Discrete Ricci curvature via convexity of the entropy

Transcription

1 Discrete Ricci curvature via convexity of the entropy Jan Maas University of Bonn Joint work with Matthias Erbar Simons Institute for the Theory of Computing UC Berkeley 2 October 2013

2 Starting point McCann 94: beautiful connection between

3 Starting point McCann 94: beautiful connection between Boltzmann-Shannon entropy

4 Starting point McCann 94: beautiful connection between Boltzmann-Shannon entropy 2-Wasserstein metric from optimal transport

5 Starting point McCann 94: beautiful connection between Boltzmann-Shannon entropy 2-Wasserstein metric from optimal transport Relative entropy Let m be a reference measure on X. ρ(x) log ρ(x) dm(x), For µ P(X ): Ent m (µ) = X +, dµ dm = ρ, otherwise.

6 Optimal transport and entropy The optimal transport problem (with quadratic cost)

7 Optimal transport and entropy The optimal transport problem (with quadratic cost) Let (X, d) be a Polish space and let µ, ν P(X ).

8 Optimal transport and entropy The optimal transport problem (with quadratic cost) Let (X, d) be a Polish space and let µ, ν P(X ). Definition of the 2-Wasserstein metric { W 2 (µ, ν) 2 := inf d(x, y) 2 dγ(x, y) : X X } γ with marginals µ and ν

9 Optimal transport and entropy The optimal transport problem (with quadratic cost) Let (X, d) be a Polish space and let µ, ν P(X ). Definition of the 2-Wasserstein metric { W 2 (µ, ν) 2 := inf d(x, y) 2 dγ(x, y) : X X } γ with marginals µ and ν { } = inf E[d(X, Y ) 2 ] : law(x ) = µ, law(y ) = ν

10 Optimal transport and entropy The optimal transport problem (with quadratic cost) Let (X, d) be a Polish space and let µ, ν P(X ). Definition of the 2-Wasserstein metric { W 2 (µ, ν) 2 := inf d(x, y) 2 dγ(x, y) : X X } γ with marginals µ and ν { } = inf E[d(X, Y ) 2 ] : law(x ) = µ, law(y ) = ν Properties W 2 defines a metric on P 2 (X ). (X, d) is a geodesic space (P 2 (X ), W 2 ) is a geodesic space.

11 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ).

12 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Construction of geodesics in the 2-Wasserstein space P 2 (R n )

13 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Construction of geodesics in the 2-Wasserstein space P 2 (R n ) Let γ be an optimal coupling of µ and ν

14 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Construction of geodesics in the 2-Wasserstein space P 2 (R n ) Let γ be an optimal coupling of µ and ν If µ is a.c., then γ is supported on the graph of a function Ψ : R n R n (by Brenier s Theorem).

15 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Construction of geodesics in the 2-Wasserstein space P 2 (R n ) Let γ be an optimal coupling of µ and ν If µ is a.c., then γ is supported on the graph of a function Ψ : R n R n (by Brenier s Theorem). For t [0, 1], set Ψ t (x) := (1 t)x + tψ(x).

16 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Construction of geodesics in the 2-Wasserstein space P 2 (R n ) Let γ be an optimal coupling of µ and ν If µ is a.c., then γ is supported on the graph of a function Ψ : R n R n (by Brenier s Theorem). For t [0, 1], set Ψ t (x) := (1 t)x + tψ(x). Let µ t be the image measure of µ under Ψ t.

17 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Construction of geodesics in the 2-Wasserstein space P 2 (R n ) Let γ be an optimal coupling of µ and ν If µ is a.c., then γ is supported on the graph of a function Ψ : R n R n (by Brenier s Theorem). For t [0, 1], set Ψ t (x) := (1 t)x + tψ(x). Let µ t be the image measure of µ under Ψ t. Then: (µ t ) is a constant speed geodesic from µ to ν.

18 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Construction of geodesics in the 2-Wasserstein space P 2 (R n ) Let γ be an optimal coupling of µ and ν If µ is a.c., then γ is supported on the graph of a function Ψ : R n R n (by Brenier s Theorem). For t [0, 1], set Ψ t (x) := (1 t)x + tψ(x). Let µ t be the image measure of µ under Ψ t. Then: (µ t ) is a constant speed geodesic from µ to ν. Highly non-linear interpolation based on geometry of the underlying space!

19 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ).

20 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Theorem (Otto Villani 00, Cordero McCann Schmuckenschläger 01, von Renesse Sturm 05) For a Riemannian manifold M, TFAE: 1. Displacement κ-convexity of the entropy: Ent(µ t ) (1 t)ent(µ 0 ) + tent(µ 1 ) κ 2 t(1 t)w 2 2 (µ 0, µ 1 ) for all 2-Wasserstein geodesics (µ t ) t [0,1] ;

21 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Theorem (Otto Villani 00, Cordero McCann Schmuckenschläger 01, von Renesse Sturm 05) For a Riemannian manifold M, TFAE: 1. Displacement κ-convexity of the entropy: Ent(µ t ) (1 t)ent(µ 0 ) + tent(µ 1 ) κ 2 t(1 t)w 2 2 (µ 0, µ 1 ) for all 2-Wasserstein geodesics (µ t ) t [0,1] ; 2. Ric κ everywhere on M.

