THEORY OF OPTIMAL TRANSPORT FOR LORENTZIAN COST FUNCTIONS STEFANSUHR Abstract. TheoptimaltransportproblemisstudiedinthecontextofLorentz- 6 Finslergeometry. ForgloballyhyperbolicLorentz-Finslerspacetimes thefirst 1 Kantorovich problem and the Monge problem are solved. Further the inter- 0 mediate regularity of the transport paths isstudied. These results generalize 2 parts of[5]and[7]. n a J 8 1. Introduction 1 This article studies optimal transportation in Lorentz-Finsler manifolds from a ] geometric point of view. Geometric in this context denotes, among others things G taking the perspective of spacetimes according to the idea, basic to the theory of D relativity,thatthereisnocanonicalisochronicity. Consideringatransportproblem h. on a space N and the interval [0,1] is in geometric terms studying a problem on t the productspaceN [0,1]withboundaryvaluesonN 0 andN 1 andthe a × ×{ } ×{ } m canonicalprojectionN [0,1] [0,1]asthetimeparameter. ThesurfacesN t × 7→ ×{ } arethussurfacesofisochronicity. InLorentziangeometrythisamountstoaspecial [ case since measures can be for example concentrated on achronal or nonsmooth 1 sets, e.g. ∂J−(p). Seen the other way around one can say that measures relevant v inLorentziangeometrymightnotonly be distributedinspace,but intime aswell. 2 ThefirstonetotakenoticeoftheproblemofoptimaltransportationinLorentzian 3 geometrywas[6]. Thereinatransportationproblemisproposed,whichonlyweakly 5 4 disguised is the problem of transportation between parallel spacelike hyperplanes 0 inMinkowskispacewithrespecttotheLorentziandistanceextendedby . Herea 1. strong form of isochronicityis assumedfor the supportof each measure.∞Following 0 this formulation [5] generalized the problem to a wider class of functions called 6 relativistic costs, andgaveinteraliaasolutiontothe Mongeproblemwhile staying 1 in the same basic geometric frame. : v The early universe reconstruction problem, studied in [7] and [10] with methods i ofoptimaltransportation,askswhetheronecanconstructthetrajectoriesofmasses X from the big bang to their present day positions in Robertson-Walker spacetimes. r a A mathematical formulation for general globaly hyperbolic spacetimes would read as follows: Given two measures, one concentrated on a Cauchy hypersurface, the other on the past cone of a point. Then what can be said about the trajectories of the minimizers in a dynamical optimal coupling of the two measure? [10] givesajustificationto why the problemcanbe studied withmethods fromoptimal transportation. The results in this article generalize the previous approaches to the problem of Lorentzianoptimaltransportation. Furthernewresultsonthestructureofminimal couplingsaregiven. Insection2thesettingandthetheoremsaredescribed. Proofs are given in section 3. Acknowledgement: TheauthorwouldliketothankVictorBangertforsuggesting the problem of optimal transportation in the context of Lorentzian geometry and Date:January 19,2016. 1 2 STEFANSUHR Albert Fathi for encouraging the pursuit of the project. The author would further liketothankPatrickBernardforprovidingtheopportunitytocarryouttheideasfor thisarticleandmakingnumeroussuggestionswhichhelpedtoshapetheexposition ofthepresentresults. ValentineRoosandRodolfoRíos-Zertuchearekindlythanked for many helpful discussions in the process of this research. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement 307062. 