Radial processes on RCD(K, N) spaces

Transcription

1 Radial processes on RCD(K, N) spaces Kazumasa Kuwada (Tohoku University) joint work with K. Kuwae (Fukuoka University) Geometric Analysis on Smooth and Nonsmooth Spaces SISSA, Jun. 2017

2 1. Introduction

3 Radial process on R m (Z t ) t 0 : a stochastic process on a metric space (X, d) d(x 0, Z t ): radial process Example (B t ) t 0 : Brownian motion on R m generated by m 2 r t := d(0, B t ) solves r t = r 0 + 2β t + (m 1) t 0 ds r s (β t : 1-dim. standard BM) ( m-dimensional Bessel process) 3 / 30

4 Radial process on R m (Z t ) t 0 : a stochastic process on a metric space (X, d) d(x 0, Z t ): radial process Example (B t ) t 0 : Brownian motion on R m generated by m 2 r t := d(0, B t ) solves r t = r 0 + 2β t + (m 1) t 0 ds r s (β t : 1-dim. standard BM) ( m-dimensional Bessel process) 3 / 30

5 Radial process on R m (Z t ) t 0 : a stochastic process on a metric space (X, d) d(x 0, Z t ): radial process Example (B t ) t 0 : Brownian motion on R m generated by m 2 r t := d(0, B t ) solves r t = r 0 + 2β t + (m 1) t 0 ds r s (β t : 1-dim. standard BM) ( m-dimensional Bessel process) 3 / 30

6 Radial process on R m (Z t ) t 0 : a stochastic process on a metric space (X, d) d(x 0, Z t ): radial process Example (B t ) t 0 : Brownian motion on R m generated by m 2 r t := d(0, B t ) solves r t = r 0 + 2β t + (m 1) t 0 ds r s (β t : 1-dim. standard BM) ( m-dimensional Bessel process) 3 / 30

7 Radial process on R m (Z t ) t 0 : a stochastic process on a metric space (X, d) d(x 0, Z t ): radial process Example (B t ) t 0 : Brownian motion on R m generated by m 2 r t := d(0, B t ) solves r t = r 0 + 2β t + (m 1) t 0 ds r s (β t : 1-dim. standard BM) ( m-dimensional Bessel process) B t : m-dim. proc. r t : 1-dim. proc. 3 / 30

8 Local time Example (B t ) t 0 : BM on R generated by r t := d(0, B t ) solves L t = 2 t 0 r t = 2β t + L t δ 0 (r s )ds : local time at 0 of r t 4 / 30

9 Local time Example (B t ) t 0 : BM on R generated by r t := d(0, B t ) solves L t = 2 t 0 r t = 2β t + L t δ 0 (r s )ds : local time at 0 of r t 4 / 30

10 Local time Example (B t ) t 0 : BM on R generated by r t := d(0, B t ) solves L t = 2 t 0 r t = 2β t + L t δ 0 (r s )ds : local time at 0 of r t 4 / 30

11 Local time Example (B t ) t 0 : BM on R generated by r t := d(0, B t ) solves L t = 2 t 0 r t = 2β t + L t δ 0 (r s )ds : local time at 0 of r t L t 0,, increases only on {s 0 r s = 0} 4 / 30

12 Radial proc. on Riem. mfds Example (B t ) t 0 : BM on a cpl. Riem. mfd of dim = m 2 d x0 (x) := d(x 0, x) r t := d x0 (B t ) solves r t = 2β t + t 0 d x0 (B s )ds L c t [Kendall 87/Cranston, Kendall & March 93] L c t : local time at cut locus of x 0 5 / 30

13 Radial proc. on Riem. mfds Example (B t ) t 0 : BM on a cpl. Riem. mfd of dim = m 2 d x0 (x) := d(x 0, x) r t := d x0 (B t ) solves r t = 2β t + t 0 d x0 (B s )ds L c t [Kendall 87/Cranston, Kendall & March 93] L c t : local time at cut locus of x 0 5 / 30

14 Radial proc. on Riem. mfds Example (B t ) t 0 : BM on a cpl. Riem. mfd of dim = m 2 d x0 (x) := d(x 0, x) r t := d x0 (B t ) solves r t = 2β t + t 0 d x0 (B s )ds L c t [Kendall 87/Cranston, Kendall & March 93] L c t : local time at cut locus of x 0 5 / 30

15 Radial proc. on Riem. mfds Example (B t ) t 0 : BM on a cpl. Riem. mfd of dim = m 2 d x0 (x) := d(x 0, x) r t := d x0 (B t ) solves r t = 2β t + t 0 d x0 (B s )ds L c t [Kendall 87/Cranston, Kendall & March 93] L c t : local time at cut locus of x 0 L c t 0,, increases only on {s B s cut(x 0 )} {s B s cut(x 0 )} = 0 a.s. 5 / 30

