Heat kernel asymptotics on the usual and harmonic Sierpinski gaskets. Cornell Prob. Summer School 2010 July 22, 2010

Transcription

1 Heat kernel asymptotics on the usual and harmonic Sierpinski gaskets Naotaka Kajino (Kyoto University) Cornell Prob. Summer School 2010 July 22, 2010 For some reasons I have slightly changed the title. JSPS Research Fellow PD ( ): Supported by Japan Society for the Promption of Science

2 Q. What happens for heat kernels on fractals? 0 Main Question 1/15 Given a Laplacian, let p t (x, y) be the heat kernel (transition density of the diffusion); e t f(x) = p t (x, y)f(y)dy. Question. How does p t (x, y) behave as t 0? cf. M n : Riem. mfd lim 2t log p M t (x, y) = d M (x, y) 2, t 0 p M t (x, x) (2πt) n/ S M (x) t + O(t 2 ) t 0 12

3 Q. What happens for heat kernels on fractals? 0 Main Question 1/15 Given a Laplacian, let p t (x, y) be the heat kernel (transition density of the diffusion); e t f(x) = p t (x, y)f(y)dy. Question. How does p t (x, y) behave as t 0? cf. M n : Riem. mfd lim 2t log p M t (x, y) = d M (x, y) 2, t 0 p M t (x, x) (2πt) n/ S M (x) t + O(t 2 ) t 0 12

4 Q. What happens for heat kernels on fractals? 0 Main Question 1/15 Given a Laplacian, let p t (x, y) be the heat kernel (transition density of the diffusion); e t f(x) = p t (x, y)f(y)dy. Question. How does p t (x, y) behave as t 0? cf. M n : Riem. mfd lim 2t log p M t (x, y) = d M (x, y) 2, t 0 p M t (x, x) (2πt) n/ S M (x) t + O(t 2 ) t 0 12

5 Q. What happens for heat kernels on fractals? 0 Main Question 1/15 Given a Laplacian, let p t (x, y) be the heat kernel (transition density of the diffusion); e t f(x) = p t (x, y)f(y)dy. Question. How does p t (x, y) behave as t 0? cf. M n : Riem. mfd lim 2t log p M t (x, y) = d M (x, y) 2, t 0 p M t (x, x) (2πt) n/ S M (x) t + O(t 2 ) t 0 12

6 Q. What happens for heat kernels on fractals? 0 Main Question 1/15 Given a Laplacian, let p t (x, y) be the heat kernel (transition density of the diffusion); e t f(x) = p t (x, y)f(y)dy. Question. How does p t (x, y) behave as t 0? cf. M n : Riem. mfd lim 2t log p M t (x, y) = d M (x, y) 2, t 0 p M t (x, x) (2πt) n/ S M (x) t + O(t 2 ) t 0 12

7 2/15 Usual and Harmonic Sierpinski gaskets Φ := h1 h 2 «h 1, h 2 : V 0 -harmonic 2E(h i, h j ) = δ ij Sierpinski gasket K Harmonic Sier. gasket K H

8 1 Heat kernel on the usual SG ν : Self-similar measure with weight ` 1, 1, 1 K K 1 K 2 K 3 1/3 each (E, F): Standard Dirichlet form on L 2 (K, ν) 3/15 1/9 each T t f(x) = E x [f(x t )] X = ({X t } t 0, {P x } x K ): ν-symm. conservative diffusion (the Brownian motion on the SG)

9 1 Heat kernel on the usual SG ν : Self-similar measure with weight ` 1, 1, 1 K K 1 K 2 K 3 1/3 each (E, F): Standard Dirichlet form on L 2 (K, ν) E(u, v) = 1 2 R 3/15 1/9 each R d u, v dx T t f(x) = E x [f(x t )] X = ({X t } t 0, {P x } x K ): ν-symm. conservative diffusion (the Brownian motion on the SG)

