aa r X i v : . [ m a t h . D S ] F e b AUBRY–MATHER THEORY ON GRAPHS
ANTONIO SICONOLFI AND ALFONSO SORRENTINO
Abstract.
We formulate Aubry–Mather theory for Hamiltonians/Lagrangiansdefined on graphs and discuss its relationship with weak KAM theory developedin [24]. Introduction
The Hamilton Jacobi equation and its wide spectrum of applications represent animportant crossroads for ideas and techniques coming from different areas of math-ematics: PDEs, calculus of variations, control theory, optimal transport, symplecticgeometry, etc... Over recent years, this beneficial synergy has resulted in the devel-opment of novel lines of research and significant scientific advances.In 1990’s this multifaceted interaction has experience a particular boost thanksto a remarkable intuition by Albert Fathi and the ensuing development of what isnowadays called weak KAM theory [10, 11]. This novel point of view shed light onthe noteworthy connection between the very successful action-minimizing methodsin Hamiltonian dynamics – in particular
Aubry-Mather theory , originated from thework by Serge Aubry [2] and John Mather [17, 18] –, and the analysis of viscositysolutions and sub-solutions to the Hamilton-Jacobi equation.Not only these ideas were particularly beneficial for enhancing our understandingof both the dynamics of these systems and the global properties of the solutions,but they also contributed to draw unexpected connections with other problems indynamics, geometry and analysis (see for instance [11, 23, 25]).These successes encouraged a very active investigation on the possibility of ex-tending these theories beyond the classical settings, either to other classes of systemsor to different ambient spaces, more suitable for some applications.In this article, it is pursued the latter direction.The paper presents the first, as far as we know, systematic detailed account ofAubry–Mather theory for Hamiltonians/Lagrangians defined on graphs , recoveringthe whole theory in this new context and relating it to weak KAM analysis carriedout in [24].
Motivations and Significance.
Over the last years there has been an increasinginterest in the study of the Hamilton-Jacobi equation on graphs and networks andrelated questions. These problems, in fact, besides having a great impact in theapplications in various fields (for example to data transmission, traffic managementproblems, etc...), they involve a number of subtle theoretical issues related to theintertwining between the local analysis of the problem and the global structure ofthe network/graph.Several reasons can be advanced for embarking on the job of settling Aubry-Mather theory in this setting.On the one hand, the need of following up the line of investigation initiated in[24], where we provided a thorough discussion of KAM theory on graphs/networks(see also [21]). Weak KAM and Aubry-Mather theories, in fact, are in a dualityrelationship: they are both intrinsically based on the study of objects that arise asminimizers of some action functionals , possibly with constraints (see for instance[11, 23, 25] for more details).On the other hand, the passage from manifolds to graphs requires a specific adap-tation of the main tools and techniques involved ( e.g. , parametrized paths, spacesof probability measures, occupation and closed probability measures, etc...) whichis by no means straightforward and, we believe, would be of potential interest formany other problems and applications.As a matter of fact, one of the initial motivation for this work, was to prove ahomogenization result for the Hamilton–Jacobi equations on networks, following thehomological approach introduced in [9]. However, in order to pursue the project,it became crucial to first develop an Aubry-Mather theory in this context, both fordetermining the limit problem and the space on which it is defined. For instance,the effective Hamiltonian appearing, as in the compact manifold case, in the limitequation is nothing else that the minimal average action (or
Mather’s α function , seesubsection 5.2), namely the value function related to one of the variational problemat the core of Aubry Mather theory.Note that, even if the approximated equations in the homogenization problem areposed on a network, the natural setting where the approximation procedure shouldtake place is the corresponding abstract graph. This is one of the reasons why thepresent contribution is focused on this issue. UBRY–MATHER 3
To avoid misunderstanding, we make it clear that the topic we are talking about– that has not been treated in the literature yet –, is different from other interestingmodels of partial homogenization on junctures considered in [12, 13], and mainlydevoted to applications to traffic theory.We remark that a central role in our construction is played by probability mea-sures, defined on a sort of tangent bundle of the graph: they constitute the relaxedframework for the variational problems under consideration. There is a broad inter-est in the recent literature on probability measures supported on graphs/networks,see for instance [7, 19]. One of the goal being, for instance, to extend mean fieldgames models to graphs (see [1, 6, 14, 15]). Passing to a related field, connectionsbetween Aubry–Mather theory and optimal transport have been pointed out by var-ious author, see [4, 5, 3].The outputs of the present paper can be seen as a first step to explore these direc-tions of research in the graph setting.
Main contributions.
In this paper we prove that Aubry–Mather theory can becompleted established in the graphs context. In particular, we discuss: the role ofoccupation and closed probability measures, existence of action minimizing mea-sures, properties of the corresponding value functions (Mather’s α and β functions),properties of Mather measures, structure of Mather sets, Mather graph property,etc....We recover the duality links with weak KAM theory as well. Weak KAM theory,as formulated in [24], is essentially a metric theory based on the notion of intrinsiclength of paths, which in turn depends on a given level of the Hamiltonian; we donot even need a Lagrangian function to be defined. The sign of the intrinsic lengthof cycles is related to the existence of subsolutions to the corresponding Hamilton-Jacobi equation. The existence of cycles with vanishing intrinsic length is attainedat the critical value of the Hamiltonian, which is the unique value for which onefinds solutions to the Hamilton–Jacobi equations. The edges forming cycles withvanishing length make up the so–called Aubry set .Aubry-Mather theory is instead a variational theory, inspired by the principle ofleast action [18, 23, 25], whose aim is to find, in suitable spaces, minimizers of the
Lagrangian action functional (possible with constraints). Our starting point is tointroduce the notion of parametrization of a path (subsection 4.1), which is in du-ality with that of intrinsic length. It is obtained by equipping every edge of a pathwith a non-negative weight, that can be interpreted as an average speed. This allowsus to introduce an action functional on the set of paths, setting up the variational
ANTONIO SICONOLFI AND ALFONSO SORRENTINO problem of interest.The usual relaxation procedure yields to pose the problem in a suitable spaceof measures, where all the minimizers can be found. To this aim we define, on anappropriate tangent bundle of the graph, the notion of closed occupation measures,which somehow correspond to parametrized cycles, and prove that they are dense,with respect to the first Wasserstein metric, in the space of all closed probabilitymeasures (Appendix B), which we believe has its own interest beyond the problemat hand. Significantly, the measures minimizing the action are supported by cycleswith vanishing intrinsic length and the minimum of the action is equal, up to a sign,to the critical value (see Theorem 8.1).1.1.
Organization of the article.
We describe hereafter how the article is orga-nized.In
Section 2 we provide a brief introduction to graph theory, in order to set theterminology and introduce the main concepts that will be needed. In particular,we define the algebraic topological notions of chains, cochains, homology and coho-mology of the graph, that are crucial importance for the full implementation of thevariational analysis.In
Section 3 we give the notion of Hamiltonian on a graph and introduce the as-sociated Lagrangian which allows us to define the action functional to be minimizedunder appropriate constraints.In
Section 4 we define the relaxed setting on which the variational analysis willoccur. Then, we introduce the notion of occupation measures and closed probabil-ity measures , and the relevant Wasserstein topology. These are central objects inAubry-Mather theory that represent useful relaxations of the notion of paths andclosed paths.
Sections 5 & 6 are the core of the development of Aubry-Mather theory in thecontext of graphs. We set, in analogy to the classical setting, a family of variationalproblems, show that they admit global minimizers and discuss their significance andtheir structural properties. Interestingly, we prove in this context the analogue ofthe celebrated Mather’s graph theorem (Proposition 6.2 and Corollary 6.4).After having recalled in
Section 7 the basic results of weak KAM theory from[24], in
Section 8 we discuss the relation between Aubry Mather theory and weakKAM theory on graphs. We show the equality between the (projected) Mather sets
UBRY–MATHER 5 and the corresponding Aubry sets, and in Theorem 8.4 we use viscosity solutionsand subsolutions to provide a more explicit description of Mather’s graph theorem.Finally, in
Appendix A we describe how to develop an Aubry–Mather theoryon networks, and look from the point of view of networks to some notions we haveintroduced on graphs. In
Appendix B we provide the proof of the density resultof closed occupation measures.
Acknowledgments.
The second author acknowledges the support of the Univer-sity of Rome Tor Vergata’s
Beyond Borders grant “
The Hamilton-Jacobi Equation:at the crossroads of Analysis, Dynamics and Geometry ” (CUP: E84I19002220005)and the Italian Ministry of Education and Research (MIUR)’s grants: PRIN Project“
Regular and stochastic behavior in dynamical systems ” (CUP: 2017S35EHN) andthe Department of Excellence grant 2018-2022 awarded to the Department of Math-ematics of University of Rome Tor Vergata (CUP: E83C18000100006).Finally, both authors with to express their gratitude to the Mathematical SciencesResearch Institute in Berkeley (USA) for its kind hospitality in Fall 2018 duringthe trimester program “
Hamiltonian systems, from topology to applications throughanalysis ”, where part of this project was carried out.2.
Prerequisites on graphs
Definition and terminology.
A graph Γ = ( V , E ) is an ordered pair ofdisjoint sets V and E , which are called, respectively, vertices and (directed) edges ,plus two functions: o : E −→ V which associates to each edge its origin (initial vertex), and − : E −→ E e e, which changes direction and is a fixed point free involution, namely − e = e and − ( − e ) = e for any e ∈ E .We define the terminal vertex of e ast( e ) := o( − e ) . We further denote by | V | , | E | , the number of vertices and edges, respectively. Forany vertex x ∈ V , we denote by E x := { e ∈ E : o( e ) = x } ANTONIO SICONOLFI AND ALFONSO SORRENTINO the set of edges originating from x ; this is sometimes called the star centered at x .An orientation of Γ is a subset E + of the edges satisfying − E + ∩ E + = ∅ and − E + ∪ E + = E . In other words, an orientation of Γ consists of a choice of exactly one edge in eachpair { e, − e } .We define a path ξ := ( e , · · · , e M ) = ( e i ) Mi =1 as a finite sequence of concatenatededges in E , namely t( e j ) = o( e j +1 ) for any j = 1 , · · · , M − length of a path as the number of its edges. We set o( ξ ) := o( e ),t( ξ ) := t( e M ). We call a path closed , or a cycle , if o( ξ ) = t( ξ ).Throughout the paper, we assume Γ to be (G1) finite , namely with | E | , | V | finite; (G2) connected , in the sense that any two vertices are linked by some path; (G3) without loops , namely for any e ∈ E o( e ) = t( e ).The first two assumptions are structural, while the last one could be removed atthe price of introducing further details in the development of the theory. For thesake of clarity of this presentation, we prefer to avoid it in the present paper.It follows from the connectedness assumption, that the functions o and t are sur-jective.We call simple a path without repetition of vertices, except possibly the initialand terminal vertex, in other terms ξ = ( e i ) Mi =1 is simple ift( e i ) = t( e j ) ⇒ i = j. Clearly, there are finitely many simple paths in a finite graph. We call circuit asimple closed path. Given any edge e , we call equilibrium circuit (based on e ) thepath ( e, − e ).2.2. Homology of a graph.