16 Radial proc. on Riem. mfds Example (B t ) t 0 : BM on a cpl. Riem. mfd of dim = m 2 d x0 (x) := d(x 0, x) r t := d x0 (B t ) solves r t = 2β t + t 0 d x0 (B s )ds L c t [Kendall 87/Cranston, Kendall & March 93] L c t : local time at cut locus of x 0 L c t 0,, increases only on {s B s cut(x 0 )} {s B s cut(x 0 )} = 0 a.s. 5 / 30

17 Laplacian comparison thm Comparison theorems A lower Ricci bound An upper bound of d x0 6 / 30

18 Laplacian comparison thm Comparison theorems A lower Ricci bound An upper bound of d x0 Ric K r t ρ t a.s. ρ t : radial proc. on (K, m)-spaceform driven by β t 6 / 30

19 Laplacian comparison thm Comparison theorems A lower Ricci bound An upper bound of d x0 Ric K r t ρ t a.s. ρ t : radial proc. on (K, m)-spaceform driven by β t Applications Stochastic completeness Exit probability of B t from a ball Heat kernel comparison theorem Liouville type theorem 6 / 30

20 Laplacian comparison thm Comparison theorems A lower Ricci bound An upper bound of d x0 Ric K r t ρ t a.s. ρ t : radial proc. on (K, m)-spaceform driven by β t Applications Stochastic completeness Exit probability of B t from a ball Heat kernel comparison theorem Liouville type theorem 6 / 30

21 Laplacian comparison thm Comparison theorems A lower Ricci bound An upper bound of d x0 Ric K r t ρ t a.s. ρ t : radial proc. on (K, m)-spaceform driven by β t Applications Stochastic completeness Exit probability of B t from a ball Heat kernel comparison theorem Liouville type theorem 6 / 30

22 Laplacian comparison thm Comparison theorems A lower Ricci bound An upper bound of d x0 Ric K r t ρ t a.s. ρ t : radial proc. on (K, m)-spaceform driven by β t Applications Stochastic completeness Exit probability of B t from a ball Heat kernel comparison theorem Liouville type theorem 6 / 30

23 Laplacian comparison thm Comparison theorems A lower Ricci bound An upper bound of d x0 Ric K r t ρ t a.s. ρ t : radial proc. on (K, m)-spaceform driven by β t Applications Stochastic completeness Exit probability of B t from a ball Heat kernel comparison theorem Liouville type theorem 6 / 30

24 Expression of r t Goal A corresponding expression of r t on RCD(K, N) sp. s (met. meas. sp. s with Ric K and dim N ) 7 / 30

25 Expression of r t Goal A corresponding expression of r t on RCD(K, N) sp. s (met. meas. sp. s with Ric K and dim N ) Outline of the proof on Riem. mfd d(b t, cut(x 0 )) > ε: Itô formula to d x0 (B t ) d(b t, cut(x 0 )) ε: Change the ref. pt. from x 0 ε 0 7 / 30

26 Expression of r t Goal A corresponding expression of r t on RCD(K, N) sp. s (met. meas. sp. s with Ric K and dim N ) Outline of the proof on Riem. mfd d(b t, cut(x 0 )) > ε: Itô formula to d x0 (B t ) d(b t, cut(x 0 )) ε: Change the ref. pt. from x 0 ε 0 Difficulty Lack of usual differentiable structure d x0 & cut(x 0 ) can be wilder 7 / 30

27 Outline of the talk 1. Introduction 2. Framework and main result 3. Outline of the proof 4. Local structure

28 1. Introduction 2. Framework and main result 3. Outline of the proof 4. Local structure

29 Met. meas. sp. & Cheeger energy (X, d, m): Polish geod. met. meas. sp. (m: loc.-finite, supp m = X) BM P t = e t Cheeger s L 2 -energy 9 / 30

30 Met. meas. sp. & Cheeger energy (X, d, m): Polish geod. met. meas. sp. (m: loc.-finite, supp m = X) BM P t = e t Cheeger s L 2 -energy { } 2Ch(f):=inf lim lip(f n ) 2 dm f n : Lip. n f n f in L 2 X 9 / 30

31 Met. meas. sp. & Cheeger energy (X, d, m): Polish geod. met. meas. sp. (m: loc.-finite, supp m = X) BM P t = e t Cheeger s L 2 -energy { } 2Ch(f):=inf lim lip(f n ) 2 dm f n : Lip. n X f n f in L 2 = Df 2 w dm X 9 / 30

