Sierpiski carpet as a Martin boundary Stefan Kohl 6th Cornell - - PowerPoint PPT Presentation

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Sierpiński carpet as a Martin boundary

Stefan Kohl 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals June 16th, 2017

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 1 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

table of contents

1 Previous approaches to a Laplacian 2 Martin boundary theory 3 Sierpiński carpet as a Martin boundary 4 harmonic functions on the Sierpiński carpet

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 2 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Previous approaches to a Laplacian

graph approximation: define a discrete Laplacian on each step and get the Laplacian in the limit (Kigami (1989/93), Kusuoka/Zhou (1992), Strichartz (2001)) Brownian motion: the Laplace operator is the infinitesimal generator of the Brownian motion (i.e. Barlow/Bass (1989/99), Lindstrøm (1990)) function spaces: using the theory for function spaces, it is possible, to define differential operators like the Laplacian (Triebel (1997)) Martin boundary: see next slides (Denker/Sato (2001/02), Ju/Lau/Wang (2011))

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 3 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Martin boundary theory

Let (Xn)n≥1 be a Markov chain with state space W and transition probability p(v, w) with v, w ∈ W Based on p(v, w) define the Martin kernel k(v, w) of (Xn)n≥0 Define the Martin space W by ρ-completion of W (ρ is a certain metric on W depending on k) M := ∂W = W\W is called the Martin boundary By Dynkin’s theorem (1969) exists for every non-negative p-harmonic function h on W a measure µh(dα) ≥ 0 on M such that h(w) =

• M

k(w, ξ)µh(dα) for all w ∈ W holds.

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 4 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Sierpiński carpet as a Martin boundary

Define the alphabet A := 1

• ,

2

• ,

1

• ,

1 1

• ,

1 2

• ,

2

• ,

2 1

• ,

2 2

• 1

1

• 1
• 2

1

• 1
• 2
• 1

2

• 2
• 2

2

• Stefan Kohl

University of Stuttgart Sierpiński carpet as a Martin boundary 5 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Define the word space W := ∞

n=1 An ∪ {∅} (∅ is the empty word)

1 2

• 2
• 11

01 10 01 12 01

• 11

00

• 12

00

• 11

02 10 02 12 02

• 11

11 10 11 12 11

• 11

10

• 12

10

• 11

12 10 12 12 12

• w =

22 10

• w

1 1

• =

221 101

• v =

2221 0211

• u =

22002 22211

• Stefan Kohl

University of Stuttgart Sierpiński carpet as a Martin boundary 6 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Define the word space W := ∞

n=1 An ∪ {∅} (∅ is the empty word)

1 2

• 2
• 11

01 10 01 12 01

• 11

00

• 12

00

• 11

02 10 02 12 02

• 11

11 10 11 12 11

• 11

10

• 12

10

• 11

12 10 12 12 12

• b

22∞ 01∞

• =

22∞ 12∞

• b

202∞ 211∞

• =

221∞ 211∞

• =

202∞ 022∞

• =

221∞ 022∞

• Stefan Kohl

University of Stuttgart Sierpiński carpet as a Martin boundary 6 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Define an equivalence relation ∼ on W ∪ A∞:

1 let w =

w1 w2

• ∈ W ∪ A∞. Define wi by:

wi :=      u(a + b)(2b)k if wi = uabk or wi = uab∞ with u ∈ {0, 1, 2}n, a, b ∈ {0, 1, 2}, a = b, k ≥ 1 wi else where the addition/multiplication is mod 3. Remark: it holds by definition, that wi = wi

2 set ˆ

w := w1 w2

• , ˇ

w := w1 w2

• and ˜

w := w1 w2

• 3 define the equivalence class of w by

[w] := { ˆ w, ˇ w, ˜ w} ∩ (W ∪ A∞)

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 7 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

illustration of the equivalence relation ∼

w

w = 11 00

• ˆ

w, ˇ w and ˜ w are equal to w

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 8 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

illustration of the equivalence relation ∼

ˆ w w

w = 02 20

• ,

ˆ w = 02 20

• =

21 20

• ˇ

w and ˜ w don’t distinguish, because ˇ w = 02 20

• =

02 20

• = w

˜ w = 02 20

• =

21 20

• = ˆ

w

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 8 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

illustration of the equivalence relation ∼

˜ w ˆ w w ˇ w

w = 12 21

• ,

ˆ w = 12 21

• =

01 21

• ,

ˇ w = 12 21

• =

12 02

• ˜

w doesn’t exist, because ˜ w = 12 21

• =

01 02

• /

∈ W

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 8 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

illustration of the equivalence relation ∼ w = 201 011

• ,

ˆ w = 201 011

• =

212 011

• ,

ˇ w = 201 011

• =

201 122

• ,

˜ w = 201 011

• =

212 122

• Stefan Kohl

University of Stuttgart Sierpiński carpet as a Martin boundary 8 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Set R(w) := #[w] = # {w∗ ∈ W : w∗ ∼ w} ∈ {1, 2, 3, 4} Define the transition probability p : W × W → [0, 1] by p(v, w) :=

• 1

8·R(v)

if ∃i ∈ A and v∗ ∼ v s.t. w = v∗i, else

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 9 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Set R(w) := #[w] = # {w∗ ∈ W : w∗ ∼ w} ∈ {1, 2, 3, 4} Define the transition probability p : W × W → [0, 1] by p(v, w) :=

• 1

8·R(v)

if ∃i ∈ A and v∗ ∼ v s.t. w = v∗i, else for example: R(v) = 1 : v vi each 1

8

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 9 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Set R(w) := #[w] = # {w∗ ∈ W : w∗ ∼ w} ∈ {1, 2, 3, 4} Define the transition probability p : W × W → [0, 1] by p(v, w) :=