Throughout the paper we will take homology andcohomology with coefficients in R . We refer to [26, Ch. 4] for a more detailed andgeneral presentation.We define the 0 –chain group as the free Abelian group on the vertices with coeffi-cients in R . We denote it by C (Γ , R ). We have C (Γ , R ) ∼ R | V | . UBRY–MATHER 7
We do the same operation with edges, making the reversed edge − e coincide withthe opposite of e with respect to the group operation, and we obtain the 1 –chaingroup , denoted by C (Γ , R ). A basis is given by any orientation E + , in the sense theany element of the 1–chain group can be uniquely expressed as a linear combinationof elements in E + with real coefficients. We consequently have C (Γ , R ) ∼ R | E | / . We define the boundary operator ∂ : C (Γ , R ) → C (Γ , R ) by setting for any edge ∂e := t( e ) − o( e )and then extending it linearly; clearly, ∂ ( − e ) = − ∂e .The ( first ) Homology group of Γ with coefficients in R is defined by H (Γ , R ) := K er ∂, Some remarks:– H (Γ , R ) is a subgroup of C (Γ , R ).– H (Γ , R ) is a free Abelian group of finite rank. The ( first ) Betti number is defined to be the rank of H (Γ , R ), it is an indicator of the topologicalcomplexity of the network.– An element of H (Γ , R ) is called a 1 –cycle . In particular a 1–chain P e ∈ E + a e e is a 1–cycle if and only if(1) X e ∈ E + , t( e )= x a e = X e ∈ E + , o( e )= x a e for any x ∈ V ;This can be considered as an analogue of Kirchhoff law for electric circuits.Due to (1), we can associate to any closed path ξ = ( e i ) Mi =1 in Γ an element of H (Γ , R ) via(2) [ ξ ] := M X i =1 e i . We call [ ξ ] the homology class of ξ . The converse is also true: every element of H (Γ , Z ) can be represented by a closed path (see [26, pp. 40–41]). ANTONIO SICONOLFI AND ALFONSO SORRENTINO
Cohomology of a graph.
Let us introduce the dual entities of chains. The0 –cochain group , denoted by C (Γ , R ), is the space of functions from V to R , andthe 1 –cochain group , denoted by C (Γ , R ), is the space of functions η : E −→ R ,satisfying the compatibility condition η ( − e ) = − η ( e ) for any e ∈ E .The algebraic structure of additive Abelian group is induced by the one in ( R , +).We introduce the differential or coboundary operator d : C (Γ , R ) −→ C (Γ , R )which is defined in the following way: for every g ∈ C (Γ , R ), the 1–cochain dg isgiven via dg ( e ) := g (t( e )) − g (o( e )) for all e ∈ E ;it clearly satisfies the compatibility condition dg ( − e ) = − dg ( e ).It is easy to check that d is a group homomorphism. Hence, the ( first ) Cohomologygroup of Γ with coefficients in R can be defined as the quotient group H (Γ , R ) := C (Γ , R ) / Im d. One can show that there exists a canonical isomorphism H (Γ , R ) ≃ Hom ( H (Γ , R ) , R ) . Pairings between chains and cochains, homology and cohomology.
Let us introduce a pairing between 0–chains and 0–cochains: h· , ·i : C (Γ , R ) × C (Γ , R ) −→ R g, X x ∈ V α x x ! X x ∈ V α x g ( x ) . Similarly, we can define the pairing between 1–chains and 1–cochains (we adoptthe same notation): h· , ·i : C (Γ , R ) × C (Γ , R ) −→ R η, X e ∈ E α e e ! X e ∈ E α e η ( e ) . UBRY–MATHER 9
The above pairings allow us to relate differential and boundary operators. Let g ∈ C (Γ , R ) and ζ = P e ∈ E α e e ∈ C (Γ , R ); then we have: h dg, ζ i = X e ∈ E α e dg ( e ) = X e ∈ E α e (cid:0) g (t( e )) − g (o( e )) (cid:1) = X e ∈ E α e h g, ∂e i = h g, X e ∈ E α e e i = h g, ∂ζ i . (3)In particular, this means that whenever ζ ∈ C (Γ , R ) is such that ∂ζ = 0, then h dg, ζ i = 0 for all g ∈ C (Γ , R ). Hence, the above pairing descends to a well-definedpairing between first homology and first cohomology groups, that we continue todenote h· , ·i : H (Γ , R ) × H (Γ , R ) −→ R . Hamiltonians and Lagrangians on graphs
Definitions and assumptions.
We call a
Hamiltonian on the graph Γ =( V , E ) a family of functions H ( e, · ) : R → R labeled by the edges, such that(4) H ( e, p ) = H ( − e, − p ) for any e ∈ E , p ∈ R .We further require that, for any e ∈ E , H ( e, · ) is (H1) strictly convex and differentiable ; (H2) superlinear at ±∞ , namelylim p →±∞ H ( e, p ) | p | = + ∞ . This implies that there exists, for any e , a unique p e = − p − e global minimizer ofboth H ( e, · ) in R . We consider in what follows H ( e, · ) mostly restricted to [ p e , + ∞ ),(resp. H ( − e, · ) restricted to [ p − e , + ∞ )), which is strictly increasing in this domainof definition. We set(5) a e = H ( e, p e ) = H ( − e, p − e ) = a − e We define σ ( e, · ) as the inverse function of H ( e, · ) in [ p e , + ∞ ). We have σ ( e, · ) : [ a e , + ∞ ) → [ p e , + ∞ ) for any e ∈ E and(6) σ ( e, a e ) = − σ ( − e, a e ) = p e = − p − e for any e. The properties summarized in the next statement are immediate.
Lemma 3.1.
Let e ∈ E . The function a σ ( e, a ) from [ a e , + ∞ ) to R is continuous,differentiable in ( a e , + ∞ ) , and strictly increasing for any e . In addition, it is strictlyconcave and satisfies lim a → + ∞ σ ( e, a ) a = 0 . We define the
Lagrangian L ( e, · ) : R → R as the convex conjugate of H ( e, · ),namely L ( e, q ) := max p ∈ R (cid:0) p q − H ( e, p ) (cid:1) . Proposition 3.2.
Let e ∈ E . The function q
7→ L ( e, q ) is strictly convex andsuperlinear as q goes to ±∞ . In addition (7) L ( e, q ) = L ( − e, − q ) for any q ∈ R . This is a consequence of (H1)–(H2) and (4) (see, for instance, [22, Theorem 26.6]).In what follows, we mostly consider L ( e, · ) restricted to [0 , + ∞ ). We have L ( e, q ) = max p ≥ σ ( e,a e ) (cid:0) p q − H ( e, p ) (cid:1) for q ≥ L ( e, q ) = max a ≥ a e (cid:0) q σ ( e, a ) − a (cid:1) for q ≥ L ( e,
0) = − a e .Given ω ∈ C (Γ , R ), we further consider the ω – modified Hamiltonian H ω ( e, p ) := H ( e, p + h ω, e i ) , which clearly still satisfies assumptions (H1) , (H2) . It is therefore invertible on theright of its minimizer and the inverse is(9) σ ω ( e, a ) := σ ( e, a ) − h ω, e i . The corresponding ω –modified Lagrangian is given by L ω ( e, q ) := L ( e, q ) − h ω, qe i . Remark 3.3.
Note that a e does not depend on ω , i.e. , it is the same for H ω ( e, · ).In fact by (6) a e is characterized by the relation σ ( e, a e ) + σ ( − e, a e ) = 0 UBRY–MATHER 11 and by (9) σ ( e, a e ) + σ ( − e, a e ) = σ ω ( e, a e ) + σ ω ( − e, a e ) for any 1–cochain ω .4. Probability measures on edges
Preamble: parametrized paths.
The notion of parametrized path is centralin the paper and it will be essential to define occupation measures.Intuitively speaking, a parametrized path is a path where it is assigned to anyedge a non-negative average speed and a time needed to go through it. The time isthe inverse of the speed, if the latter is positive, while it can be any possible positivenumber if the speed is zero. We motivate this choice in Section A.2 in the casewhere Γ is the abstract graph associated to a network.
Definition 4.1.
We say that ξ = ( e i , q i , T i ) Mi =1 is a parametrized path if (i) ( e i ) Mi =1 is a family of concatenated edges which is called the support of ξ ; (ii) the q i are non-negative numbers and T i = (cid:26) q i if q i >
0a positive constant if q i = 0;we denote by T ξ := P i T i the total time of the parametrization of ξ ; (iii) if all the q ′ i s vanish then o( ξ ) = t( ξ ); (iv) if q i = 0 and e i +1 = − e i then q i +1 = 0; (v) if q i = 0, i >
1, theno( e i ) = t( e j ) with j = max { k < i, q k = 0 } .We call a parametrized cycle , a parametrized path supported on a closed path (orcycle). We call a parametrized circuit , a parametrized path supported on a circuit. Remark 4.2.
Intuitively, a parametrized path can be thought as a concatenation oftriples with non-zero average velocity, and pairs of triples ( i.e. , equilibrium circuits )of the form { ( e, , T ) , ( − e, , S ) } for some e ∈ E and T, S >
0. In particular, condi-tion (iv) reads that there cannot be consecutive equilibrium circuits correspondingto different edges.Equilibrium circuits represent steady states, interpreted as floating with zero aver-age speed along an edge and its opposite. Therefore, if all speeds vanish (item (iii) )then initial and final position must coincide. Items (iv) , (v) further prescribe thatan object possessing vanishing speed on an edge e starts floating back and forth along e and − e , and exits the swinging state from the same vertex it entered, onlywhen the speed becomes positive.We deduce from the definition the following properties. Proposition 4.3.
Let ξ = ( e i , q i , T i ) Mi =1 be a parametrized path. (i) If some speed q i is non-vanishing, and i , · · · , i K is the increasing sequenceof indices corresponding to edges with positive speed, then ¯ ξ := ( e i j , q i j , T i j ) Kj =1 is still a parametrized path with all average velocities different from andsuch that o( ¯ ξ ) = o( ξ ) , t( ¯ ξ ) = t( ξ ) . (ii) If a parametrized path has all average speeds equal to zero, then it is supportedon an edge and its opposite. (iii)
A parametrized circuit with some vanishing speed consists of an equilibriumcircuit { ( e, , T ) , ( − e, , S ) } for some e ∈ E and T, S > . Basic definitions.
In this section we introduce a notion of tangent bundle T Γ of Γ and define suitable sets of probability measures that we will use to build aversion of Mather theory on graphs.
Definition 4.4.
The tangent bundle of Γ is defined as T Γ := E × R + / ∼ , where R + := [0 , + ∞ ) and ∼ is the identification ( e, ∼ ( − e, R + e := { e } × R + .We endow T Γ := E × R + with a distance defined as: d (( e , q ) , ( e , q )) := q + q + 1 if e = ± e q + q if e = e | q − q | if e = − e . This makes T Γ a Polish space. A set A is open in T Γ in the induced topology ifand only if A ∩ R + e is open in the natural topology of R + for any e . Accordingly, F is a Borelian set on T Γ if and only if F ∩ R + e is Borelian in R + e for any edge e . Definition 4.5.
Given µ a Borel probability measure on T Γ, we define the supportof µ as the set supp E µ = { e ∈ E | µ ( R + e ) > } . UBRY–MATHER 13
Proposition 4.6.
