Stochastic Processes on Fractals. Takashi Kumagai (RIMS, Kyoto University, Japan)

Transcription

1 Stochastic Processes on Fractals Takashi Kumagai (RIMS, Kyoto University, Japan) Stochastic Analysis and Related Topics July 2006 at Marburg

2 Plan (L1) Brownian motion on fractals Construction of BM on the Sierpinski gasket using Dirichlet forms, Some basic properties (L2) Properties of Brownian motion on fractals Some spectral properties, Characterization of the domain of Dirichlet forms (L3) Jump type processes on d-sets (Alfors d-regular sets) Relations of some jump-type processes on d-sets, Heat kernel estimates

3 1 Brownian motion on fractals 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 C 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 i 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. 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) u 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) f 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) f, g F. (E, F ): quadratic form on R V. Further, f R V m, 1P 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} p, 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), f 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), x.) By (1.2), f F can be extended conti. to K. F: the set of functions in F extended to K F C(K) 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) f F, p, q K (1.3) E(f, g) = 5 X E(f F i, g F i ) 3 f, g F (1.4) i I {B t }: corresponding diffusion process (Brownian motion) : corresponding self-adjoint operator on L 2 (K, µ).

10 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,... Unbounded Sierpinski gaskets ˆK := n 1 2 n K: the unbdd Sierpinski gasket We can construct Brownian motion similarly to Thm 1.2.

11 2 Properties of Brownian motion on fractals (A) Spectral properties (Fukushima-Shima 92) on K 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. (2.1) many localized eigenfunctions that produce eigenvalues with high multiplicities u: a localized eigenfunction Def u: is an eigenfunction of s.t. Supp u K \ V 0. d s = 2 log 3/ log 5 = 2d f /d w : spectral dimension Kigami-Lapidus, Lindstrøm, Mosco, Strichartz, Teplyaev,...

12 (B) Heat kernel estimates (Barlow-Perkins 88) p t (x, y): jointly continuous sym. transition density of {X t } w.r.t. µ (P t f(x) = R K p t(x, 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 ) dw 1) 1 p t (x, y) c 3 t d s 2 exp( c 4 ( d(x, y)d w ) dw 1). 1 (HK(d w )) t t Barlow-Bass, Hambly-K, Grigor yan-telcs,... By integrating (HK(d w )), 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

13 Proof of on-diagonal upper bound on K. Theorem 2.1 Let d s = 2 log 3/ log 5. Then kuk 2+4/d s 2 c(e(u, u) + kuk 2 2)kfk 4/d s 1, u F. (2.2) Note. This is equivalent to p t (x, y) c 0 t d s/2 exp(t), t > 0, x, y K. (CKS 87) Proof. By integrating (1.3), kuk 2 2 c(e(u, u) + kuk 2 1) ( ). So, for m N, kuk 2 S.S. 2 = X ( 1 Z 3 )m (u w ) 2 dµ (*) c X ( 1 w I m K 3 )m {E(u w ) + ku w k 2 1} w = c X ( 1 5 )m ( 5 3 )m E(u w ) + c X 3 m {( 1 Z 3 )m u w dµ} 2 C{( 1 w w K 5 )m E(u, u) + 3 m kuk 2 1} where u w := u F w. We thus obtain kuk 2 2 C{ 2/d s E(u, u) + 1 kuk 2 1}, (0, 1). If E(u, u) > kuk 2 1, then taking 2/d s+1 = kuk 2 1/E(u, u), we obtain (2.2). If E(u, u) kuk 2 1, then by (*), kuk 2 2 2ckuk 2 1. Thus (2.2) holds.

14 Let Q = Q(x 0, T, R) = (0, 4T ) B(x 0, 2R), Q (T, 2T ) B(x 0, R) and Q + = (3T, 4T ) B(x 0, R). Parabolic Harnack inequality (P HI(d w )): c 1 > 0 s.t. the following holds. Let R > 0, T = R d w, and u = u(t, x) : Q R = u in Q. Then, sup Q u c 1 inf Q + u. (P HI(d w ))

15 (HK(d w )) (P HI(d w )) Various properties of the process. (i) c 1 t 1/d w E x [d(x, X t )] c 2 t 1/d w (d w > 2: subdiffusive) (ii) Law of the iterated logarithm i.e. lim sup t 1 d(x t,x 0 ) t 1/d w(log log t) 1 1/d w = C, P x -a.s. (iii) Hölder continuity of the sol. of the heat equation (iv) Elliptic Harnack inequality: (EHI) (v) Liouville property (i.e. positive harm. function on X is const.) Indeed, if m u := inf X u, then by (EHI), sup B (u m u ) c inf B (u m u ) 0 as B 1. So u m u, µ-a.e. (vi) Estimates of the Green kernel etc. *Note that (ii), (v) are consequences of (HK(d w )) for all t > 0 (i.e. on ˆK).