• 1

8·R(v)

if ∃i ∈ A and v∗ ∼ v s.t. w = v∗i, else for example: R(v) = 1 : v vi each 1

8

R(v) = 3 : v vi2 ˆ v ˆ vi1 ˇ v ˇ vi3

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 9 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Set R(w) := #[w] = # {w∗ ∈ W : w∗ ∼ w} ∈ {1, 2, 3, 4} Define the transition probability p : W × W → [0, 1] by p(v, w) :=

• 1

8·R(v)

if ∃i ∈ A and v∗ ∼ v s.t. w = v∗i, else for example: R(v) = 1 : v vi each 1

8

R(v) = 3 : v vi2 ˆ v ˆ vi1 ˇ v ˇ vi3

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 9 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Set R(w) := #[w] = # {w∗ ∈ W : w∗ ∼ w} ∈ {1, 2, 3, 4} Define the transition probability p : W × W → [0, 1] by p(v, w) :=

• 1

8·R(v)

if ∃i ∈ A and v∗ ∼ v s.t. w = v∗i, else for example: R(v) = 1 : v vi each 1

8

R(v) = 3 : v vi2 ˆ v ˆ vi1 ˇ v ˇ vi3 each with probability

1 8·3 = 1 24

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 9 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

the way it goes on and a short outlook

p defines a Markov chain (Xn)n≥0. Futher define the n-step probability, the Green function, the Martin kernel, a metric ρ on W and the Martin boundary M = W\W. Denote by K the Sierpiński carpet, which is generated by the IFS {R2; S1, . . . , S8} and which fulfills K = 8

i=1 Si(K)

Our aim is to prove, that K

?

∼ = A∞/∼

?

∼ = M holds. Problems: choice of the metric on A∞/∼

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 10 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

harmonic functions on the Sierpiński carpet

First hard question: What is the boundary of the Sierpiński carpet? And why?

01 21 12 21 12 02 11 11 01 11 21 11 11 01 11 21 21 21 10 11 00 11 20 11 10 01 20 01 10 21 00 21 20 21 12 11 02 11 22 11 12 01 22 01 02 21 22 21 11 10 01 10 21 10 11 00 21 00 11 20 01 20 21 20 12 10 02 10 22 10 12 00 22 00 12 20 02 20 11 12 01 12 11 02 21 02 11 22 01 22 21 22 10 12 00 12 20 12 10 02 20 02 10 22 00 22 20 22 12 12 22 12 22 02 12 22 02 22 22 22 22 20 02 12 21 01 21 12 Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 11 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

harmonic functions on the Sierpiński carpet

First hard question: What is the boundary of the Sierpiński carpet? And why?

01 21 12 21 12 02 11 11 01 11 21 11 11 01 11 21 21 21 10 11 00 11 20 11 10 01 20 01 10 21 00 21 20 21 12 11 02 11 22 11 12 01 22 01 02 21 22 21 11 10 01 10 21 10 11 00 21 00 11 20 01 20 21 20 12 10 02 10 22 10 12 00 22 00 12 20 02 20 11 12 01 12 11 02 21 02 11 22 01 22 21 22 10 12 00 12 20 12 10 02 20 02 10 22 00 22 20 22 12 12 22 12 22 02 12 22 02 22 22 22 22 20 02 12 21 01 21 12 Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 11 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

harmonic functions on the Sierpiński carpet

First hard question: What is the boundary of the Sierpiński carpet? And why?

01 21 12 21 12 02 11 11 01 11 21 11 11 01 11 21 21 21 10 11 00 11 20 11 10 01 20 01 10 21 00 21 20 21 12 11 02 11 22 11 12 01 22 01 02 21 22 21 11 10 01 10 21 10 11 00 21 00 11 20 01 20 21 20 12 10 02 10 22 10 12 00 22 00 12 20 02 20 11 12 01 12 11 02 21 02 11 22 01 22 21 22 10 12 00 12 20 12 10 02 20 02 10 22 00 22 20 22 12 12 22 12 22 02 12 22 02 22 22 22 22 20

Should be a boundary cell, because the cell is out!

02 12 21 01 21 12

Should be no boundary cells, because the absence

• f the cell

comes from the structure of the frac- tal!

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 11 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

harmonic functions on the Sierpiński carpet

First hard question: What is the boundary of the Sierpiński carpet? And why?

01 21 12 21 12 02 11 11 01 11 21 11 11 01 11 21 21 21 10 11 00 11 20 11 10 01 20 01 10 21 00 21 20 21 12 11 02 11 22 11 12 01 22 01 02 21 22 21 11 10 01 10 21 10 11 00 21 00 11 20 01 20 21 20 12 10 02 10 22 10 12 00 22 00 12 20 02 20 11 12 01 12 11 02 21 02 11 22 01 22 21 22 10 12 00 12 20 12 10 02 20 02 10 22 00 22 20 22 12 12 22 12 22 02 12 22 02 22 22 22 22 20

Should be a boundary cell, because the cell is out!

02 12 21 01 21 12

Should be no boundary cells, because the absence

• f the cell

comes from the structure of the frac- tal!

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 11 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

harmonic functions on the Sierpiński carpet

First hard question: What is the boundary of the Sierpiński carpet? And why?

Should be a boundary cell, because the cell is out! Should be no boundary cells, because the absence

• f the cell

comes from the structure of the frac- tal!

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 11 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

a short numerical approach

using this definition of a boundary, we get for example: 0.5 0.5 1 with the boundary function: (x, y) → x + y 2 and values: 1

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 12 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

a short numerical approach

using this definition of a boundary, we get for example: 0.5 0.5 1 with the boundary function: (x, y) → x + y 2 and values: 1

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 12 / 13

Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary

• harm. fct. on SC

Thank you for your attention!

Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 13 / 13