Any Borel probability measure in T Γ can be decomposed as theconvex combination of Borel probability measures in each fiber, namely (10) µ ( F ) = X e ∈ E λ e µ e ( F ∩ R + e ) for any Borelian set F ⊆ T Γ , where µ e are Borel probability measures on R + e and λ e ≥ such that P e ∈ E λ e = 1 .In particular, supp E µ = { e ∈ E | λ e = 0 } . Proof:
We distinguish two cases, according to whether µ ( e,
0) = 0 or µ ( e, > λ e := ( µ ( R + e )): if λ e = 0 ( i.e. , e supp E µ ), then the choiceof µ e is irrelevant; otherwise we define µ e as the restriction of µ on R + e , normalizedin order to be a probability measure.If ( µ ( e, >
0, then µ e is not uniquely determined since we have a degree offreedom in sharing the contribute of µ ( e,
0) = µ ( − e,
0) between e and − e . For, weintroduce two positive constant m e and m − e , such that m e + m − e = 1, and denoteby ˆ µ e the restriction of µ to R + e \ { } . Then, we define µ e := 1ˆ µ e ( R + e ) + m e µ ( e,
0) ˆ µ e + m e δ ( e, λ e := ˆ µ e ( R + e ) + m e µ ( e, , where δ ( e,
0) denotes Dirac delta at ( e, (cid:3) Note that a Borel probability measure µ = P e ∈ E λ e µ e has finite first momentumif and only such property holds for any µ e , namely Z + ∞ q dµ e < + ∞ for any e ∈ E .We denote by P the family of Borel probability measures on T Γ with finite firstmomentum and we endow it with the (first) Wasserstein distance (see, for example,[27]). The corresponding convergence of measures can be expressed in duality withcontinuous functions F ( e, q ) on T Γ possessing linear growth at infinity; namely,given a sequence { µ n } n and µ in M µ n → µ ⇐⇒ Z F ( e, q ) dµ n → Z F ( e, q ) dµ n for any function F continuous in T Γ such that | F ( e, q ) | ≤ a e q + b e for any q ≥ a e , b e ∈ R . Closed probability measures on T Γ . Let us observe that for any ω ∈ C (Γ , R ), the function ( e, q ) ω, q e i is continuous with linear growth on T Γ. Given µ = P e λ e µ e ∈ P , we consequentlydefine Z ω dµ := X e ∈ E λ e Z + ∞ h ω, q e i dµ e (11) = * ω, X e ∈ E (cid:20) λ e Z + ∞ q dµ e (cid:21) e + . This associates to µ a 1–chain(12) ρ ( µ ) := X e ∈ E (cid:20) λ e Z + ∞ q dµ e (cid:21) e ∈ C (Γ , R ) . Definition 4.7.
We say that µ is a closed measure if Z df dµ = 0 for any f ∈ C (Γ , R ).We set M := { µ ∈ P : µ is closed } Remark 4.8. (i)
Given µ ∈ P , we have for any g ∈ C (Γ , R ) Z dg dµ = h dg, ρ ( µ ) i , hence µ is closed ⇐⇒ ∂ρ ( µ ) = 0 ⇐⇒ ρ ∈ H (Γ , R ) , namely ρ ( µ ) is a 1–cycle. We call it rotation vector (or Schartzman asymptoticcycle ) of µ . This should be compared with the corresponding classical definitions inAubry–Mather theory (see [8], [25]). (ii) Given µ ∈ M and ω ∈ C (Γ , R ), it follows from the definition of closed measureand (11) that Z ω dµ = h [ ω ] , ρ ( µ ) i , i.e. , it only depends on the cohomology class [ ω ] ∈ H (Γ , R ). UBRY–MATHER 15
Proposition 4.9.
The subset M ⊂ P is convex and closed in the Wasserstein topol-ogy. Proof:
The convexity property is obvious. Let µ n be a sequence of closed probabilitymeasures converging in the Wasserstein sense to µ . We consider g ∈ C (Γ , R ),then associating to dg the continuous function on T Γ with linear growth ( e, q ) dg, q e i and taking into account (11), we get Z dg dµ n → Z dg dµ. This concludes the proof. (cid:3)
Let us define the map ρ : M −→ H (Γ , R ) that to any closed probability measure µ associates its rotation vector ρ ( µ ) (see Remark 4.8 (i) ). One proves the followingproperties. Proposition 4.10.
The map ρ is continuous and affine (for convex combinations),i.e., for every λ ∈ [0 , and µ , µ ∈ M ρ ( λµ + (1 − λ ) µ ) = λρ ( µ ) + (1 − λ ) ρ ( µ ) . In particular, it is surjective.
Proof:
Let us first prove continuity. If µ n → µ in M and ω is any element of C (Γ , R ) with cohomology class c , then associating to ω the continuous function on T Γ with linear growth ( e, q ) ω, q e i and taking into account (11), we have thatif µ n converges to µ in the Wasserstein sense then h c, ρ ( µ n ) i = Z ω dµ n −→ Z ω dµ = h c, ρ ( µ ) i . Since c has been arbitrarily chosen in H (Γ , R ), ρ ( µ n ) −→ ρ ( µ ) as n → + ∞ , whichproves continuity.The fact that the map ρ is affine (under convex combination) is an immediateconsequence of the definition of the rotation vector.Finally, let us prove surjectivity. Let h ∈ H (Γ , R ) given by h = P Ni =1 a i e i , with ∂ ( h ) = 0; we can assume that a i > e i with − e i ). Then,it is sufficient to consider the measure µ = P Ni =1 1 N δ ( e i , N a i ) – where δ ( e, q ) denotesDirac delta at ( e, q ) – and use (12) to check that ρ ( µ ) = N X i =1 N a i N e i = N X i =1 a i e i = h. (cid:3) Occupation measures.
Let us introduce the notion of occupation measure ,which can the thought as a measure representation of a parametrized path.
Definition 4.11.
Given a parametrized path ξ = ( e i , q i , T i ) Mi =1 , the associated oc-cupation measure is defined as(13) µ ξ := 1 T ξ M X i =1 T i δ ( e i , q i ) , where T ξ = P Mi =1 T i and δ ( e, q ) denotes Dirac delta concentrated on the point ( e, q ). Remark 4.12. (i)
Taking into account that an edge e can be equal to e i for differentvalues of the index i , we see that an occupation measure restricted to any edge isthe convex combination of Dirac measures. (ii) For any e ∈ E , δ ( e,
0) is a closed occupation measure corresponding to theequilibrium circuit based on e with vanishing speed and any pair of positive numbersas time parametrization. Proposition 4.13.
Let µ ξ be an occupation measure associated to a parametrizedpath ξ = { ( e i , q i , T i ) } Mi =1 . Then, µ ξ is closed if and only if ξ is a parametrized cycle. Proof:
Let g ∈ C (Γ , R ). Observe that for every e ∈ E Z dg dδ ( e,
0) = 0since we are integrating the function h dg, qe i with respect to δ ( e, q i vanish (see Proposition 4.3). Let us assume that some q i = 0; then,recalling Definition 4.1 and Proposition 4.3: Z dg dµ ξ = 1 T ξ M X i =1 T i Z dg dδ ( e i , q i ) = 1 T ξ X i | q i =0 T i h dg, q i e i i = 1 T ξ X i | q i =0 (cid:0) g (t( e i )) − g (o( e i )) (cid:1) = 1 T ξ (cid:0) g (t( ξ )) − g (o( ξ )) (cid:1) . Therefore, µ ξ is closed if and only if g (t( ξ )) = g (o( ξ )) for every g ∈ C (Γ , R ), whichis equivalent to t( ξ ) = o( ξ ), i.e. , ξ is a parametrized cycle. (cid:3) UBRY–MATHER 17
Remark 4.14.
Given a parametrized cycle ξ , we have (see (2) for the definition of[ ξ ]) ρ ( µ ξ ) = 1 T ξ M X i =1 e i = [ ξ ] T ξ . We close this section with a density result. This theorem is well known for mea-sures on the tangent bundle of a manifold, a piece of folklore according to [3]. Wewill not use it in the rest of the paper, however we include it for two reasons: firstly,it somehow validates our previous definition of occupation measures, secondly be-cause the proof, which follows the same lines of [3, Theorem 31], is simple andilluminating, and represents a nice application of weak KAM theory on graphs tothe analysis of closed probability measures.
Theorem 4.15.
The set of closed occupation measures is dense in M . The proof is in Appendix B.5.
Mather’s theory on graphs
Mather theory is about the minimization of the action functional µ Z L ω dµ on suitable subsets of closed probability measures. Results and definitions of thissection are inspired by the corresponding ones in the classical Mather theory, see[8], [25]. We provide full details to make the text self–contained.5.1. Existence of minimizers.
We recall the main compactness criterion in theWasserstein space P (see, for example, [27]).– A subset K ⊂ P is relatively compact if and only for any ε > K ε of T Γ such that Z K cε q dµ < ε for any µ ∈ K ,where K cε stands for the complement of K ε in T Γ.From the superlinearity property of L , we derive the following property. Proposition 5.1.
Given a ∈ R , the set K a := (cid:26) µ ∈ M | Z L dµ ≤ a (cid:27) is compact in M . Proof:
Assume that K a = ∅ , otherwise there is nothing to prove. According tothe compactness criterion in the Wasserstein space M and the definition of T Γ, it isenough to prove that, given ε >
0, there exists M ε > Z R + e ∩ ( M ε , + ∞ ) q dµ < ε for any e ∈ E , µ ∈ K a .If this is not the case, we find ε > e ∈ E , a sequence of positively divergingnumbers M n and a sequence of measures µ ( n ) = X e ∈ E λ ( n ) e µ ( n ) e ∈ K a such that Z + ∞ M n q dµ ( n ) e ≥ ε for any n .Taking into account that L ( e , · ) is superlinear, we find another positively divergingsequence h n satisfying L ( e , q ) ≥ h n q for q ≥ M n .Since edges are finitely many, we can find a constant b such that a ≥ Z L ( e, q ) dµ ( n ) ≥ Z + ∞ M n L ( e , q ) dµ ( n ) e + b ≥ h n Z + ∞ M n q dµ ( n ) e + b ≥ h n ε + b, which, as n goes to + ∞ , leads to a contradiction. (cid:3) As a consequence:
Corollary 5.2.
The action functional µ R L dµ is lower semicontinuous on M . This in turn implies:
Theorem 5.3.(i)
The action functional admits minimum in M ; (ii) Given h ∈ H (Γ , R ) , the action functional admits minimum in ρ − ( h ) . UBRY–MATHER 19
Proof:
Recall that a lower-semicontinuous function admits minimum on compactsets. Therefore, ( i) follows from Proposition 5.1 and Corollary 5.2. Similarly, (ii) follows from Proposition 5.1, Corollary 5.2, and the fact that ρ − ( h ) is closed in M (the map ρ : M → H (Γ , R ) is continuous in force of Proposition 4.10). (cid:3) Mather’s minimal average actions and Mather measures.
We define
Mather’s β –function as: β : H (Γ , R ) −→ R h min µ ∈ ρ − ( h ) Z L dµ. The above minimum does exist in force of Theorem 5.3 (ii) . Definition 5.4.
We say that a measure µ ∈ M is a Mather measure with homology h if R L d µ = β ( h ). We denote the subset of these measures by M h .We define the Mather set of homology h as(14) f M h := [ µ ∈ M h supp µ ⊂ T Γ , where supp µ denotes the support of µ in T Γ.Properties of β : • β is convex. In fact, let h , h ∈ H (Γ , R ), λ ∈ [0 ,
1] and let us consider µ i ∈ M h i for i = 1 ,
2. If follows from Proposition 4.10 that ρ ( λµ + (1 − λ ) µ ) = λh + (1 − λ ) h . Moreover, using the linearity of the integral and the definition of β , we obtain: β ( λh + (1 − λ ) h ) ≤ Z L d ( λµ + (1 − λ ) µ )= λ Z L dµ + (1 − λ ) Z L dµ = λβ ( h ) + (1 − λ ) β ( h ) . • β is superlinear. This could be proved directly by using the superlinearity of L ; however, we deduce it from the finiteness of its convex conjugate α (see(15) and Remark 5.5). We consider the convex conjugate of β , that we shall call Mather’s α -function : α : H (Γ , R ) −→ R c max h ∈ H (Γ , R ) ( h c, h i − β ( h )) , where h c, h i denotes the pairing between H (Γ , R ) and H (Γ , R ) defined in section2.4.One can also characterize α in a variational way, which shows that it is finiteeverywhere: α ( c ) = max h ∈ H (Γ , R ) ( h c, h i − β ( h ))(15) = max h ∈ H (Γ , R ) (cid:18) h c, h i − min µ ∈ ρ − ( h ) Z L dµ (cid:19) = − min h ∈ H (Γ , R ) (cid:18) min µ ∈ ρ − ( h ) (cid:18)Z L dµ − h c, ρ ( µ ) i (cid:19)(cid:19) = − min µ ∈ M Z L ω dµ, where ω ∈ C (Γ , R ) has cohomology class c . Due to the superlinearity of L ω , we see,arguing as in Proposition 5.1, that the sublevels of L ω are compact in the Wassersteintopology, and consequently by Proposition 4.9 the minimum in the above formuladoes exist. Therefore α is finite, convex with convex conjugate equal to β . Remark 5.5.
The fact that α is finite, convex with convex conjugate equal to β ,implies that β has superlinear growth. In fact, a convex function on finite dimen-sional vector spaces possess a finite convex conjugate if and only if it has superlineargrowth, see [22]. Definition 5.6.
Given c in H (Γ , R ) and ω in the class c , we say that a measure µ ∈ M is a Mather measure with cohomology c if R L ω dµ = − α ( c ) (observe thatbeing µ closed, this notion does not depend on the choice of the representative ω ,but only on its cohomology class). We denote the subset of these measures by M c .We define the Mather set of cohomology c as(16) f M c := [ µ ∈ M c supp µ ⊂ T Γ , where supp µ denotes the support of µ in T Γ. UBRY–MATHER 21
As a consequence of Proposition 5.1, we have
Proposition 5.7.
For any h ∈ H (Γ , R ) , c ∈ H (Γ , R ) , the sets of Mather measures M h , M c are compact, convex subsets of M . Next proposition will help clarify the relation between the two notions of Mathermeasures in Definitions 5.4 and 5.6. To state it, recall that, like any convex functionon a finite-dimensional space, β admits a subdifferential at each point h ∈ H (Γ , R ), i.e. , we can find c ∈ H (Γ , R ) such that β ( h ′ ) − β ( h ) ≥ h c, h ′ − h i for any h ∈ H (Γ , R ). We will denote by ∂β ( h ) the set of c ∈ H (Γ , R ) that are subdifferentialsof β at h . Similarly, we will denote by ∂α ( c ) the set of subdifferentials of α at c .Fenchel’s duality implies an easy characterization of subdifferentials (see for example[25, Proposition 3.3.3]):(17) c ∈ ∂β ( h ) ⇐⇒ h ∈ ∂α ( c ) ⇐⇒ h c, h i = α ( c ) + β ( h ) . The next proposition can be proven as the corresponding ones in the classicalMather theory, with obvious adaptations (we omit the proof, see for example [25,Proposition 3.3.4]).
Proposition 5.8.(i) µ ∈ M is a Mather measure with homology h if and only if µ ∈ M c for any c ∈ ∂β ( h ) . (ii) For every c ∈ H (Γ , R ) ∂α ( c ) = { ρ ( µ ) | µ ∈ M c } . Corollary 5.9. If c ∈ ∂β ( h ) , then f M h ⊆ f M c . In particular: f M c = [ h ∈ ∂α ( c ) f M h . Remark 5.10.
We will say that µ is a Mather measure tout court , if it is a Mathermeasure for some cohomology c , or equivalently it is a Mather measure of homology ρ ( µ ). Properties of Mather measures
Structural properties and Mather’s graph property.
Exploiting the strictconvexity of L ( e, · ), we can derive this first property of Mather measures, namelythat they consist of a finite convex combinations of Dirac deltas, in particular eachedge appears at most once. Proposition 6.1.
The restriction of any Mather measure to an edge of its supportis concentrated on a point.
Proof:
Let µ = P e ∈ E λ e µ e be a Mather measure. We set ν := X e ∈ E λ e δ (cid:18) e, Z + ∞ q dµ e (cid:19) . Thanks to the convexity of L ( e, · ) for each e ∈ E , we can apply Jensen inequalityto µ e and get Z L ( e, q ) dµ = X e ∈ E λ e Z + ∞ L ( e, q ) dµ e ≥ X e ∈ E λ e L (cid:18) e, Z + ∞ q dµ e (cid:19) = Z L ( e, q ) dν. Observe that ρ ( µ ) = ρ ( ν ); hence, due to the strict convexity of L ( e, · ) for each e ∈ E and the fact that µ is a Mather measure, we conclude that equality must prevail inthe above formula, and this is possible if and only if µ = ν . (cid:3) Proposition 6.2.
Let c ∈ H (Γ , R ) and h ∈ H (Γ , R ) . (i) If ( f, q ) , ( f, q ) ∈ f M c (resp. f M h ) for some f ∈ E , then q = q . (ii) If ( f, q ) , ( − f, q ) ∈ f M c (resp. f M h ) for some f ∈ E , then q = q = 0 and α ( c ) = min α . Proof:
Since, by Corollary 5.9, f M h is contained in some f M c , then it suffices toprove the property for the latter.Let ( f, q ), ( f, q ) ∈ f M c ; then, by Proposition 6.1 there are two Mather measures µ = P e ∈ E λ e µ e , ν = P e ∈ E τ e ν e in M c such that λ f > , τ f > µ f = δ ( f, q ) , ν f = δ ( f, q ) . Due to the convexity of M c (see Proposition 5.7), we have that µ + ν is in M c ,and the restriction of it on f is a convex combination with positive coefficients of UBRY–MATHER 23 δ ( f, q ) and δ ( f, q ). We then derive, again from Proposition 6.1, that q = q , whichconcludes the proof of item ( i) .We proceed by proving (ii) . Let ( f, q ), ( − f, q ) ∈ f M c ; then, there exists µ ∈ M c such that f, − f ∈ supp E µ ; in fact, by Definition 5.6, there exist µ , µ ∈ M c such that ( f, q ) ∈ supp µ and ( − f, q ) ∈ supp µ , hence it suffices to consider µ = µ + µ , which still belongs to M c (due to convexity, see Proposition 5.7).Let us define ˜ µ := 11 − ( λ + λ ) (cid:0) µ − λ δ ( f, q ) − λ δ ( − f, q ) (cid:1) with λ , λ ∈ (0 , q , q ≥
0, so that µ can be written as µ = λ δ ( f, q ) + λ δ ( − f, q ) + (1 − λ − λ )˜ µ. Note that ± f supp E ˜ µ because of Proposition 6.1.Assume, without any loss of generality, that λ q ≥ λ q (otherwise, invert the rolesof f and − f ) and define(18) q := λ q − λ q λ + λ = λ λ + λ q + λ λ + λ ( − q ) ≥ . Consider the new measure ν := ( λ + λ ) δ ( f, q ) + (1 − ( λ + λ ))˜ µ. Clearly, ν is a probability measure and it is also closed; in fact: ρ (cid:0) ( λ + λ ) δ ( f, q ) (cid:1) = (cid:0) λ + λ (cid:1) q f = (cid:0) λ q − λ q (cid:1) f = ρ (cid:0) λ δ ( f, q ) + λ δ ( − f, q ) (cid:1) , hence, ρ ( ν ) = ρ ( µ ) is a 1-cycle, which implies that ν is closed (see Remark 4.8 (i) ).In order to get a contradiction, we want to prove that the action of ν is less thanthe action of µ , thus contradicting minimality of µ . In fact: Z L dν − Z L dµ = ( λ + λ ) L ( f, q ) − λ L ( f, q ) − λ L ( − f , q )= ( λ + λ ) (cid:18) L ( f, q ) − λ λ + λ L ( f, q ) − λ λ + λ L ( f, − q ) (cid:19) ≤ , (19)where in the last inequality we have used the convexity of L ( f, · ); taking into accountthat L ( f, · ) is in addition strictly convex, we see that a strict inequality prevails in (19), leading to a contradiction, unless q = q = − q ⇐⇒ q = q = 0 . The property that α ( c ) = min α follows from the fact that δ ( f,
0) belongs to M c ,hence 0 ∈ ∂α ( c ) (see Proposition 5.8 (ii)). Being α convex implies that α ( c ) is theminimum of α . (cid:3) We can now derive a central property that can be read as an instance of thecelebrated
Mather’s graph theorem (see [18, Theorem 2]) in the graph setting .To state it more precisely, let us introduce the projection π E : T Γ → E defined as π E ( e, q ) := (cid:26) e if q > { e, − e } if q = 0. Remark 6.3.
Observe that the projection π E that we have defined is multivaluedat some points: this is needed in order to cope with the fact that the elements ( e, − e,
0) are identified in T Γ, for any e ∈ E .Alternatively, one could consider π E + : T Γ → E + , denoting the projection on agiven orientation E + of the graph (namely, π E + ( ± e, q ) = e for any e ∈ E + ). In thelight of Proposition 6.2, the graph property in Corollary 6.4 continues to hold withsuch a projection and all related results can be suitably restated. Corollary 6.4. (Mather graph property)
The restriction of π E to f M c and f M h is injective for every c ∈ H (Γ , R ) , h ∈ H (Γ , R ) . Proof:
Since, by Corollary 5.9, f M h is contained in some f M c , then it suffices toprove the property for the latter. The result then follows from Proposition 6.2 (i) . (cid:3) Remark 6.5.
It follows from Corollary 6.4 that for any c ∈ H (Γ , R ) (cid:0) π E | f M c (cid:1) − : π E (cid:0) f M c (cid:1) −→ f M c is a well-defined map. In Section 8 we will describe this function more explicitly(see Theorem 8.4).Next result is an important step in our analysis. It puts in relation, via Proposition6.1, Mather and occupation measures. Ironically, the term graph appearing twice in this sentence, is used with two completely distinctmeanings.
UBRY–MATHER 25
Theorem 6.6.
A closed probability measure, whose restriction on any edge is con-centrated on a point, is a convex combination of occupation measures based on cir-cuits.
Proof:
Let(20) µ = X e ∈ E λ e δ ( e, q e )with λ e ≥ P λ e = 1, be a measure as indicated in the statement. We firstassume that q e = 0 for any e . We argue by finite induction on the cardinality ofsupp E µ indicated by | supp E µ | . By taking the function which is equal to 1 at agiven vertex x and 0 elsewhere, and exploiting that µ is closed, we deduce that therelation(21) X e ∈ E x λ e q e = X e ∈− E x λ e q e ∀ x ∈ V . If | supp E µ | = 2, set supp E µ = { e, f } . By applying (21) to x = o( e ), x = t( e ), werealize that ( e, f ) makes up a circuit and λ q e = (1 − λ ) q f for some λ ∈ (0 , λ = q f q e + q f = 1 q e q e q f q e + q f = 1 /q e q e + q f and 1 − λ = 1 /q f q e + q f . This implies that µ is the occupation measure corresponding to the parameterizedcircuit (( e, q e , /q e ) , ( f, q f , , /q f )).Let us now assume the assertion true for measures with support of cardinality lessthan a given M , and assume | supp E µ | = M ≥
3. Starting by any edge e ∈ supp E µ ,we choose one of the edges f ∈ supp E µ witht( e ) = o( f )and we call it π ( e ). This choice is possible, for any initial e , because of (21). Weiterate the procedure starting from π ( e ) to define π ( e ). Taking again into account(21), we see that we can go on until we reach π k ( e ) witht( π k ( e )) = o( π h ( e )) for some h ≤ k .The edges { π h ( e ) , π h +1 ( e ) , · · · , π k ( e ) } make up a circuit contained in supp E µ . We set M ′ = k + 1 − h , e i = π h + i − ( e ) , λ i = λ e i q i = q e i for i = 1 , · · · , M ′ and consider the parametrized circuit ξ = ( e i , q i , /q i ) M ′ i =1 . The associated occupationmeasure is(22) µ ξ = 1 T ξ M ′ X i =1 q i δ ( e i , q i ) , where T ξ = (cid:16)P M ′ i =1 1 q i (cid:17) . We distinguish two cases:– If M = M ′ we show that µ = µ ξ , which proves the claim. In fact, in thiscase for any vertex x of the graph there is an alternative: either no edge insupp E µ is incident on it or there are exactly two incident edges, one with x as initial point and the other with x as terminal point. By applying (21) wededuce(23) λ i q i = λ j q j =: A for any i, j ∈ { , · · · , M ′ } .This implies that λ i = Aq i for any i , and, since P i λ i = 1 we obtain A = X i q i ! − = 1 T ξ . By exploiting the above relation plus (20), (22), (23) we obtain µ ξ = 1 T ξ M X i =1 q i δ ( e i , q i ) = M X i =1 T ξ q i δ ( e i , q i )= M X i =1 Aq i δ ( e i , q i ) = M X i =1 λ i δ ( e i , q i ) = X e ∈ supp E µ λ e δ ( e, q e ) = µ. – Let us assume now that M ′ < M and define λ = T min i q i λ i . Observe that λT q i ≤ λ i for any i ∈ { , · · · , M ′ } and consequently λ = λ X i T q i ≤ X i λ i < , UBRY–MATHER 27 where the rightmost strict inequality comes from the fact that M ′ < M . Letus define the following probability measure ν = 11 − λ M ′ X i =1 (cid:18) λ i − λ T q i (cid:19) δ ( e i , q i ) + X e supp E µ ξ λ e δ ( e, q e ) . This is actually a probability measure since X i (cid:18) λ i − λ T q i (cid:19) + X e supp E µ ξ λ e = X e ∈ supp E µ λ e − λ T X i q i = 1 − λ. Moreover λ µ ξ + (1 − λ ) ν (24) = λ " T X i q i δ ( e i , q i ) + X i (cid:18) λ i − λ T q i (cid:19) δ ( e i , q i ) + X e supp E µ ξ λ e δ ( e, q e )= X e ∈ supp E µ λ e δ ( e, q e ) = µ. We see from (24) that ν is closed since both µ and µ ξ are closed. In addition,some of the coefficients λ i − λ T q i must vanish by the very definition of λ . Thesupport of ν has then cardinality less than M , and by inductive assumption ν is the convex combination of occupation measures based on circuits. Thesame holds true for µ in force of (24).Let us now discuss the case in which some of the q e ’s vanish. Let µ be as in (20)and define E = { e ∈ supp E µ | q e > } , F = { f ∈ supp E µ | q f = 0 } , λ F = X f ∈ F λ f . If E = ∅ , then µ = δ ( e,
0) for a suitable e ∈ E and this measure is supported bythe equilibrium circuit based on e , so that the assertion is proved. We then assumethat both E and F are nonempty. We consider the probability measure ν = X e ∈ E λ e − λ F δ ( e, q e )and derive µ = (1 − λ F ) ν + X f ∈ F λ f δ ( f, . By the first part of the proof there exist occupation measures µ ξ i corresponding tocircuits ξ i with ν = X i σ i µ ξ i σ i > , X i σ i = 1 . Summing, up we have µ = (1 − λ F ) X i σ i ν i + X f ∈ F λ f δ ( f, . This concludes the proof. (cid:3)
Irreducible Mather measures.
A point in a convex set is called extremal ifit cannot be obtained as convex combination of two distinct elements of the set.A closed probability measure is said to be irreducible if it is extremal in M . Proposition 6.7.
A Mather measure is irreducible if and only if it is an occupationmeasure corresponding to a parametrized circuit.
Proof:
Let µ be a Mather measure. If it is not an occupation measure supported bya parametrized circuit, then by Theorem 6.6 and Proposition 6.1 it must the convexcombination of distinct occupation measures supported on parametrized circuits.This proves that it is not irreducible.Conversely, assume for the purpose of contradiction that µ is an occupation mea-sure supported on a parametrized circuit and that it is not irreducible. Hence, thereexist µ = µ in M , λ ∈ (0 ,
1) such that µ = (1 − λ ) µ + λ µ . This implies by Proposition 4.10 that ρ ( µ ) = (1 − λ ) ρ ( µ ) + λ ρ ( µ ) . We thus have β ( ρ ( µ )) = Z L dµ = (1 − λ ) Z L dµ + λ Z L dµ ≥ (1 − λ ) β ( ρ ( µ )) + λ β ( ρ ( µ ) , due to the convex character of β , equality must prevail in the above formula, sothat both µ and µ are Mather measures. Taking again into account Theorem 6.6and Proposition 6.1, we find an occupation measure ν supported on a parametrizedcircuit with supp E ν proper subset of supp E µ . This is in contrast with µ beingsupported on a circuit. (cid:3) UBRY–MATHER 29
Remark 6.8.
It follows from Remark 4.14 and Proposition 6.7, that the rotationvector of an occupation measure µ must have a special form:(25) λ X e ∈ E + τ e e, where λ > τ e ∈ { , ± } , and E + denotes an orientation of the graph. Theorem 6.9.
For any c ∈ H (Γ , R ) , the set of Mather measures M c is the convexhull of the irreducible Mather measures with cohomology c , which are finitely many. Proof:
We know from Proposition 5.7 that M c is a convex set. We claim that µ ∈ M c is irreducible if and only if it is an extremal point of M c . It is trivial that ifit is irreducible then it is extremal in M c . Conversely, let µ be extremal in M c , andassume that there exist µ , µ in M , λ ∈ (0 ,
1) with µ = (1 − λ ) µ + λ µ . If If ω ∈ C (Γ , R ) is of cohomology c , we have − α ( c ) = Z L ω dµ = (1 − λ ) Z L ω dµ + λ Z L ω dµ which implies, by the minimality property of α ( · ) that both µ and µ are Mathermeasures of cohomology c , which is impossible. This proves the claim.Let µ ∈ M c then by Theorem 6.6 it is convex combination of occupation measuressupported on parametrized circuits. Arguing as in the first part of the proof, we seethat all the measures forming the convex combination are in M c , and consequentlyby Proposition 6.7 they are irreducible Mather measures in M c . This shows that M c is the convex hull of its extremal points. These extremal measures are finitelymany since – by the graph property in Corollary 6.4 – a circuit identifies the Mathermeasures supported on it, if any, and the set of circuits in Γ is finite. (cid:3) As shown in the previous result, any M c , for c ∈ H (Γ , R ), contains some irre-ducible measure. The situation is rather different for the sets M h . In fact, we knowRemark 6.8 that if M h contains irreducible Mather measures, then h must be as in(25); hence, not all M h do contain them. We can get some information on which M h ’s contain irreducible Mather measures by looking at the extremal points of theepigraph of β . We recall that the epigraph of β is given byepi( β ) := { ( h, t ) ∈ H (Γ , R ) × R : t ≥ β ( h ) } As in the classical ergodic theory, we have:
Proposition 6.10.
Let h ∈ H (Γ , R ) . If ( h, β ( h )) is an extremal point of epi( β ) ,then there exist irreducible Mather measures of rotation vector h . Proof:
Let µ be a Mather measure with rotation vector h ; then, according to The-orem 6.6 µ = M X i =1 λ i µ i with λ > P i λ i = 1 and µ i occupation measures supported on parametrizedcircuits. Let us define h i = ρ ( µ i ) for any i = 1 , . . . , M .We have β (cid:0) M X i =1 h i (cid:1) = β ( h ) = Z L dµ = M X i =1 λ i Z L dµ i ≥ M X i =1 λ i β ( h i ) . Due to the convex character of β , we see that equality must prevail in the abovesequence of inequalities, so that all the µ i ’s must be Mather measures. In addition,thanks to Proposition 6.7, they are irreducible Mather measures. We in additionhave that ( h, β ( h )) = M X i =1 λ i ( h i , β ( h i ))Since ( h, β ( h )) is an extremal point of epi( β ), we must necessarily have h i = h for any i . Hence, all the µ i ’s are irreducible Mather measures with rotation vector h . (cid:3) Weak KAM facts
We pause the exposition of Aubry Mather theory on graphs, to recall some basicresults of weak KAM theory that we will use in the following section. Note thatcoercivity and convexity of the Hamiltonian are sufficient for these results to holdtrue. All the material is taken from [24], which contains a comprehensive treatmentof the topic.
UBRY–MATHER 31
We consider a 1–cochain ω with cohomology class c , and the family of discreteHamilton–Jacobi equations on Γ(HJ ωa ) max − e ∈ E x H ω ( e, h du, e i ) = a for x ∈ V , a ∈ R which can be equivalently written as u ( x ) = min − e ∈ E x (cid:0) u (o( e )) + σ ω ( e, a ) (cid:1) . A function u : V → R is called solution if equality in (HJ ωa ) holds for every vertex x .If instead the left hand–side is less than or equal to a , we say that u is a subsolution of (HJ ωa ).We set a := max e ∈ E a e . Remark 7.1.
It is clear that equation (HJ ωa ) does not even make sense if a < a ,because in this case the a –sublevels of H ( e, · ) are empty for some edge e .Given a path ξ = ( e i ) Mi =1 in Γ, we define for a ≥ a (see (9)) σ ω ( ξ, a ) := M X i =1 σ ω ( e i , a ) . Note that this definition only depends on the concatenated edges making up ξ , noparametrization is involved. We sometimes refer to σ ω ( ξ, a ) as the intrinsic length of the path ξ related to the Hamiltonian H ω and the level a . Proposition 7.2.(i)
Equation (HJ ωa ) admits subsolutions if and only if σ ω ( ξ, a ) ≥ for any closed path ξ . (ii) A function u : V → R is a subsolution of (HJ ωa ) if and only if u ( x ) − u ( y ) ≤ σ ω ( ξ, a ) for any path ξ with o( ξ ) = y , t( ξ ) = x . (iii) There is one and only one value of a , called critical value of H ω , for whichthe corresponding equation has solutions on the whole Γ . It is given by (26) min { a ∈ R : (HJ ωa ) admits subsolutions } . For a proof of these claims see [24, Propositions 6.5, 6.8 and Theorem 6.16]Clearly the Hamiltonian H ω is not invariant by change of representative in theclass c , however its critical value does not depend on the chosen representative, but only on the cohomology class c . If, in fact, we replace ω by ω ′ = ω + dw , for some w ∈ C (Γ , R ), then, given any (sub)solution u to the equation associated to H ω , thefunction u − w will be a (sub)solution to the equation associated to H ω ′ .We can therefore define a function e α : H (Γ , R ) → R associating to any cohomology class the critical value of H ω , as defined in (26) (itonly depends on the cohomology class of ω ). We call critical the equationmax e ∈− E x H ω ( e, h du, e i ) = e α ( c )and qualify as critical its (sub)solutions. According to Remarks 3.3 and 7.1 e α ( c ) ≥ a for any c ∈ H (Γ , R ). Proposition 7.3.
