Function spaces and stochastic processes on fractals I. Takashi Kumagai. (RIMS, Kyoto University, Japan)

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1 Function spaces and stochastic processes on fractals I Takashi Kumagai (RIMS, Kyoto University, Japan) International workshop on Fractal Analysis September 11-17, 2005 in Eisenach

2 Aim Consider Besov-Lipschtiz spaces which appear as domains of Dirichlet forms on fractals. Properties of the Besov spaces $ Properties of the corresponding stochastic proc. Plan (L1) Dirichlet forms on fractals and their domains Short survey for D-forms on fractals, characterization of their domains (L2) Jump type processes on d-sets (Alfors d-regular sets) Relations of various jump-type processes on d-sets, heat kernel estimates (L3) Trace theorem for Dirichlet forms on fractals and an application Trace theorem for self-smilar D-forms on self-smilar sets to self-similar subsets, Application: diffusion processes penetrating fractals

3 1 Dirichlet forms on fractals and their domains 1.1 A quick view of the theory of Dirichlet forms General Theory (see Fukushima-Oshima-Takeda 94 etc.) {X t } t : Sym. Hunt proc. on (K, µ) cont. path (diffusion), : non-neg. def. self-adj. op. on L 2 s.t. P t := exp(t ) Markovian local P t f(x) = E x [f(x t )], lim t 0 (P t I)/t =, (E, F) : regular Dirichlet form (i.e. sym. closed Markovian form) on L 2 Z E(u, v) = u vdµ, F = D( ) local K (E, F): regular, Def 9C Ω F C 0 (K) linear space which is dense i) in F w.r.t. E 1 -norm and ii) in C 0 (K) w.r.t. k k 1 -norm. (E, F): local Def, (u, v F, Supp u Supp v = ; ) E(u, v) = 0).

4 Example BM on R n, Laplace op. on R n, E(f, f) = 1 2 R f 2 dx, F = H 1 (R n ) 1.2 Sierpinski gaskets {p 0, p 1,, p n }: vertices of the n-dimensional simplex, p 0 : the origin. F i (z) = (z p i 1 )/2 + p i 1, z R n, i = 1, 2,, n non-void compact set K s.t. K = n+1 i=1 F i(k). K:(n-dimensional) Sierpinski gasket. When n = 1, K = [p 0, p 1 ]. For simplicity, we will consider the 2-dimensional gasket.

5 V 0 = {p 0, p 1, p 2 }, V n = i1,,i n IF i1 i n (V 0 ) where I := {1, 2, 3} and F i1 i n := F i1 F in. Let V = n NV n, where N := N {0}. Then K = Cl(V ). d f := log 3/ log 2: Hausdorff dimension of K (w.r.t. the Euclidean metric) µ: (normalized) Hausdorff measure on K, i.e. a Borel measure on K s.t. µ(f i1 i n (K)) = 3 n 8i 1,, i n I.

6 E [σ ] = 5 E [σ ] = Construction of Brownian motion on the gasket (Ideas) (Goldstein 87, Kusuoka 87) X n : simple random walk on V n X n ([5 n t]) n 1 B t : Brownian motion on K

7 1.4 Construction of Dirichlet forms on the gaskets For f, g R V n := {h : h is a real-valued function on V n }, define E n (f, g) := b n 2 X i 1 i n I X x,y V 0 (f F i1 i n (x) f F i1 i n (y))(g F i1 i n (x) g F i1 i n (y)), where {b n } is a sequence of positive numbers with b 0 = 1 (conductance on each bond). Choose {b n } s.t. 9 nice relations between the E n s Elementary computations yield inf{e 1 (f, f) : f R V 1, f V0 = u} = 3 5 b 1E 0 (u, u) 8u R V 0. (1.1) So, taking b n = (5/3) n, we have ( =, f is harmonic on V n+1 \ V n ). E n (f Vn, f Vn ) E n+1 (f, f) 8f R V n+1

