Lecture 7. Further topics

In this lecture let X be a locally compact and complete metric space equipped with a Radon measure μ supported on X. Let (ℰ, ℱ = dom(ℰ)) be a Dirichlet form on X. We assume throughout that the heat semigroup P_t is stochastically complete, i.e., P_t1 = 1 for every t ≥ 0.

Regular Dirichlet forms, Energy measures
Hunt process associated with a regular Dirichlet form
Intrinsic metric
Further reading

Regular Dirichlet forms, Energy measures

We denote by C_c(X) the vector space of all continuous functions with compact support in X and C₀(X) its closure with respect to the supremum norm.

A core for (X,μ,ℰ,ℱ) is a subset 𝒞 of C_c(X) ∩ ℱ which is dense in C_c(X) in the supremum norm and dense in ℱ in the norm

$(||f||^2_{L^2(X,\mu)} + \mathcal{E}(f,f))^{1/2}.$

Definition. The Dirichlet form ℰ is called regular if it admits a core.

Recall that for any f,g ∈ ℱ, we have

$\mathcal{E}(f,g)=\lim_{t\to 0}\frac1t\langle (I-P_t)f,g\rangle=\lim_{t\to 0}\frac1{t}\int_X \int_X(f(x)-f(y)) g(x) p_t(x,dy) d\mu(x),$

where p_t(x,·) are the heat kernel measures associated to the Dirichlet form (ℰ, ℱ).
From the symmetry property \eqref{heat kernel measure symmetry} of the heat kernel measure one also has

$\mathcal{E}(f, g)=\lim_{t\to 0}\frac1{2t}\int_X \int_X(f(x)-f(y))(g(x)-g(y)) p_t(x,dy) d\mu(x).$

Lemma. For f,g ∈ ℱ ∩ L^∞(X,μ), fg ∈ ℱ and

$\mathcal{E}(fg)^{1/2} \le ||f||_\infty \mathcal{E}(g)^{1/2} +||g||_\infty \mathcal{E}(f)^{1/2}.$

Proof. For f, g ∈ ℱ ∩ L^∞(X,μ) such that fg ∈ ℱ,

$\mathcal{E}(fg)=\lim_{t\to 0}\frac1{2t}\int_X \int_X(f(x)g(x)-f(y)g(y))^2 p_t(x,dy) d\mu(x).$

Write f(x)g(x) – f(y)g(y) = f(x)(g(x) – g(y)) + g(y)(f(x) – f(y)), then by Minkowski’s inequality

$\left(\int_X \int_X(f(x)g(x)-f(y)g(y))^2 p_t(x,dy) d\mu(x)\right)^{1/2}$

$\le ||f||_{\infty}\left(\int_X \int_X (g(x)-g(y))^2 p_t(x,dy) d\mu(x)\right)^{1/2}$

$+||g||_{\infty}\left(\int_X \int_X(f(x)-f(y))^2 p_t(x,dy) d\mu(x)\right)^{1/2}.$

We conclude the expected inequality by multiplying by 1/√2t and taking the limit t → 0 for both sides above.

Theorem (Energy measures) Assume that ℰ is regular. For f ∈ ℱ ∩ L^∞(X,μ), there exists a unique Radon measure on X, denoted by dΓ(f), so that for every ϕ ∈ ℱ ∩ C_c(X),

$\int_X\phi\, d\Gamma(f)=\frac{1}{2}[2\mathcal{E}(\phi f,f)-\mathcal{E}(\phi, f^2)]$

$= \lim_{t \to 0} \frac{1}{2t} \int_X \int_X \phi(x) (f(x)-f(y))^2 p_t(x,dy) d\mu(x).$

The Radon measure dΓ(f) is called the energy measure of f (and is therefore the weak * limit of (1/(2t)) ∫_X (f(x) – f(y))² p_t(x,dy)).

Proof. Let f ∈ ℱ ∩ L^∞(X,μ). For any ϕ ∈ ℱ ∩ C_c(X),

$\frac1{2t} \int_X \int_X \phi(x) (f(x)-f(y))^2 p_t(x,dy) d\mu(x)=-\frac1{2t} \langle (I-P_t)f^2 ,\phi\rangle+\frac1{t} \langle (I-P_t)f ,f\phi\rangle.$

Letting t → 0, the right-hand side converges to

$\frac{1}{2}[2\mathcal{E}(\phi f,f)-\mathcal{E}(\phi, f^2)]$ .

