Entropic curvature-dimension condition and Bochner s inequality

Transcription

1 Entropic curvature-dimension condition and Bochner s inequality Kazumasa Kuwada (Ochanomizu University) joint work with M. Erbar and K.-Th. Sturm (Univ. Bonn) German-Japanese conference on stochastic analysis (Univ. Leipzig) Sep. 9 13, 2013

2 1. Introduction

3 Framework (X, d, m): Polish geodesic metric measure sp., m: loc. finite, σ-finite, supp m = X, P t f = e t f: gradient curve in L 2 (m) of Cheeger s L 2 -Dirichlet energy functional Ch(f) = f 2 w dm 3 / 21

4 Framework (X, d, m): Polish geodesic metric measure sp., m: loc. finite, σ-finite, supp m = X, P t f = e t f: gradient curve in L 2 (m) of Cheeger s L 2 -Dirichlet energy functional Ch(f) = f 2 w dm = inf lim inf f n 2 dm f n : Lip. n f n f in L 2 3 / 21

5 Framework (X, d, m): Polish geodesic metric measure sp., m: loc. finite, σ-finite, supp m = X, P t f = e t f: gradient curve in L 2 (m) of Cheeger s L 2 -Dirichlet energy functional Ch(f) = f 2 w dm = inf lim inf f n 2 dm f n : Lip. n f n f in L 2 (X, d, m): infinitesimally Hilbertian def Ch: quadratic form ( Ch: Dirichlet form & P t, : linear) 3 / 21

6 Purpose/History Purpose Unify the study of Ric K and dim N in terms of optimal transportation / P t 4 / 21

7 Purpose/History Purpose Unify the study of Ric K and dim N in terms of optimal transportation / P t Established for Ric K & infin. Hilb. ( Riem. mfd.: [von Renesse & Sturm 05] mm-sp.: [Ambrosio, Gigli & Savaré et al.] 4 / 21

8 Purpose/History Purpose Unify the study of Ric K and dim N in terms of optimal transportation / P t Established for Ric K & infin. Hilb. ( Riem. mfd.: [von Renesse & Sturm 05] mm-sp.: [Ambrosio, Gigli & Savaré et al.] Study via optimal transport & Study via P t 4 / 21

9 Purpose/History Purpose Unify the study of Ric K and dim N in terms of optimal transportation / P t Established for Ric K & infin. Hilb. ( Riem. mfd.: [von Renesse & Sturm 05] mm-sp.: [Ambrosio, Gigli & Savaré et al.] Study via optimal transport & Study via P separated t 4 / 21

10 Purpose/History via P t (or ; Bochner inequality): [Bakry & Émery 84, Bakry & Ledoux 06] via Optimal transport: [Sturm 06, Lott & Villani, 09, Sturm & Bacher 10] 5 / 21

11 Purpose/History via P t (or ; Bochner inequality): [Bakry & Émery 84, Bakry & Ledoux 06] via Optimal transport: [Sturm 06, Lott & Villani, 09, Sturm & Bacher 10] Key fact for Ric K P t µ: gradient curve of Ent on (P(X), W 2 ) ( ) Ent(µ) := ρ log ρ dm (µ = ρm) X 5 / 21

12 Purpose/History via P t (or ; Bochner inequality): [Bakry & Émery 84, Bakry & Ledoux 06] via Optimal transport: [Sturm 06, Lott & Villani, 09, Sturm & Bacher 10] in terms of the Rényi entropy (NOT Ent) Key fact for Ric K P t µ: gradient curve of Ent on (P(X), W 2 ) ( ) Ent(µ) := ρ log ρ dm (µ = ρm) X 5 / 21

13 Purpose/History via P t (or ; Bochner inequality): [Bakry & Émery 84, Bakry & Ledoux 06] via Optimal transport: [Sturm 06, Lott & Villani, 09, Sturm & Bacher 10] in terms of the Rényi entropy (NOT Ent) Another approach via porous medium eq. (gradient curve of the Rényi ent.) [Ambrosio, Savaré & Mondino] 5 / 21

14 Goal Characterize Ric K & dim N by Ent Find missing conditions { optimal transport approach connecting & P t approach Establish the equivalence 6 / 21

15 Goal Characterize Ric K & dim N by Ent (K, N)-convexity of Ent Find missing conditions { optimal transport approach connecting & P t approach (K, N)-evolution variational inequality Space-time W 2 -control Establish the equivalence 6 / 21

