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 ( X n ) 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 ( X n ) 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 µ h ( d α ) ≥ 0 on M such that � for all w ∈ W h ( w ) = k ( w , ξ ) µ h ( d α ) M 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 �� 0 � � 0 � � 1 � � 1 � � 1 � � 2 � � 2 � � 2 �� A := , , , , , , , 1 2 0 1 2 0 1 2 � 1 � � 0 � � 2 � 2 2 2 � 1 � � 2 � 0 0 � 1 � � 0 � � 2 � 1 1 1 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 n = 1 A n ∪ {∅} ( ∅ is the empty word) Define the word space W := � ∞ � 1 � � 0 � � 22002 � u = 22211 2 2 � 11 �� 10 �� 12 � � 2221 � 02 02 02 v = � 11 � � 12 � 0211 00 00 � 11 �� 10 �� 12 � 01 01 01 � 11 �� 10 �� 12 � � 22 � w = 12 12 12 10 � 11 � � 12 � 10 10 � 1 � � 221 � � 11 �� 10 �� 12 � w = 1 101 11 11 11 Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 6 / 13
b b Previous approaches to a Laplacian Martin boundary theory SC as Martin boundary harm. fct. on SC n = 1 A n ∪ {∅} ( ∅ is the empty word) Define the word space W := � ∞ � 1 � � 0 � � 202 ∞ � 221 ∞ 2 2 � � = = 211 ∞ 211 ∞ � 202 ∞ � 221 ∞ � � � 11 �� 10 �� 12 � = 02 02 02 022 ∞ 022 ∞ � 11 � � 12 � 00 00 � 22 ∞ � � 11 �� 10 �� 12 � = 01 ∞ 01 01 01 � 22 ∞ � � 11 �� 10 �� 12 � 12 12 12 12 ∞ � 11 � � 12 � 10 10 � 11 �� 10 �� 12 � 11 11 11 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 ∞ : � w 1 � ∈ W ∪ A ∞ . Define w i by: 1 let w = w 2 if w i = uab k or w i = uab ∞ with u ( a + b )( 2 b ) k u ∈ { 0 , 1 , 2 } n , a , b ∈ { 0 , 1 , 2 } , a � = b , k ≥ 1 w i := w i else where the addition/multiplication is mod 3. Remark: it holds by definition, that w i = w i � w 1 � � w 1 � � w 1 � 2 set ˆ w := , ˇ w := and ˜ w := w 2 w 2 w 2 3 define the equivalence class of w by w } ∩ ( W ∪ A ∞ ) [ w ] := { ˆ w , ˇ w , ˜ 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 ∼ � 11 � w = 00 w , ˇ ˆ w and ˜ w are equal to w 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 ∼ � 02 � , w = 20 ˆ w w � 02 � � 21 � w = ˆ = 20 20 w and ˜ ˇ w don’t distinguish, because � 02 � � 02 � w = ˇ = = w 20 20 � 02 � � 21 � w = ˜ = = ˆ w 20 20 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 ∼ � 12 � , w = 21 � 12 � � 01 � w = ˆ = , ˆ w w 21 21 ˇ ˜ w w � 12 � � 12 � w = ˇ = 21 02 w doesn’t exist, because ˜ � 12 � � 01 � w = ˜ = ∈ W / 21 02 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 ∼ � 201 � , w = 011 � 201 � � 212 � w = ˆ = , 011 011 � 201 � � 201 � w = ˇ = , 011 122 � 201 � � 212 � w = ˜ = 011 122 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 if ∃ i ∈ A and v ∗ ∼ v s.t. w = v ∗ i , � 1 8 · R ( v ) p ( v , w ) := 0 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 if ∃ i ∈ A and v ∗ ∼ v s.t. w = v ∗ i , � 1 8 · R ( v ) p ( v , w ) := 0 else for example: R ( v ) = 1 : v each 1 8 vi 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 if ∃ i ∈ A and v ∗ ∼ v s.t. w = v ∗ i , � 1 8 · R ( v ) p ( v , w ) := 0 else for example: R ( v ) = 1 : R ( v ) = 3 : v v v ˇ ˆ v each 1 8 vi 2 ˇ vi 3 vi vi 1 ˆ 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 if ∃ i ∈ A and v ∗ ∼ v s.t. w = v ∗ i , � 1 8 · R ( v ) p ( v , w ) := 0 else for example: R ( v ) = 1 : R ( v ) = 3 : v v v ˇ ˆ v each 1 8 vi 2 ˇ vi 3 vi vi 1 ˆ 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 if ∃ i ∈ A and v ∗ ∼ v s.t. w = v ∗ i , � 1 8 · R ( v ) p ( v , w ) := 0 else for example: R ( v ) = 1 : R ( v ) = 3 : v v v ˇ ˆ v each with probability each 1 8 · 3 = 1 1 8 24 vi 2 ˇ vi 3 vi vi 1 ˆ 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 ( X n ) 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 { R 2 ; S 1 , . . . , S 8 } and which fulfills K = � 8 i = 1 S i ( K ) Our aim is to prove, that ? ? ∼ ∼ = A ∞ / ∼ = M K holds. Problems: choice of the metric on A ∞ / ∼ Stefan Kohl University of Stuttgart Sierpiński carpet as a Martin boundary 10 / 13
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