Boundary Representation of Dirichlet Forms on Canonically - PowerPoint PPT Presentation

Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs Michael Schwarz 18.12.2016 Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Dirichlet Forms Definition ( X , m ) measure space Q bilinear form with D ( Q ) ⊆ L 2 ( X , m ) Q is called Dirichlet form in the wide sense if: D ( Q ) with �· , ·� Q := Q ( · , · ) + �· , ·� L 2 is a Hilbert space. Q ( u ∧ 1 ) ≤ Q ( u ) , u ∈ D ( Q ) . A Dirichlet form is a densely defined Dirichlet form in the wide sense. Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Dirichlet Forms Definition Let Q 1 , Q 2 be Dirichlet forms in the wide sense. Then, Q 1 ≥ Q 2 if D ( Q 1 ) ⊆ D ( Q 2 ) and Q 1 ( u ) ≥ Q 2 ( u ) , u ∈ D ( Q 1 ) . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Boundary Representation of Dirichlet Forms on R n Let Ω ⊆ R n be an open set with smooth boundary ∂ Ω . Let ∆ : C ∞ c (Ω) ⊆ L 2 (Ω) → L 2 (Ω) . Denote the forms in correspondence with the Dirichlet- and Neumann extensions by Q ( D ) and Q ( N ) . Then, every Dirichlet form Q in correspondence with an extension of ∆ satisfies Q ( D ) ≥ Q ≥ Q ( N ) . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Boundary Representation of Dirichlet Forms on R n Theorem (A.Posilicano 2012, Special case) Let Q be a Dirichlet form in correspondence with an extension of ∆ . Then, there is a Dirichlet form in the wide sense q on L 2 ( ∂ X , σ ) such that for every u ∈ D ( Q ) the equality Q ( u ) = Q ( D ) ( P D ( Q ( D ) ) u ) + q ( Tr u ) holds. Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Graphs Definition Let X be a countably infinite set. A graph over X is a pair ( b , c ) consisting of a symmetric function b : X × X → [ 0 , ∞ ) that vanishes on the diagonal and satisfies � b ( x , y ) < ∞ , x ∈ X , y ∈ X and c : X → [ 0 , ∞ ) . For x , y ∈ X we write x ∼ y if b ( x , y ) > 0. Let ( b , c ) be a graph over X . We call ( b , c ) connected if for every x , y ∈ X there are x 1 , . . . , x n ∈ X such that x ∼ x 1 ∼ . . . ∼ x n ∼ y . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Graphs Definition Let X be a countably infinite set. A graph over X is a pair ( b , c ) consisting of a symmetric function b : X × X → [ 0 , ∞ ) that vanishes on the diagonal and satisfies � b ( x , y ) < ∞ , x ∈ X , y ∈ X and c : X → [ 0 , ∞ ) . For x , y ∈ X we write x ∼ y if b ( x , y ) > 0. Let ( b , c ) be a graph over X . We call ( b , c ) connected if for every x , y ∈ X there are x 1 , . . . , x n ∈ X such that x ∼ x 1 ∼ . . . ∼ x n ∼ y . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Graphs Definition Let X be a countably infinite set. A graph over X is a pair ( b , c ) consisting of a symmetric function b : X × X → [ 0 , ∞ ) that vanishes on the diagonal and satisfies � b ( x , y ) < ∞ , x ∈ X , y ∈ X and c : X → [ 0 , ∞ ) . For x , y ∈ X we write x ∼ y if b ( x , y ) > 0. Let ( b , c ) be a graph over X . We call ( b , c ) connected if for every x , y ∈ X there are x 1 , . . . , x n ∈ X such that x ∼ x 1 ∼ . . . ∼ x n ∼ y . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Quadratic Form � Q Let C ( X ) = { f : X → R } . Define � Q : C ( X ) → [ 0 , ∞ ] , � � Q ( u ) = 1 b ( x , y )( u ( x ) − u ( y )) 2 + � c ( x ) u ( x ) 2 2 x , y ∈ X x ∈ X and D := { u ∈ C ( X ): � � Q ( u ) < ∞} . By polarization � Q induces a bilinear form on � D , which will also be denoted by � Q . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Dirichlet Forms on Graphs Let m : X → ( 0 , ∞ ) . Then m defines via � m ( A ) := m ( x ) x ∈ A a measure on X . Two natural Dirichlet forms via Q ( N ) := � Q | D ( Q ( N ) ) , D ( Q ( N ) ) = � D ∩ ℓ 2 ( X , m ) and �·� Q ( N ) . Q ( D ) := � Q | D ( Q ( D ) ) , D ( Q ( D ) ) = C c ( X ) These satisfy Q ( D ) ≥ Q ( N ) . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Dirichlet Forms on Graphs Let m : X → ( 0 , ∞ ) . Then m defines via � m ( A ) := m ( x ) x ∈ A a measure on X . Two natural Dirichlet forms via Q ( N ) := � Q | D ( Q ( N ) ) , D ( Q ( N ) ) = � D ∩ ℓ 2 ( X , m ) and �·� Q ( N ) . Q ( D ) := � Q | D ( Q ( D ) ) , D ( Q ( D ) ) = C c ( X ) These satisfy Q ( D ) ≥ Q ( N ) . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Dirichlet Forms on Graphs Let m : X → ( 0 , ∞ ) . Then m defines via � m ( A ) := m ( x ) x ∈ A a measure on X . Two natural Dirichlet forms via Q ( N ) := � Q | D ( Q ( N ) ) , D ( Q ( N ) ) = � D ∩ ℓ 2 ( X , m ) and �·� Q ( N ) . Q ( D ) := � Q | D ( Q ( D ) ) , D ( Q ( D ) ) = C c ( X ) These satisfy Q ( D ) ≥ Q ( N ) . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Dirichlet Forms on Graphs Let m : X → ( 0 , ∞ ) . Then m defines via � m ( A ) := m ( x ) x ∈ A a measure on X . Two natural Dirichlet forms via Q ( N ) := � Q | D ( Q ( N ) ) , D ( Q ( N ) ) = � D ∩ ℓ 2 ( X , m ) and �·� Q ( N ) . Q ( D ) := � Q | D ( Q ( D ) ) , D ( Q ( D ) ) = C c ( X ) These satisfy Q ( D ) ≥ Q ( N ) . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Goal Dirichlet form Q on ℓ 2 ( X , m ) such that Q ( D ) ≥ Q ≥ Q ( N ) . Wanted: Decomposition Q ( u ) = Q ( D ) ( v ) + q ( w ) for a Dirichlet form in the wide sense q on the boundary. Questions: How to define a boundary of a graph? How to define a “good” measure on the boundary? How to define q ? Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Goal Dirichlet form Q on ℓ 2 ( X , m ) such that Q ( D ) ≥ Q ≥ Q ( N ) . Wanted: Decomposition Q ( u ) = Q ( D ) ( v ) + q ( w ) for a Dirichlet form in the wide sense q on the boundary. Questions: How to define a boundary of a graph? How to define a “good” measure on the boundary? How to define q ? Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Goal Dirichlet form Q on ℓ 2 ( X , m ) such that Q ( D ) ≥ Q ≥ Q ( N ) . Wanted: Decomposition Q ( u ) = Q ( D ) ( v ) + q ( w ) for a Dirichlet form in the wide sense q on the boundary. Questions: How to define a boundary of a graph? How to define a “good” measure on the boundary? How to define q ? Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Goal Dirichlet form Q on ℓ 2 ( X , m ) such that Q ( D ) ≥ Q ≥ Q ( N ) . Wanted: Decomposition Q ( u ) = Q ( D ) ( v ) + q ( w ) for a Dirichlet form in the wide sense q on the boundary. Questions: How to define a boundary of a graph? How to define a “good” measure on the boundary? How to define q ? Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Royden Compactification Theorem There is a compact Hausdorff space K such that: X can be embedded in K as an open and dense subset � D ∩ ℓ ∞ ( X ) ⊆ C ( K ) . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Royden Compactification Idea of the construction of K : � D ∩ ℓ ∞ ( X ) is a commutative algebra with respect to pointwise multiplication. �·� ∞ A := � D ∩ ℓ ∞ ( X ) is a commutative Banach algebra with respect to � · � ∞ . A + := “smallest algebra containing A and the constant 1” is an unitary commutative Banach algebra. The complexification of A + is a commutative C ∗ -algebra. Therefore, it is isomorphic to the set of continuous functions on a compact Hausdorff space K . X can be embedded in K by x �→ δ x . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Royden Compactification Idea of the construction of K : � D ∩ ℓ ∞ ( X ) is a commutative algebra with respect to pointwise multiplication. �·� ∞ A := � D ∩ ℓ ∞ ( X ) is a commutative Banach algebra with respect to � · � ∞ . A + := “smallest algebra containing A and the constant 1” is an unitary commutative Banach algebra. The complexification of A + is a commutative C ∗ -algebra. Therefore, it is isomorphic to the set of continuous functions on a compact Hausdorff space K . X can be embedded in K by x �→ δ x . Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs