Lecture 3. Markov semigroups and Dirichlet forms

Let (X, ℬ) be a measurable space. We say that (X, ℬ) is a good measurable space if there is a countable family generating ℬ and if every finite measure γ on (X × X, ℬ ⊗ ℬ) can be decomposed as

γ(dx dy) = k(x,dy) γ₁(dx)

where γ₁ is the projection of γ on the first coordinate and k is a kernel, i.e. k(x,·) is a finite measure on (X, ℬ) and x → k(x,A) is measurable for every A ∈ ℬ.

For instance, if X is a Polish space (or a Radon space) equipped with its Borel σ-field, then it is a good measurable space.

Throughout the lecture, we will consider (X, ℬ, μ) to be a good measurable space equipped with a σ-finite measure μ.

Markovian semigroups
Dirichlet forms

Markovian semigroups

Definition Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction semigroup on L²(X,μ). The semigroup (P_t)_t≥0 is called Markovian if and only if for every f ∈ L²(X,μ) and t ≥ 0:

f ≥ 0, a.e. ⇒ P_tf ≥ 0, a.e.
f ≤ 1, a.e. ⇒ P_tf ≤ 1, a.e.

We note that if (P_t)_t≥0 is Markovian, then for every f ∈ L²(X,μ) ∩ L^∞(X,μ),

||P_tf||_L^∞(X,μ) ≤ ||f||_L^∞(X,μ).

As a consequence (P_t)_t≥0 can be extended to a contraction semigroup defined on all of L^∞(X,μ).

Definition A transition function {p_t,t ≥ 0} on X is a family of kernelsnp_t : X × ℬ → [0,1]

such that:

For t ≥ 0 and x ∈ X, p_t(x, ·) is a finite measure on X;
For t ≥ 0 and A ∈ ℬ the application x → p_t(x,A) is measurable;
For s,t ≥ 0, a.e. x ∈ X and A ∈ ℬ,
p_t+s(x,A) = ∫_X p_t(y,A) p_s(x,dy).

The relation above is often called the Chapman-Kolmogorov relation.

Theorem (Heat kernel measure)nLet (P_t)_t≥0 be a strongly continuous self-adjoint contraction Markovian semigroup on L²(X,μ).nThere exists a transition function {p_t,t ≥ 0} on X such that for every f ∈ L^∞(X,μ) and a.e. x ∈ X

P_tf(x) = ∫_X f(y) p_t(x,dy) , t > 0.

This transition function is called the heat kernel measure associated to (P_t)_t≥0.

The proof relies on the following lemma sometimes called the bi-measure theorem. A set function ν : ℬ ⊗ ℬ → [0,+∞) is called a bi-measure, if for every A ∈ ℬ, ν(A, ·) and ν(·, A) are measures.

Lemma If ν : ℬ ⊗ ℬ → [0,+∞) is a bi-measure, then there exists a measure γ on ℬ ⊗ ℬ such that for every A,B ∈ ℬ,

γ(A × B) = ν(A,B).

Proof of Theorem 7.3

We assume that μ is finite and let as an exercise the extension to σ-finite measures. For t > 0, we consider the set function

ν_t(A,B) = ∫_X 1_A P_t 1_B dμ.

Since P_t is supposed to be Markovian, it is a bi-measure. From the bi-measure theorem, there exists a measure γ_t on ℬ ⊗ ℬ such that for every A,B ∈ ℬ,

γ_t(A × B) = ν_t(A,B) = ∫_X 1_A P_t 1_B dμ.

The projection of γ_t on the first coordinate is (P_t1) dμ, thus from the measure decomposition theorem, γ_t can be decomposed as

γ_t(dx dy) = p_t(x,dy) μ(dx)

for some kernel p_t. One has then for every A,B ∈ ℬ

∫_X 1_A P_t 1_B dμ = ∫_A ∫_B p_t(x,dy) μ(dx),

from which it follows that for every f ∈ L^∞(X,μ), and a.e. x ∈ X

P_tf(x) = ∫_X f(y) p_t(x,dy).

