Conference in honor of Professor Amari Riemannian interpretation of - PowerPoint PPT Presentation

Conference in honor of Professor Amari Riemannian interpretation of Wasserstein geometry Felix Otto Max Planck Institute for Mathematics in the Sciences Leipzig, Germany

Arnol’d ’66: Geometrization of fluid dynamics Euler’s equations for incompressible inviscid fluid, x ∈ M = T d : u = u ( t, x ) ∈ R d ∇ · u = 0 , Eulerian velocity ∂ t u + u · ∇ u + ∇ p = 0 , p = p ( t, x ) ∈ R pressure (Formal) Riemannian manifold: M := { Φ diffeomorphism | Φ# dx = dx } ⊂ L 2 ( T d , R d ) For curve Φ( t, · ) in M , consider vector field u ( t, · ) given by ∂ t Φ( t, · ) = u ( t, · ) ◦ Φ( t, · ), then Φ is geodesic in M ⇐ ⇒ u satisfies Euler’s equations

Arnol’d ’66: an easy calculation Euler’s equations: ∇ · u = 0, ∂ t u + u · ∇ u + ∇ p = 0. M := { Φ diffeomorphism | Φ# dx = dx } = { Φ diffeomorphism | det D Φ ≡ 1 } ⊂ L 2 ( T d , R d ). Φ is geodesic in M ⇐ ⇒ u satisfies Euler’s equations, where ∂ t Φ ( t ) = u ( t ) ◦ Φ ( t ) . Liouville: ∂ t det D Φ ( t ) = ( ∇ · u ) ( t ) ◦ φ ( t ) det D Φ ( t ) Acceler. Lagrange vs Euler : ∂ 2 t Φ ( t ) = ( ∂ t + u · ∇ u ) ( t ) ◦ Φ ( t ) .

Arnol’d ’66: curvature can get very negative ... u satisfies Euler’s equations ⇐ ⇒ Φ is geodesic in M where ∂ t Φ( t ) = u ( t ) ◦ Φ( t ). M := { Φ diffeom. | Φ# dx = dx } = { Φ diffeom. | det D Φ ≡ 1 } ⊂ L 2 ( T d , R d ). Tangent space in Φ: T Φ M = { u ◦ Φ |∇· u = 0 } � = { u |∇ · u = 0 } Liouville: ∂ t det D Φ( t ) = ( ∇ · u )( t ) ◦ φ ( t ) det D Φ( t ) Sectional curvature of M in plane u 1 − u 2 � A ( u 1 , u 1 ) · A ( u 2 , u 2 ) − | A ( u 1 , u 2 ) | 2 dx R Φ ( u 1 , u 2 ) = where A ( u, u ) := ∇ p with p solving ∇· ( u ·∇ u + ∇ p ) = 0 ... geodesics diverge, effective unpredictability of Euler

Brenier ’91: Projection onto M ... M = ( R d , dµ ) so that M := { Φ diffeomorphism | Φ# dµ = dµ } ⊂ L 2 µ ( R d , R d ). Given g ∈ L 2 µ ( R d , R d ) consider inf Φ ∈M � Φ − g � L 2 µ . Existence & uniqueness, solution is of the form g = ∇ ψ ◦ Φ ψ convex. with multi − d � 1 − d : amounts to monotone rearrangement nonlinear � linear : amounts to Helmholtz projection ... � “polar factorization”

Brenier ’91: Connection to optimal transportation Set ρ := g # µ , then Φ ∈M � Φ − g � 2 inf L 2 µ � � � � � � Φ: R d → R d , Φ# µ = µ R d | g − Φ | 2 dµ � = inf � � � � R d | Ψ ( x ) − x | 2 µ ( dx ) � = inf � � Ψ: R d → R d , Ψ# µ = ρ Monge � � � � � R d × R d | x − y | 2 π ( dxdy ) � = inf � π has marginals µ, ρ � � � � � 2 | y | 2 − ϕ ( y ) ) ρ ( dy ) + 2 | x | 2 − ψ ( x ) ) µ ( dx ) ( 1 ( 1 � = sup � � ψ, ϕ : R d → R , ϕ ( y ) + ψ ( x ) ≥ x · y Kantorowicz = W 2 ( ρ, µ ) Wasserstein metric