16 (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 2.2 (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).

17 Proof. Proof of F 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), (2.3) 2t x y c 0 t 1/d w where (HK(d w )) was used in the last inequality. Take t = L md w and use d s /2 = d f /d w (2.3)= 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).

18 Proof of F Λ d w/2 2,1. Set = 1/(d w 1), diam (K) = 1. Then, g 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 md w) L m(d w+d f ) a m 1 (d w /2, g) 2, (2.4) m=1 where (HK(d w )) 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. x t > 0 s.t. Φ t (x) (0 x < x t ), Φ t (x) (x t < x < 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 (2.4) 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.

19 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).

20 3 Jump type processes on Alfors d-regular set K: compact Alfors d-regular set in R n (n 2, 0 < d n). I.e., K R n, c 1, c 2 > 0 s.t. c 1 r d µ(b(x, r)) c 2 r d for all x K, 0 < r 1, (3.1) B(x, r): ball center x, radius r w.r.t. Euclidean metric. d: Hausdorff dimension of K, µ: Hausdorff measure on K. ˆK: unbounded Alfors d-regular set in R n, i.e. (3.1) holds for all r > 0. For 0 < α < 2, let E Y (α)(u, u) = Z Z K K c(x, y) u(x) u(y) 2 x y d+α µ(dx)µ(dy), where c(x, y) is jointly measurable, c(x, y) = c(y, x) and c(x, y) 1. We denote E Ŷ (α) if we integrate over ˆK.

21 A Besov space Λ α/2 2,2 (K) is defined as follows, Z Z ku Λ α/2 2,2 (K)k = kuk L 2 + ( (K,µ) Λ α/2 2,2 K K u(x) u(y) 2 µ(dx)µ(dy))1/2 x y d+α (K) = {u : u is measurable, ku Λα/2(K)k < 1}. Theorem 3.1 (E Y (α), Λ α/2 2,2 (K)) is a regular Dirichlet space on L2 (K, µ). Denote {Y (α) t } t 0 the corresponding Hunt process on K. 2,2 Examples c(x, y) 1 (Fukushima-Uemura 03, Stós 00) *K = R n {Y (α) t } is a α-stable process on R n. *K: an open n-set {Y (α) t } is a reflected α-stable process on K.

22 Proof of Λ α/2 2,2 (K) C 0(K) dense in C 0 (K). First, note that (using α < 2) Z Z µ(dy) sup z z y z y 2 Z µ(dy) r d+α cr α, sup c s 1 α ds c 0 r 2 α. ( ) z z y d+α B(z,r) c B(z,r) For x 6= y K, let r := x y and (ξ) = 1 ξ x r r. Then, C 0 (K), supp[ ] B(x, r) =: B and (ξ) (η) η ξ /r. So, using (*), Z Z ( (ξ) (η)) 2 Z Z (η) 2 E(, ) = µ(dξ)µ(dη) + 2 µ(dξ) B B ξ η d+α B c B ξ η d+αµ(dη) 1 Z Z ξ η 2 Z r 2 B B ξ η d+αµ(dξ)µ(dη) + cr α (η) 2 µ(dη) B c 0 r α µ(b) < 1. Thus Λ α/2 2,2 (K) C 0(K). Since this holds for all x 6= y K, using Stone-Weierstrass theorem we see that Λ α/2 2,2 (K) C 0(K) is dense in C 0 (K). 0

23 Cf. Jump process as a subordination of a diffusion on th gasket (Restrictive) K: the Sierpinski gasket, {B K t } t 0 : Brownian motion on K. Recall c 1 t dw d exp( c 2 ( x y d w )) dw 1 1 p t (x, y) c 3 t dw d exp( c 4 ( x y d w )) dw 1. 1 t t (HK(d w )) (Other examples: nested fractals, Sierpinski carpets) {ξ t } t>0 : strictly (α/2)-stable subordinator (0 < α < 2). I.e., 1-dim. non-neg. Lévy process, indep. of {B K t } t 0, E[exp( uξ t )] = exp( tu α/2 ). {η t (u) : t > 0, u 0}: distribution density of {ξ t } t>0. Define q t (x, y) := Z 1 0 p u (x, y)η t (u)du for all t > 0, x, y K. (3.2)