8 Define F := {f R V : lim n 1 E n (f, f) < 1}, E(f, g) := lim n 1 E n (f, g) 8f, g F. (E, F ): quadratic form on R V. Further, 8f R V m, 91P m f F s.t. E(P m f, P m f) = E m (f, f). Want: to extend this form to a form on L 2 (K, µ). Define R(p, q) 1 := inf{e(f, f) : f V, f(p) = 1, f(q) = 0} 8p, q V, p 6= q. R(p, q): effective resistance between p and q. Set R(p, p) = 0 for p V. Proposition 1.1 1) R(, ) is a metric on V. It can be extended to a metric on K, which gives the same topology on K as the one from the Euclidean metric. 2) For p 6= q V, R(p, q) = sup{ f(p) f(q) 2 /E(f, f) : f F, f(p) 6= f(q)}. So, f(p) f(q) 2 R(p, q)e(f, f), 8f F, p, q V. (1.2)

9 Remark. R(p, q) kp qk d w d f, where d w = log 5/ log 2 (Walk dimension). (Here f(x) g(x), c 1 f(x) g(x) c 2 f(x), 8x.) By (1.2), f F can be extended conti. to K. F: the set of functions in F extended to K ) F Ω C(K) Ω L 2 (K, µ). Theorem 1.2 (Kigami) (E, F) is a local regular D-form on L 2 (K, µ). f(p) f(q) 2 R(p, q)e(f, f) 8f F, 8p, q K (1.3) E(f, g) = 5 X E(f F i, g F i ) 3 8f, g F (1.4) i I {B t }: corresponding diffusion process (Brownian motion) : corresponding self-adjoint operator on L 2 (K, µ). Uniqueness (Barlow-Perkins 88) Any self-similar diffusion process on K whose law is invariant under local translations and reflections of each small triangle is a constant time change of {B t }. Metz, Peirone, Sabot,...

10 1.5 Properties of the Dirichlet forms on the gaskets (A) Spectral properties (Fukushima-Shima 92) has a compact resolvent. Set ρ(x) = ]{ x : is an eigenvalue of }. Then 0 < lim inf x 1 (Barlow-Kigami 97) < above is because ρ(x) < lim sup x d s/2 x 1 ρ(x) x d s/2 < 1. (1.5) 9 many localized eigenfunctions that produce eigenvalues with high multiplicities u: a localized eigenfunction, Def u: is an eigenfunction of s.t. d s = 2 log 3/ log 5 = 2d f /d w : spectral dimension Supp u Ω O, 9 open set O Ω Int K. Kigami-Lapidus, Lindstrøm, Mosco, Strichartz, Teplyaev,...

11 (B) Heat kernel estimates (Barlow-Perkins 88) 9p t (x, y): jointly continuous sym. transition density of {X t } w.r.t. µ (P t f(x) = R K p t(x, y)f(y)µ(dy) p t(x 0, x) = x p t (x 0, x) ) s.t. c 1 t d s/2 exp( c 2 ( d(x, y)d w t ) dw 1) 1 p t (x, y) c 3 t ds/2 exp( c 4 ( d(x, y)d w ) dw 1). 1 (1.6) t Barlow-Bass, Hambly-K, Grigor yan-telcs,... By integrating (1.6), we have E 0 [d(0, X t )] t 1/d w. d w = log 5/ log 2 > 2, d s = 2 log 3/ log 5 = 2d f /d w < 2 As d w > 2, we say the process is sub-diffusive. n-dim. Sierpinski gasket (n 2) d f = log(n + 1)/ log 2, d w = log(n + 3)/ log 2> 2, d s = 2 log(n + 1)/ log(n + 3)< 2

