecture Notes in Mathematics 2184,Springer, 2017.[43] Y. Ollivier,
Ricci curvature of Markov chains on metric spaces.
J. Funct. Anal. (2009), 810–864.[44] Y. Ollivier,
A survey of Ricci curvature for metric spaces and Markov chains.
Probabilistic approach to geometry,343–381, Adv. Stud. Pure Math., , Math. Soc. Japan, Tokyo, 2010.[45] Y. Ollivier and C. Villani, A curved Brunn-Minkowski inequality on the discrete hypercube, or: what is the Riccicurvature of the discrete hypercube?
SIAM J. Discrete Math. (2012), 983–996.[46] S-H. Paeng, Volume and diametrer of a grap anf Olivier’s Ricci curvature . European Journal of Combinatorics (2012), 1808–1819.[47] G. Polya and S. Szego, Isoperimetric Inequalities in Mathematical Physics,
Annals of Math. Studies, no. 27, PrincetonUniversity Press, (1951).[48] M. Reed and B. Simon. Functional Analalysis I. Academic Press 1980.[49] M.-K von Renesse. K-L. Sturm,
Transport Inequalities, Gradient Estimates, Entropy, and Ricci Curvature , Commu-nications on Pure and Applied Mathematics, Vol. LVIII, (2005), 0923–0940.[50] M. Schmuckenschlager,
Curvature of Nonlocal Markov Generators . In Convex Geometric Analysis (Berkeley, CA, 1996),Cambrigge Univ. Press. 1998, pp. 189–197.[51] K-L. Sturm,
On the geometry of metric measure spaces. I, II.
Acta Math. (2006), 65–131 and 133–177.[52] A. Szlam and X. Bresson,
Total Variation and Cheeger Cuts . Proceedings of the 27 th International Confer- ence onMachine Learning, Haifa, Israel, 2010.[53] M. Talagrand,
Transportation cost for Gaussian and other product measures . Geom. Funct. Anal., (1996), 587–600.[54] C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics. Vol. 58, 2003.[55] C. Villani. Optimal transport. old and new, Grundlehren der Mathematischen Wissenschaften, vol. 338, Springer-Verlag,Berlin, 2009. J. M. Maz´on: Departamento de An´alisis Matem´atico, Univ. Valencia, Dr. Moliner 50, 46100 Burjassot,Spain. [email protected]
M. Solera: Departamento de An´alisis Matem´atico, Univ. Valencia, Dr. Moliner 50, 46100 Burjassot, Spain. [email protected]