24 {X (α) t } t 0 : the subordinate process (with the transition density q t (x, y)). Pt X(α) f := E (ξ) [Pξ BK t f] = Z 1 0 Ps BK f η t (s)ds. Then, {X (α) t } t 0 is a µ-symmetric Hunt process. (Stós 00, Bogdan-Stós-Sztonyk 02) (E X (α), F X (α)): the corresponding Dirichlet form on L 2 (K, µ). Remark. 1) If we start from the BM on R n, the resulting process is the α- stable process on R n. 2) For d-sets on R n, we can also construct jump-type processes by a time change of the α-stable process on R n. (Triebel, K, Zähle etc.)

25 Comparison of the forms K: Sierpinski gasket, ᾱ =: αd w /2 Proposition 3.2 For 0 < α < 2, In particular, F X (α) = Λᾱ/2 2,2 (K). E X (α)(f, f) E Y (ᾱ)(f, f) for all f L 2 (K, µ). Further, the densities of the Levy measures are also compatible. Note. On the gasket, the two Dirichlet forms introduced are different and the corresponding processes cannot be obtained by time changes of others by PCAFs.

26 Heat kernel estimates The HK estimates for X (α) is easy to obtain using (3.2). So, for the gasket case, we have the sharp HK estimates for Y (α) as well. How about the general d-set case? Recall that for 0 < α < 2, E Y (α)(u, u) := Z Z K K c(x, y) u(x) u(y) 2 x y d+α µ(dx)µ(dy), where c(x, y) is jointly measurable, c(x, y) = c(y, x) and c(x, y) 1. Theorem 3.3 (Chen-K 03) For all 0 < α < 2, p Y (α) t (x, y): jointly continuous heat kernel s.t. c 1 (t d/α t (α) x y d+α) py t (x, y) c 2 (t d/α Parabolic Harnack inequalities hold. Related works: Bass-Levin ( 02) t x y d+α).

27 Corollary 3.4 (Transience, recurrence) For ˆK, Ŷ (α) is transient iff d > α, point recurrent iff d < α. For d = α, P x (σ y < 1) = 0, P x (σ B(y,r) < 1) = 1 x, y ˆK, r > 0. Application Hausdorff dim. for the range of the process Proposition 3.5 dim H {Ŷ (α) t : 0 < t < 1} = d α µ a.e. *More general version Y. Xiao ( 04), R. Schilling-Y.Xiao ( 05).

28 Theorem 3.6 (Nash ineq.) Let r 0 = diamk. Then, kuk 2 (1+ α d) 2 C (r α 0 kuk E(u, u)) kuk 2α d 1, u F. (3.3) Proof. Define the average function u r of u by Z 1 u r (x) := µ(b(x, r)) Then we have Thus ku r k 2 2 = B(x,r) u(z) dµ(z), x K. ku r k 1 c 0 r d kuk 1,, ku r k 1 C kuk 1, 0 < r < r 0. Z K On the other hand, u r (x) 2 dµ(x) ku r k 1 ku r k 1 C r d kuk 2 1, 0 < r < r 0. (3.4)

29 Z Z ku u r k = (u(x) u(y)) dµ(y) 2 dµ(x) K µ(b(x, r)) B(x,r) Z Z Schwarz c 0 r d u(x) u(y) 2 dµ(y)dµ(x) K B(x,r) Z Z = c 0 r d u(x) u(y) 2 x y d+α dµ(y)dµ(x) x y d+α K B(x,r) c 0 r α E(u, u), 0 < r < r 0. (3.5) Therefore, it follows from (3.4) and (3.5) that kuk 2 2 2(ku r k ku u r k 2 2) C (r d kuk r α E(u, u)), 0 < r < r 0. (3.6) Noting that kuk 2 2 ( r r 0 ) α kuk 2 2 for r r 0, we see from (3.6) that kuk 2 2 C (r d kuk r α r α 0 kuk E(u, u) ) (3.7) for all r > 0. We obtain (3.3) by minimizing the right-hand side of (3.7).

30 d w := sup{α : (E Y (α), Λ α/2 2,2 (K)) is regular in L2 } Then, we can prove the former theorems for all α < d w if d w > d (strongly recurrent case). (Open Prob.) Does Them 3.3 hold α < d w when d w d? Remark: dw := sup{α : Λ α/2 2,2 (K) is dense in L2 }. (cf. Paluba 00, Stós 00) Then d w d w. When there is a fractional diffusion on K, then d w = d w.