2. The results LetM beasmoothmanifold. Throughoutthearticleafixedchoiceofacomplete RiemannianmetrichonM ismade. ConsiderasmoothandfunctionL: TM 0 R (here 0 denotes the zero section in TM) positive homogenous of degree 2\su→ch that the second fiber derivative is nondegenerate with index dimM 1. A cone structure on M is a subset of TM such that π ( ) = M, R − = and TM + := TCM convexforallp M. Acausal structure Cof(M,L)ist·hCenaCchoice p p oCf a coCn∩e structure such that∈L >0 and ∂ 0 isCa connected component of intC L=0 . CausalstrCuturesarekno|wntoexistupCto\afinitecover,see[12]. Actually {more is}true: Every tangent space TM is intersected by L > 0 in a nonempty p set. FurthereveryconnectedcomponentofTM L>0 {isastri}ctlyconvexcone p ∩{ } and belongs to a unique causal structure up to a finite covering of M. Fix a causal structure for (M,L). Define a new LagrangianL on by setting C C L(v), for v , L(v):= − ∈C (∞p, otherwise. L is a fiberwise convex function, finite on its domain and positive homogeneous of degree one. Note that L is smooth on int . The function L has the features of a C Finsler metric of Lorentzian type. This justifies to call the pair (M,L) a Lorentz- Finsler manifold. The generality of Lorentz-Finsler geometry is chosen in view of recentdevelopmentsinthearea,seee.g. [11],[12],[13],[14],andthegoaltoachieve a scope comparable to the one of Tonelli-Lagrangian systems, see e.g. [3], [4] and [9]. Onecallsanabsolutelycontinuouscurveγ: I M ( -)causalifγ˙ whenever → C ∈C the tangentvectorexists. Acausalcurveγ: I M is timelike if for alls I there → ∈ existsε,δ >0suchthatdist(γ˙(t),∂ ) εγ˙(t),foreveryt I forwhichγ˙(t)exists C ≥ | | ∈ and s t <δ. Define the Lagrangianaction relative to L: | − | L(γ˙)dt, if γ is -causal, (γ):= C A (∞R , else. Denote with J+(p) the setof points q M suchthat there exists a causalcurve ∈ withinitialandterminalpointpandq,respectively. J−(p)isdefinedwiththeroles ofp andq exchangedas initial andterminalpoint. I+(p) and I−(p)have the same definition as J±(p) with causal replaced by timelike. Define the set J+ := (p,q) M M q J+(p) . { ∈ × | ∈ } For an open set U M define J± and I± as before for the restriction (U, ). ⊆ U U C|U A Lorentz-Finsler manifold is said to be causal if it does not admit a closed causal curve. Definition 2.1. A causal Lorentz-Finsler manifold (M,L) is globally hyperbolic if the sets J+(p) J−(q) are compact for all p,q M. ∩ ∈ OPTIMAL TRANSPORTION FOR LORENTZIAN COST FUNCTIONS 3 From[17]itfollowsthatforgloballyhyperbolicLorentz-Finslerspacetimesthere exists a diffeomorphism (called a splitting) M = R N such that the projection τ: R N R, (θ,v) θ satisfies dτ L.∼Not×e that this diffeomorphism is × → 7→ − ≤ by far not unique. In the spirit of the present approach all results are formulated withasittle referencetoasplitting aspossible. Notethatforcompactlysupported measures all results are indeed independent of the splitting. The following result is provenin the same fashion as in the Lorentziancase (see [13] for the results on local minimizers). Proposition 2.2. Let (M,L) be globally hyperbolic. Then for every pair of points p,q with q J+(p) there exists a minimizer of with finite action connecting the two poin∈ts. The minimizer γ solves the EuleAr-Lagrange equation of L up to monotone reparameterization and one has γ˙ everywhere. ∈C For a globally hyperbolic Lorentz-Finsler manifold define the cost function relative to L: c (x,y):=min (γ) γ connects x and y R L {A | }∈ ∪{∞} The following proposition is a direct consequence of [18], Theorem 4.1. Proposition 2.3. Let µ,ν be probability measures on M such that there exists a splitting with τ L1(µ) L1(ν). Then there exists a coupling π of µ and ν which minimizes the to∈tal cost∩c dπ R . L ∈ ∪{∞} Denote with C (µ,ν)Rthe minimal costs relative to c of couplings between µ L L and ν, i.e. C (µ,ν):=inf c dπ π Π(µ,ν) R . L L ∈ ∈ ∪{∞} (cid:26)Z (cid:12) (cid:27) (cid:12) TheabstractexistenceresultinPropo(cid:12)sition2.3immediatelyraisesthequestion: (cid:12) Under what assumptions does a finite coupling exists? The simplest case is that of two Dirac measures µ=δ and ν =δ . Then a finite coupling exists iff y J+(x) x y ∈ iff ν(J+(A)) µ(A) and µ(J−(A)) ν(A) for all measurable A M where ≥ ≥ ⊆ J±(A) := y y J±(x) for some x A . The necessity of the condition was { | ∈ ∈ } noticed in [5] for relativistic cost functions and general measures. The problem though can be formulated in a more abstract setting. Let ( ,d ) X X and ( ,d ) be locally compact Polish spaces. Denote with π : and Y 1 Y X ×Y → X π : the canonicalprojectionsandconsideraclosedsetJ . For 2 x X×deYfin→eJY+(x):=π (π−1(x) J)andfory defineJ−(y):=π⊆(πX−×1(yY) J). ∈X 2 1 ∩ ∈Y 1 2 ∩ Further set J+(A):=π (π−1(A) J) for A andJ−(B):=π (π−1(B) J) for 2 1 ∩ ⊆X 1 2 ∩ B . ⊆Y Definition 2.4. Two probability measures are J-related if there exists a coupling π with π(J)=1. For = =M, J :=J+ and two probability measures µ,ν (M) such that X Y ∈P there exists a splitting with τ L1(µ) L1(ν) the J-relation is equivalent to the ∈ ∩ finiteness of the minimal cost, i.e. C (µ,ν)< . L ∞ Theorem 2.5. Let ( ,d ) and ( ,d ) be Polish spaces and J closed. X Y X Y ⊆ X ×Y Further let µ ( ),ν ( ) be probability measures. Then the following are ∈ P X ∈ P Y equivalent: (1) µ and ν are J-related. (2) ν(J+(A)) µ(A) and µ(J−(B)) ν(B) for all measurable A and ≥ ≥ ⊆ X B . ⊆Y 4 STEFANSUHR After adressing the existence problem of minimal couplings attention turns to- wards the structure of the minimal couplings. Recall that a set A M M is ⊆ × c -cyclically monotone if for all (x ,y ) A one has L i i 1≤i≤n { } ⊆ c (x ,y ) c (x ,y ) L i i L i σ(i) ≤ for all σ S(n). From [2]Xthen follows: X ∈ Proposition 2.6. Let µ,ν be two J+-related probability measures on M such that there exists a splitting with τ L1(µ) L1(ν). ∈ ∩ (1) One has C (µ,ν)=sup ϕ(y)dν(y) ψ(x)dµ(x) L − (cid:18)ZM ZM (cid:19) where the supremum is taken over the functions ψ L1(µ),ϕ L1(ν) with ∈ ∈ ϕ(y) ψ(x) c (x,y). L − ≤ (2) Every optimal transport plan π is concentrated on a c -cyclic monotone L Borel subset of M M. × DenotedenotewithΓ(τ)thesetofminimizersγof suchthatdτ(γ˙) const(γ). A ≡ For (x,y) J+ consider the subspace ∈ Γ := γ Γ(τ) ev (γ)=x, ev (γ)=y x→y 0 1 { ∈ | } and Γ := Γ . Set ev: Γ [0,1] M, (γ,t) γ(t) and ev := ev(.,t). ∪(x,y)∈J+ x→y × → 7→ t Recall the definition of optimal dynamical coupling from [18]. Definition 2.7. A dynamical optimal coupling is a probability measure Π on Γ such that π := (ev ,ev ) Π is a minimal coupling between µ := (ev ) Π and 0 1 ♯ 0 0 ♯ µ :=(ev ) Π. 1 1 ♯ Proposition 2.8. Let µ and µ be J+-related probability measures with τ 0 1 ∈ L1(µ ) L1(µ ) for some splitting. Then there exists a dynamical optimal cou- 0 1 ∩ pling Π for µ and µ with suppΠ Γ. 0 1 ⊆ Definethemap[∂ ev]: Γ [0,1] PTM,(γ,t) [γ˙(t)] PTM wherePTM t γ(t) × → 7→ ∈ denotestheprojectivetangentbundle. ForthecanonicalprojectionP: PTM M → one has ev=P [∂ ev]. t ◦ Theorem 2.9. Let µ,ν (M) be J+-related with disjoint supports such that ∈ P τ L1(µ) L1(ν) for some splitting. Then every dynamical optimal coupling Π ∈ ∩ has the following property: The canonical projection P restricted to the image of T :=[∂ ev](suppΠ ]0,1[)isinjective. Furthertheinverse(P )−1 islocallyHölder t T × | continuous with exponent 1/2. Example. It is not difficult to construct examples that show the optimality of the Hölder continuity in the claim. Consider Minkowski space (R3, .,. ), i.e. the 1 h i Lorentz-Finsler metric L = z2 x2 y2 on the cone structure C | − − − := px2+y2 z2 0,z 0 . C { − ≤ ≥ } The intersection of J+(x,0,0) z = 1 for (x,0,0) are discs of radius 1 around ∩{ } (x,0,1). The union of the boundaries of these discs form a foliation near (x+ { 1,0,1) 0 x 1 transverse to (x+1,0,1) 0 x 1 . In other words there | ≤ ≤ } { | ≤ ≤ } exists a map Φ: [0,1] ( ε,ε) z = 1 , a diffeomorphism onto its image, with × − → { } Φ(x,0)=(x+1,0,1) and Φ(x,.) traces the boundary of J+(x,0,0) z =1 near ∩{ } (x+1,0,1). Consider µ to be the 1-dimensional Lebesgue measure concentrated on [0,1] × 0 0 . Choose any measureable map f: [0,1] ( ε,ε) and consider ν := { }× { } → − (Φ (id,f,0) µ. It is easy to see that µ and ν are causally related by the obvious ♯ ◦ OPTIMAL TRANSPORTION FOR LORENTZIAN COST FUNCTIONS 5 coupling. Further the coupling is unique. For f chosen appropriately one sees that the intermediate regularity is at most 1/2-Hölder. The map P−1 in the theorem is always Lipschitz for dimM = 2, i.e. M is a surface. This is a well know fact for positive definite Lagrangians relying on the fact that trajectories (1) solve a differential equation with smooth coefficients and (2) have codimension 1 in a surface. These facts carry over readily to this case. Theorem 2.10. Let µ,ν (M) be J+-related with disjoint supports and τ ∈ P ∈ L1(µ) L1(ν) for some splitting. Further let K be a compact subset of int . Then ∩ C the canonical projection P restricted to the image of [∂ ev](suppΠ ]0,1[) K is t × ∩ Lipschitz for every dynamical optimal coupling Π. Recall that a set X is called achronal if every timelike curve meets it at most once. Using a splitting one sees that X can be written as the graph of a function f over a subset of N. With the same proof as for Proposition 14.25 in [15], one X seesthatf islocallyLipschitz. NowonecanuseaLipschitz-continuousextension X of f to N to say that X is the subset of a locally Lipschitz hypersurface. X A locally Lipschitz hypersurface X has a tangent space almost everywhere and with the induced Riemannian metric defines a Lebesgue measure on X. A X L measure concentrated on X is called absolutely continuous with respect to the Lebesgue measure if it is absolutely continuous with respect to . Note that X L this definition is independent of the chosen Riemannian metric since any pair of Lebesgue measures induced by Riemannian metrics are absolutely continuous with respect to each other. CallahypersurfaceY locallyuniformlyspacelikeifforone(henceevery)splitting there exists a locally Lipschitz continuous function f : N R with Y being the Y → graphoff andforallcompactK M thereexistsε>0,suchthatthe Hausdorff Y ⊆ distance between TY T1M and 1 = T1M is bounded below by ε for all y ∩ C C ∩ y K Γ such that TY exists. T1M denotes the unit tangent bundle of h. With y ∈ ∩ these notions the following generalizationof Theorem 4.3 in [5] can be proven. Theorem 2.11. Let µ,ν (M) be J+-related such that τ L1(µ) L1(ν) ∈ P ∈ ∩ for some splitting. Assume that µ and ν are concentrated on a locally uniformly spacelike hypersurface A and an achronal set B respectively. Further assume that µ is absolutely continuous with respect to the Lebesgue measure on A. Then there exists a unique optimal coupling π and a Borel map F: M M such that π = → (id,F) µ. ♯ ThetheorembecomesfalseifbothAandB areallowedtobeachronalonly. E.g. consider subsets of ∂J−(p) in Minkowski space for some p Rn+1. For suitable ∈ choices of A and B not every minimal coupling is supported on a graph. More precisely every causalcoupling has vanishing cost, but not everycausalcoupling is supported on a graph. Further the theorem is in general not true if µ is assumed to be absolutely continuous with respect to the Lebesgue measure on M. Again examples can be constructed in Minkowski space. Note that in the present formulation the problem bears greater resemblance to the classical Monge problem for distance functions. 3. The proofs 3.1. Proof of Theorem 2.5. (1) (2): For any (measurable) set B one has ⇒ ⊆Y π−1(B) J π−1(J−(B)). 2 ∩ ⊆ 1 Since ν(B) = π(π−1(B)) = π(π−1(B) J) and µ(J−(B)) = π(π−1(J−(B))), the 2 2 ∩ 1 claim follows. The other inclusion is analogous. (2) (1): For this part of the proof one needs two lemmata. ⇒ 6 STEFANSUHR Lemma 3.1. Let µ and ν be J-related. If there exists a measurable set A ⊆ X such that 0<µ(A)=ν(J+(A))<1 then the pairs 1 1 (µ ,ν ):= µ , ν A A µ(A) |A µ(A) |J+(A) (cid:18) (cid:19) and 1 1 (µ ,ν ):= µ , ν Ac Ac µ(Ac) |Ac µ(Ac) |J+(A)c (cid:18) (cid:19) satisfy the condition in Theorem 2.5 (2). If 0<ν(B)=µ(J−(B))<1 for B measureable the pairs ⊆Y 1 1 (µB,νB):= ν(B)µ|J−(B),ν(B)ν|B (cid:18) (cid:19) and 1 1 (µBc,νBc):= ν(Bc)µ|J−(B)c,ν(Bc)ν|Bc (cid:18) (cid:19) satisfy the condition in Theorem 2.5 (2). Proof. It suffices to consider the first case. The second case follows by exchange of and . So assume µ(A) = ν(J+(A)) for some measurable set A with X Y ⊆ X µ(A) (0,1). Firstnotethatallfourmeasuresµ ,ν ,µ andν areprobability A A Ac Ac ∈ measures by the assumption. One has 1 1 µ (B)= µ(B A) ν(J+(B A)) A µ(A) ∩ ≤ µ(A) ∩ 1 = ν(J+(B) J+(A))=ν (J+(B)) A µ(A) ∩ which shows µ (B) ν (J+(B)). A A ≤ Next note that µ(Ac) = ν(J+(A)c). Assume that there exists a measurable set C with ν (J+(C))<µ (C), i.e. Ac Ac ⊆X ν(J+(C) J+(A)c)=ν (J+(C))<µ (C)=µ(C Ac). ∩ |J+(A)c |Ac ∩ Then a contradiction follows from µ(C A)=µ(C Ac)+µ(A)>ν(J+(C) J+(A)c)+ν(J+(A))=ν(J+(C A)) ∪ ∩ ∩ ∪ since J+(C) J+(A) = J+(C A). Therefore one has µ (C) ν (J+(C)) for Ac Ac ∪ ∪ ≤ all measurable C . This shows the first set of inequalities. ⊆X It remains to show µ (J−(D)) ν (D) and µ (J−(D)) ν (D) for D A A Ac Ac ≥ ≥ ⊆Y measureable. If µ (J−(D))<ν (D) one has A A µ (J−(D)c)=1 µ (J−(D))>1 ν (D) ν (J+(J−(D)c)) A A A A − − ≥ sinceJ+(J−(D)c)andDaredisjoint. Thiscontradictsthefirstpart. Theinequality µ (J−(D)) ν (D) follows analogously. (cid:3) Ac Ac ≥ Lemma 3.2. Consider the product 1,...,N 1,...,N with the canonical pro- { }×{ } jections π onto the first and second factor, respectively. Let J 1,...,N 1,2 ⊆ { }× 1,...,N have the property that { } (1) ♯π (π−1(A) J) ♯A and ♯π (π−1(A) J) ♯A 1 2 ∩ ≥ 2 1 ∩ ≥ for all A 1,...,N . Then J contains the graph of a permutation σ S(N). ⊆{ } ∈ Proof. The proof is carried out by induction over N. If N = 1 the claim is trivial since J = 1 1 . Now assume that the claim has been shown for numbers less { }×{ } thanN. Firstassumethat ♯π (π−1(A) J)>♯A and♯π (π−1(A) J)>♯Afor all 1 2 ∩ 2 1 ∩ nonempty proper subsets A. Choose 1 j N with (N,j) J. By renumbering ≤ ≤ ∈ one can assume j = N. Now consider I := J 1,...,N 1 1,...,N 1 . ∩{ − }×{ − } OPTIMAL TRANSPORTION FOR LORENTZIAN COST FUNCTIONS 7 Define ρ and ρ to be the canonical projections from I onto the first and second 1 2 factor, respectively. Since ρ (ρ−1(A) J′) π (π−1(A) J) 1 ♯A 1 2 ∩ ≥ 1 2 ∩ − ≥ and vice versa for all A 1,...,N 1 one obtains a permutation o S(N 1) ⊆{ − } ∈ − whose graph is contained in I. o extends to a permutation σ S(N) whose graph ∈ is a subset of J by setting σ(N):=N and σ o. {1,...,N−1} | ≡ Ifthere existanonemptypropersubsetAof 1,...,N with♯π (π−1(A) J)= { } 1 2 ∩ ♯A or ♯π (π−1(A) J) = ♯A the idea of the previous