McCann ’97: displacement convexity M = R d . For densities ρ 1 and ρ 0 related via ρ 1 = Ψ# ρ 0 with Ψ = ∇ ψ , ψ convex , see Brenier consider curve ρ s := ( s Ψ + (1 − s ) id)# ρ 0 , s ∈ [0 , 1]. It is a metric geodesic in arc length wrt Wasserstein: W ( ρ 0 , ρ s ) = sW ( ρ 0 , ρ 1 ) and W ( ρ s , ρ 1 ) = (1 − s ) W ( ρ 0 , ρ 1 ) � Consider functional on densities ρ of form E ( ρ ) := R d U ( ρ ) dx . If U such that (0 , ∞ ) ∋ λ �→ λ d U ( λ − d ) convex & decreasing then E is convex along these geodesics 1 since A symmetric positive semi-definite �→ (det A ) is concave d

Barenblatt ’52: nonlinear diffusions ∂ t ρ − △ ρ m = 0, Fix m > 0. Consider ρ ( t, x ) ≥ 0 solution of � ρdx = 1. wlog 1 ρ ∗ ( x Admits self-similar solution ρ ∗ ( t, x ) = t dα ˆ t α ) 1 with α := 2+( m − 1) d . ρ ∗ describes asymptotic behavior of any solution ρ : x ) t ↑∞ t dα ρ ( t, t α ˆ → ˆ ρ ∗ (ˆ x ) Friedman & Kamin ’80 based on Caffarelli & Friedman ’79

Otto ’01: Formal Riemannian structure on space of probability measures � = { ρ : M → [0 , ∞ ) | M ρdx = 1 } with metric tensor P � � g ρ ( δρ 1 , δρ 2 ) = M ∇ ϕ 1 · ∇ ϕ 2 dρ where ϕ i solves elliptic equation −∇ · ρ ∇ ϕ i = δρ i Connection to Arnol’d for M = T d : The map Π: L 2 ( T d , R d ) → P , Φ �→ ρ = Φ# dx is Riemannian submersion, Π − 1 { dx } = M . Sectional curvature of P in plane ∇ ϕ 1 , ∇ ϕ 2 � T d | [ ∇ ϕ 1 , ∇ ϕ 2 ] − ∇ p | 2 dρ R ρ ( ∇ ϕ 1 , ∇ ϕ 2 ) = where p solves ∇ · ρ ([ ∇ ϕ 1 , ∇ ϕ 2 ] − ∇ p ) = 0 (O’Neill formula) . Note R ≥ 0 and ≡ 0 if and only if d = 1.

Connections to Brenier and McCann � ρdx = 1 } endowed with = { ρ : M → [0 , ∞ ) | P � � g ρ ( δρ 1 , δρ 2 ) = M ∇ ϕ 1 · ∇ ϕ 2 dρ where ϕ i solves −∇ · ρ ∇ ϕ i = δρ i Connection to Brenier for M = R d : Wasserstein distance W = induced distance on P (Benamou-Brenier ’00) Connection to McCann for M = R d : displacement convexity = (geodesic) convexity

Nonlinear diffusion = contraction in Wasserstein Connection to Barenblatt for M = R d : nonlinear diffusion ∂ t ρ − △ ρ m = 0 is gradient flow on P   m − 1 ρ m m � = 1 1 �   of E ( ρ ) = R d U ( ρ ) dx with U ( ρ ) :=   ρ ln ρ m = 1 (Jordan-Kinderlehrer-O.’97) λ �→ λ d U ( λ − d ) convex m ≥ 1 − 1 ⇐ ⇒ ⇐ ⇒ E convex on P d Hence if ρ i , i = 1 , 2, solve ∂ t ρ i − △ ρ m i = 0 then d dtW 2 ( ρ 1( t, · ) , ρ 2( t, · ) ) ≤ 0. � ρ ∗ ) ≤ t − 2 α R d | x | 2 dρ ( t = 0) In particular W ( t dα ρ ( t, t d · ) , ˆ