22 Optimal transport and entropy Theorem (McCann 94) The entropy is convex along geodesics in (P 2 (R n ), W 2 ). Theorem (Otto Villani 00, Cordero McCann Schmuckenschläger 01, von Renesse Sturm 05) For a Riemannian manifold M, TFAE: 1. Displacement κ-convexity of the entropy: Entropy Ent(µ t ) (1 t)ent(µ 0 ) + tent(µ 1 ) κ 2 t(1 t)w 2 2 (µ 0, µ 1 ) for all 2-Wasserstein geodesics (µ t ) t [0,1] ; 2. Ric κ everywhere on M.

23 Optimal transport and Ricci curvature II Definition (Sturm 06, Lott Villani 09) A metric measure space (X, d, m) satisfies Ric(X ) κ if Ent(µ t ) (1 t) Ent(µ 0 ) + t Ent(µ 1 ) κ 2 t(1 t)w 2 2 (µ 0, µ 1 ) along all W 2 -geodesics (µ t ) t [0,1] in P(X ).

24 Optimal transport and Ricci curvature II Definition (Sturm 06, Lott Villani 09) A metric measure space (X, d, m) satisfies Ric(X ) κ if Ent(µ t ) (1 t) Ent(µ 0 ) + t Ent(µ 1 ) κ 2 t(1 t)w 2 2 (µ 0, µ 1 ) along all W 2 -geodesics (µ t ) t [0,1] in P(X ). Crucial features

25 Optimal transport and Ricci curvature II Definition (Sturm 06, Lott Villani 09) A metric measure space (X, d, m) satisfies Ric(X ) κ if Ent(µ t ) (1 t) Ent(µ 0 ) + t Ent(µ 1 ) κ 2 t(1 t)w 2 2 (µ 0, µ 1 ) along all W 2 -geodesics (µ t ) t [0,1] in P(X ). Crucial features Applicable to a wide class of metric measure spaces

26 Optimal transport and Ricci curvature II Definition (Sturm 06, Lott Villani 09) A metric measure space (X, d, m) satisfies Ric(X ) κ if Ent(µ t ) (1 t) Ent(µ 0 ) + t Ent(µ 1 ) κ 2 t(1 t)w 2 2 (µ 0, µ 1 ) along all W 2 -geodesics (µ t ) t [0,1] in P(X ). Crucial features Applicable to a wide class of metric measure spaces Many geometric, analytic and probabilistic consequences logarithmic Sobolev, Talagrand, Poincaré inequalities; Brunn-Minkowski.

27 Optimal transport and Ricci curvature II Definition (Sturm 06, Lott Villani 09) A metric measure space (X, d, m) satisfies Ric(X ) κ if Ent(µ t ) (1 t) Ent(µ 0 ) + t Ent(µ 1 ) κ 2 t(1 t)w 2 2 (µ 0, µ 1 ) along all W 2 -geodesics (µ t ) t [0,1] in P(X ). Crucial features Applicable to a wide class of metric measure spaces Many geometric, analytic and probabilistic consequences logarithmic Sobolev, Talagrand, Poincaré inequalities; Brunn-Minkowski. Stability under measured Gromov Hausdorff convergence

28 Optimal transport and Ricci curvature II Definition (Sturm 06, Lott Villani 09) A metric measure space (X, d, m) satisfies Ric(X ) κ if Ent(µ t ) (1 t) Ent(µ 0 ) + t Ent(µ 1 ) κ 2 t(1 t)w 2 2 (µ 0, µ 1 ) along all W 2 -geodesics (µ t ) t [0,1] in P(X ). Crucial features Applicable to a wide class of metric measure spaces Many geometric, analytic and probabilistic consequences logarithmic Sobolev, Talagrand, Poincaré inequalities; Brunn-Minkowski. Stability under measured Gromov Hausdorff convergence But... what about discrete spaces?

29 What about discrete spaces? Example: 2-point space X = {0, 1}.

30 What about discrete spaces? Example: 2-point space X = {0, 1}. Set µ α := (1 α)δ 0 + αδ 1 for α [0, 1].

31 What about discrete spaces? Example: 2-point space X = {0, 1}. Set µ α := (1 α)δ 0 + αδ 1 for α [0, 1]. Then: W 2 (µ α, µ β ) = α β.

32 What about discrete spaces? Example: 2-point space X = {0, 1}. Set µ α := (1 α)δ 0 + αδ 1 for α [0, 1]. Then: W 2 (µ α, µ β ) = α β. Suppose that ( µ α(t) ) is a constant speed geodesic.

33 What about discrete spaces? Example: 2-point space X = {0, 1}. Set µ α := (1 α)δ 0 + αδ 1 for α [0, 1]. Then: W 2 (µ α, µ β ) = α β. Suppose that ( µ α(t) ) is a constant speed geodesic. Then: W 2 (µ α(t), µ α(s) ) = c t s.