32 Met. meas. sp. & Cheeger energy (X, d, m): Polish geod. met. meas. sp. (m: loc.-finite, supp m = X) BM P t = e t Cheeger s L 2 -energy { } 2Ch(f):=inf lim lip(f n ) 2 dm f n : Lip. n X f n f in L 2 = Df 2 w dm X Definition 1 ([Ambrosio, Gigli & Savaré 14]) (X, d, m): infinitesimally Hilbertian def Ch: quadratic form ( P t : linear : linear) 9 / 30

33 Met. meas. sp. & Cheeger energy (X, d, m): Polish geod. met. meas. sp. BM P t = e t Cheeger s L 2 -energy { } 2Ch(f):=inf lim lip(f n ) 2 dm f n : Lip. n X f n f in L 2 = Df 2 w dm X Definition 1 ([Ambrosio, Gigli & Savaré 14]) (X, d, m): infinitesimally Hilbertian def Ch: quadratic form ( P t : linear : linear) D, D w bilinear s.t. Df, Df w = Df 2 w 9 / 30

34 RCD(K, N) spaces (N < ) RCD(K, N): infin. Hilb., some regularity ass ns & 1 2 Df 2 w Df, D f w K Df 2 w + 1 N f 2 10 / 30

35 RCD(K, N) spaces (N < ) RCD(K, N): infin. Hilb., some regularity ass ns & 1 2 Df 2 w Df, D f w K Df 2 w + 1 N f 2 Original definition: a convexity of some entropy f nal on (P 2 (X), W 2 ) 10 / 30

36 RCD(K, N) spaces (N < ) RCD(K, N): infin. Hilb., some regularity ass ns & 1 2 Df 2 w Df, D f w K Df 2 w + 1 N f 2 Original definition: a convexity of some entropy f nal on (P 2 (X), W 2 ) Without R : [Sturm 06] [Lott & Villani 09] [Bacher & Sturm 10] N = : [Ambrosio, Gigli & Savaré 14] [Ambrosio, Gigli, Mondino & Rajala 15] (Equiv.): [Ambrosio, Gigli & Savaré 15] N < : [Gigli 15][Gigli] (Equiv.): [Erbar, K. & Sturm 15] [Ambrosio, Mondino & Savare] [Cavalletti & Milman] 10 / 30

37 RCD(K, N) spaces (N < ) RCD(K, N): infin. Hilb., some regularity ass ns & 1 2 Df 2 w Df, D f w K Df 2 w + 1 N f 2 Original definition: a convexity of some entropy f nal on (P 2 (X), W 2 ) Without R : [Sturm 06] [Lott & Villani 09] [Bacher & Sturm 10] N = : [Ambrosio, Gigli & Savaré 14] [Ambrosio, Gigli, Mondino & Rajala 15] (Equiv.): [Ambrosio, Gigli & Savaré 15] N < : [Gigli 15][Gigli] (Equiv.): [Erbar, K. & Sturm 15] [Ambrosio, Mondino & Savare] [Cavalletti & Milman] 10 / 30

38 RCD(K, N) spaces (N < ) RCD(K, N): infin. Hilb., some regularity ass ns & 1 2 Df 2 w Df, D f w K Df 2 w + 1 N f 2 Original definition: a convexity of some entropy f nal on (P 2 (X), W 2 ) Without R : [Sturm 06] [Lott & Villani 09] [Bacher & Sturm 10] N = : [Ambrosio, Gigli & Savaré 14] [Ambrosio, Gigli, Mondino & Rajala 15] (Equiv.): [Ambrosio, Gigli & Savaré 15] N < : [Gigli 15][Gigli] (Equiv.): [Erbar, K. & Sturm 15] [Ambrosio, Mondino & Savare] [Cavalletti & Milman] 10 / 30

39 Brownian motion and heat kernel (X, d): loc. cpt. ( Bishop-Gromov ineq.) E(f, f) := 2Ch(f), F := D(Ch)( L 2 (m)) (E, F): reg. str. loc. Dirichlet form on L 2 (m) (E, F) ((B t ) t 0, (P x ) x X ): diffusion process ( Brownian motion ) P t admits a heat kernel p t (x, y), c d(x, y)2 p t (x, y) exp ( m(b t (x)) Ct ) + λt [Jiang, Li & Zhang 16] 11 / 30

40 Brownian motion and heat kernel (X, d): loc. cpt. ( Bishop-Gromov ineq.) E(f, f) := 2Ch(f), F := D(Ch)( L 2 (m)) (E, F): reg. str. loc. Dirichlet form on L 2 (m) (E, F) ((B t ) t 0, (P x ) x X ): diffusion process ( Brownian motion ) P t admits a heat kernel p t (x, y), c d(x, y)2 p t (x, y) exp ( m(b t (x)) Ct ) + λt [Jiang, Li & Zhang 16] 11 / 30

