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Some Geometric Calculations on Wasserstein Space

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Some Geometric Calculations on Wasserstein Space Commun. Math. Phys. 277, 423–437 (2008) Communications in Digital Object Identifier (DOI) 10.1007/s00220-007-0367-3 Mathematical Physics Some Geometric Calculations on Wasserstein Space John Lott Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA. E-mail: [email protected] Received: 5 January 2007 / Accepted: 9 April 2007 Published online: 7 November 2007 – © Springer-Verlag 2007 Abstract: We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold. 1. Introduction If M is a smooth compact Riemannian manifold then the Wasserstein space P2(M) is the space of Borel probability measures on M, equipped with the Wasserstein metric W2.We refer to [21] for background information on Wasserstein spaces. The Wasserstein space originated in the study of optimal transport. It has had applications to PDE theory [16], metric geometry [8,19,20] and functional inequalities [9,17]. Otto showed that the heat flow on measures can be considered as a gradient flow on Wasserstein space [16]. In order to do this, he introduced a certain formal Riemannian metric on the Wasserstein space. This Riemannian metric has some remarkable proper- n ties. Using O’Neill’s theorem, Otto gave a formal argument that P2(R ) has nonnegative sectional curvature. This was made rigorous in [8, Theorem A.8] and [19, Prop. 2.10] in the following sense: M has nonnegative sectional curvature if and only if the length space P2(M) has nonnegative Alexandrov curvature. In this paper we study the Riemannian geometry of the Wasserstein space. In order to write meaningful expressions, we restrict ourselves to the subspace P∞(M) of absolutely continuous measures with a smooth positive density function. The space P∞(M) is a smooth infinite-dimensional manifold in the sense, for example, of [7]. The formal calculations that we perform can be considered as rigorous calculations on this smooth manifold, although we do not emphasize this point. In Sect. 3 we show that if c is a smooth immersed curve in P∞(M) then its length in P2(M), in the sense of metric geometry, equals its Riemannian length as computed with Otto’s metric. In Sect. 4 we compute the Levi-Civita connection on P∞(M).We use it to derive the equation for parallel transport and the geodesic equation. This research was partially supported by NSF grant DMS-0604829. 424 J. Lott In Sect. 5 we compute the Riemannian curvature of P∞(M). The answer is relatively simple. As an application, if M has sectional curvatures bounded below by r ∈ R, one can ask whether P∞(M) necessarily has sectional curvatures bounded below by r.This turns out to be the case if and only if r = 0. There has been recent interest in doing Hamiltonian mechanics on the Wasserstein space of a symplectic manifold [1,4,5]. In Sect. 6 we briefly describe the Poisson geo- metry of P∞(M). We show that if M is a Poisson manifold then P∞(M) has a natural Poisson structure. We also show that if M is symplectic then the symplectic leaves of the Poisson structure on P∞(M) are the orbits of the group of Hamiltonian diffeomorphisms, thereby making contact with [1,5]. This approach is not really new; closely related re- sults, with applications to PDEs, were obtained quite a while ago by Alan Weinstein and collaborators [10,11,22]. However, it may be worth advertising this viewpoint. 2. Manifolds of Measures In what follows, we use the Einstein summation convention freely. Let M be a smooth connected closed Riemannian manifold of positive dimension. We denote the Riemannian density by dvolM .LetP2(M) denote the space of Borel probability measures on M, equipped with the Wasserstein metric W2. For relevant results about optimal transport and the Wasserstein metric, we refer to [8, Sects. 