12 (C) Domains of the Dirichlet forms For 1 p < 1, 1 q 1, β 0 and m N, set Z Z a m (β, f) := L mβ (L md f f(x) f(y) p dµ(x)dµ(y)) 1/p, f L p (K, µ), x y <c 0 L m where 1 < L < 1, 0 < c 0 < 1. Λ β p,q(k): a set of f L p (K, µ) s.t. ā(β, f) := {a m (β, f)} 1 m=0 l q. Λ β p,q(k) is a Besov-Lipschitz space. It is a Banach space. p = 2 Λ β 2,q (Rn ) = B β 2,q (Rn ) if 0 < β < 1, = {0} if β > 1. p = 2, β = 1 Λ 1 2,1(R n ) = H 1 (R n ), Λ 1 2,2(R n ) = {0}. Theorem 1.3 (Jonsson 96, K, Paluba, Grigor yan-hu-lau, K-Sturm) Let (E, F) be the Dirichlet form on the gasket. Then, F = Λ d w/2 2,1 (K).

13 Proof. Proof of F Ω Λ d w/2 2,1. Let E t(f, f) := (f P t f, f) L 2/t, f L 2 (K, µ). Then, E t (f, f) = 1 Z Z (f(x) f(y)) 2 p t (x, y)µ(dx)µ(dy) 2t K K 1 Z Z (f(x) f(y)) 2 p t (x, y)µ(dx)µ(dy) 2t x y c 0 t 1/d w c Z Z 1 t ds/2 (f(x) f(y)) 2 µ(dx)µ(dy), (1.7) 2t x y c 0 t 1/d w where (1.6) was used in the last inequality. Take t = L md w and use d s /2 = d f /d w ) (1.7)= c 1 a m (d w /2, f) 2. E t (f, f) % E(f, f) as t 0. So we obtain sup m a m (d w /2, f) c 2 p E(f, f).

14 Proof of F æ Λ d w/2 2,1. Set = 1/(d w 1), diam (K) = 1. Then, 8g Λ d w/2 2,1 E t (g, g) = 1 Z Z (g(x) g(y)) 2 p t (x, y)µ(dx)µ(dy) 2t 1 2t x,y K x y 1 1X Z c 3 t ds/2 e c 4(tL md w) ZL (g(x) g(y)) 2 µ(dx)µ(dy) m < x y L m+1 m=1 c 3 t (1+d s/2) 1X e c 4(tL mdw ) L m(d w+d f ) a m 1 (d w /2, g) 2, (1.8) m=1 where (1.6) was used in the first inequality. Let Φ t (x) = e c 4(tL xd w) L x(d w+d f ). Φ t (0) > 0, lim x 1 Φ t (x) = 0 and R 1 0 Φ t (x)dx = c 5 t 1+d s/2. 9x t > 0 s.t. Φ t (x) (0 8x < x t ), Φ t (x) (x t < 8x < 1), and Φ t (x t ) = c 6 t 1+d s/2. Thus, P 1 m=1 Φ t(m) R 1 0 Φ t (x)dx + 2Φ t (x t ) c 7 t 1+d s/2. Since (1.8) c 3 t (1+d s/2) (sup m a m (d w /2, f)) 2 P 1 m=1 Φ t(m), we conclude that sup t>0 E t (g, g) = lim t 0 E t (g, g) c 8 (sup m a m (d w /2, f)) 2.

15 1.6 Unbounded Sierpinski gaskets ˆK := n 1 2 n K: the unbdd Sierpinski gasket Construction of D-forms, as in Thm 1.2. Heat kernel estimates : (1.6) holds for all x, y ˆK, 0 < t < 1. Domains of the D-forms: Thm 1.3 holds. 1.7 More general fractals Nested fractals (Lindstrøm 90): Similar constructions, similar results. P.c.f. self-similar sets (Kigami 93): Under the existence of the reg. harm. structure, similar constructions, generalized versions for (A), (B) and (C). Sierpinski carpets: Construction of D-forms, much harder, but possible (Barlow-Bass etc). Similar results for (B) and (C).