16 Outline of the talk 1. Introduction 2. Entropic curvature-dimension condition 3. Connection with Bakry-Émery theory 4. Applications 7 / 21

17 1. Introduction 2. Entropic curvature-dimension condition 3. Connection with Bakry-Émery theory 4. Applications 7 / 21

18 Entropic curvature-dimension condition (K, N)-convexity of Ent Hess Ent 1 N Ent 2 K Hess U N K N U N, U N := exp ( 1N ) Ent 8 / 21

19 Entropic curvature-dimension condition (K, N)-convexity of Ent Hess U N K N U N, U N := exp ( 1N ) Ent 8 / 21

20 Entropic curvature-dimension condition (K, N)-convexity of Ent Hess U N K N U N, U N := exp ( 1N ) Ent Entropic curvature-dimension cond. CD e (K, N): µ 0, µ 1 P 2 (M), (µ t ) t [0,1] : W 2 -geod. s.t. U N (µ t ) σ (1 t) K/N (W 2(µ 0, µ 1 ))U N (µ 0 ) + σ (t) K/N (W 2(µ 0, µ 1 ))U N (µ 1 ), s κ (r) := sin( κr), σ (s) κ κ (r) := s κ(sr) s κ (r) 8 / 21

21 EVI K,N and Riemannian CD e (K, N) A (strong) formulation of t µ t = Ent(µ t ) (K, N)-Evolution Variational Inequality of Ent: 9 / 21

22 EVI K,N and Riemannian CD e (K, N) A (strong) formulation of t µ t = Ent(µ t ) (K, N)-Evolution Variational Inequality of Ent: µ 0, (µ t ) t 0 : abs. conti. s.t. for ν, ( ) d dt s W2 (µ t, ν) 2 ( ) W2 (µ t, ν) 2 K/N + K s K/N 2 2 N ( 1 U ) N(ν) 2 U N (µ t ) ( s κ (r) := sin( κr), U N := exp ( 1N ) ) Ent κ 9 / 21

23 CD (K, N): reduced curvature-dimension cond. [Bacher & Sturm 10] For Riem. mfds, (i) (iii) Ric K and dim N 10 / 21 EVI K,N and Riemannian CD e (K, N) Theorem 1 (RCD (K, N) cond.) For K R and N > 0, TFAE: (i) CD (K, N) & infinitesimally Hilbertian (ii) CD e (K, N) & infinitesimally Hilbertian (iii) µ 0, (µ t ) t 0 : EVI K,N -curve & (V)

24 EVI K,N and Riemannian CD e (K, N) Theorem 1 (RCD (K, N) cond.) For K R and N > 0, TFAE: (i) CD (K, N) & infinitesimally Hilbertian (ii) CD e (K, N) & infinitesimally Hilbertian (iii) µ 0, (µ t ) t 0 : EVI K,N -curve & (V) (V) X ) exp ( c d(x 0, x) 2 m(dx) < 10 / 21

25 1. Introduction 2. Entropic curvature-dimension condition 3. Connection with Bakry-Émery theory 4. Applications 11 / 21

26 Space-time W 2 -control EVI K,N -curve ( ) d dt s W2 (µ t, ν) 2 ( ) W2 (µ t, ν) 2 K/N + K s K/N 2 2 N ( 1 U ) N(ν) 2 U N (µ t ) 12 / 21

27 Space-time W 2 -control EVI K,N -curve ( ) d dt s W2 (µ t, ν) 2 ( ) W2 (µ t, ν) 2 K/N + K s K/N 2 2 N ( 1 U ) N(ν) 2 U N (µ t ) Space-time W 2 -control: ( ) W2 (µ t, ν s ) 2 ( s K/N e K(s+t) W2 (µ 0, ν 0 ) s K/N N 1 e K(s+t) 2 K(s + t) ( t s) 2 ) 2 12 / 21

28 Space-time W 2 -control EVI K,N -curve ( ) d dt s W2 (µ t, ν) 2 ( ) W2 (µ t, ν) 2 K/N + K s K/N 2 2 N ( 1 U ) N(ν) 2 U N (µ t ) Space-time W 2 -control (NOTE: µ t = P t µ 0 ): ( ) W2 (µ t, ν s ) 2 ( s K/N e K(s+t) W2 (µ 0, ν 0 ) s K/N N 1 e K(s+t) 2 K(s + t) ( t s) 2 ) 2 12 / 21