34 What about discrete spaces? Example: 2-point space X = {0, 1}. Set µ α := (1 α)δ 0 + αδ 1 for α [0, 1]. Then: W 2 (µ α, µ β ) = α β. Suppose that ( µ α(t) ) is a constant speed geodesic. Then: α(t) α(s) = W2 (µ α(t), µ α(s) ) = c t s.

35 What about discrete spaces? Example: 2-point space X = {0, 1}. Set µ α := (1 α)δ 0 + αδ 1 for α [0, 1]. Then: W 2 (µ α, µ β ) = α β. Suppose that ( µ α(t) ) is a constant speed geodesic. Then: α(t) α(s) = W2 (µ α(t), µ α(s) ) = c t s. (α(t)) is 2-Hölder, hence constant.

36 What about discrete spaces? Example: 2-point space X = {0, 1}. Set µ α := (1 α)δ 0 + αδ 1 for α [0, 1]. Then: W 2 (µ α, µ β ) = α β. Suppose that ( µ α(t) ) is a constant speed geodesic. Then: α(t) α(s) = W2 (µ α(t), µ α(s) ) = c t s. (α(t)) is 2-Hölder, hence constant. Thus: there are no non-trivial W 2 -geodesics. In fact:

37 What about discrete spaces? Example: 2-point space X = {0, 1}. Set µ α := (1 α)δ 0 + αδ 1 for α [0, 1]. Then: W 2 (µ α, µ β ) = α β. Suppose that ( µ α(t) ) is a constant speed geodesic. Then: α(t) α(s) = W2 (µ α(t), µ α(s) ) = c t s. (α(t)) is 2-Hölder, hence constant. Thus: there are no non-trivial W 2 -geodesics. In fact: (P 2 (X ), W 2 ) is a geodesic space (X, d) is a geodesic space.

38 What about discrete spaces? Example: 2-point space X = {0, 1}. Set µ α := (1 α)δ 0 + αδ 1 for α [0, 1]. Then: W 2 (µ α, µ β ) = α β. Suppose that ( µ α(t) ) is a constant speed geodesic. Then: α(t) α(s) = W2 (µ α(t), µ α(s) ) = c t s. (α(t)) is 2-Hölder, hence constant. Thus: there are no non-trivial W 2 -geodesics. In fact: (P 2 (X ), W 2 ) is a geodesic space (X, d) is a geodesic space. Conclusion: LSV-Definition does not apply to discrete spaces.

39 Ricci curvature of discrete spaces Many approaches to discrete Ricci curvature: W 1 -contraction à la Dobrushin Ollivier ( 09); Sammer ( 05), Joulin ( 09), Jost, Bauer, Hua, Liu ( ) approximate W 2 -geodesics Bonciocat, Sturm ( 09), Ollivier, Villani ( 12) modified Bakry-Émery criterion Lin, Lu, S.-T. Yau ( ), Bauer, Horn, Lin, Lippner, Mangoubi, S.-T. Yau ( 13) discrete displacement interpolation Gozlan, Roberto, Samson, Tetali ( 12) [see next lecture!], Hillion ( 12)

40 Ricci curvature of discrete spaces Many approaches to discrete Ricci curvature: W 1 -contraction à la Dobrushin Ollivier ( 09); Sammer ( 05), Joulin ( 09), Jost, Bauer, Hua, Liu ( ) approximate W 2 -geodesics Bonciocat, Sturm ( 09), Ollivier, Villani ( 12) modified Bakry-Émery criterion Lin, Lu, S.-T. Yau ( ), Bauer, Horn, Lin, Lippner, Mangoubi, S.-T. Yau ( 13) discrete displacement interpolation Gozlan, Roberto, Samson, Tetali ( 12) [see next lecture!], Hillion ( 12) Our goal: Find a notion of discrete Ricci curvature in the spirit of Lott-Sturm-Villani which allows to prove sharp functional inequalities.

41 Why 2-Wasserstein?

42 Why 2-Wasserstein? Theorem [Jordan Kinderlehrer Otto 98] The heat flow is the gradient flow of the entropy w.r.t W 2

43 Why 2-Wasserstein? Theorem [Jordan Kinderlehrer Otto 98] The heat flow is the gradient flow of the entropy w.r.t W 2 How to make sense of gradient flows in metric spaces?

44 Why 2-Wasserstein? Theorem [Jordan Kinderlehrer Otto 98] The heat flow is the gradient flow of the entropy w.r.t W 2 How to make sense of gradient flows in metric spaces? Gradient flows in R n Let ϕ : R n R smooth and convex. For u : R + R n TFAE: 1. u solves the gradient flow equation u (t) = ϕ(u(t)). 2. u solves the evolution variational inequality 1 d 2 dt u(t) y 2 ϕ(y) ϕ(u(t)) y. (De Giorgi 93, Ambrosio Gigli Savaré 05)

45 Why 2-Wasserstein? Theorem [Jordan Kinderlehrer Otto 98] The heat flow is the gradient flow of the entropy w.r.t W 2, i.e., t µ = µ 1 d 2 dt W 2(µ t, ν) 2 Ent(ν) Ent(µ t ) ν. How to make sense of gradient flows in metric spaces? Gradient flows in R n Let ϕ : R n R smooth and convex. For u : R + R n TFAE: 1. u solves the gradient flow equation u (t) = ϕ(u(t)). 2. u solves the evolution variational inequality 1 d 2 dt u(t) y 2 ϕ(y) ϕ(u(t)) y. (De Giorgi 93, Ambrosio Gigli Savaré 05)