41 Brownian motion and heat kernel (X, d): loc. cpt. ( Bishop-Gromov ineq.) E(f, f) := 2Ch(f), F := D(Ch)( L 2 (m)) (E, F): reg. str. loc. Dirichlet form on L 2 (m) (E, F) ((B t ) t 0, (P x ) x X ): diffusion process ( Brownian motion ) P t admits a heat kernel p t (x, y), c d(x, y)2 p t (x, y) exp ( m(b t (x)) Ct ) + λt [Jiang, Li & Zhang 16] 11 / 30

42 Brownian motion and heat kernel (X, d): loc. cpt. ( Bishop-Gromov ineq.) E(f, f) := 2Ch(f), F := D(Ch)( L 2 (m)) (E, F): reg. str. loc. Dirichlet form on L 2 (m) (E, F) ((B t ) t 0, (P x ) x X ): diffusion process ( Brownian motion ) P t admits a heat kernel p t (x, y), c d(x, y)2 p t (x, y) exp ( m(b t (x)) Ct ) + λt [Jiang, Li & Zhang 16] 11 / 30

43 Fukushima decomposition d x0 C Lip (X) F loc C(X) d x0 (B t ) d x0 (B 0 ) = M t + N t P x -a.s. q.e. x X [ ] Mt : martingale additive f nal locally of finite energy N t : continuous additive f nal locally of zero energy 12 / 30

44 Fukushima decomposition d x0 C Lip (X) F loc C(X) d x0 (B t ) d x0 (B 0 ) = M t + N t P x -a.s. q.e. x X [ ] Mt : martingale additive f nal locally of finite energy N t : continuous additive f nal locally of zero energy Q. M t = 2β t, N t = t 0 d x0 (B s )ds L c t? 12 / 30

45 Fukushima decomposition d x0 C Lip (X) F loc C(X) d x0 (B t ) d x0 (B 0 ) = M t + N t P x -a.s. q.e. x X [ ] Mt : martingale additive f nal locally of finite energy N t : continuous additive f nal locally of zero energy Q. M t = 2β t, N t = Q. Can q.e. be removed? t 0 d x0 (B s )ds L c t? (q.e.: exceptional set of zero capacity) 12 / 30

46 Revuz correspondence Positive continuous additive functional A t Smooth measure µ S For f, h 0 & t > 0, [ t ] E x f(b s )da s h(x)m(dx) X 0 t ( ) = fp s h dµ ds Example A t = t 0 ϕ(b s )ds µ = ϕm 0 X 13 / 30

47 Revuz correspondence Positive continuous additive functional A t Smooth measure µ S For f, h 0 & t > 0, [ t ] E x f(b s )da s h(x)m(dx) X 0 t ( ) = fp s h dµ ds Example A t = t 0 ϕ(b s )ds µ = ϕm 0 X 13 / 30

48 Revuz correspondence Positive continuous additive functional A t Smooth measure µ S For f, h 0 & t > 0, [ t ] E x f(b s )da s h(x)m(dx) X 0 t ( ) = fp s h dµ ds Example A t = t 0 ϕ(b s )ds µ = ϕm 0 X 13 / 30

49 Revuz correspondence II r t r 0 = M t + N t M t µ dx0, f dµ ϕ := 2E(ϕ, fϕ) E(ϕ 2, f) X µ dx0 = 2 Dd x0 2 w m = 2m E(d x0, v) = X v d ν 14 / 30

50 Revuz correspondence II r t r 0 = M t + N t M t µ dx0, f dµ ϕ := 2E(ϕ, fϕ) E(ϕ 2, f) X µ dx0 = 2 Dd x0 2 w m = 2m E(d x0, v) = X v d ν 14 / 30

51 Revuz correspondence II r t r 0 = M t + N t M t µ dx0, f dµ ϕ := 2E(ϕ, fϕ) E(ϕ 2, f) X µ dx0 = 2 Dd x0 2 w m = 2m E(d x0, v) = X v d ν 14 / 30

52 Revuz correspondence II r t r 0 = M t + N t M t µ dx0, f dµ ϕ := 2E(ϕ, fϕ) E(ϕ 2, f) X µ dx0 = 2 Dd x0 2 w m = 2m E(d x0, v) = X v d ν 14 / 30

53 Revuz correspondence II r t r 0 = M t + N t M t µ dx0, f dµ ϕ := 2E(ϕ, fϕ) E(ϕ 2, f) X µ dx0 = 2 Dd x0 2 w m = 2m M t = 2β t (Lévy s thm) E(d x0, v) = v d ν X 14 / 30

54 Revuz correspondence II r t r 0 = M t + N t M t µ dx0, f dµ ϕ := 2E(ϕ, fϕ) E(ϕ 2, f) X µ dx0 = 2 Dd x0 2 w m = 2m M t = 2β t (Lévy s thm) E(d x0, v) = v d ν X 14 / 30

