Uniqueness of the Brownian motion on the Sierpinski carpet

Richard Bass University of Connecticut www.math.uconn.edu/∼bass

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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Co-authors

This is joint work with Martin Barlow, University of British Columbia Takashi Kumagai, Kyoto University Alexander Teplyaev, University of Connecticut

Richard Bass (University of Connecticut)

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Sierpinski carpets We consider bounded fractals such as the Sierpinski carpet.

Richard Bass (University of Connecticut)

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Generalized Sierpinski carpets

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Brownian motion on the Sierpinski carpet

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The Menger sponge

Richard Bass (University of Connecticut)

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Sierpinski gaskets Unlike the Sierpinski gasket, GSC’s are not finitely ramified.

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Existence

(a) Barlow-Bass (1989) (b) Kusuoka-Zhou (1992) (c) Barlow-Bass (1999) See also Osada. Also: Hambly, K. Hattori, T. Hattori, Hino, Hu, Pietruska-Paluba, St´os, Watanabe, and others

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Brownian motion on the Sierpinski carpet

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The ’89 paper had 4 questions, and one was added in the ’99 paper: (a) Higher dimensional SCs (b) Local times (essentially, good estimates on heat kernels) (c) Uniqueness (d) Random state spaces (e) (1999) Characterize the spectral dimension On p. 256 of the ’89 paper, last line, regarding uniqueness, ”This problem seems quite hard.”

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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Definitions

Let E denotes the set of (i) non-zero; (ii) local; (iii) regular (iv) conservative Dirichet forms, which are (v) “invariant under all local symmetries.” Note elements of E do not have to be scale invariant.

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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A Dirichlet form E with domain D(E) × D(E) is Symmetric: E(f , g ) = E(g , f ). Bilinear: E(af + g , h) = aE(f , h) + E(g , h). Closed: The space D(E) with norm 

(f , f ) + E(f , f )

1/2

is complete.

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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Markovian: If

  |t| ≤ 1, t, ϕ(t) = 1, t > 1,   −1, t < −1,

and f ∈ D(E), then ϕ(f ) ∈ D(E) and E(ϕ(f ), ϕ(f )) ≤ E(f , f ).

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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Regular: If C is the set of continuous functions with compact support, then D(E) ∩ C is dense in C with respect to the sup norm, and dense in D(E) with respect to the norm that we used to define what it means to be closed. Conservative: If Pt is the associated semigroup, then Pt 1 = 1 Local: If f , g have disjoint supports and are in the domain of the Dirichlet form, then E(f , g ) = 0.

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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An example

Look at a domain in Rd and let E(f , g ) =

1 2

Z ∇f (x) · ∇g (x) dx, D

where the domain is some suitable function space. The domain makes a difference!

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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If one takes the infinitesimal generator of Brownian motion, namely, one half the Laplacian, then Z Z 1 1 f (x)∆g (x) dx = − 2 ∇f (x) · ∇g (x) dx = −E(f , g ). 2 D

D

I ignored the boundary term, a dangerous thing to do!

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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The main theorem

Theorem 1. (a) E is non-empty (in fact, the B-B and K-Z processes belong) (b) Up to scalar multiples, E consists of one element.

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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Corollaries

(a) EBB = cEKZ , (b) The EBB processes are scale invariant after all. (c) There is a well-defined Laplacian on F .

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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A process formulation

Theorem 2. Let Xt be (i) strong Markov; (ii) non-degenerate; (iii) continuous paths; (iv) state space F ; (v) “invariant under all local symmetries.” Then the law of X under Px is uniquely defined. Two caveats: Unique up to deterministic time change There is always a null set of x’s to deal with, one way or another.

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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What does invariant mean?

Suppose when constructing F , we divide the unit cube into md subcubes and remove some of them. If S1 and S2 are subcubes constructed at the nth stage, so that S1 , S2 have side lengths m−n , and both S1 , S2 contain points of F , then the processes reflected on the boundaries of Si , i = 1, 2, have the same law.

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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To be more precise, let RS be the restriction operator, and US the unfolding operator. Define ES (g , g ) =

1 E(US g , US g ). mn

Suppose Φ : S1 → S2 is an isometry. Then ES1 (f ◦ Φ, f ◦ Φ) = ES2 (f , f ) and E(f , f ) =

X

ES (RS f , RS f ).

S

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Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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The example revisited

F is [0, 1]2 and we look at ordinary Brownian motion with reflection on the boundary of F . Then Z 1 |∇f (x)|2 dx, E(f , f ) = 2 F

and in this case ES (f , f ) =

1 2

Z

|∇f (x)|2 dx.

S

P The condition E(f , f ) = S ES (RS f , RS f ) is there to guarantee that nothing surprising happens on the boundary.

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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Skeleton of the proof

If A, B ∈ E, let λ = sup{r > 0 : A ≥ r B}. Then (a) C = A − λB ∈ E (not quite true) (b) All elements of E have the same domain and are comparable. (c) So C ≥ εB for some ε. But then A ≥ (λ + ε)B, a contradiction to the definition of λ.

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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First complication

C need not be closed. We look at Cδ = (1 + δ)A − λB, get estimates independent of δ, and that is good enough.

Richard Bass (University of Connecticut)

Brownian motion on the Sierpinski carpet

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Where most of the work is

To show that all elements of E have the same domain and are comparable, this comes down to heat kernel estimates for p E (t, x, y ) for arbitrary E in E. The ingredients necessary to getting heat kernel estimates are (a) Corner moves and slides (b) A coupling argument (c) Elliptic Harnack inequality (d) Resistance estimates

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A R

A

R

A R

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A

R

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Richard Bass (University of Connecticut)

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A major difficulty What to do when starting in the middle of a L-shaped pattern. How do processes behave starting at such a point? The situation can be quite complicated in higher dimensions, but even in 2 dimensions it is a serious problem. This problem was avoided in Barlow-Bass because our approximations were reflecting Brownian motions, which do not hit points. For arbitrary E in E we can’t make any such approximation.

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