46 Heat flow as gradient flow of the entropy Many extensions have been proved: R n Jordan Kinderlehrer Otto Riemannian manifolds Villani, Erbar Hilbert spaces Ambrosio Savaré Zambotti Finsler spaces Ohta Sturm Wiener space Fang Shao Sturm Heisenberg group Juillet Alexandrov spaces Gigli Kuwada Ohta Metric measures spaces Ambrosio Gigli Savaré

47 Heat flow as gradient flow of the entropy Many extensions have been proved: R n Jordan Kinderlehrer Otto Riemannian manifolds Villani, Erbar Hilbert spaces Ambrosio Savaré Zambotti Finsler spaces Ohta Sturm Wiener space Fang Shao Sturm Heisenberg group Juillet Alexandrov spaces Gigli Kuwada Ohta Metric measures spaces Ambrosio Gigli Savaré Question Is there a version of the JKO-Theorem for discrete spaces?

48 Setting Discrete setting X : finite set K : X X R + Markov kernel, x : y K(x, y) = 1 π: reversible measure, x, y : K(x, y)π(x) = K(y, x)π(y)

49 Setting Discrete setting X : finite set K : X X R + Markov kernel, x : y K(x, y) = 1 π: reversible measure, x, y : K(x, y)π(x) = K(y, x)π(y) Heat flow Discrete Laplacian: ψ(x) := y K(x, y)(ψ(y) ψ(x)) Continuous time Markov semigroup: P t = e t

50 Setting Discrete setting X : finite set K : X X R + Markov kernel, x : y K(x, y) = 1 π: reversible measure, x, y : K(x, y)π(x) = K(y, x)π(y) Heat flow Discrete Laplacian: ψ(x) := y K(x, y)(ψ(y) ψ(x)) Continuous time Markov semigroup: P t = e t Relative Entropy { P(X ) := ρ : X R + } x X ρ(x)π(x) = 1, Ent(ρ) := x X ρ(x) log ρ(x) π(x), ρ P(X ).

51 Simplest non-trivial example: 2-point space X = {0, 1} K(0, 1) = K(1, 0) = 1 π(0) = π(1) = 1 2

52 Simplest non-trivial example: 2-point space X = {0, 1} K(0, 1) = K(1, 0) = 1 π(0) = π(1) = 1 2 Recall the notation: µ α = (1 α)δ 0 + αδ 1.

53 Simplest non-trivial example: 2-point space X = {0, 1} K(0, 1) = K(1, 0) = 1 π(0) = π(1) = 1 2 Recall the notation: µ α = (1 α)δ 0 + αδ 1. Question Is the heat flow on {0, 1} the gradient flow of the entropy w.r.t. the L 2 -Wasserstein metric?

54 Simplest non-trivial example: 2-point space X = {0, 1} K(0, 1) = K(1, 0) = 1 π(0) = π(1) = 1 2 Recall the notation: µ α = (1 α)δ 0 + αδ 1. Question Is the heat flow on {0, 1} the gradient flow of the entropy w.r.t. the L 2 -Wasserstein metric? Answer NO! Reason: W 2 (µ α, µ β ) = α β.

55 Simplest non-trivial example: 2-point space X = {0, 1} K(0, 1) = K(1, 0) = 1 π(0) = π(1) = 1 2 Recall the notation: µ α = (1 α)δ 0 + αδ 1. Question Is the heat flow on {0, 1} the gradient flow of the entropy w.r.t. some other metric on P({ 1, 1})?

56 Simplest non-trivial example: 2-point space X = {0, 1} K(0, 1) = K(1, 0) = 1 π(0) = π(1) = 1 2 Recall the notation: µ α = (1 α)δ 0 + αδ 1. Question Is the heat flow on {0, 1} the gradient flow of the entropy w.r.t. some other metric on P({ 1, 1})? Answer YES!

57 Simplest non-trivial example: 2-point space X = {0, 1} K(0, 1) = K(1, 0) = 1 π(0) = π(1) = 1 2 Recall the notation: µ α = (1 α)δ 0 + αδ 1. Question Is the heat flow on {0, 1} the gradient flow of the entropy w.r.t. some other metric on P({ 1, 1})? Answer YES! Proposition [M. 2011] The heat flow is the gradient flow of Ent w.r.t. the metric W, where W(µ α, µ β ) := β arctanh(2r 1) 2 dr, 0 α β 1. α 2r 1

58 Simplest non-trivial example: 2-point space X = {0, 1} K(0, 1) = K(1, 0) = 1 π(0) = π(1) = 1 2 Recall the notation: µ α = (1 α)δ 0 + αδ 1. Question Is the heat flow on {0, 1} the gradient flow of the entropy w.r.t. some other metric on P({ 1, 1})? Answer YES! Proposition [M. 2011] The heat flow is the gradient flow of Ent w.r.t. the metric W, where W(µ α, µ β ) := β arctanh(2r 1) 2 dr, 0 α β 1. α 2r 1 How to generalise this to the general discrete case?