55 Revuz correspondence II r t r 0 = M t + N t M t µ dx0, f dµ ϕ := 2E(ϕ, fϕ) E(ϕ 2, f) X µ dx0 = 2 Dd x0 2 w m = 2m M t = 2β t (Lévy s thm) E(d x0, v) = v d ν N t = A ν t X 14 / 30

56 Laplacian comparison thm κ := K N 1, s κ(r) := sin( κr), ct κ := s κ (r) κ s κ (r) 15 / 30

57 Laplacian comparison thm κ := K N 1, s κ(r) := sin( κr), ct κ := s κ (r) κ s κ (r) Theorem 2 ([Gigli 15]/[K. & Kuwae]) Radon meas. ν on X \ {x 0 } s.t. E(d x0, v) + (N 1) ct κ (d x0 )v dm = for v C Lip c,+(x \ {x 0 }) X X v dν 15 / 30

58 Laplacian comparison thm κ := K N 1, s κ(r) := sin( κr), ct κ := s κ (r) κ s κ (r) Theorem 2 ([Gigli 15]/[K. & Kuwae]) Radon meas. ν on X \ {x 0 } s.t. E(d x0, v) + (N 1) ct κ (d x0 )v dm = for v C Lip c,+(x \ {x 0 }) X X v dν Under (R1) below, X \ {x 0 } X & ν({x 0 }) = 0 15 / 30

59 Main thm σ x0 := inf{t 0 B t = x 0 } Suppose that X is not 1-dimensional ( N > 1) (cf. [Kitabeppu & Lakzian 16]) 16 / 30

60 Main thm σ x0 := inf{t 0 B t = x 0 } Suppose that X is not 1-dimensional ( N > 1) (cf. [Kitabeppu & Lakzian 16]) Theorem 3 ([K. & Kuwae]) (1) A ν t : PCAF on X \ {x 0} in the strict sense ν & β: 1-dim. std. BM s.t. r t r 0 = β t + (N 1) t 0 ct κ (r s )ds A ν t for t [0, σ x0 ) P x -a.s. for any x X \ {x 0 } 16 / 30

61 Main thm σ x0 := inf{t 0 B t = x 0 } Suppose that X is not 1-dimensional ( N > 1) (cf. [Kitabeppu & Lakzian 16]) Theorem 3 ([K. & Kuwae]) (1) A ν t : PCAF on X \ {x 0} in the strict sense ν & β: 1-dim. std. BM s.t. r t r 0 = β t + (N 1) t 0 ct κ (r s )ds A ν t for t [0, σ x0 ) P x -a.s. for any x X \ {x 0 } 16 / 30

62 Main thm σ x0 := inf{t 0 B t = x 0 } Suppose that X is not 1-dimensional ( N > 1) (cf. [Kitabeppu & Lakzian 16]) Theorem 3 ([K. & Kuwae]) (1) A ν t : PCAF on X \ {x 0} in the strict sense ν & β: 1-dim. std. BM s.t. r t r 0 = β t + (N 1) t 0 ct κ (r s )ds A ν t for t [0, σ x0 ) P x -a.s. for any x X \ {x 0 } (2) Under (R2) below, X \ {x 0 } X & σ x0 16 / 30

63 (R1) B 1 (x 0 ) dm d x0 Assumptions < (R2) 1 m(b r (x 0 )) B r (x 0 ) dm d x0 C r for r 1 17 / 30

64 (R1) B 1 (x 0 ) dm d x0 Assumptions < (R2) 1 m(b r (x 0 )) B r (x 0 ) dm d x0 C r for r 1 (R2) (R1) 17 / 30

65 (R1) B 1 (x 0 ) dm d x0 Assumptions < (R2) 1 m(b r (x 0 )) B r (x 0 ) dm d x0 C r for r 1 (R2) (R1) m(b r (x 0 )) Cr α for α > 1 (R1) 17 / 30

66 (R1) B 1 (x 0 ) dm d x0 Assumptions < (R2) 1 m(b r (x 0 )) B r (x 0 ) dm d x0 C r for r 1 (R2) (R1) m(b r (x 0 )) Cr α for α > 1 (R1) m(b r (x 0 )) Cr N (R2) 17 / 30

67 (R1) B 1 (x 0 ) dm d x0 Assumptions < (R2) 1 m(b r (x 0 )) B r (x 0 ) dm d x0 C r for r 1 (R2) (R1) m(b r (x 0 )) Cr α for α > 1 (R1) m(b r (x 0 )) Cr N (R2) (R2) holds for m-a.e. x 0 17 / 30

