Periodic and non-periodic aspects of the heat kernel asymptotics on - PowerPoint PPT Presentation

Periodic and non-periodic aspects of the heat kernel asymptotics on Sierpi´ nski carpets Naotaka Kajino (Universit¨ at Bielefeld) http://www.math.uni-bielefeld.de/~nkajino/ Advances on Fractals and Related Topics @ Chinese Univ. Hong Kong December 11, 2012 16:25 –16:45

1/11 0 Main Question 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, x ) behave as t ↓ 0 ? cf. M d : Riem. mfd t ↓ 0 1 + S M ( x ) ⇒ p M = (4 πt ) − d/ 2 ` t + O ( t 2 ) ´ = t ( x, x ) , 6 vol d ( M ) n e − λM n t = t ↓ 0 M p M R M cpt ⇒ Z M ( t ) := P t ( x, x ) ∼ (4 πt ) d/ 2 . Q. What happens for the heat kernels on fractals?

1/11 0 Main Question 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, x ) behave as t ↓ 0 ? cf. M d : Riem. mfd t ↓ 0 1 + S M ( x ) ⇒ p M = (4 πt ) − d/ 2 ` t + O ( t 2 ) ´ = t ( x, x ) , 6 vol d ( M ) n e − λM n t = t ↓ 0 M p M R M cpt ⇒ Z M ( t ) := P t ( x, x ) ∼ (4 πt ) d/ 2 . Q. What happens for the heat kernels on fractals?

1/11 0 Main Question 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, x ) behave as t ↓ 0 ? cf. M d : Riem. mfd t ↓ 0 1 + S M ( x ) ⇒ p M = (4 πt ) − d/ 2 ` t + O ( t 2 ) ´ = t ( x, x ) , 6 vol d ( M ) n e − λM n t = t ↓ 0 M p M R M cpt ⇒ Z M ( t ) := P t ( x, x ) ∼ (4 πt ) d/ 2 . Q. What happens for the heat kernels on fractals?

1/11 0 Main Question 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, x ) behave as t ↓ 0 ? cf. M d : Riem. mfd t ↓ 0 1 + S M ( x ) ⇒ p M = (4 πt ) − d/ 2 ` t + O ( t 2 ) ´ = t ( x, x ) , 6 vol d ( M ) n e − λM n t = t ↓ 0 M p M R M cpt ⇒ Z M ( t ) := P t ( x, x ) ∼ (4 πt ) d/ 2 . Q. What happens for the heat kernels on fractals?

1/11 0 Main Question 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, x ) behave as t ↓ 0 ? cf. M d : Riem. mfd t ↓ 0 1 + S M ( x ) ⇒ p M = (4 πt ) − d/ 2 ` t + O ( t 2 ) ´ = t ( x, x ) , 6 vol d ( M ) n e − λM n t = t ↓ 0 M p M R M cpt ⇒ Z M ( t ) := P t ( x, x ) ∼ (4 πt ) d/ 2 . Q. What happens for the heat kernels on fractals?

2/11 the Sierpi´ nski carpet ∂ (SC) = ∂ R 2 [0 , 1] 2 !

2/11 the Sierpi´ nski carpet generalized SCs ∂ (SC) = ∂ R 2 [0 , 1] 2 !

r 3/11 r r r Examples of nested fractals Solid circles:“Boundary” V 0 • # V 0 < ∞ • highly symmetric

4/11 1 Dirichlet form and B.M. on Sierpi´ nski carpets K 7 K 6 K 5 ⊲ µ : Self-similar measure K K 8 K 4 ` 1 1 ´ K 1 K 2 K 3 with weight N , . . . , N 1 /N each 1 /N 2 each 1 ⊲ ∃ 1 ( E , F ) : canonical self-sim. Dirich. form on L 2 ( K, µ ) � T t f ( x ) = E x [ f ( X t )] X =( { X t } t ≥ 0 , { P x } x ∈ K ) : µ -symm. conservative diffusion