59 Back to R n : W 2 as Riemannian metric (Otto 01) If t ρ t is smooth, the continuity equation t ρ + div(ρψ) = 0 holds for some velocity vector field Ψ(t, x).

60 Back to R n : W 2 as Riemannian metric (Otto 01) If t ρ t is smooth, the continuity equation t ρ + div(ρψ) = 0 holds for some velocity vector field Ψ(t, x). Ψ t ρ0 ρ t ρ 1

61 Back to R n : W 2 as Riemannian metric (Otto 01) If t ρ t is smooth, the continuity equation t ρ + div(ρψ) = 0 holds for some velocity vector field Ψ(t, x). For a.e. t, a unique gradient Ψ t = ψ t solving cont.eq. Ψ t ρ0 ρ t ρ 1

62 Back to R n : W 2 as Riemannian metric (Otto 01) If t ρ t is smooth, the continuity equation t ρ + div(ρψ) = 0 holds for some velocity vector field Ψ(t, x). For a.e. t, a unique gradient Ψ t = ψ t solving cont.eq. Regard ψ t as tangent vector at ρ t. Ψ t ρ0 ρ t ρ 1

63 Back to R n : W 2 as Riemannian metric (Otto 01) If t ρ t is smooth, the continuity equation t ρ + div(ρψ) = 0 holds for some velocity vector field Ψ(t, x). For a.e. t, a unique gradient Ψ t = ψ t solving cont.eq. Regard ψ t as tangent vector at ρ t. For ρ P(R n ), define an inner product by ϕ, ψ ρ = ϕ(x), ψ(x) ρ(x)dx. R n Ψ t ρ0 ρ t ρ 1

64 Back to R n : W 2 as Riemannian metric (Otto 01) If t ρ t is smooth, the continuity equation t ρ + div(ρψ) = 0 holds for some velocity vector field Ψ(t, x). For a.e. t, a unique gradient Ψ t = ψ t solving cont.eq. Regard ψ t as tangent vector at ρ t. For ρ P(R n ), define an inner product by ϕ, ψ ρ = ϕ(x), ψ(x) ρ(x)dx. R n Associated Riemannian distance (Benamou-Brenier formula) { 1 inf ψ t 2 ρ t dt : t ρ + div(ρ ψ) = 0, ρ,ψ 0 } ρ t=0 = ρ 0, ρ t=1 = ρ 1.

65 Back to R n : W 2 as Riemannian metric (Otto 01) If t ρ t is smooth, the continuity equation t ρ + div(ρψ) = 0 holds for some velocity vector field Ψ(t, x). For a.e. t, a unique gradient Ψ t = ψ t solving cont.eq. Regard ψ t as tangent vector at ρ t. For ρ P(R n ), define an inner product by ϕ, ψ ρ = ϕ(x), ψ(x) ρ(x)dx. R n Associated Riemannian distance (Benamou-Brenier formula) { 1 W 2 (ρ 0, ρ 1 ) 2 = inf ψ t 2 ρ t dt : t ρ + div(ρ ψ) = 0, ρ,ψ 0 } ρ t=0 = ρ 0, ρ t=1 = ρ 1.

66 Definition of the metric W Benamou-Brenier formula in R n { 1 } W2 2(ρ 0, ρ 1 ) = inf ψ t (x) 2 ρ t (x) dx dt ρ,ψ 0 R n s.t. t ρ+div(ρ ψ) = 0 and ρ t=0 = ρ 0, ρ t=1 = ρ 1.

67 Definition of the metric W Benamou-Brenier formula in R n { 1 } W2 2(ρ 0, ρ 1 ) = inf ψ t (x) 2 ρ t (x) dx dt ρ,ψ 0 R n s.t. t ρ+div(ρ ψ) = 0 and ρ t=0 = ρ 0, ρ t=1 = ρ 1. Definition in the discrete case (M. 2011) W(ρ 0, ρ { 1 ) 2 } := inf ρ,ψ s.t.

68 Definition of the metric W Benamou-Brenier formula in R n { 1 } W2 2(ρ 0, ρ 1 ) = inf ψ t (x) 2 ρ t (x) dx dt ρ,ψ 0 R n s.t. t ρ+div(ρ ψ) = 0 and ρ t=0 = ρ 0, ρ t=1 = ρ 1. Definition in the discrete case (M. 2011) W(ρ 0, ρ { 1 ) 2 1 := inf ρ,ψ 0 } dt s.t.

69 Definition of the metric W Benamou-Brenier formula in R n { 1 } W2 2(ρ 0, ρ 1 ) = inf ψ t (x) 2 ρ t (x) dx dt ρ,ψ 0 R n s.t. t ρ+div(ρ ψ) = 0 and ρ t=0 = ρ 0, ρ t=1 = ρ 1. Definition in the discrete case (M. 2011) W(ρ 0, ρ { 1 ) 2 1 := inf ρ,ψ x,y X } K(x, y)π(x) dt s.t.

