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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) ⊆ L2(X, m) Q is called Dirichlet form in the wide sense if: D(Q) with ·, ·Q := Q(·, ·) + ·, ·L2 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 Q1, Q2 be Dirichlet forms in the wide sense. Then, Q1 ≥ Q2 if D(Q1) ⊆ D(Q2) and Q1(u) ≥ Q2(u), u ∈ D(Q1).

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Boundary Representation of Dirichlet Forms on Rn

Let Ω ⊆ Rn be an open set with smooth boundary ∂Ω. Let ∆ : C ∞

c (Ω) ⊆ L2(Ω) → L2(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 Rn

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 L2(∂X, σ) such that for every u ∈ D(Q) the equality Q(u) = Q(D)(PD(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

• y∈X

b(x, y) < ∞, x ∈ 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 x1, . . . , xn ∈ X such that x ∼ x1 ∼ . . . ∼ xn ∼ 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

• y∈X

b(x, y) < ∞, x ∈ 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 x1, . . . , xn ∈ X such that x ∼ x1 ∼ . . . ∼ xn ∼ 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

• y∈X

b(x, y) < ∞, x ∈ 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 x1, . . . , xn ∈ X such that x ∼ x1 ∼ . . . ∼ xn ∼ 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

2

• x,y∈X

b(x, y)(u(x) − u(y))2 +

• x∈X

c(x)u(x)2 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) :=

• x∈A

m(x) a measure on X. Two natural Dirichlet forms via Q(N) := Q|D(Q(N)), D(Q(N)) = D ∩ ℓ2(X, m) and Q(D) := Q|D(Q(D)), D(Q(D)) = Cc(X)

·Q(N).

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) :=

• x∈A

m(x) a measure on X. Two natural Dirichlet forms via Q(N) := Q|D(Q(N)), D(Q(N)) = D ∩ ℓ2(X, m) and Q(D) := Q|D(Q(D)), D(Q(D)) = Cc(X)

·Q(N).

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) :=

• x∈A

m(x) a measure on X. Two natural Dirichlet forms via Q(N) := Q|D(Q(N)), D(Q(N)) = D ∩ ℓ2(X, m) and Q(D) := Q|D(Q(D)), D(Q(D)) = Cc(X)

·Q(N).

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) :=

• x∈A

m(x) a measure on X. Two natural Dirichlet forms via Q(N) := Q|D(Q(N)), D(Q(N)) = D ∩ ℓ2(X, m) and Q(D) := Q|D(Q(D)), D(Q(D)) = Cc(X)

·Q(N).

These satisfy Q(D) ≥ Q(N).

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Goal

Dirichlet form Q on ℓ2(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 ℓ2(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 ℓ2(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 ℓ2(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

• n a compact Hausdorff space K.

X can be embedded in K by x → δx.

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

• n a compact Hausdorff space K.

X can be embedded in K by x → δx.

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

• n a compact Hausdorff space K.

X can be embedded in K by x → δx.

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

• n a compact Hausdorff space K.

X can be embedded in K by x → δx.

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

• n a compact Hausdorff space K.

X can be embedded in K by x → δx.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Royden Boundary

Definition The set ∂X := K \ X is called Royden boundary of X. By continuity, for u ∈ D ∩ ℓ∞(X) we can define u|∂X. Lemma (u ∧ 1)|∂X = u|∂X ∧ 1, u ∈ D ∩ ℓ∞(X)

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Royden Boundary

Definition The set ∂X := K \ X is called Royden boundary of X. By continuity, for u ∈ D ∩ ℓ∞(X) we can define u|∂X. Lemma (u ∧ 1)|∂X = u|∂X ∧ 1, u ∈ D ∩ ℓ∞(X)

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Canonically Compactifiable Graphs

Recall D = {u ∈ C(X): Q(u) < ∞}. Definition A connected graph (b, c) is called canonically compactifiable if

• D ⊆ ℓ∞(X).

From now on let (b, c) be a canonically compactifiable graph.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Canonically Compactifiable Graphs

Lemma D(Q(D)) = {u ∈ D(Q(N)): u|∂X ≡ 0}

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Goal

Dirichlet form Q on ℓ2(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?( D ⊆ ℓ∞(X)) How to define a “good” measure on the boundary? How to define q?

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Harmonic Functions

Definition Let u ∈

• D. Then, u harmonic if

Lu(x) :=

• y∈X

b(x, y)(u(x) − u(y)) + c(x)u(x) = 0, x ∈ X. Theorem (Maximum Principle) Let u ∈ D be harmonic. Then, u∞ = u|∂X∞.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Harmonic Functions

Definition Let u ∈

• D. Then, u harmonic if

Lu(x) :=

• y∈X

b(x, y)(u(x) − u(y)) + c(x)u(x) = 0, x ∈ X. Theorem (Maximum Principle) Let u ∈ D be harmonic. Then, u∞ = u|∂X∞.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Harmonic Functions

Theorem (Royden decomposition) Let Q be a Dirichlet form such that Q(D) ≥ Q ≥ Q(N). Let u ∈ D(Q) be arbitrary. Then, u = u0 + uh with unique u0 ∈ D(Q(D)) and unique harmonic uh ∈ D(Q). Moreover, Q(u) = Q(D)(u0) + Q(uh).