lemma (see below) reduces 2 1 ∩ the problem to constructing two separate permutations on A and Ac. Thus again the induction gives separate permutations on A and Ac which together form a permutation σ whose graph is contained in J. One only needs to consider the case ♯π (π−1(A) J) = ♯A. The other case 1 2 ∩ follows by exchanging the order. Further by renumbering one can assume that A = π (π−1(A) J). Set J := J A A, J := J Ac Ac and denote with 1 2 ∩ A ∩ × Ac ∩ × π and π the canonical projections onto A and Ac for i=1,2. The goal is to i,A i,Ac show that JA and JAc satisfy the assumptions of the lemma. It is clear that ♯π (π−1(B) J )=♯π (π−1(B) J) ♯B 1,A 2,A ∩ A 1 2 ∩ ≥ for all B A since π (π−1(B) J) A. If however there exists C Ac with ⊆ 1 2 ∩ ⊆ ⊆ ♯π (π−1 (C) J )<♯C then♯π (π−1(A C) J) <♯(A C)whichcontradicts 1,Ac 2,Ac ∩ Ac 1 2 ∪ ∩ ∪ the initial assumption. Assume now that there exists a set D A with ♯π (π−1(D) J )<♯D. Set ⊆ 2,A 1,A ∩ A E := A π (π−1(D) J ). Then D and π (π−1(E) J ) are disjoint. This \ 2,A 1,A ∩ A 1,A 2,A ∩ A can be seen as follows. If i π (π−1(E) J ) then there exists j E such that ∈ 1,A 2,A ∩ A ∈ (i,j) J . If i D then for all (i,j) J one has j π (π−1(D) J ). Thus ∈ A ∈ ∈ A ∈ 2,A 1,A ∩ A the sets are disjoint. It follows that ♯E =♯A ♯π (π−1(D) J )>♯A ♯D ♯π (π−1(E) J ) − 2,A 1,A ∩ A − ≥ 1,A 2,A ∩ A which clearly contradicts the first part of the argument. Now the same argument applies to subsets of Ac. (cid:3) Without loss of generality one can assume that the support of both µ and ν are finite. By covering suppµ and suppν with sequences of locally finite, disjoint and measurablecoveringsonecanapproximatebothmeasuresintheweak- topologyby ∗ finitemeasureswhosesupportiscontainedinagivenneighborhoodofthesupports of µ and ν. By considering B (J) for ε> 0 instead of J gives at every step of the ε approximation a pair of finite measures satisfying the assumptions in (2). Below it will be shown how to construct a finite coupling in this case. By construction the approximations of µ and ν form precompact sets in the weak- topology. This ∗ implies that the set of couplings is precompact in the weak- topology as well, see ∗ [18],ch. 4. TheclaimfollowswhenpassingtothelimitusingthatJ suppµ suppν ∩ × is closed. So far the problem has been reduced to the case of measures µ and ν having finite support, i.e. are a finite sum of weighted Dirac measures. Using an in- ductive argument one can assume by the Lemma 3.1 that ν(J+(A)) > µ(A) and µ(J−(B)) > ν(B) for all nonempty proper subsets A of suppµ and B of suppν respectively. Thusonecanassumebyaslightperturbationthattheweightsforthe points in the support of the measures are rational and still one has ν(J+(A)) > µ(A) and µ(J−(A)) > ν(A) for all A. Thus there exist N,α ,β N such that i i ∈ 8 STEFANSUHR µ= 1 m α δ and ν = 1 n β δ . Counting the points x and y multiple N i=1 i xi N j=1 j yj i j times the measures take the form µ= 1 N δ and ν = 1 N δ . P P N i=1 xi N j=1 yj Identify x ,...,x and y ,...,y with 1,...,N . Define the set 1 N 1 N { } { }P { } P J := (i,j)(x ,y ) J 1,...,N 1,...,N . i j { | ∈ }⊆{ }×{ } Denotewithπ andπ thecanonicalprojectionsfrom 1,...,N 1,...,N onto 1 2 { }×{ } the first and second factor, respectively. Then the assumptions become ♯π (π−1(A) J) ♯A and ♯π (π−1(A) J) ♯A 1 2 ∩ ≥ 2 1 ∩ ≥ for all A 1,...,N . Lemma 3.2 now gives a permutation σ whose graph is ⊆ { } contained in J. Reversing the identifications one obtains (by abuse of notation) a bijective map σ: x ,...,x y ,...,y with (x ,σ(x )) J for all i. Since 1 N 1 N i i { } → { } ∈ µ and ν are counting measures the measure concentrated on the graph of σ is the desired coupling. 