Given c ∈ H (Γ , R ) and ω of cohomology class c , the criticalvalue e α ( c ) is characterized by the following properties: (i) σ ω ( ξ, e α ( c )) ≥ for all cycles ξ in Γ ; (ii) there exists a cycle ζ with σ ω ( ζ , e α ( c )) = 0 . For a proof of these claims see [24, Lemma 6.7, Corollary 6.9, Proposition 6.15and Theorem 6.16].We define the Aubry sets as follows: A c := { e ∈ E | belonging to some cycle with σ ω ( ξ, e α ( c )) = 0 } . Remark 7.4.
Given an arbitrary path ξ , the intrinsic length σ ω ( ξ, e α ( c )) is not in-variant for the change of representative, however invariance is valid if ξ is a cycle.This is the reason why the Aubry set only depends on c and not on the representa-tive ω .We state in the next proposition a relevant property of the Aubry sets (see [24,Lemma 7.3]). Proposition 7.5.
Let c ∈ H (Γ , R ) and ω ∈ C (Γ , R ) be of cohomology class c .Then, any subsolution u of H ω = e α ( c ) satisfies h du, e i = σ ω ( e, e α ( c )) and H ω ( e, h du, e i ) = e α ( c ) for e ∈ A c . Consequently, the differentials of all such subsolutions coincide on e ∈ A c . UBRY–MATHER 33
The value of du on the Aubry set A c is clearly not invariant for change of represen-tative in c , however the element ∂∂p H ω ( e, h du, e i ), namely the element characterizedby the equality (27) ∂∂p H ω ( e, h du, e i ) h du, e i = L ω ( e, ∂∂p H ω ( e, h du, e i ) + H ω ( e, h du, e i ) ∀ e ∈ A c possesses such an invariance, as made precise by the following result. Lemma 7.6.
Let ω, ω ′ ∈ C (Γ , R ) be in the same cohomology class c , and let u , v be subsolutions to (HJ ω e α ( c ) ) and (HJ ω ′ e α ( c ) ) , respectively; then (28) ∂∂p H ω ( e, h du, e i ) = ∂∂p H ω ′ ( e, h dv, e i ) for any e ∈ A c . Proof:
We set q e := ∂∂p H ω ( e, h du, e i ) for e ∈ A c .We have that ω ′ = ω + dw for some w ∈ C (Γ . R ), and consequently dv = du − dw. Let e ∈ A c , then keeping in mind (27) we have q e h dv, e i = q e h du, e i − q e h dw, e i = L ω ( e, q e ) + H ω ( e, h du, e i ) − q e h dw, e i = L ( e, q e ) − q e h ω, e i + H ( e, h du − dw + dw + ω, e i ) − q e h dw, e i = L ω ′ ( e, q e ) + H ω ′ ( e, dv ) . This proves (28). (cid:3)
We denote by Q c : A c → R the function(29) e ∂∂p H ω ( e, h du, e i ) . by the monotonicity properties of H ω ( e, · ), Q c ( e ) is non–negative for any e ∈ A c .8. Weak KAM and Aubry–Mather
In this section we put in relation weak KAM theory and Aubry Mather theory ongraphs.
Mather’s α function and critical value.Theorem 8.1. Given c ∈ H (Γ , R ) and ω ∈ C (Γ , R ) of cohomology class c , wehave: (i) e α ( c ) and α ( c ) coincide, i.e., the critical value of H ω and the minimal actionof Mather measures of cohomology class c are the same; (ii) an irreducible measure belongs to M c if and only if it is supported on a circuit ζ such that σ ω ( ζ , α ( c )) = 0 , equipped with a suitable parametrization. Proof:
We denote by u a subsolution to (HJ ω e α ( c ) ). Taking into account the definitionof Lagrangian, we get for any closed probability measure µ Z L ω ( e, q ) dµ ≥ Z (cid:2) q h du, e i − H ω ( e, h du, e i ) (cid:3) dµ = − e α ( c ) , which shows that(30) − α ( c ) ≥ − e α ( c ) . Let ξ = ( e i ) Mi =1 be a circuit with σ ω ( ξ, e α ( c )) = X i σ ω ( e i , e α ( c )) = 0so that ξ is contained in A c . We have by Proposition 7.5 and (27) that e α ( c ) = H ω ( e i , h du, e i i ) = σ ω ( e i , e α ( c )) Q c ( e i ) − L ω ( e i , Q c ( e i )) . We first assume that Q c ( e i ) = 0 for every i , then we get σ ω ( e i , e α ( c )) = 1 Q c ( e i ) (cid:0)e α ( c ) + L ω ( e i , Q c ( e i )) (cid:1) . By summing over i , we further obtain(31) 0 = M X i =1 Q c ( e i ) L ω ( e i , Q c ( e i )) + M X i =1 Q c ( e i ) ! e α ( c ) . We denote by µ ξ the occupation measure associated with the parametrized circuit( e i , Q c ( e i ) , / Q c ( e i )) Mi =1 , and deduce from (31) Z L ω dµ ξ = − e α ( c )which together with (30) proves the item (i) , in the case Q c ( e i ) = 0 for every i . Ifsome Q c ( e i ) vanishes, then according to Proposition 4.3, ξ is an equilibrium circuit UBRY–MATHER 35 based on some edge e , namely ξ = (( e, , T ) , ( − e, , S )) for some T, S >
0. In thiscase we have e α ( c ) = a = a e and L ω ( e,
0) = L ω ( − e,
0) = − a e = − e α ( c ) . The occupation measure related to ξ is δ ( e, Z L ω dδ ( e,
0) = L ω ( e,
0) = − e α ( c ) . This ends the proof of item (i) . We have shown at the same time that the convex hullof occupation measures supported by circuits ζ with σ ω ( ζ , α ( c )) = 0 is contained in M c . Conversely, let µ ∈ M c be the occupation measure related to the parametrizedcycle ξ = ( e i , q i , /q i ) Mi =1 , with T = P Mi =1 1 q i , then − α ( c ) = Z L ω dµ = 1 T M X i =1 q i L ω ( e i , q i ) ≥ T [ σ ω ( ξ, α ( c )) − T α ( c )] ≥ − α ( c ) , (32)which implies σ ω ( ξ, α ( c )) = 0. If instead µ ∈ M c is equal to δ ( e, e , we find α ( c ) = a e and σ ω ( e, α ( c )) + σ ω ( − e, α ( c )) = 0which concludes the proof. (cid:3) We deduce:
Corollary 8.2.
Let c ∈ H (Γ , R ) , for any ( e, q ) ∈ f M c we have L ω ( e, q ) = σ ω ( e, α ( c )) q − α ( c ) . Recalling the definition of the Aubry set A c ⊂ E , we further derive: Corollary 8.3.
Given c ∈ H (Γ , R ) , we have π E (cid:16) f M c (cid:17) = A c Next theorem refines the information provided in Corollary 6.4 and Remark 6.5.
Theorem 8.4.
Given c ∈ H (Γ , R ) , f M c = { ( e, Q c ( e )) | e ∈ A c } . Proof:
Let ω be of cohomology c . We know from Proposition 7.5 that the differen-tials of all subsolutions u to (HJ ωα ( c ) ) coincide on A c and satisfy(33) h du, e i = σ ω ( e, α ( c )) , H ω ( e, h du, e i ) = α ( c ) . Let µ be an irreducible occupation measure in M c , and assume that it correspondsto a parametrized circuit ξ = ( e i , q i , T i ) Mi =1 . We derive from Corollary 8.2 that L ω ( e i , q i ) = σ ω ( e i , α ( c )) q i − α ( c ) for i = 1 , . . . , M .This implies by (33) L ω ( e i , q i ) + H ω ( e, h du, e i ) = h du, e i q i , which yields q i = Q c ( e i ), for i = 1 , · · · , M , in view of (27). (cid:3) Minimizers of Mather’s α function.Proposition 8.5. The minimum of the function α is equal to a . Proof:
The function α admits minimum because of its coercive character. Assume c to be a minimizer of α and denote by ω ∈ C (Γ , R ) a representative of the cohomologyclass c . Then there exists µ ∈ M c with ρ ( µ ) = 0 in view of Proposition 5.8 (ii) .Taking into account the definition of rotation vector, we derive that for some edge f , both f and − f belong to supp E µ . This implies by Proposition 6.2 (ii) that Q c ( f ) = Q c ( − f ) = 0 and α ( c ) = min α . Since Q c ( f ) = 0, then: ∂∂p H ω ( f, h du, f i ) = Q c ( f ) = 0 , where u is a subsolution to (HJ ωα ( c ) ). We deduce that h du, f i is a minimizer of H ω ( f, · ) and consequently α ( c ) = H ω ( f, h du, f i ) = a f ≤ a ≤ min α, which implies that α ( c ) = min α = a . (cid:3) Corollary 8.6.
An element c ∈ H (Γ , R ) is a minimizer of α if and only if thefunction Q c vanishes at some e ∈ A c . Proof:
The fact that if Q c ( e ) = 0 for some e ∈ A c then c is a minimizer of α , hasbeen proved in Proposition 8.5.Conversely, if c is a minimizer of α , then α ( c ) = a f for some f ∈ E , by Proposition UBRY–MATHER 37 f ∈ A c , moreover, if u is a subsolution to (HJ ωa f ), where ω isa representative of c , we get H ω ( f, h du, f i ) = a f . Taking into account that a f is the minimum of H ω ( f, · ), we finally have Q c ( f ) = ∂∂p H ω ( f, h du, f i ) = 0 . (cid:3) Appendix A. From networks to graphs
In this appendix, we describe how it is possible to develop Aubry-Mather theoryon networks, by means of the discrete theory that we have developed on graphs.Let us start by recalling the definition of network, as given in [24]. We consider afinite collection E of regular simple oriented curves in R N parametrized over [0 , γ ∈ E , we denote by − γ ∈ E the curve − γ ( s ) = γ (1 − s ) for s ∈ [0 , γ and opposite orientation. We further assume(34) γ ((0 , ∩ γ ′ ((0 , ∅ whenever γ = ± γ ′ .A network G is a subset of R N of the form G = [ γ ∈E γ ([0 , ⊂ R N , the curves in E are called arcs of the network.We call vertices the initial and terminal points of the arcs, and denote by V thesets of all such vertices. We assume that the network is finite and connected, namelythe number of arcs and vertices is finite and there is a finite concatenation of arcslinking any pair of vertices. Remark A.1.
This setting can be naturally extended to the case in which G is em-bedded in a Riemannian manifold ( M, g ) (for example by means of Nash embeddingtheorem [20]).