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Harmonic Functions

Theorem For every ϕ ∈ D(Q(N))|∂X there is an unique harmonic Hϕ ∈ D(Q(N)) such that Hϕ|∂X = ϕ. The mapping H· : D(Q(N))|∂X → D(Q(N)), ϕ → Hϕ is linear. Moreover, Hϕ∧1 = (Hϕ ∧ 1)h, ϕ ∈ D(Q(N))|∂X.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Harmonic Functions

Theorem For every ϕ ∈ D(Q(N))|∂X there is an unique harmonic Hϕ ∈ D(Q(N)) such that Hϕ|∂X = ϕ. The mapping H· : D(Q(N))|∂X → D(Q(N)), ϕ → Hϕ is linear. Moreover, Hϕ∧1 = (Hϕ ∧ 1)h, ϕ ∈ D(Q(N))|∂X.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Measure on the Boundary

From now on let m be a finite measure. Theorem There is a measure m on B(∂X), such that the equality

• x∈X

u(x)m(x) =

• ∂X

u|∂X d m holds for every harmonic u ∈ D(Q(N)). Idea: Apply the theorem of Riesz-Markov to ϕ →

x∈X Hϕ(x)m(x). . .

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Measure on the Boundary

From now on let m be a finite measure. Theorem There is a measure m on B(∂X), such that the equality

• x∈X

u(x)m(x) =

• ∂X

u|∂X d m holds for every harmonic u ∈ D(Q(N)). Idea: Apply the theorem of Riesz-Markov to ϕ →

x∈X Hϕ(x)m(x). . .

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Measure on the Boundary

Lemma Hϕℓ2(X,m) ≤ ϕL2(∂X,

m), ϕ ∈ D(Q(N))|∂X ⊆ L2(∂X,

m) Theorem There is a C > 0, such that for every u ∈ D(Q) u|∂XL2(∂X,

m) ≤ CuQ.

In particular, · Q-convergence implies L2(∂X, m)-convergence.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

The Measure on the Boundary

Lemma Hϕℓ2(X,m) ≤ ϕL2(∂X,

m), ϕ ∈ D(Q(N))|∂X ⊆ L2(∂X,

m) Theorem There is a C > 0, such that for every u ∈ D(Q) u|∂XL2(∂X,

m) ≤ CuQ.

In particular, · Q-convergence implies L2(∂X, m)-convergence.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Goal

Dirichlet form Q on ℓ2(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?( D ⊆ ℓ∞(X)) How to define a “good” measure on the boundary? (m(X) < ∞) How to define q?

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Forms on the Boundary

Definition Define the boundary form q by D(q) := D(Q)|∂X and q(ϕ, ψ) = Q(Hϕ, Hψ), ϕ, ψ ∈ D(q).

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Forms on the Boundary

Lemma (D(q), · q) is complete. Proof. (ϕn) · q-Cauchy. Then, (Hϕn) is · Q-Cauchy and, thus, it converges to u ∈ D(Q). Since ϕn − u|∂XL2(∂X,

m) ≤ CHϕn − uQ → 0 we infer

ϕn − u|∂Xq → 0.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Forms on the Boundary

Lemma (D(q), · q) is complete. Proof. (ϕn) · q-Cauchy. Then, (Hϕn) is · Q-Cauchy and, thus, it converges to u ∈ D(Q). Since ϕn − u|∂XL2(∂X,

m) ≤ CHϕn − uQ → 0 we infer

ϕn − u|∂Xq → 0.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Forms on the Boundary

Lemma q(ϕ ∧ 1) ≤ q(ϕ), ϕ ∈ D(q) Proof. Let u ∈ D(Q) such that u|∂X = ϕ. Then, u ∧ 1 ∈ D(Q) and (u ∧ 1)|∂X = u|∂X ∧ 1 = ϕ ∧ 1 holds. Hence, ϕ ∧ 1 ∈ D(q). Furthermore, q(ϕ ∧ 1) = Q(Hϕ∧1) = Q((Hϕ ∧ 1)h) ≤ Q((Hϕ ∧ 1)h) + Q(D)((Hϕ ∧ 1)0) = Q(Hϕ ∧ 1) ≤ Q(Hϕ) = q(ϕ).

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Forms on the Boundary

Lemma q(ϕ ∧ 1) ≤ q(ϕ), ϕ ∈ D(q) Proof. Let u ∈ D(Q) such that u|∂X = ϕ. Then, u ∧ 1 ∈ D(Q) and (u ∧ 1)|∂X = u|∂X ∧ 1 = ϕ ∧ 1 holds. Hence, ϕ ∧ 1 ∈ D(q). Furthermore, q(ϕ ∧ 1) = Q(Hϕ∧1) = Q((Hϕ ∧ 1)h) ≤ Q((Hϕ ∧ 1)h) + Q(D)((Hϕ ∧ 1)0) = Q(Hϕ ∧ 1) ≤ Q(Hϕ) = q(ϕ).

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Forms on the Boundary

Theorem q is a Dirichlet form in the wide sense.

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Goal

Dirichlet form Q on ℓ2(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?( D ⊆ ℓ∞(X)) How to define a “good” measure on the boundary? (m(X) < ∞) How to define q?

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Boundary Representation of Dirichlet Forms

Theorem Let Q be a Dirichlet form on ℓ2(X, m) that satisfies Q(D) ≥ Q ≥ Q(N). Then, for every u ∈ D(Q) the equality Q(u) = Q(D)(u0) + q(u|∂X) holds. Proof. Q(u) = Q(D)(u0) + Q(uh) = Q(D)(u0) + q(uh|∂X) = Q(D)(u0) + q(u|∂X).

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs

Boundary Representation of Dirichlet Forms

Theorem Let Q be a Dirichlet form on ℓ2(X, m) that satisfies Q(D) ≥ Q ≥ Q(N). Then, for every u ∈ D(Q) the equality Q(u) = Q(D)(u0) + q(u|∂X) holds. Proof. Q(u) = Q(D)(u0) + Q(uh) = Q(D)(u0) + q(uh|∂X) = Q(D)(u0) + q(u|∂X).

Michael Schwarz Boundary Representation of Dirichlet Forms on Canonically Compactifiable Graphs