29 Bakry-Ledoux gradient estimate Space-time W 2 -control: ( ) W2 (µ t, ν s ) 2 ( s K/N e K(s+t) W2 (µ 0, ν 0 ) s K/N N 1 e K(s+t) 2 K(s + t) ( t s) 2 ) 2 13 / 21

30 Bakry-Ledoux gradient estimate Space-time W 2 -control: ( ) W2 (µ t, ν s ) 2 ( s K/N e K(s+t) W2 (µ 0, ν 0 ) s K/N N 1 e K(s+t) 2 K(s + t) ( t s) 2 Derivative in space & time ) 2 13 / 21

31 Bakry-Ledoux gradient estimate Space-time W 2 -control: ( ) W2 (µ t, ν s ) 2 ( s K/N e K(s+t) W2 (µ 0, ν 0 ) s K/N N 1 e K(s+t) 2 K(s + t) ( t s) 2 Derivative in space & time Bakry-Ledoux gradient est.: For f W 1,2, m-a.e., P t f 2 w e 2Kt P t ( f 2 w ) 2tC(t) N P tf 2, ) 2 (C(t) = 1 + O(t) (t 0)) 13 / 21

32 RCD (K, N) Bakry-Ledoux Theorem 2 For K R & N > 0, (iii) (iv) & (v) (iii) µ 0, EVI K,N -curve (µ t ) t & (V) (iv) Space-time W 2 -control (v) Bakry-Ledoux gradient estimate 14 / 21

33 Bakry-Ledoux Bochner Theorem 3 Suppose (M, d, m): infinitesimally Hilbertian For K R & N > 0, (v) (vi) (v) P t f 2 w e 2Kt P t ( f 2 w ) 2C(t) N P tf 2 (vi) f W 1,2 with f W 1,2 & g D( ) L with g 0 & g L ( ) 1 M 2 g f 2 w g f, f dm g (K f 2w + 1N ) f 2 dm M 15 / 21

34 Bakry-Ledoux Bochner Theorem 3 Suppose (M, d, m): infinitesimally Hilbertian For K R & N > 0, (v) (vi) (v) P t f 2 w e 2Kt P t ( f 2 w ) 2C(t) N P tf 2 (vi) f W 1,2 with f W 1,2 & g D( ) L with g 0 & g L ( ) 1 M 2 g f 2 w g f, f dm g (K f 2w + 1N ) f 2 dm M 15 / 21

35 Theorem 4 K R, N > 0 Summary (1) TFAE (i) CD (K, N) & infin. Hilb. (ii) CD e (K, N) & infin. Hilb. (iii) EVI K,N -curves & (V) (2) Under (M, d, m): infin. Hilb. & Ass. 1 below, either (iv) (vi) is also equiv. to (i) (iii) (iv) Space-time W 2 -control (v) Bakry-Ledoux gradient estimate (vi) Bochner inequality 16 / 21

36 Assumption 1 The volume growth bound (V) f w 1 m-a.e. f: 1-Lip. Remark (M, d, m): infin. Hilb., Ass. 1 & P t f 2 w e 2Kt P t ( f 2 w ) CD(K, ) (Bakry-Émery s gradient estimate) [Ambrosio, Gigli & Savaré] 17 / 21

37 1. Introduction 2. Entropic curvature-dimension condition 3. Connection with Bakry-Émery theory 4. Applications 18 / 21

38 Properties of RCD (K, N) Stability under mgh (or Sturm s D)-conv. Tensorization From local to global Measure contraction property MCP(K, N) (via CD (K, N); [Cavalletti & Sturm 12]) (sharp) Bishop-Gromov volume comparison volume doubling property (local unif.) Poincaré ineq. [Rajala 11] heat kernel, two-sided Gaussian bound Ultracontractivity of P t 19 / 21

39 Properties of RCD (K, N) N-precision of f nal ineq. s N-HWI ineq. N-log Sobolev ineq. N-Talagrand ineq. Lipschitz regularity of the heat kernel/eigenfn. s Lichnerowicz bound of λ 1 [Ketterer] Li-Yau s ineq. [Garofalo & Mondino] 20 / 21

40 Examples N log s K/N (x) or N log s K/N (x) are (K, N)-convex fn s on R (or an interval) When (M, d, e V vol): cpl. Riem. mfd, M =, RCD (K, N) 1 Ric + Hess V N m V 2 K (m = dim M N) More generally, (M, d, m): RCD (K, N) & V : (K, N )-convex on M (M, d, e V m): RCD (K + K, N + N ) 21 / 21