We can associate to any network G a finite and connected abstract graph Γ =( V , E ) with the same vertices of the network and edges corresponding to the arcs.More precisely, we consider an abstract set E with a bijection(35) Ψ : E −→ E . This induces maps o : E −→ V , − : E −→ E viao( e ) = Ψ( e )(0) and − e = Ψ − ( − Ψ( e )) , satisfying the properties in the definition of graph, see Section 2.A.1. Hamiltonians and Lagrangians on networks.
An Hamiltonian on G is acollection of Hamiltonians H γ : [0 , × R → R ; ( s, p ) H γ ( s, p )labeled by the arcs. We assume the compatibility conditions(36) H − γ ( s, p ) = H γ (1 − s, − p ) for any γ ∈ E .As we will discuss with more detail hereafter, we can associate to the family H γ anHamiltonian H ( e, · ) on Γ. Exploiting the results of [24], we see that the correspond-ing Hamilton–Jacobi equations H γ ( s, ( u ◦ γ ) ′ ) = a on (0 ,
1) for γ ∈ E ,and max − e ∈ E x H ( e, h du, e i ) = a for x ∈ V , a ∈ R are equivalent, in the sense that if u : G → R is a (sub)solution of the former thenits trace on V solves the latter, and, conversely, any function w : V → R solution ofthe latter can be uniquely extended on G in such a way that the extended functionis solution of the former equation. In addition, in [24] we developed in parallel weakKAM results for the two equations, proved that the two critical values coincide,define the corresponding Aubry sets, etc. ...The aim of this appendix to determine a set of rather natural assumptions onthe H γ ’s such that the corresponding Hamiltonian on the graph Γ satisfies (H1) , (H2) . This will allow to take advantage of the output of this paper to provide anAubry–Mather theory on networks.We require the H γ ’s to satisfy the following properties: UBRY–MATHER 39 (H1 ′ ) H γ is continuous in ( s, p ), differentiable in p for any fixed s , and such thatthe function ( s, p ) ∂∂p H γ ( s, p )is continuous; (H1 ′ ) H γ is superlinear in p , uniformly in [0 , r → + ∞ min (cid:26) H γ ( s, p ) p | p > r, s ∈ [0 , (cid:27) = + ∞ ; (H3 ′ ) H γ is strictly convex in p ; (H4 ′ ) the map s min p ∈ R H γ ( s, p ) is constant in [0 , γ ∈ E .We define a γ = a − γ as the value of the constant function appearing in the as-sumption (H4’) , in other terms the sublevel of the Hamiltonian H γ correspondingto a γ is a singleton for any s ; we further denote by p γs the minimizer of H γ ( s, · ).Therefore (H4’) reads H γ ( s, p γs ) = a γ for any s ∈ [0 , Remark A.2.
Actually condition (H4’) is required only for γ ∈ E such that a γ =max { a λ : λ ∈ E } . We refer to [24, Remark 3.3] for an explanation of the role ofthis condition.We fix γ ∈ E , e ∈ E with γ = Ψ( e ). The procedure to pass from H γ to H ( e, · )consists in the following three steps:– consider, for any s , the inverse, with respect to the composition, of H γ ( s, · )in [ p γs , + ∞ ), denoted by σ + γ ( s, · );– for any fixed a ≥ a γ , integrate σ + γ ( · , a ) in [0 ,
1] obtaining σ ( e, a ), where σ + γ ( s, a ) := max { p | H γ ( s, p ) = a } σ ( e, a ) := Z σ + γ ( s, a ) ds ;– define(38) H ( e, p ) := (cid:26) σ − ( e, p ) for p ≥ σ ( e, a γ ) σ − ( − e, − p ) for p ≤ σ ( e, a γ ) , where the inverse is with respect the composition. It is easy to see that if H γ is independent of s , then H γ ( · ) and H ( e, · ) coincide. Proposition A.3.
If assumptions (H1 ′ ) – (H4 ′ ) hold, then H ( e, · ) : R → R satisfies (H1) – (H2) . Moreover, a e = a γ and p e = σ ( e, a γ ) , as defined in (5) . We need a preliminary result.
Lemma A.4.
The function a σ ( e, a ) from [ a γ , + ∞ ) to R is: (i) continuous and strictly increasing; (ii) strictly concave with lim a → + ∞ σ ( e,a ) a = 0 ; (iii) differentiable in ( a γ , + ∞ ) with lim a → a γ ∂∂a σ ( e, a ) = + ∞ . Proof:
The claimed continuity and monotonicity properties in item (i) have beenalready proved in [24, Lemma 5.15]. Exploiting the strict convexity assumption on H γ , we deduce that, for any s ∈ [0 , λ ∈ (0 , a , b in [ a γ , + ∞ ) H γ (cid:0) s, σ + γ ( s, (1 − λ ) a + λb ) (cid:1) = (1 − λ ) a + λ b = (1 − λ ) H γ ( s, σ + γ ( s, a )) + λ H γ ( s, σ + γ ( s, b ))(39) > H γ ( s, (1 − λ ) σ + γ ( s, a ) + λ σ + γ ( s, b )) . Since H γ ( s, · ) is increasing in the interval ( p s , + ∞ ), the inequality in (39) yields σ + γ ( s, (1 − λ ) a + λb ) > (1 − λ ) σ + γ ( s, a ) + λ σ + γ ( s, b ) . By integrating the above relation over [0 , σ ( e, (1 − λ ) a + λb ) > (1 − λ ) σ ( e, a ) + λ σ ( e, b ) , which shows the strictly concave character of σ ( e, · ).To prove the limit relation in (ii) , we exploit the uniform superlinearity assumption (H2 ′ ) on H γ . Assume by contradiction that there is a sequence a n → ∞ and apositive M such that lim n → + ∞ σ ( e, a n ) a n > M. It follows from the definition of σ ( e, a n ) that there exist, for any n , s n ∈ [0 , p n ∈ R such that H γ ( s n , p n ) = a n and p n a n > M. Hence, we derive p n → + ∞ and H γ ( s n , p n ) p n < M ,
UBRY–MATHER 41 which is in contrast with (37). We deduce from (H1 ′ ) that the inverse function a σ + γ ( s, a ) is differentiable in ( a γ , + ∞ ). By differentiating under the integralsign, we further get that a σ ( e, a ) is differentiable in ( a γ , + ∞ ) and ∂∂a σ ( e, a ) = Z ∂∂a σ + γ ( s, a ) ds. We denote by ω ( · ) a uniform continuity modulus of ( s, a ) σ + γ ( s, a ) in [0 , × [ a γ , a γ + 1] and of ( s, p ) ∂∂p H γ ( s, p ) in K (see assumption (H1 ′ ) ), where K = { ( s, p ) | s ∈ [0 , , p ∈ [ p s , + ∞ ) , H γ ( s, p ) ≤ a γ + 1 } is compact by the superlinearity assumption (H2 ′ ) . Then(40) 0 ≤ ∂∂p H γ ( s, p ) ≤ ω ( p − p s ) for ( s, p ) ∈ K .Observe that a = H γ ( s, σ + a ( s, a )) = ⇒ ∂∂p H γ ( s, σ + a ( s, a )) ∂∂a σ + a ( s, a ) . This and (40) imply that for a ∈ ( a e , a e + 1) we have ∂∂a σ ( e, a ) = Z ∂∂p H γ ( s, σ + γ ( s, a )) ds ≥ Z ω ( σ + γ ( s, a ) − p s ) ds ≥ ω ◦ ω ( a − a γ ) . From this we derive item (iii) , and conclude the proof. (cid:3)
Proof of Proposition A.3:
We derive from (39) and Lemma A.4 that H ( e, · ) iscontinuous in R and differentiable in R \ { σ ( e, a γ ) } . Taking into account that ∂∂p H ( e, p ) = 1 ∂∂a σ ( e, σ − ( e, p )) for p > σ ( e, a γ ) ∂∂p H ( e, p ) = − ∂∂a σ ( − e, σ − ( − e, p )) for p < σ ( e, a γ )we derive from Lemma A.4 (iii) thatlim p → σ ( e,a γ ) ∂∂p H ( e, p ) = 0 , which implies that H ( e, · ) is differentiable in σ ( e, a γ ) with vanishing derivative.Strict convexity is straight forward from the previous discussion. Let us prove (H2) ,namely that lim p →±∞ H ( e,p ) | p | = + ∞ . Recalling (39) and using (ii) in Lemma A.4:lim p → + ∞ H ( e, p ) p = lim p → + ∞ σ − ( e, p ) p = lim a → + ∞ aσ ( e, a ) = + ∞ . (41)Similarly for p → −∞ , considering σ ( − e, a ).Easily follows that a e = a γ and p e = σ ( e, a γ ) (see (5)). (cid:3) For every γ ∈ E , consider the Lagrangian associated to H γ , namely its convexconjugate L γ : [0 , × R −→ R defined as(42) L γ ( s, q ) := sup p ∈ R (cid:0) p q − H γ ( s, p ) (cid:1) , where equality is achieved for p such that ∂H γ ∂p ( s, p ) = q .Since H γ satisfies (H1 ′ )–(H3 ′ ) , then it follows (see for example [22]) that L γ is continuous in ( s, q ), differentiable , superlinear and strictly convex in q .Using (36) we see that the L γ ’s satisfy the following compatibility condition: L − γ ( s, q ) = L γ (1 − s, − q ) ∀ s ∈ [0 , , q ∈ R . A.2.
How to develop Aubry-Mather theory on networks.
In this section welook, from the point of view of networks, at some of the notions that we have in-troduced in the previous sections. This will help clarify and validate the settingthat we have proposed, and it will outline the ideas and the tools that are neededin order to transfer the previous construction to the network setting.In this section we assume the Hamiltonian { H γ } γ ∈E to be Tonelli , namely, besides (H1 ′ ) – (H4 ′ ) , we further require that for any γ ∈ E (H5 ′ ) L γ ( s, q ) is of class C in ( s, q ) and ∂ ∂q L γ is positive definite.We consider the network G and its corresponding abstract graph Γ. We fix an arc γ and an edge e with Ψ( e ) = γ .Given a parametrization ( q e , T e ) of the edge e ∈ E , we provide an interpretationof it on the corresponding arc γ . We first assume q >
0, so that, according to the
UBRY–MATHER 43 definition of parametrized path, T e = q e . Then, due to the strict convexity of L ( e, · ),there exists a unique p q e ≥ p e such that(43) L ( e, q ) = p q e q e − H ( e, p q e ) = q σ ( e, a q e ) − a q e , where a q e > a e is such that p q e = σ ( e, a q e ) (it is uniquely defined because of thecontinuity and strict monotonicity of σ ( e, · ) stated in Lemma A.4). This also impliesthe relation q e = ∂∂p H ( e, p q e ) = ∂∂p H ( e, σ ( e, a q e )) . We consider the solution to H γ ( s, w ′ ( s )) = a q e in (0 ,
1) given by w ( s ) = Z s σ + γ ( t, a q e ) dt, and the orbit of the Hamiltonian flow related to H γ in [0 , × R with initial datum(0 , σ + γ (0 , a q e )) = (0 , w ′ (0)), contained in the energy level a q e . This orbit has as firstcomponent the curve ξ q e with ξ q e (0) = 0 and˙ ξ q e = ∂∂p H γ ( ξ q e ( t ) , w ′ ( ξ q e ( t ))) , while the second component is given by w ′ ( ξ q e ( t )). We have in fact0 = ddt H γ ( ξ q e ( t ) , w ′ ( ξ q e ( t )))= ∂∂s H γ ( ξ q e ( t ) , w ′ ( ξ q e ( t ))) ˙ ξ q e ( t ) + ˙ ξ q e ( t ) ddt w ′ ( ξ q e ( t )) , and accordingly ddt w ′ ( ξ q e ( t )) = − ∂∂s H γ ( ξ q e ( t ) , w ′ ( ξ q e ( t ))) . The orbit is defined in [0 , T q e ], where T q e is the time in which ξ q e reaches the boundarypoint s = 1. Proposition A.5.