The relation

p_t+s(x,A) = ∫_X p_t(y,A) p_s(x,dy)

follows from the semigroup property.

Exercise Prove Theorem 7.3 if μ is σ-finite.

Exercise Show that for every non-negative measurable function F : X × X → ℝ,

∫_X ∫_X F(x,y) p_t(x,dy) dμ(x) = ∫_X ∫_X F(x,y) p_t(y,dx) dμ(y).

Definition Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction Markovian semigroup on L²(X,μ). We say that the semigroup {P_t}_t∈[0,∞) admits a heat kernel if the heat kernel measures have a density with respect to μ, i.e. there exists a measurable function p : ℝ_>0 × X × X → ℝ_≥0, such that for every t > 0, a.e. x,y ∈ X, f ∈ L^∞(X,μ),

P_tf(x) = ∫_X p_t(x,y) f(y) dμ(y).

If the heat kernel exists, we will often denote p(t,x,y) as p_t(x,y) for t > 0 and a.e. x,y ∈ X.

Dirichlet forms

Definition A function v on X is called a normal contraction of the function u if for almost every x, y ∈ X,

|v(x)-v(y)| ≤ |u(x) – u(y)| and |v(x)| ≤ |u(x)|.

Definition Let (ℰ,ℱ = dom(ℰ)) be a densely defined closed quadratic form on L²(X,μ). The form ℰ is called a Dirichlet form if it is Markovian, that is, has the property that if u ∈ ℱ and v is a normal contraction of u then v ∈ ℱ and

ℰ(v,v) ≤ ℰ(u,u).

Exercise Show that a densely defined closed quadratic form on L²(X,μ) is Markovian if and only if for every u ∈ ℱ, (0 ∨ u) ∧ 1 ∈ ℱ and ℰ( (0 ∨ u) ∧ 1, (0 ∨ u) ∧ 1 ) ≤ ℰ(u,u).

Theorem Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction semigroup on L²(X,μ). Then, (P_t)_t≥0 is a Markovian semigroup if and only if the associated closed symmetric form on L²(X,μ) is a Dirichlet form.

Proof

Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction Markovian semigroup on L²(X,μ). There exists a transition function {p_t, t ≥ 0} on X such that for every u ∈ L^∞(X,μ) and a.e. x ∈ X

P_tu(x) = ∫_X u(y) p_t(x,dy), t > 0.

Denote

k_t(x) = P_t1(x) = ∫_X p_t(x,dy).

We observe that from the Markovian property of P_t, we have 0 ≤ k_t ≤ 1 a.e.
We have then

1/2 ∫_X ∫_X (u(x) – u(y))² p_t(x,dy) dμ(x) = ∫_X u(x)² k_t(x)dμ(x) – ∫_X u(x) P_tu(x) dμ(x).

Therefore,

⟨u – P_tu, u⟩ = 1/2 ∫_X ∫_X (u(x) – u(y))² p_t(x,dy) dμ(x) + ∫_X u(x)² (1 – k_t(x)) dμ(x).

Let us now assume that u ∈ ℱ and that v is a normal contraction of u. One has

∫_X ∫_X (v(x) – v(y))² p_t(x,dy) dμ(x) ≤ ∫_X ∫_X (u(x) – u(y))² p_t(x,dy) dμ(x)

and

∫_X v(x)² (1 – k_t(x)) dμ(x) ≤ ∫_X u(x)² (1 – k_t(x)) dμ(x).

Therefore,

⟨v – P_tv,v⟩ ≤ ⟨u – P_tu,u⟩

Since u ∈ ℱ, one knows that (1/t)⟨u – P_tu,u⟩ converges to ℰ(u) when t → 0. Since (1/t)⟨v – P_tv,v⟩ is non-increasing and bounded it does converge when t → 0. Thus v ∈ ℱ and

ℰ(v) ≤ ℰ(u).

One concludes that ℰ is Markovian.