10 1 Heat kernel on the usual SG ν : Self-similar measure with weight ` 1, 1, 1 K K 1 K 2 K 3 1/3 each (E, F): Standard Dirichlet form on L 2 (K, ν) (E, F) Dirichlet form E (u,v )= Au,v ν E ( Au, Av ) A selfad, 0 Laplacian T t =e ta T A=lim t I t 0 t T t f(x) = E x [f(x t )] 3/15 1/9 each {T t } t Markov semigr. X = ({X t } t 0, {P x } x K ): ν-symm. conservative diffusion (the Brownian motion on the SG)

11 1 Heat kernel on the usual SG ν : Self-similar measure with weight ` 1, 1, 1 K K 1 K 2 K 3 1/3 each (E, F): Standard Dirichlet form on L 2 (K, ν) (E, F) Dirichlet form E (u,v )= Au,v ν E ( Au, Av ) A selfad, 0 Laplacian T t =e ta T A=lim t I t 0 t T t f(x) = E x [f(x t )] 3/15 1/9 each {T t } t Markov semigr. X = ({X t } t 0, {P x } x K ): ν-symm. conservative diffusion (the Brownian motion on the SG)

12 Heat kernel p ν t T t f (x) = E x [f (X t )] = 4/15 (x, y) and its sub-gaussian bound Z K p ν t (x, y)f (y)dν(y) Thm (Barlow-Perkins 88). For t (0, 1], x, y K, p ν t (x, y) c 1 t d f /d w exp ( c 2 ( x y d w t ) 1 dw 1 ). d f := dim H,Euc K = log 2 3 d w := log 2 5 > 2 = c 3 t log 5 3 p ν t (x, x) c 4, t (0, 1], x K.

13 Heat kernel p ν t T t f (x) = E x [f (X t )] = 4/15 (x, y) and its sub-gaussian bound Z K p ν t (x, y)f (y)dν(y) Thm (Barlow-Perkins 88). For t (0, 1], x, y K, p ν t (x, y) c 1 t d f /d w exp ( c 2 ( x y d w t ) 1 dw 1 ). d f := dim H,Euc K = log 2 3 d w := log 2 5 > 2 = c 3 t log 5 3 p ν t (x, x) c 4, t (0, 1], x K.

14 Heat kernel p ν t T t f (x) = E x [f (X t )] = 4/15 (x, y) and its sub-gaussian bound Z K p ν t (x, y)f (y)dν(y) Thm (Barlow-Perkins 88). For t (0, 1], x, y K, p ν t (x, y) c 1 t d f /d w exp ( c 2 ( x y d w t ) 1 dw 1 ). d f := dim H,Euc K = log 2 3 d w := log 2 5 > 2 = c 3 t log 5 3 p ν t (x, x) c 4, t (0, 1], x K.

15 5/15 Main Theorem for Usual SG: Oscillatory behavior Recall: c 3 t log 5 3 p ν t (x, x) c 4, t (0, 1], x K. Thm (K., in preparation). For any x K, lim t 0 t log 5 3 p ν t (x, x) does not exist. `For x V0 : Essentially implied by Grabner-Woess 97 V 0

16 5/15 Main Theorem for Usual SG: Oscillatory behavior Recall: c 3 t log 5 3 p ν t (x, x) c 4, t (0, 1], x K. Thm (K., in preparation). For any x K, lim t 0 t log 5 3 p ν t (x, x) does not exist. `For x V0 : Essentially implied by Grabner-Woess 97 V 0

17 5/15 Main Theorem for Usual SG: Oscillatory behavior Recall: c 3 t log 5 3 p ν t (x, x) c 4, t (0, 1], x K. Thm (K., in preparation). For any x K, lim t 0 t log 5 3 p ν t (x, x) does not exist. `For x V0 : Essentially implied by Grabner-Woess 97 V 0

18 5/15 Main Theorem for Usual SG: Oscillatory behavior Recall: c 3 t log 5 3 p ν t (x, x) c 4, t (0, 1], x K. Thm (K., in preparation). For any x K, lim t 0 t log 5 3 p ν t (x, x) does not exist. `For x V0 : Essentially implied by Grabner-Woess 97 V 0