Connections with Ricci curvature Theorem. M (compact) d -dim. Riemannian manifold with Ric ≥ 0. For m ≥ 1 − 1 ∂ t ρ i − △ ρ m d consider i = 0 , i = 1 , 2 . d dtW 2 ( ρ 1( t, · ) , ρ 2( t, · ) ) ≤ 1. Then O.’01 for M = R d , O.&Villani ’00 for general M , m = 1 (heuristics), ager’01 , Cordero&McCann&Schmuckenschl¨ Sturm&v.Renesse ’05 for general M , m = 1 (necessity), O.&Westdickenberg ’05

Calculus from differential geometry Generalize to ∂ t ρ − △ π ( ρ ) = 0. Induced distance � energy of curves: Given one-parameter family { ρ ( s, · ) } s ∈ [0 , 1] of solutions ∂ t ρ ( s, · ) −△ π ( ρ ( s, · )) = 0. � 1 d Show 0 g ρ ( s, · ) ( ∂ s ρ ( s, · ) , ∂ s ρ ( s, · )) ds ≤ 0. dt Infinitesimal version: Suppose ∂ t ρ − △ π ( ρ ) = 0 and ∂ t δρ − △ ( π ′ ( ρ ) δρ ) = 0 . d Show dtg ρ ( δρ, δρ ) ≤ 0.

Reduction to single formula Infinitesimal version: Suppose ∂ t ρ − △ π ( ρ ) = 0 and ∂ t δρ − △ ( π ′ ( ρ ) δρ ) = 0 . d Show dtg ρ ( δρ, δρ ) ≤ 0. ∂ t δρ − △ ( π ′ ( ρ ) δρ ) = 0 Explicit formula: For ∂ t ρ − △ π ( ρ ) = 0 , and δρ = −∇ · ( ρ ∇ ϕ ) have � 1 d 2 |∇ ϕ | 2 dρ dt � ( ρπ ′ ( ρ ) − π ( ρ )) ( △ ϕ ) 2 + π ( ρ ) ( | D 2 ϕ | 2 + ∇ ϕ · Ric ∇ ϕ ) dx = − Use ( △ ϕ ) 2 ≤ d | D 2 ϕ | 2 , need ρπ ′ ( ρ ) − π ( ρ ) ≥ 1 d π ( ρ ) ≥ 0

An easy calculation � 1 d 2 |∇ ϕ | 2 dρ eliminate ∂ t ∇ ϕ dt � ϕ∂ t δρ − 1 2 |∇ ϕ | 2 ∂ t ρdx = eliminate ∂ t δρ, ∂ t ρ � π ′ ( ρ ) δρ △ ϕ − π ( ρ ) △ 1 2 |∇ ϕ | 2 dx = eliminate δρ � ρπ ′ ( ρ ) ( △ ϕ ) 2 + π ( ρ ) ( △ 1 2 |∇ ϕ | 2 − ∇ · ( △ ϕ ∇ ϕ )) dx = − 2 |∇ ϕ | 2 − ∇ · ( △ ϕ ∇ ϕ ) △ 1 Use Bochner’s formula = | D 2 ϕ | 2 + ∇ ϕ · Ric ∇ ϕ − ( △ ϕ ) 2 Reminiscent of Γ 2 -calculus of Bakry-Emery ’84

Past – present Use Wasserstein contraction to give “synthetic” definition of Ric ≥ 0 on metric spaces M (Sturm, Lott-Villani, Ambrosio-Gigli-Savar´ e, ...) Connections with Ricci flow (McCann-Topping, ...) Regularity of Brenier map on smooth manifolds M (Caffarelli+, Trudinger+, Kim, Loeper, Figalli+, ...) Large deviation principle of underlying particle system selects the good gradient flow structure (Dawson&G¨ artner, Peletier, Mielke, ...)