4/11 1 Dirichlet form and B.M. on Sierpi´ nski carpets K 7 K 6 K 5 ⊲ µ : Self-similar measure K K 8 K 4 ` 1 1 ´ K 1 K 2 K 3 with weight N , . . . , N 1 /N each 1 /N 2 each 1 ⊲ ∃ 1 ( E , F ) : canonical self-sim. Dirich. form on L 2 ( K, µ ) R “ E ( u, v ) = R d �∇ u, ∇ v � dx ” Existence: Barlow-Bass ’89, ’99, Kusuoka-Zhou ’92 Uniqueness: Barlow-Bass-Kumagai-Teplyaev ’10 � T t f ( x ) = E x [ f ( X t )] X =( { X t } t ≥ 0 , { P x } x ∈ K ) : µ -symm. conservative diffusion

4/11 1 Dirichlet form and B.M. on Sierpi´ nski carpets K 7 K 6 K 5 ⊲ µ : Self-similar measure K K 8 K 4 ` 1 1 ´ K 1 K 2 K 3 with weight N , . . . , N 1 /N each 1 /N 2 each 1 ⊲ ∃ 1 ( E , F ) : canonical self-sim. Dirich. form on L 2 ( K, µ ) T t = e tA E ( u,v )= �− Au,v � µ − − − − − − − − − − − − → − − − − − − − − → ( E , F ) { T t } t A ← − − − − − − − − − − − − − ← − − − − − − − − E ( √ − Au, √ Tt − I selfad, ≤ 0 Dirichlet Markov − Av ) A =lim t t ↓ 0 form semigr. “Laplacian” � T t f ( x ) = E x [ f ( X t )] X =( { X t } t ≥ 0 , { P x } x ∈ K ) : µ -symm. conservative diffusion

4/11 1 Dirichlet form and B.M. on Sierpi´ nski carpets K 7 K 6 K 5 ⊲ µ : Self-similar measure K K 8 K 4 ` 1 1 ´ K 1 K 2 K 3 with weight N , . . . , N 1 /N each 1 /N 2 each 1 ⊲ ∃ 1 ( E , F ) : canonical self-sim. Dirich. form on L 2 ( K, µ ) T t = e tA E ( u,v )= �− Au,v � µ − − − − − − − − − − − − → − − − − − − − − → ( E , F ) { T t } t A ← − − − − − − − − − − − − − ← − − − − − − − − E ( √ − Au, √ Tt − I selfad, ≤ 0 Dirichlet Markov − Av ) A =lim t t ↓ 0 form semigr. “Laplacian” � T t f ( x ) = E x [ f ( X t )] X =( { X t } t ≥ 0 , { P x } x ∈ K ) : µ -symm. conservative diffusion (the “Brownian motion” on K )

5/11 K 7 K 6 K 5 ⊲ µ : Self-similar measure K K 8 K 4 ` 1 1 ´ K 1 K 2 K 3 with weight N , . . . , N 1 /N each 1 /N 2 each 1 ⊲ ∃ 1 ( E , F ) : canonical self-sim. Dirich. form on L 2 ( K, µ ) T t = e tA E ( u,v )= �− Au,v � µ − − − − − − − − − − − − → − − − − − − − − → ( E , F ) { T t } t A ← − − − − − − − − − − − − − ← − − − − − − − − E ( √ − Au, √ Tt − I selfad, ≤ 0 Dirichlet Markov − Av ) A =lim t t ↓ 0 form semigr. “Laplacian” � T t f ( x ) = E x [ f ( X t )] X =( { X t } t ≥ 0 , { P x } x ∈ K ) : µ -symm. conservative diffusion (the “Brownian motion” on K )

5/11 K 7 K 6 K 5 ⊲ µ : Self-similar measure K K 8 K 4 ` 1 1 ´ K 1 K 2 K 3 with weight N , . . . , N 1 /N each 1 /N 2 each 1 ⊲ ∃ 1 ( E , F ) : canonical self-sim. Dirich. form on L 2 ( K, µ ) T t = e tA E ( u,v )= �− Au,v � µ − − − − − − − − − − − − → − − − − − − − − → ( E , F ) { T t } t A ← − − − − − − − − − − − − − ← − − − − − − − − E ( √ − Au, √ Tt − I selfad, ≤ 0 Dirichlet Markov − Av ) A =lim t t ↓ 0 form semigr. “Laplacian” � T t f ( x ) = E x [ f ( X t )] X =( { X t } t ≥ 0 , { P x } x ∈ K ) : µ -symm. conservative diffusion ⊲ p t ( x, y ) : Heat kernel R T t f ( x ) = E x [ f ( X t )] = K p t ( x, y ) f ( y ) dµ ( y )