19 2 Harmonic SG and its heat kernel 6/15 Kigami 93: Harmonic embedding Φ : K K H Φ := h1 h 2 «h 1, h 2 : V 0 -harmonic 2E(h i, h j ) = δ ij Sierpinski gasket K Harmonic Sier. gasket K H

20 7/15 Energy measures µ u, u F (note: F C(K)) K fdµ u = 2E(fu, u) E(f, u 2 ), f F. dµ u = u 2 dx µ := µ h1 + µ h2 : Kusuoka measure (the energy of the harmonic map Φ) p µ t (x, y): Heat kernel assoc. with (K, µ, E, F) (i.e. the L 2 -inner product is changed to, L 2 (K,µ) )

21 7/15 Energy measures µ u, u F (note: F C(K)) K fdµ u = 2E(fu, u) E(f, u 2 ), f F. dµ u = u 2 dx µ := µ h1 + µ h2 : Kusuoka measure (the energy of the harmonic map Φ) p µ t (x, y): Heat kernel assoc. with (K, µ, E, F) (i.e. the L 2 -inner product is changed to, L 2 (K,µ) )

22 7/15 Energy measures µ u, u F (note: F C(K)) K fdµ u = 2E(fu, u) E(f, u 2 ), f F. dµ u = u 2 dx µ := µ h1 + µ h2 : Kusuoka measure (the energy of the harmonic map Φ) p µ t (x, y): Heat kernel assoc. with (K, µ, E, F) (i.e. the L 2 -inner product is changed to, L 2 (K,µ) )

23 8/15 Geodesic metric d H and Gaussian estimate of p µ t (J. Kigami, Math. Ann. 340 (2008), ) Def. For x, y K, define d H (x, y) := inf j Φ γ Euc ff γ : [0, 1] K cont.. γ(0) = x, γ(1) = y

24 9/15 d H (x, y) : Geodesic metric in K H y x

25 10/15 Geodesic metric d H and Gaussian estimate of p H t Def. For x, y K, define d H (x, y) := inf j Φ γ Euc ff γ : [0, 1] K cont.. γ(0) = x, γ(1) = y Thm (Kigami 2008). For t > 0, x, y K, p µ t (x, y) ( c 5 ( µ B ) exp d H(x, y) 2 t (x, d H) c 6 t ). Q. How manifold-like is p µ t (x, y), as t 0?

26 10/15 Geodesic metric d H and Gaussian estimate of p H t Def. For x, y K, define d H (x, y) := inf j Φ γ Euc ff γ : [0, 1] K cont.. γ(0) = x, γ(1) = y Thm (Kigami 2008). For t > 0, x, y K, p µ t (x, y) ( c 5 ( µ B ) exp d H(x, y) 2 t (x, d H) c 6 t ). Q. How manifold-like is p µ t (x, y), as t 0?

27 11/15 Main Thm for h.s.g. : 1. Varadhan s asymptotics Thm (K.). For any x, y K, (Vrd) lim t 0 2t log p µ t (x, y) = d H (x, y) 2. d µ (x, y) := sup{u(x) u(y) u F, µ u µ} (note that µ u µ e u 1 µ-a.e.).

28 11/15 Main Thm for h.s.g. : 1. Varadhan s asymptotics Thm (K.). For any x, y K, (Vrd) lim t 0 2t log p µ t (x, y) = d H (x, y) 2. Cf. (Kumagai 97) For any x, y K, x y, lim t 0 t 1 dw 1 log p ν (x, y) does not exist. t d µ (x, y) := sup{u(x) u(y) u F, µ u µ} (note that µ u µ e u 1 µ-a.e.).

29 11/15 Main Thm for h.s.g. : 1. Varadhan s asymptotics Thm (K.). For any x, y K, (Vrd) lim t 0 2t log p µ t (x, y) = d H (x, y) 2. By Sturm 95, Ramírez 01, it suffices to show: Thm (K.). d H = d µ, d µ (x, y) := sup{u(x) u(y) u F, µ u µ} (note that µ u µ e u 1 µ-a.e.).