70 Definition of the metric W Benamou-Brenier formula in R n { 1 } W2 2(ρ 0, ρ 1 ) = inf ψ t (x) 2 ρ t (x) dx dt ρ,ψ 0 R n s.t. t ρ+div(ρ ψ) = 0 and ρ t=0 = ρ 0, ρ t=1 = ρ 1. Definition in the discrete case (M. 2011) W(ρ 0, ρ { 1 ) 2 1 := inf ρ,ψ 2 s.t. 1 0 x,y X } (ψ t (x) ψ t (y)) 2 K(x, y)π(x) dt

71 Definition of the metric W Benamou-Brenier formula in R n { 1 } W2 2(ρ 0, ρ 1 ) = inf ψ t (x) 2 ρ t (x) dx dt ρ,ψ 0 R n s.t. t ρ+div(ρ ψ) = 0 and ρ t=0 = ρ 0, ρ t=1 = ρ 1. Definition in the discrete case (M. 2011) W(ρ 0, ρ { 1 ) 2 1 := inf ρ,ψ 2 s.t. 1 0 x,y X } (ψ t (x) ψ t (y)) 2 K(x, y)π(x) dt Problem: ρ is defined on vertices, ψ is defined on edges No canonical way to define the product ρ ψ!

72 Definition of the metric W Benamou-Brenier formula in R n { 1 } W2 2(ρ 0, ρ 1 ) = inf ψ t (x) 2 ρ t (x) dx dt ρ,ψ 0 R n s.t. t ρ+div(ρ ψ) = 0 and ρ t=0 = ρ 0, ρ t=1 = ρ 1. Definition in the discrete case (M. 2011) W(ρ 0, ρ { 1 ) 2 1 := inf ρ,ψ 2 s.t. 1 0 x,y X } (ψ t (x) ψ t (y)) 2 K(x, y)π(x) dt Use the logarithmic mean as the density on an edge! ˆρ(x, y) := 1 0 ρ(x) 1 α ρ(y) α dα = ρ(x) ρ(y) log ρ(x) log ρ(y)

73 Definition of the metric W Benamou-Brenier formula in R n { 1 } W2 2(ρ 0, ρ 1 ) = inf ψ t (x) 2 ρ t (x) dx dt ρ,ψ 0 R n s.t. t ρ+div(ρ ψ) = 0 and ρ t=0 = ρ 0, ρ t=1 = ρ 1. Definition in the discrete case (M. 2011) W(ρ 0, ρ { 1 ) 2 1 := inf ρ,ψ 2 s.t. 1 0 x,y X } (ψ t (x) ψ t (y)) 2 ˆρ t (x, y)k(x, y)π(x) dt Use the logarithmic mean as the density on an edge! ˆρ(x, y) := 1 0 ρ(x) 1 α ρ(y) α dα = ρ(x) ρ(y) log ρ(x) log ρ(y)

74 Definition of the metric W Benamou-Brenier formula in R n { 1 } W2 2(ρ 0, ρ 1 ) = inf ψ t (x) 2 ρ t (x) dx dt ρ,ψ 0 R n s.t. t ρ+div(ρ ψ) = 0 and ρ t=0 = ρ 0, ρ t=1 = ρ 1. Definition in the discrete case (M. 2011) W(ρ 0, ρ { 1 ) 2 1 := inf ρ,ψ 2 s.t. 1 0 x,y X } (ψ t (x) ψ t (y)) 2 ˆρ t (x, y)k(x, y)π(x) dt d dt ρ t(x) + ˆρ t (x, y)(ψ t (x) ψ t (y))k(x, y) = 0 y X x Use the logarithmic mean as the density on an edge! ˆρ(x, y) := 1 0 ρ(x) 1 α ρ(y) α dα = ρ(x) ρ(y) log ρ(x) log ρ(y)

75 Basic properties of W (M. 11) Results W defines a metric on P(X ).

76 Results Basic properties of W (M. 11) W defines a metric on P(X ). W is induced by a Riemannian structure on P >0 (X ).

77 Results Basic properties of W (M. 11) W defines a metric on P(X ). W is induced by a Riemannian structure on P >0 (X ). The tangent space at ρ is the set of discrete gradients with ψ 2 ρ = 1 ( ) 2 ψ(x) ψ(y) ˆρ(x, y)k(x, y)π(x). 2 x,y X

78 Results Basic properties of W (M. 11) W defines a metric on P(X ). W is induced by a Riemannian structure on P >0 (X ). The tangent space at ρ is the set of discrete gradients with ψ 2 ρ = 1 ( ) 2 ψ(x) ψ(y) ˆρ(x, y)k(x, y)π(x). 2 x,y X Theorem [Discrete JKO] (M. 11) The heat flow is the gradient flow of the entropy w.r.t. W.

79 Results Basic properties of W (M. 11) W defines a metric on P(X ). W is induced by a Riemannian structure on P >0 (X ). The tangent space at ρ is the set of discrete gradients with ψ 2 ρ = 1 ( ) 2 ψ(x) ψ(y) ˆρ(x, y)k(x, y)π(x). 2 x,y X Theorem [Discrete JKO] (M. 11) The heat flow is the gradient flow of the entropy w.r.t. W. Independent works by Chow Huang Li Zhou 12 and Mielke 11.