68 (R1) B 1 (x 0 ) dm d x0 Assumptions < (R2) 1 m(b r (x 0 )) B r (x 0 ) dm d x0 C r for r 1 (R2) (R1) m(b r (x 0 )) Cr α for α > 1 (R1) m(b r (x 0 )) Cr N (R2) (R2) holds for m-a.e. x / 30

69 Remarks When X: Riem. mfd, t A ν t d x0 (B s ) ds 0 +(N 1) t 0 ct κ (r s ) ds + L c t σ x0 < may happen even under (R2), while P x [σ x0 = ] = 1 for any x m-a.e. x 0 18 / 30

70 Remarks When X: Riem. mfd, t A ν t d x0 (B s ) ds 0 +(N 1) t 0 ct κ (r s ) ds + L c t Q. The same decomposition on RCD spaces? (Can one extract L c t?) σ x0 < may happen even under (R2), while P x [σ x0 = ] = 1 for any x m-a.e. x 0 18 / 30

71 Remarks When X: Riem. mfd, t A ν t d x0 (B s ) ds 0 +(N 1) t 0 ct κ (r s ) ds + L c t Q. The same decomposition on RCD spaces? (Can one extract L c t?) σ x0 < may happen even under (R2), while P x [σ x0 = ] = 1 for any x m-a.e. x 0 18 / 30

72 1. Introduction 2. Framework and main result 3. Outline of the proof 4. Local structure

73 Notations E α (u, v) := E(u, v) + α u, v m (u, v F; α > 0) R α f := 0 e αt P t f dt E α (R α f, g) = f, g m (f L 2 (m), g F) r α (x, y) := 0 e αt p t (x, y)dt 20 / 30

74 Notations E α (u, v) := E(u, v) + α u, v m (u, v F; α > 0) R α f := 0 e αt P t f dt E α (R α f, g) = f, g m (f L 2 (m), g F) r α (x, y) := 0 e αt p t (x, y)dt 20 / 30

75 Notations E α (u, v) := E(u, v) + α u, v m (u, v F; α > 0) R α f := 0 e αt P t f dt E α (R α f, g) = f, g m (f L 2 (m), g F) r α (x, y) := 0 e αt p t (x, y)dt 20 / 30

76 Notations E α (u, v) := E(u, v) + α u, v m (u, v F; α > 0) R α f := 0 e αt P t f dt E α (R α f, g) = f, g m (f L 2 (m), g F) r α (x, y) := 0 e αt p t (x, y)dt 20 / 30

77 Notations E α (u, v) := E(u, v) + α u, v m (u, v F; α > 0) R α f := 0 e αt P t f dt E α (R α f, g) = f, g m (f L 2 (m), g F) r α (x, y) := e αt p t (x, y)dt 0 R α f(x) = r α (x, y)f(y)m(dy), X R α µ for a meas. µ can be defined 20 / 30

78 Smooth measures in the strict sense A ν t : PCAF in the strict sense (i.e., without q.e. ) ν S 1 ϕ C Lip c,+(x), ϕν S 00 (similarly for X \ {x 0 }) In our case, µ S 00 if (i) µ(x) < (ii) R α µ < for some α > 0 (iii) R α µ F & E α (R α µ, v) = X v dµ ( v F C c (X)) 21 / 30

79 Smooth measures in the strict sense A ν t : PCAF in the strict sense (i.e., without q.e. ) ν S 1 ϕ C Lip c,+(x), ϕν S 00 (similarly for X \ {x 0 }) In our case, µ S 00 if (i) µ(x) < (ii) R α µ < for some α > 0 (iii) R α µ F & E α (R α µ, v) = X v dµ ( v F C c (X)) 21 / 30

80 Smooth measures in the strict sense A ν t : PCAF in the strict sense (i.e., without q.e. ) ν S 1 ϕ C Lip c,+(x), ϕν S 00 (similarly for X \ {x 0 }) In our case, µ S 00 if (i) µ(x) < (ii) R α µ < for some α > 0 (iii) R α µ F & E α (R α µ, v) = X v dµ ( v F C c (X)) 21 / 30

81 Smooth measures in the strict sense A ν t : PCAF in the strict sense (i.e., without q.e. ) ν S 1 ϕ C Lip c,+(x), ϕν S 00 (similarly for X \ {x 0 }) In our case, µ S 00 if (i) µ(x) < (ii) R α µ < for some α > 0 (iii) R α µ F & E α (R α µ, v) = X v dµ ( v F C c (X)) 21 / 30

82 Smooth measures in the strict sense A ν t : PCAF in the strict sense (i.e., without q.e. ) ν S 1 ϕ C Lip c,+(x), ϕν S 00 (similarly for X \ {x 0 }) In our case, µ S 00 if (i) µ(x) < (ii) R α µ < for some α > 0 (iii) R α µ F & E α (R α µ, v) = X v dµ ( v F C c (X)) 21 / 30