Now, consider a Dirichlet form ℰ and denote by P_t the associated semigroup in L²(X,μ) and by A its generator.
The main idea is to first prove that for λ > 0, the resolvent operator (λId – A)^-1 preserves the positivity of function. Then, we may conclude by the fact that for f ∈ L²(X,μ), in the L²(X,μ) sense

P_tf = lim_{n → +∞} (Id – t/n A)^-nf.

Let λ > 0. We consider on ℱ the norm

||f||²_λ = ||f||²_L²(X,μ) + λℰ(f,f).

From the Markovian property of ℰ, if u ∈ ℱ, then |u| ∈ ℱ and

ℰ(|u|, |u|) ≤ ℰ(u, u).

We consider the bounded operator

R_λ = (Id – λA)^-1

that goes from L²(X,μ) to 𝒟(A) ⊂ ℱ. For f ∈ ℱ and g ∈ L²(X,μ) with g ≥ 0, we have

⟨|f| , R_λ g⟩_λ = ⟨|f| , R_λg⟩_L²(X,μ) – λ⟨|f| , AR_λ g⟩_L²(X,μ)

= ⟨|f|, (Id – λA) R_λ g⟩_L²(X,μ)

=⟨|f|, g⟩_L²(X,μ)

≥ |⟨f, g⟩_L²(X,μ)|

≥ |⟨f , R_λg⟩_λ|.

Moreover, from inequality above, for f ∈ ℱ,

|||f|||_λ² = |||f| ||²_L²(X,μ) + λℰ(|f|,|f|)

≤ ||f||²_L²(X,μ) + λℰ(f,f)

≤ ||f||_λ².

By taking f = R_λ g in the two above sets of inequalities, we draw the conclusion

|⟨R_λg, R_λg⟩_λ| ≤ ⟨|R_λg| , R_λg⟩_λ ≤ |||R_λg|||_λ ||R_λg||_λ ≤ |⟨R_λg, R_λg⟩_λ|.

The above inequalities are therefore equalities which implies

R_λg = |R_λg|.

As a conclusion if g ∈ L²(X,μ) is a.e. ≥ 0, then for every λ > 0, (Id – λA)^-1g ≥ 0 a.e.. Thanks to the spectral theorem, in L²(X,μ),

P_tg = lim_{n → +∞} (Id – t/n A)^-ng.

By passing to a subsequence that converges pointwise almost surely, we deduce that P_tg ≥ 0 almost surely.

The proof of

f ≤ 1, a.e. ⇒ P_tf ≤ 1, a.e.

follows the same lines:

The first step is to observe that if 0 ≤ f ∈ ℱ, then 1 ∧ f ∈ ℱ and moreover

ℰ(1 ∧ f, 1 ∧ f) ≤ ℰ(f,f).

2. Let f ∈ L²(X,μ) satisfy 0 ≤ f ≤ 1 and set g = R_λf = (Id – λA)^-1f ∈ ℱ and h = 1 ∧ g. According to the first step, h ∈ ℱ and ℰ(h,h) ≤ ℰ(g,g). Now, we observe that:

||g – h||_λ² = ||g||_λ² – 2⟨g,h⟩_λ + ||h||_λ²

= ⟨R_λf,f⟩_L²(X,μ) – 2⟨f,h⟩_L²(X,μ) + ||h||²_L²(X,μ) + λℰ(h,h)

= ⟨R_λf,f⟩_L²(X,μ) – ||f||²_L²(X,μ) + ||f – h||²_L²(X,μ) + λℰ(h,h)

≤ ⟨R_λf,f⟩_L²(X,μ) – ||f||²_L²(X,μ) + ||f – g||²_L²(X,μ) + λℰ(g,g) = 0.

As a consequence g = h, that is 0 ≤ g ≤ 1.

3. The previous step shows that if f ∈ L²(X,μ) satisfies 0 ≤ f ≤ 1 then for every λ > 0, 0 ≤ (Id – λL)^-1 f ≤ 1. Thanks to the spectral theorem, in L²(X,μ),

P_tf = lim_{n → +∞} (Id – t/n L)^-nf.

By passing to a subsequence that converges pointwise almost surely, we deduce that 0 ≤ P_tf ≤ 1 almost surely.