30 11/15 Main Thm for h.s.g. : 1. Varadhan s asymptotics Thm (K.). For any x, y K, (Vrd) lim t 0 2t log p µ t (x, y) = d H (x, y) 2. By Sturm 95, Ramírez 01, it suffices to show: Thm (K.). d H = d µ, where d µ (x, y) := sup{u(x) u(y) u F, µ u µ} (note that µ u µ e u 1 µ-a.e.).

31 12/15 Main Thm for h.s.g. : 2. One-dim. asymptotics β := log 5/3 3 (= ). V := m 0 V m V 1 Thm (K.). For x V, as t 0, p µ t (x, x) = 1 ξ x 2πt ( ( 1 + O t β)), V 2 moreover lim r 0 µ(b r (x, d H )) r = 2ξ x, where ξ x > 0 is an explicit constant.

32 12/15 Main Thm for h.s.g. : 2. One-dim. asymptotics β := log 5/3 3 (= ). V := m 0 V m V 1 Thm (K.). For x V, as t 0, p µ t (x, x) = 1 ξ x 2πt ( ( 1 + O t β)), V 2 moreover lim r 0 µ(b r (x, d H )) r = 2ξ x, where ξ x > 0 is an explicit constant.

33 12/15 Main Thm for h.s.g. : 2. One-dim. asymptotics β := log 5/3 3 (= ). V := m 0 V m V 1 Thm (K.). For x V, as t 0, p µ t (x, x) = 1 ξ x 2πt ( ( 1 + O t β)), V 2 moreover lim r 0 µ(b r (x, d H )) r = 2ξ x, where ξ x > 0 is an explicit constant.

34 12/15 Main Thm for h.s.g. : 2. One-dim. asymptotics β := log 5/3 3 (= ). V := m 0 V m V 1 Thm (K.). For x V, as t 0, p µ t (x, x) = 1 ξ x 2πt ( ( 1 + O t β)), V 2 moreover lim r 0 µ(b r (x, d H )) r = 2ξ x, where ξ x > 0 is an explicit constant.

35 13/15 1-dim. asymptotics at x V x

36 Cf. Non-integer dim. behavior of p µ t 14/15 at a.e. point Thm (K.). 1 < d loc S dim H(K, d H ) (< 1.52), lim t 0 2 log p µ t (x, x) log t = d loc S µ-a.e. x K, i.e. p µ t (x, x) t dloc S /2. References for Harmonic S.G. case (available on my website): [1] N. K., Heat kernel asymptotics for the measurable Riemannian structure on the Sierpinski gasket, preprint. [2] N. K., Time changes of local Dirichlet spaces by energy measures of harmonic functions, to appear in Forum Math.

37 Cf. Non-integer dim. behavior of p µ t 14/15 at a.e. point Thm (K.). 1 < d loc S dim H(K, d H ) (< 1.52), lim t 0 2 log p µ t (x, x) log t = d loc S µ-a.e. x K, i.e. p µ t (x, x) t dloc S /2. References for Harmonic S.G. case (available on my website): [1] N. K., Heat kernel asymptotics for the measurable Riemannian structure on the Sierpinski gasket, preprint. [2] N. K., Time changes of local Dirichlet spaces by energy measures of harmonic functions, to appear in Forum Math.

38 Cf. Non-integer dim. behavior of p µ t 14/15 at a.e. point Thm (K.). 1 < d loc S dim H(K, d H ) (< 1.52), lim t 0 2 log p µ t (x, x) log t = d loc S µ-a.e. x K, i.e. p µ t (x, x) t dloc S /2. References for Harmonic S.G. case (available on my website): [1] N. K., Heat kernel asymptotics for the measurable Riemannian structure on the Sierpinski gasket, preprint. [2] N. K., Time changes of local Dirichlet spaces by energy measures of harmonic functions, to appear in Forum Math.

39 15/15 Big Open Problem. Can we do similar analysis on the Sierpinski carpet?