83 Basic idea (when x 0 / supp ϕ) R α (ϕν) L (supp ϕ) < R α (ϕν) < R α (ϕν)(x) X 0 ce (α λ)t m(b t (x)) ) d(x, y)2 exp ( ϕ(y)ν(dy)dt Ct 22 / 30

84 Basic idea (when x 0 / supp ϕ) R α (ϕν) L (supp ϕ) < R α (ϕν) < R α (ϕν)(x) X 0 ce (α λ)t m(b t (x)) ) d(x, y)2 exp ( ϕ(y)ν(dy)dt Ct 22 / 30

85 Basic idea (when x 0 / supp ϕ) R α (ϕν) L (supp ϕ) < R α (ϕν) < R α (ϕν)(x) X 0 ce (α λ)t m(b t (x)) ) d(x, y)2 exp ( ϕ(y)ν(dy)dt Ct Reduce to the following two estimates: e α ( ) 0t sup x K 1 m(b t (x)) exp d2 x dmdt <, K C t 1 ( ) 1 sup exp d2 x dmdt < x K tm(b t (x)) C t 0 K 22 / 30

86 Basic idea (when x 0 / supp ϕ) R α (ϕν) L (supp ϕ) < R α (ϕν) < R α (ϕν)(x) X 0 ce (α λ)t m(b t (x)) ) d(x, y)2 exp ( ϕ(y)ν(dy)dt Ct Reduce to the following two estimates: e α ( ) 0t sup x K 1 m(b t (x)) exp d2 x dmdt <, K C t 1 ( ) 1 sup exp d2 x dmdt < x K tm(b t (x)) C t 0 K 22 / 30

87 Basic idea (when x 0 / supp ϕ) R α (ϕν) L (supp ϕ) < R α (ϕν) < R α (ϕν)(x) X 0 ce (α λ)t m(b t (x)) ) d(x, y)2 exp ( ϕ(y)ν(dy)dt Ct Reduce to the following two estimates: e α ( ) 0t sup x K 1 m(b t (x)) exp d2 x dmdt <, K C t 1 ( ) 1 sup exp d2 x dmdt < x K tm(b t (x)) C t 0 K 22 / 30

88 1 sup x K 0 Idea B R (x) Basic idea (when x 0 / supp ϕ) II ( ) 1 exp d2 x dmdt < tm(b t (x)) K C t ( ) exp d2 x dm C t ) 2u = ( X d x C t exp u2 du dm C t = (integral in u involving m(b u (x))) & use the Bishop-Gromov ineq. 23 / 30

89 1 sup x K 0 Idea B R (x) Basic idea (when x 0 / supp ϕ) II ( ) 1 exp d2 x dmdt < tm(b t (x)) K C t ( ) exp d2 x dm C t ) 2u = ( X d x C t exp u2 du dm C t = (integral in u involving m(b u (x))) & use the Bishop-Gromov ineq. 23 / 30

90 1 sup x K 0 Idea B R (x) Basic idea (when x 0 / supp ϕ) II ( ) 1 exp d2 x dmdt < tm(b t (x)) K C t ( ) exp d2 x dm C t ) 2u = ( X d x C t exp u2 du dm C t = (integral in u involving m(b u (x))) & use the Bishop-Gromov ineq. 23 / 30

91 1 sup x K 0 Idea B R (x) Basic idea (when x 0 / supp ϕ) II ( ) 1 exp d2 x dmdt < tm(b t (x)) K C t ( ) exp d2 x dm C t ) 2u = ( X d x C t exp u2 du dm C t = (integral in u involving m(b u (x))) & use the Bishop-Gromov ineq. 23 / 30

92 1 sup x K 0 Idea B R (x) Basic idea (when x 0 / supp ϕ) II ( ) 1 exp d2 x dmdt < tm(b t (x)) K C t ( ) exp d2 x dm C t ) 2u = ( X d x C t exp u2 du dm C t = (integral in u involving m(b u (x))) & use the Bishop-Gromov ineq. 23 / 30

93 Reduce to Basic idea (when x 0 supp ϕ) ( ) 1Br (x sup R 0 ) α (x) < (r 1) x B r (x 0 ) d x0 24 / 30

94 Reduce to Basic idea (when x 0 supp ϕ) ( ) 1Br (x sup R 0 ) α (x) < (r 1) x B r (x 0 ) d x0 Lemma 4 Suppose (R2): 1 m(b r (x 0 )) B r (x 0 ) dm d x0 C r B u (x) B r (x 0 ) dm d x0 C m(b 5u (x)) u for x B r (x 0 ), r 1 and u < 2r 24 / 30