Let q e > and let ξ q e and T q e be defined as above. Then: (i) The time T q e is equal to q e ; (ii) q e is the average speed of ξ q e in the time interval [0 , T q e ] ; (iii) L ( e, q e ) = T qe R T qe L γ ( ξ q e , ˙ ξ q e ) dt ; (iv) L ( e, q e ) = T qe min nR T qe L γ ( ζ ( t ) , ˙ ζ ( t )) dt o , where the minimum is taken inthe family of absolutely continuous curves ζ : [0 , T q e ] −→ [0 , with ζ (0) = 0 , ζ ( T q e ) = 1 . Proof:
We have that ˙ ξ q e ( t ) and w ′ ( ξ q e ( t )) are conjugate in [0 , T q e ], in the sense that˙ ξ q e ( t ) w ′ ( ξ q e ( t )) = L γ ( ξ q e ( t ) , ˙ ξ q e ( t )) + H γ ( ξ q e ( t ) , w ′ ( ξ q e ( t ))) . which implies(44) L γ ( ξ q e ( t ) , ˙ ξ q e ( t )) = ˙ ξ q e ( t ) σ + γ ( ξ q e ( t ) , a q e ) − a q e . In addition, it follows from the definition of L γ that(45) L γ ( ξ q e ( t ) , ˙ ξ q e ( t )) ≥ ˙ ξ q e ( t )) σ + γ ( ξ q e ( t ) , b ) − b for any b ≥ a e . By integrating (44), (45) over [0 , T q e ] we further get Z T qe L γ ( ξ q e ( t ) , ˙ ξ q e ( t )) = σ ( e, a q e ) − T q e a q e (46) Z T qe L γ ( ξ q e ( t ) , ˙ ξ q e ( t )) ≥ σ ( e, b ) − T q e b for any b ≥ a e . Taking into account (8), we derive(47) L ( e, /T q e ) = 1 T q e σ ( e, a q e ) − a q e . This implies by (43) and the strict convexity of L ( e, · ) T q e = 1 q e and q e = 1 T q e Z T qe ˙ ξ q e ( t ) dt, showing items (i) and (ii) . By combining (46) and (47), we get (iii) .Finally, to obtain item (iv) , it is enough to observe that for any absolutely contin-uous curve ζ in [0 ,
1] with ζ (0) = 0 and ζ ( T q e ) = 1, one has Z T qe L γ ( ζ ( t ) , ˙ ζ ( t )) ≥ σ ( e, b ) − T q e b. (cid:3) Remark A.6.
The equality in item (iii) of Proposition A.5 can be interpreted bysaying that the action functional on the graph computed in δ ( e, q ) equals the actionfunctional on the network computed in the occupation measure corresponding tothe speed curve ( ξ q e ( t ) , ˙ ξ q e ( t )) in [0 , T q e ]. The latter measure is obtained by pushingforward through ( ξ q e ( t ) , ˙ ξ q e ( t )) the 1–dimensional Lebesgue measure restricted to[0 , T q e ] and normalize it. UBRY–MATHER 45
In particular, item (iv) of Proposition A.5 reads that the curve ξ q e defined on[0 , T q e ] is action minimizing for L γ . This sheds light on the reason why Mathermeasures on the graph consist of convex combinations of Dirac deltas (see Propo-sition 6.1), in analogy with what happens in the classical theory, in which Mathermeasures are supported on action-minimizing curves (see [18, 25]). Remark A.7.
In the case where e ∈ A and q e = Q ( e ) > σ + γ ( s, α (0)) is not just the derivative of a local (in(0 , H γ = α (0), but we also have that σ + γ ( s, α (0)) = dds u ◦ γ ( s )for any critical subsolution u of the Hamilton–Jacobi equation on the network, see[24]. Remark A.8.
To discuss the case when the speed q e vanishes for some e ∈ E , theequilibrium circuit ( e, − e ) with the parametrization (( e, , T ) , ( − e, , T )), with T , T positive constants. We set γ = Ψ( e ) and consequently − γ = Ψ( − e ). We have L ( e,
0) = L ( − e,
0) = −H ( e, p e ) = H ( − e, p − e ) = − a e = a − e . In addition we have by assumption (H4 ′ ) L γ ( s,
0) = L − γ ( s,
0) = a e = a − e for every s ∈ [0 , ∂∂s L γ ( s,
0) = − ∂∂s H γ ( s, σ + γ ( s, a e ))0 = ∂∂s L − γ ( s,
0) = − ∂∂s H − γ ( s, σ + − γ ( s, a − e )) . This implies that all the points ( s, σ + a e ( γ, s )), ( s, σ + a − e ( − γ, s )) are equilibria of theHamiltonian flows related to H γ , H − γ , respectively. We can put in relation themeasures δ ( e,
0) = δ ( − e,
0) with the Dirac measures concentrated at all points ofthe arcs γ , − γ , which – in analogy with what we did in the graph – can be identified. Appendix B. Proof of Theorem 4.15
We need a preliminary result, see [3, Proposition 42].
Lemma B.1.
Let K be a closed convex subset of P , we set C + = (cid:26) F : T Γ → R continuous with linear growth | Z F dµ ≥ ∀ µ ∈ K (cid:27) . Then: K = (cid:26) ν ∈ P | Z F dν ≥ ∀ F ∈ C + (cid:27) . The proof of this lemma is based on a separation result in Wasserstein spaces thatwe take from [16]. We state it below with slight changes to adapt it to our notationand setting.
Lemma B.2. [16, Theorem 2.9]
Let K be a closed convex subset of P , and ν K .Then, there exists F : T Γ → R with linear growth such that Z F dµ > Z F dν for any µ ∈ K . Proof of Lemma B.1:
Given ν K , we fix µ ∈ K and define ν λ = (1 − λ ) ν + λ µ for λ ∈ [0 , K is closed, there exists λ ∈ (0 ,
1) with ν λ K . We denote by F a functionsatisfying the statement of Lemma B.2 with respect to ν λ ; we can in additionassume, without loosing generality, that(48) Z F dν λ = 0 . Therefore F ∈ C + and(49) Z F dµ > µ ∈ K It follows from the definition of ν λ , (48), (49) that Z F dν < . Summing up, we have found that for any ν K , there exists F ∈ C + whose integralwith respect to ν is strictly negative. This proves the assertion. (cid:3) Lemma B.3.
The closure in P of the space of closed occupation measures is convex. UBRY–MATHER 47
The fact that the closure of the space of closed occupation measures is convex,stems from the property that a closed occupation measure stays unchanged underany finite repetition of the corresponding cycle. Therefore, we can connect a finitenumber of cycles through simple paths in order to make a unique cycle. We canthen repeat n times the cycles leaving unaffected the connecting paths and obtaina sequence of closed occupation measures indexed by n converging, as n → + ∞ , toa measure which does not “see” the connecting simple paths and is a convex com-bination of the occupation measures corresponding to the cycles with repetitions.A formal argument can be found [3, Lemma 30] for measures on the tangent bundleof a compact manifold. It can be adapted with minor modifications to our setting.We can now prove the main result of this appendix. Proof of Theorem 4.15:
In view of Lemma B.1, it is enough to show that if acontinuous function F with linear growth in T Γ satisfies(50) Z F dµ ≥ µ ,then it also satisfies(51) Z F dν ≥ ν ∈ M .Let F satisfy (50). By integration with respect to the closed occupation measures δ ( e, e ∈ E , we get F ( e, ≥ e ∈ E .Thanks to the above inequality, we can modify F in [0 , /n ] ∪ [ n, + ∞ ) ⊂ R + e , forany e ∈ E , constructing a sequence of continuous functions F n defined on T Γ suchthat(52) F n ( e, > , F n ( e, · ) has superlinear growth at + ∞ for any e ∈ E and in such a way that for any n , for each e ∈ E , q ≥ F n +1 ( e, q ) ≤ F n ( e, q )(53) F n ( e, q ) ≥ F ( e, q )(54) F n ( e, q ) → F ( e, q ) as n → + ∞ .(55)We define G n ( e, p ) := max q ≥ (cid:0) p q − F n ( e, q ) (cid:1) ; the function G n ( e, · ) is finite by the superlinear growth of F n , convex and superlinearat + ∞ ; in addition G n ( e, · ) is increasing in p and by (52)inf p ∈ R G n ( e, p ) = lim p →−∞ G n ( e, p ) = − F n ( e, < . Therefore, the value 0 is attained by G n ( e, · ) and is above the infimum. We denoteby ϕ ne , for any e ∈ E , the unique element such that G n ( e, ϕ ne ) = 0 . The quantity ϕ ne must be understood as an intrinsic length of the edge e related tothe Hamiltonian G n and the value 0. We have(56) ϕ ne q ≤ F n ( e, q ) for any q ≥ q e > G n ( e, ϕ e ) = ϕ ne q e − F n ( e, q e ) . We consider the discrete Hamilton–Jacobi equation on Γ(58) max − e ∈ E x G n ( e, h du, e i ) = 0 for x ∈ V .We know from Proposition 7.2 (i) that in order (58) to have subsolutions it isnecessary and sufficient that for any cycle ξ = ( e i ) Mi =1 in Γ the intrinsic length ϕ n ( ξ ) := M X i =1 ϕ ne i ≥ . We deduce from (57) that(59) ϕ ne i = 1 q i F n ( e i , q i ) , where q i := q e i (see (56), (57)). We consider the parametrized version of ξ given by( e i , q i , /q i ) Mi =1 and denote by µ ξ the corresponding closed occupation measure. Wehave by (54) and the assumption that0 ≤ Z F n dµ ξ = 1 P Mi =1 1 q i M X i =1 q i F n ( e i , q i ) , which finally implies, using (59), that ϕ n ( ξ ) ≥
0. If u : V → R is a subsolution of(58), we have(60) h du, e i ≤ ϕ ne for any e ∈ E . UBRY–MATHER 49
Let ν = P e ∈ E λ e ν e be a closed measure on T Γ, then by (56), (60)0 = X e ∈ E λ e Z q h du, e i dν e ≤ X e ∈ E λ e Z ϕ ne q dν e ≤ X e ∈ E λ e Z F n ( e, q ) dν e = Z F n dν. Taking into account (53), (55) and passing to the limit as n → + ∞ in the aboveinequality, we obtain Z F dν ≥ , which shows that F satisfies (51). This concludes the proof. (cid:3) References [1] Y. Achdou, M. Dao, O. Ley, N. Tchou. A class of infinite horizon mean field games on networks.
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UBRY–MATHER 51
Dipartimento di Matematica, Sapienza Universit`a di Roma, Italy.
Email address : [email protected] Dipartimento di Matematica, Universit`a degli Studi di Roma “Tor Vergata”,Rome, Italy.
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