80 Why the logarithmic mean? Formal proof of the JKO-Theorem 1. If (ρ t, ψ t ) satisfy the cont. eq. t ρ + div(ρ ψ) = 0, then d dt Ent(ρ t) = log ρ t, div(ρ t ψ t ) = log ρ t, ρ t ψ t. grad W2 Ent(ρ) = log ρ

81 Why the logarithmic mean? Formal proof of the JKO-Theorem 1. If (ρ t, ψ t ) satisfy the cont. eq. t ρ + div(ρ ψ) = 0, then d dt Ent(ρ t) = log ρ t, div(ρ t ψ t ) = log ρ t, ρ t ψ t. grad W2 Ent(ρ) = log ρ 2. If ρ t solves the heat equation in R n, then t ρ = div( ρ) = div(ρ ψ). provided ψ = log ρ. Tangent vector along the heat flow is log ρ.

82 Why the logarithmic mean? Formal proof of the JKO-Theorem 1. If (ρ t, ψ t ) satisfy the cont. eq. t ρ + div(ρ ψ) = 0, then d dt Ent(ρ t) = log ρ t, div(ρ t ψ t ) = log ρ t, ρ t ψ t. grad W2 Ent(ρ) = log ρ 2. If ρ t solves the heat equation in R n, then t ρ = div( ρ) = div(ρ ψ). provided ψ = log ρ. Tangent vector along the heat flow is log ρ. Logarithmic mean compensates for the lack of a discrete chain rule: ρ(x) ρ(y) = ˆρ(x, y) ( log ρ(x) log ρ(y) )

83 Ricci curvature of Markov chains Discrete analogue of Lott Sturm Villani: Definition (Erbar, M. 2012) We say that (X, K, π) has Ricci curvature bounded from below by κ R if the entropy is κ-convex along geodesics in (P(X ), W). Ent P(X) ϱ 1 ϱ 1/2 ϱ 0

84 Consequences: Sharp functional inequalities I Let (X, K, π) be a reversible Markov chain.

85 Consequences: Sharp functional inequalities I Let (X, K, π) be a reversible Markov chain. Discrete analogue of the Fisher information I(ρ) = 1 (ρ(x) ρ(y)) (log ρ(x) log ρ(y)) K(x, y)π(x) 2 x,y

86 Consequences: Sharp functional inequalities I Let (X, K, π) be a reversible Markov chain. Discrete analogue of the Fisher information I(ρ) = 1 (ρ(x) ρ(y)) (log ρ(x) log ρ(y)) K(x, y)π(x) 2 x,y Discrete Bakry-Émery Theorem (Erbar, M. 12) If Ric(K) κ > 0, then the modified log-sobolev inequality holds: Ent(ρ) 1 2κ I(ρ) (mlsi(κ)) for all ρ P >0 (X ).

87 Consequences: Sharp functional inequalities I Let (X, K, π) be a reversible Markov chain. Discrete analogue of the Fisher information I(ρ) = 1 (ρ(x) ρ(y)) (log ρ(x) log ρ(y)) K(x, y)π(x) 2 x,y Discrete Bakry-Émery Theorem (Erbar, M. 12) If Ric(K) κ > 0, then the modified log-sobolev inequality holds: Ent(ρ) 1 2κ I(ρ) (mlsi(κ)) for all ρ P >0 (X ). mlsi(κ) has been extensively studied in discrete settings (Bobkov Tetali 06,... )

88 Consequences: Sharp functional inequalities I Let (X, K, π) be a reversible Markov chain. Discrete analogue of the Fisher information I(ρ) = 1 (ρ(x) ρ(y)) (log ρ(x) log ρ(y)) K(x, y)π(x) 2 x,y Discrete Bakry-Émery Theorem (Erbar, M. 12) If Ric(K) κ > 0, then the modified log-sobolev inequality holds: Ent(ρ) 1 2κ I(ρ) (mlsi(κ)) for all ρ P >0 (X ). mlsi(κ) has been extensively studied in discrete settings (Bobkov Tetali 06,... ) mlsi(κ) is equivalent to the exponential decay estimate Ent(P t ρ) e 2κt Ent(ρ).

89 Consequences: Sharp functional inequalities II Let (X, K, π) be a reversible Markov chain. Let κ > 0. Discrete Otto-Villani Theorem (Erbar, M. 12) If mlsi(κ) holds, then the modified Talagrand inequality holds, i.e., W(ρ, 1) 2 2 κ Ent(ρ). (mtal(κ))

90 Consequences: Sharp functional inequalities II Let (X, K, π) be a reversible Markov chain. Let κ > 0. Discrete Otto-Villani Theorem (Erbar, M. 12) If mlsi(κ) holds, then the modified Talagrand inequality holds, i.e., W(ρ, 1) 2 2 κ Ent(ρ). (mtal(κ)) The analogous inequality with W 2 never holds in discrete settings!