6/11 Sub-Gaussian bound of p t ( x, y ) Thm (Barlow-Bass ’92, ’99) . For t ∈ (0 , 1] , x, y ∈ K , ( | x − y | d w 1 c 1 ( ) ) d w − 1 p t ( x, y ) ≍ − c 2 t d s / 2 exp . t • d s := 2 d f /d w , d f := dim H , Euc K • d w > 2 (Barlow-Bass ’90, ’92, ’99) c 3 ≤ t d s / 2 p t ( x, x ) ≤ c 4 , ⇒ t ∈ (0 , 1] , x ∈ K . t ↓ 0 t d s / 2 p t ( x, x ) ? If not, HOW it oscillates? Q. ∃ lim cf. M d : Riem. mfd ⇒ lim t ↓ 0 t d/ 2 p M t ( x, x ) = (4 π ) − d/ 2 .

6/11 Sub-Gaussian bound of p t ( x, y ) Thm (Barlow-Bass ’92, ’99) . For t ∈ (0 , 1] , x, y ∈ K , ( | x − y | d w 1 c 1 ( ) ) d w − 1 p t ( x, y ) ≍ − c 2 t d s / 2 exp . t • d s := 2 d f /d w , d f := dim H , Euc K • d w > 2 (Barlow-Bass ’90, ’92, ’99) c 3 ≤ t d s / 2 p t ( x, x ) ≤ c 4 , ⇒ t ∈ (0 , 1] , x ∈ K . t ↓ 0 t d s / 2 p t ( x, x ) ? If not, HOW it oscillates? Q. ∃ lim cf. M d : Riem. mfd ⇒ lim t ↓ 0 t d/ 2 p M t ( x, x ) = (4 π ) − d/ 2 .

6/11 Sub-Gaussian bound of p t ( x, y ) Thm (Barlow-Bass ’92, ’99) . For t ∈ (0 , 1] , x, y ∈ K , ( | x − y | d w 1 c 1 ( ) ) d w − 1 p t ( x, y ) ≍ − c 2 t d s / 2 exp . t • d s := 2 d f /d w , d f := dim H , Euc K • d w > 2 (Barlow-Bass ’90, ’92, ’99) c 3 ≤ t d s / 2 p t ( x, x ) ≤ c 4 , ⇒ t ∈ (0 , 1] , x ∈ K . t ↓ 0 t d s / 2 p t ( x, x ) ? If not, HOW it oscillates? Q. ∃ lim cf. M d : Riem. mfd ⇒ lim t ↓ 0 t d/ 2 p M t ( x, x ) = (4 π ) − d/ 2 .

6/11 Sub-Gaussian bound of p t ( x, y ) Thm (Barlow-Bass ’92, ’99) . For t ∈ (0 , 1] , x, y ∈ K , ( | x − y | d w 1 c 1 ( ) ) d w − 1 p t ( x, y ) ≍ − c 2 t d s / 2 exp . t • d s := 2 d f /d w , d f := dim H , Euc K • d w > 2 (Barlow-Bass ’90, ’92, ’99) c 3 ≤ t d s / 2 p t ( x, x ) ≤ c 4 , ⇒ t ∈ (0 , 1] , x ∈ K . t ↓ 0 t d s / 2 p t ( x, x ) ? If not, HOW it oscillates? Q. ∃ lim cf. M d : Riem. mfd ⇒ lim t ↓ 0 t d/ 2 p M t ( x, x ) = (4 π ) − d/ 2 .

7/11 2 Thm 1. p t ( x, x ) NOT vary reg. & non-periodic Thm (K.). ∃ c 5 ∈ (0 , ∞ ) , ∃ N ⊂ K Borel, ν q ( N ) = 0 for any self-similar measure ν q , and ∀ x ∈ K \ N : (NRV) p ( · ) ( x, x ) does NOT vary regularly at 0 , and hence � ∃ lim t ↓ 0 t d s / 2 p t ( x, x ) . ⊲ ν q : Self-similar measure q 7 q 6 q 5 with weight q = ( q i ) N 1 q 8 q 4 i =1 ( q i > 0 , P N q 1 q 2 q 3 i =1 q i = 1 ) 1 on K q i on K i qiqj on Kij