3.2. Dynamical Optimal Coupling. Proposition 3.3. There exists a Borel map S: J+ C0([0,1],M) such that → S(x,y) Γ . x→y ∈ Proof. The argument is a slight modification of the proof to Proposition 7.16 (vi) in [18]. For every (x,y) J+ the set Γ is nonempty and compact in every x→y ∈ Ck-topology,i.e. nonempty and closed. The evaluationmap ev ev is Lipschitz. 0 1 × This implies that the correspondence (for the definition see [1], page 4) (ev ev )−1: J+ ։Γ 0 1 × is weakly measurable in the sense of Definition 18.1. in [1]. Now Theorem 8.13 in [1] implies that (ev ev )−1 has a measurable selection S, i.e. (ev ev ) S 0 1 0 1 id . × × ◦ ≡(cid:3) |J+ Proof of Proposition 2.8. The argument is taken from the proof of Theorem 7.21 in [18]. Let π be an optimal coupling with marginals µ and µ for the cost c . 0 1 L Consider Π:=S π. Since (ev ,ev ) S id, the claim follows from the definition ♯ 0 1 of optimal dynamical couplings. ◦ ≡ (cid:3) Corollary 3.4. Let Π be a dynamical optimal coupling between causally related measures µ and µ and σ : Γ [0,1] measurable functions with σ σ . Then 0 1 1,2 1 2 → ≤ the restriction π :=(ev (id σ ),ev (id σ )) Π σ1,σ2 ◦ × 1 ◦ × 2 ♯ is an optimal coupling of µ := (ev (id σ )) Π and µ := (ev (id σ )) Π. If σ1 ◦ × 1 ♯ σ2 ◦ × 2 ♯ furthermore (σ ,σ )=(0,1)Π-almost everywhere then π is the uniqueoptimal 1 2 6 σ1,σ2 coupling of µ and µ . σ1 σ2 Proof. By the triangle inequality for c and the parameterization invariance of L A one has C (µ ,µ ) C (µ ,µ )+C (µ ,µ )+C (µ ,µ ) L 0 1 ≤ L 0 σ1 L σ1 σ2 L σ2 1 and c dπ = c dπ + c dπ + c dπ . L 0,1 L 0,σ1 L σ1,σ2 L σ2,1 Z Z Z Z Since c dπ =C (µ ,µ ) andC is the minimal action,the three terms onthe L 0,1 L 0 1 L right hand sides must individually coincide. The second statement follows directly from tRhe triangle inequality for c . (cid:3) L Theorem7.30in[18]reformulatestothepresentcaseasthefollowingstatement. OPTIMAL TRANSPORTION FOR LORENTZIAN COST FUNCTIONS 9 Corollary 3.5. Let µ and µ be finitely separated and causally related probability 0 1 measures. Further let Π be a dynamical optimal coupling. If Ξ is a measure on Γ0,1, such that Ξ Π and Ξ(Γ)>0, set ≤ Ξ Ξ′ := and ν :=(ev ) Ξ′. 0,1 0,1 ♯ Ξ(Γ) Then Ξ′ is a dynamical optimal coupling between ν and ν . 0 1 3.3. Intermediate regularity of dynamical optimal couplings. Forthesplit- ting M = R N choose a smooth vector field X with dτ(X ) 1. Define a ∼ × τ τ ≡ Lagrange function L : R TN R , L (t,v):=L(X +v). τ τ τ × → ∪{∞} Denote with R TN the domain of L . L is continuous on and smooth τ τ τ τ D ⊆ × D on int . τ D Lemma 3.6. (i) := t TN is acompact strictlyconvexdomain (t,x) τ x with smooth boDundary foDr a∩ll{(t},×x) R N. (ii) For all K R N compact there e∈xists×δ >0 such that ⊆ × δ (∂2L ) id v τ (t,v) ≥ L (t,v) · τ | | for all (t,x) K and v int . (t,x) ∈ ∈ D Proof. (i) is compact since dτ int ∗ . It is further smooth since D(t,x) (t,x) ∈ C(t,x) ∂ issmoothandkerdτ = 0 . Finallythestrictconvexityfollowsfromthe (t,x) C ∩C { } fact that ∂2L is semidefinite at a nonzero boundary point with kernel equal to the v radial direction, i.e. definite on any hyperplane transversalto the radial direction. (ii) Recall the formula for the second fibre derivative of L 1 1∂ L ∂ L ∂2L= v ⊗ v ∂2L . v 2√L 2 L − v (cid:18) (cid:19) As before one has ∂2L >0. v |T∂D×T∂D Thus one can choose N < and δ >0 such that 1 ∞ N ∂ L ∂ L ∂2L >δ id 2 v ⊗ v − v 1· (cid:18) (cid:19)(cid:12)TD×TD (cid:12) on a neighborhood U of ∂ over K. This (cid:12)implies the claim on the i.g. smaller neighborhood U L<1/ND . (cid:12) For the