1 and 2] and references therein. Put ∞ ∞ P (M) ={ρ dvolM : ρ ∈ C (M), ρ > 0, ρ dvolM = 1}. (2.1) M ∞ ∞ Then P (M) is a dense subset of P2(M), as is the complement of P (M) in P2(M). ∞ We do not claim that P (M) is necessarily a totally convex subset of P2(M), i.e. that if ∞ µ0,µ1 ∈ P (M) then the minimizing geodesic in P2(M) joining them necessarily lies in P∞(M). However, the absolutely continuous probability measures on M do form a ∞ totally convex subset of P2(M) [12]. For the purposes of this paper, we give P (M) the smooth topology. (This differs from the subspace topology on P∞(M) coming from its ∞ inclusion in P2(M).) Then P (M) has the structure of an infinite-dimensional smooth manifold in the sense of [7]. The formal calculations in this paper can be rigorously justified as being calculations on the smooth manifold P∞(M). However, we will not belabor this point. ∞ ∞ ∞ Given φ ∈ C (M), define Fφ ∈ C (P (M)) by Fφ(ρ dvolM ) = φρ dvolM . (2.2) M ∞ ∞ ∗ This gives an injection P (M) → (C (M)) , i.e. the functions Fφ separate points in ∞ ∞ P (M). We will think of the functions Fφ as “coordinates” on P (M). ∞ ∞ Given φ ∈ C (M), define a vector field Vφ on P (M) by saying that for F ∈ C∞(P∞(M)), d i (Vφ F)(ρ dvol ) = = F ρ dvol − ∇ (ρ∇ φ)dvol . (2.3) M d 0 M i M Some Geometric Calculations on Wasserstein Space 425 φ → ∞( )/R → ∞( ) The map Vφ passes to an isomorphism C M Tρ dvolM P M .This ∞( ) parametrization of Tρ dvolM P M goes back to Otto’s paper [16]; see [2] for further discussion. Otto’s Riemannian metric on P∞(M) is given [16]by , (ρ ) = ∇φ , ∇φ ρ Vφ1 Vφ2 dvolM 1 2 dvolM M i =− φ1∇ (ρ∇i φ2) dvolM . (2.4) M i In view of (2.3), we write δVφ ρ =−∇(ρ∇i φ). Then Vφ , Vφ (ρ dvol ) = φ δ φ ρ dvol = φ δ φ ρ dvol . (2.5) 1 2 M 1 V 2 M 2 V 1 M M M 2 2( ,ρ ) 1 ( ,ρ ) In terms of the weighted L -spaces L M dvolM and 2 M dvolM ,letd ∗ L be the usual differential on functions and let dρ be its formal adjoint. Then (2.4) can be written as , (ρ ) = φ , φ ρ = φ ∗ φ ρ . Vφ1 Vφ2 dvolM d 1 d 2 dvolM 1 dρ d 2 dvolM (2.6) M M We now relate the function Fφ and the vector field Vφ. Lemma 1. The gradient of Fφ is Vφ. ∞ Proof. Letting ∇ Fφ denote the gradient of Fφ, for all φ ∈ C (M) we have i ∇ Fφ, Vφ (ρ dvolM ) = (Vφ Fφ)(ρ dvolM ) =− φ ∇ (ρ∇i φ ) dvolM M =Vφ, Vφ (ρ dvolM ). (2.7) This proves the lemma. 3. Lengths of Curves In this section we relate the Riemannian metric (2.4) to the Wasserstein metric. One such relation was given in [17], where it was heuristically shown that the geodesic distance coming from (2.4) equals the Wasserstein metric. To give a rigorous relation, we recall that a curve c :[0, 1]→P2(M) has a length given by J L(c) = sup sup W2 c(t j−1), c(t j ) . (3.1) ∈N = ≤ ≤...≤ = J 0 t0 t1 tJ 1 j=1 J ( ), ( ) From the triangle inequality, the expression j=1 W2 c t j−1 c t j is nondecreasing under a refinement of the partition 0 = t0 ≤ t1 ≤ ...≤ tJ = 1. ∞ ∞ If c :[0, 1]→P (M) is a smooth curve in P (M) then we write c(t) = ρ(t) dvolM φ( ) ∂ρ =−∇i (ρ∇ φ) φ and let t satisfy dt i , where we normalize by requiring for example φρ = ∇φ( ) = that M dvolM 0. If c is immersed then t 0. The Riemannian length of c, as computed using (2.4), is 1 1 1 2 1 2 c (t), c (t) 2 dt = |∇φ(t)| (m)ρ(t) dvolM dt. (3.2) 0 0 M The next proposition says that this equals the length of c in the metric sense. 426 J. Lott Proposition 1. If c :[0, 1]→P∞(M) is a smooth immersed curve then its length L(c) in the Wasserstein space P2(M) satisfies 1 1 L(c) = c (t), c (t) 2 dt. (3.3) 0 |∇φ( )|2 ρ( ) > Proof. We can parametrize c so that M t t dvolM is a constant C 0 with respect to t. Let {St }t∈[0,1] be the one-parameter family of diffeomorphisms of M given by ∂ S (m) t = (∇φ(t))(S (m)) (3.4) ∂t t with S0(m) = m. Then c(t) = (St )∗(ρ(0) dvolM ). Given a partition 0 = t0 ≤ t1 ≤ ...≤ tJ = 1of[0, 1], a particular transference plan −1 ( − ) ( ) ◦ from c t j 1 to c t j comes from the Monge transport St j St j−1 . Then 2 −1 2 W c(t − ), c(t ) ≤ d(m, S (S (m))) ρ(t − ) dvol 2 j 1 j t j t j−1 j 1 M M = ( ( ), ( ))2 ρ( ) d St j−1 m St j m 0 dvolM M 2 t j ≤ |∇φ(t)|(St (m)) dt ρ(0) dvolM M t − j 1 t j 2 ≤ (t j − t j−1) |∇φ(t)| (St (m)) dt ρ(0) dvolM M tj−1 t j 2 = (t j − t j−1) |∇φ(t)| (m)ρ(t) dvolM dt, (3.5) t j−1 M so 1 t j 2 1 2 W2 c(t j−1), c(t j ) ≤ (t j − t j−1) 2 |∇φ(t)| (m)ρ(t) dvolM dt t j−1 M 1 2 = ( − ) |∇φ( )|2( )ρ( ) t j t j−1 t j m t j dvolM (3.6) M ∈[ , ] for some t j t j−1 t j .
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