95 Reduce to Basic idea (when x 0 supp ϕ) ( ) 1Br (x sup R 0 ) α (x) < (r 1) x B r (x 0 ) d x0 Lemma 4 Suppose (R2): 1 m(b r (x 0 )) B r (x 0 ) dm d x0 C r B u (x) B r (x 0 ) dm d x0 C m(b 5u (x)) u for x B r (x 0 ), r 1 and u < 2r 24 / 30

96 1. Introduction 2. Framework and main result 3. Outline of the proof 4. Local structure

97 Regular sets Goal: Verify σ x0 = P x -a.s. and/or (R2) for x 0 26 / 30

98 Regular sets Goal: Verify σ x0 = P x -a.s. and/or (R2) for x 0 E k X: k-regular sets (x E k has unique measured tangent cone R k ) convergence of normalized measure 26 / 30

99 Regular sets Goal: Verify σ x0 = P x -a.s. and/or (R2) for x 0 E k X: k-regular sets (x E k has unique measured tangent cone R k ) convergence of normalized measure Theorem 5 ([Mondino & Naber]) m(x \ j N E j) = 0 26 / 30

100 Regular sets Goal: Verify σ x0 = P x -a.s. and/or (R2) for x 0 E k X: k-regular sets (x E k has unique measured tangent cone R k ) convergence of normalized measure Theorem 5 ([Mondino & Naber]) m(x \ j N E j) = 0 Excluding 1-dim. sp s E 1 = [Kitabeppu & Lakzian 16] 26 / 30

101 Measured rectifiability Theorem 6 ([Mondino & Naber]) R j X, k j N[1, N] s.t. m(x \ j R j) = 0 R j : bi-lip. to a m ble set R k j m(r j \ E kj ) = 0 27 / 30

102 Measured rectifiability Theorem 6 ([Mondino & Naber]) R j X, k j N[1, N] s.t. m(x \ j R j) = 0 R j : bi-lip. to a m ble set R k j m(r j \ E kj ) = 0 m Rj H k j [Gigli & Pasqualetto] / [Kell & Mondino] 27 / 30

103 Verification of (R2) Lemma 7 For x E k, lim r 0 m(b αr (x)) m(b r (x)) = αk (α > 0) 28 / 30

104 Verification of (R2) Lemma 7 For x E k, lim r 0 m(b αr (x)) m(b r (x)) = αk (α > 0) Lemma 8 x 0 E k, k 2 (R2) 1 m(b r (x 0 )) B r (x 0 ) dm d x0 C r 28 / 30

105 Verification of σ x0 = Lemma 9 x 0 E k, k 3 σ x0 = P x -a.s. (x X) For a.e. x 0 E 2, σ x0 = P x -a.s. (x X) 29 / 30

106 Verification of σ x0 = Lemma 9 x 0 E k, k 3 σ x0 = P x -a.s. (x X) For a.e. x 0 E 2, σ x0 = P x -a.s. (x X) For the proof, use a sufficient contidion in [Sturm 95]: r1 r dr m(b r (x 0 )) = conclusion 0 29 / 30

107 Verification of σ x0 = Lemma 9 x 0 E k, k 3 σ x0 = P x -a.s. (x X) For a.e. x 0 E 2, σ x0 = P x -a.s. (x X) For the proof, use a sufficient contidion in [Sturm 95]: r1 r dr m(b r (x 0 )) = conclusion 0 When k = 2, m Rj = ρh 2 Rj (k j = 2), a.e. x 0 R j, lim r 0 m(b r (x 0 )) r 2 c 1 ρ(x 0 ) < 29 / 30

108 Strange example (X, d, m) = (D 2 1/ e, d E, e V L 2 ), V (x) = log((log x ) 2 log x ) 30 / 30

109 Strange example (X, d, m) = (D 2 1/ e, d E, e V L 2 ), V (x) = log((log x ) 2 log x ) ( m(b r (0)) = πr 2 log 1 ) 2 r 30 / 30

110 Strange example (X, d, m) = (D 2 1/ e, d E, e V L 2 ), V (x) = log((log x ) 2 log x ) ( m(b r (0)) = πr 2 log 1 ) 2 r lim r 0 m(b αr (0)) m(b r (0)) = α2, 1/ e 0 rdr m(b r (0)) < 30 / 30

111 Strange example (X, d, m) = (D 2 1/ e, d E, e V L 2 ), V (x) = log((log x ) 2 log x ) ( m(b r (0)) = πr 2 log 1 ) 2 r lim r 0 m(b αr (0)) m(b r (0)) = α2, & m(b R(0)) m(b r (0)) ( R r ) 2 1/ e 0 rdr m(b r (0)) < 30 / 30