remain∩in{g points o}utside of U L<1/N note that ∩{ } 1∂ L ∂ L v ⊗ v ∂2L 0 2 L − v ≥ with kernel equal to the radial direction. Thus one can find δ > 0 such that on 2 U L<1/N over K one has D\ ∩{ } 1∂ L ∂ L v ⊗ v ∂2L>δ id. 2 L − v 2· (cid:3) Lemma 3.7. A curve η: I N is a trajectory of the Euler-Lagrange flow Φ of τ → L if, and only if its graph η : t (t,η(t)) solves the Euler-Lagrange equations of τ Γ 7→ L. 10 STEFANSUHR Proof. Let η: I N be a Φ -trajectory. Consider a smooth variation H : I τ Γ ( ε,ε) R N→of η with fixed endpoints. Since H is smooth one can assum×e, Γ Γ − → × by diminishing ε if necessary, that ∂ (τ H ) > 0 everywhere. Thus one can t Γ ◦ smoothly reparameterize H to satisfy ∂ (τ H ) = 1, i.e. H consists of graphs Γ t Γ Γ ◦ of curves in N over the same interval I. This does not affect the value of on A the variation. Since any variationby graphs is a variationof the underlying curve, it follows that the first variation of η vanishes, i.e. η solves the Euler-Lagrange Γ Γ equations of L. The converse is obvious since any variation of η lifts to a variation of η . (cid:3) Γ Proposition 3.8. Φ extends unique smoothly to the boundary ∂ and the ex- τ τ D tension is complete. Further int and ∂ are Φ -invariant. τ τ τ D D Proof. BythepreviouslemmathegraphsofΦ -trajectoriessolvetheEuler-Lagrange τ equation of L. Reparameterizing to preserve L yields solutions to the Euler- Lagrange equations of L. The local Euler-Lagrange flow of L extends beyond the boundary 0. Parameterizingthosetrajectoriestosatisfydτ 1yieldstheunique C\ ≡ smooth extension. The completeness follows since (M,L) is globally hyperbolic and dτ L. The invariance follows from the fact that L is preserved along its local E−uler-≤Lagrange flow and Φ is induced by a reparameterizationof its trajectories. (cid:3) τ Denote with the action of L , i.e. for η: [s,t] N set τ τ A → t (η):= L (σ,η˙(σ))dσ R . τ τ A ∈ ∪{∞} Zs Call a pair (t,y) R N reachable from (s,x) if there exists an absolutely con- ∈ × tinuous curve η: [s,t] N from x to y with η˙ whenever the tangent exists. τ → ∈ D Note that (t,y) is reachable from (s,x) if, and only if (t,y) J+(s,x) for the cone ∈ structure . With this analogy one defines J±(s,x) and I±(s,x) according to the C definitions for cone structures. Lemma 3.9. For every (s,x) R N exists a neighborhood U R N such that ∈ × ⊆ × for every (t,y) U J+(s,x) the unique Φ -trajectory γ: [s,t] N from x to y τ ∈ ∩ → strictly minimizes among all curves η: [s,t] N from x to y. τ A → Proof. This follows from Lemma 3.7 and the observation that solutions to the Euler-Lagrange equations of L reparameterize to solutions of the Euler-Lagrange equations of L, which minimize among all causal curves, see [13]. (cid:3) A Remark 3.10. Under the identification M = R N follows: For (s,x), (t,y) ∼ × ∈ U J+(s,x) as in Lemma 3.9 and γ: [s,t] N the unique Φ -trajectory from x τ ∩ → to y one has A (γ)=L(v), τ −1 L L where v = exp (t,y) where exp denotes the exponential map with (s,x) (s,x) respecttoL.(cid:16)Further(cid:17)denotewith t(x,y)the minimalactionofacurveη: [s,t] Ss → N from x to y. Then the previous equalities and Lemma 3.9 imply t(x,y)=L(v) Ss with v as before. Proposition 3.11. Let ε > 0 and I K R N a compact subset. Then there × ⊆ × exist δ,κ > 0 and C < such that for a b c I with b a,c b ε and ∞ ≤ ≤ ∈ − − ≥ Φ -trajectories x : (a,c) N, i=1,2, with dist(x (b),x (b)) δ, x (b) K and τ i 1 2 i → ≤ ∈ dist(x˙ (b),x˙ (b))2 Cdist(x (b),x (b)) 1 2 1 2 ≥

Theory of optimal transport for Lorentzian cost functions

Similar Theory of optimal transport for Lorentzian cost functions