91 Consequences: Sharp functional inequalities II Let (X, K, π) be a reversible Markov chain. Let κ > 0. Discrete Otto-Villani Theorem (Erbar, M. 12) If mlsi(κ) holds, then the modified Talagrand inequality holds, i.e., W(ρ, 1) 2 2 κ Ent(ρ). (mtal(κ)) The analogous inequality with W 2 never holds in discrete settings! If mtal(κ) holds, then the Poincaré inequality holds: ψ(x) 2 π(x) 1 (ψ(x) ψ(y)) 2 K(x, y)π(x) 2κ x x,y whenever x ψ(x)π(x) = 0.

92 Consequences: Sharp functional inequalities II Let (X, K, π) be a reversible Markov chain. Let κ > 0. Discrete Otto-Villani Theorem (Erbar, M. 12) If mlsi(κ) holds, then the modified Talagrand inequality holds, i.e., W(ρ, 1) 2 2 κ Ent(ρ). (mtal(κ)) The analogous inequality with W 2 never holds in discrete settings! If mtal(κ) holds, then the Poincaré inequality holds: ψ(x) 2 π(x) 1 (ψ(x) ψ(y)) 2 K(x, y)π(x) 2κ x whenever x ψ(x)π(x) = 0. the T 1 -inequality holds: W 1 (ρ, 1) 2 1 κ Ent(ρ). x,y

93 Ricci bounds: examples

94 Theorem (Mielke 2012) Ricci bounds: examples For every finite reversible Markov chain: κ R such that Ric(K) κ.

95 Theorem (Mielke 2012) Ricci bounds: examples For every finite reversible Markov chain: κ R such that Ric(K) κ. Finite volume discretisations of Fokker-Planck equations in 1D.

96 Theorem (Mielke 2012) Ricci bounds: examples For every finite reversible Markov chain: κ R such that Ric(K) κ. Finite volume discretisations of Fokker-Planck equations in 1D. Theorem (Erbar, M. 2012) Let (X i, K i, π i ) be reversible finite Markov chains and let (X, K, π) be the product chain. Then: Ric(X i, K i, π i ) κ i = Ric(X, K, π) 1 n min κ i i

97 Theorem (Mielke 2012) Ricci bounds: examples For every finite reversible Markov chain: κ R such that Ric(K) κ. Finite volume discretisations of Fokker-Planck equations in 1D. Theorem (Erbar, M. 2012) Let (X i, K i, π i ) be reversible finite Markov chains and let (X, K, π) be the product chain. Then: Ric(X i, K i, π i ) κ i = Ric(X, K, π) 1 n min κ i i Dimension-independent bounds

98 Theorem (Mielke 2012) Ricci bounds: examples For every finite reversible Markov chain: κ R such that Ric(K) κ. Finite volume discretisations of Fokker-Planck equations in 1D. Theorem (Erbar, M. 2012) Let (X i, K i, π i ) be reversible finite Markov chains and let (X, K, π) be the product chain. Then: Ric(X i, K i, π i ) κ i = Ric(X, K, π) 1 n min κ i i Dimension-independent bounds Sharp bounds for the discrete hypercube { 1, 1} n

99 Gromov-Hausdorff convergence Let T d N = (Z/NZ)d be the discrete torus. Let W N be the normalised transportation metric for simple random walk on T d N.

100 Gromov-Hausdorff convergence Let T d N = (Z/NZ)d be the discrete torus. Let W N be the normalised transportation metric for simple random walk on T d N. Theorem (Gigli, M. 2012) (P(T d N ), W N) (P(T d ), W 2 ) in the sense of Gromov Hausdorff.

101 Gromov-Hausdorff convergence Let T d N = (Z/NZ)d be the discrete torus. Let W N be the normalised transportation metric for simple random walk on T d N. Theorem (Gigli, M. 2012) (P(T d N ), W N) (P(T d ), W 2 ) in the sense of Gromov Hausdorff. Compatibility between W 2 and W.

102 Gromov-Hausdorff convergence Let T d N = (Z/NZ)d be the discrete torus. Let W N be the normalised transportation metric for simple random walk on T d N. Theorem (Gigli, M. 2012) (P(T d N ), W N) (P(T d ), W 2 ) in the sense of Gromov Hausdorff. Compatibility between W 2 and W. Main ingredient for proving convergence of gradient flows.

103 Further developments Closely related gradient flow structures have been discovered for

104 Further developments Closely related gradient flow structures have been discovered for Systems of chemical reactions (Mielke) non-linear generalisation of continuous time Markov chains

105 Further developments Closely related gradient flow structures have been discovered for Systems of chemical reactions (Mielke) non-linear generalisation of continuous time Markov chains Non-local equations on general state spaces (Erbar) fractional heat equations

106 Further developments Closely related gradient flow structures have been discovered for Systems of chemical reactions (Mielke) non-linear generalisation of continuous time Markov chains Non-local equations on general state spaces (Erbar) fractional heat equations Discrete porous medium equations (Erbar-M.) allows for structure-preserving discretisations of PDEs

107 Further developments Closely related gradient flow structures have been discovered for Systems of chemical reactions (Mielke) non-linear generalisation of continuous time Markov chains Non-local equations on general state spaces (Erbar) fractional heat equations Discrete porous medium equations (Erbar-M.) allows for structure-preserving discretisations of PDEs Dissipative quantum mechanics (Carlen-M., Mielke) non-commutative analogue of W for density matrices

108 Thank you!