On the other hand, observing that

$\frac1{2t} \int_X \int_X |\phi(x)| (f(x)-f(y))^2 p_t(x,dy) d\mu(x) \le \|\phi\|_{\infty} \frac1{2t} \int_X \int_X (f(x)-f(y))^2 p_t(x,dy) d\mu(x),$

we deduce

$\left|\frac{1}{2}[2\mathcal{E}(\phi f,f)-\mathcal{E}(\phi, f^2)]\right|\le \|\phi\|_{\infty}\mathcal E(f).$

Therefore we conclude the proof by applying the Riesz-Markov representation theorem.

One can actually define dΓ(f,f) for every f ∈ ℱ using the following lemmas.

Lemma. Let f ∈ ℱ. Then f_n = min(n, max(-n, f)) ∈ ℱ and ℰ(f – f_n) → 0.

Proof. Let f ∈ ℱ. For every x, y ∈ X, we have

$|f_n(x) - f_n(y)| \leq |f(x) - f(y)|$

and |f_n(x)| ≤ |f(x)|. So f_n is a normal contraction of f and ℰ(f_n) ≤ ℰ(f).

Recall that

$\mathcal{E}(f-f_n)=\lim_{t\to 0}\frac1{2t} \int_X \int_X(f(x)-f_n(x)-f(y)+f_n(y))^2 p_t(x,dy) d\mu(x).$

Expanding the square inside the integral gives that

$\mathcal{E}(f-f_n)=\mathcal E(f)+\mathcal E(f_n)-2\mathcal E(f_n,f)\le 2\mathcal E(f)-2\mathcal E(f_n,f).$

Letting n → ∞, we have ℰ(f_n,f) → ℰ(f). Therefore ℰ(f – f_n) converges to 0 as n → ∞.

Lemma. For f,g ∈ ℱ ∩ L^∞(X,μ) and nonnegative ϕ ∈ ℱ ∩ C_c(X),

$\left| \sqrt{\int_X\phi\, d\Gamma(f)} -\sqrt{\int_X\phi\, d\Gamma(g) } \right|^2 \le \int_X\phi\, d\Gamma(f-g) \le ||\phi||_{L^\infty(X,\mu)}\mathcal{E}(f-g)$

Proof. The second inequality follows from the proof in Theorem 16.3. For the first inequality, it suffice to show that for any f, g ∈ ℱ ∩ L^∞(X,μ) and any nonnegative ϕ ∈ ℱ ∩ C_c(X),

$\sqrt{\int_X\phi\, d\Gamma(f)} \le \sqrt{\int_X\phi\, d\Gamma(g) } +\sqrt{ \int_X\phi\, d\Gamma(f-g) }.$

Indeed, this inequality follows from a similar proof as in Lemma 16.2 by noting that f = f – g + g and using Minkowski’s inequality.

Thanks to the previous lemmas, by approximation, one can define dΓ(f) for every f ∈ ℱ by

dΓ(f) = sup{dΓ(f_n) : f_n = min(n, max(-n, f)), n = 1, 2, …}.

For f,g ∈ ℱ, one can define dΓ(f,g) by polarization

dΓ(f,g) = (1/4)(dΓ(f + g) – dΓ(f – g)).

The following representation theorem for regular Dirichlet forms then holds.

Theorem (Beurling-Deny) Assume that ℰ is regular. For u,v ∈ ℱ, ℰ(u,v) = ∫_X dΓ(u,v).

Hunt process associated with a regular Dirichlet form

Definition. A Hunt process with state space X is a family of stochastic process (X_t)_t≥0 and probability measures (ℙ_x)_x∈X defined on a measure space (Ω, ℱ), such that (X_t)_t≥0 is adapted w.r.t. the right-continuous minimal completed admissible filtration (ℱ_t)_t≥0, X₀ = x, ℙ_x-a.s. and the following hold:

(i) x → ℙ_x(X_t ∈ B) is measurable for all t > 0 and B ∈ ℬ(X),
(ii) X is a strong Markov process, i.e. for every stopping time T, X_T is ℱ_T-measurable and for every B ∈ ℬ(X)
$\mathbb{P}_x(X_{T+t} \in B | \mathcal{F}_T) = \mathbb{P}_{X_T}(X_t \in B) \quad \mathbb{P}_x\text{-a.s. on } \{T < \infty\},$
(iii) X is right-continuous, i.e.
$\lim_{s \downarrow t} X_s = X_t, \quad \forall t \quad \mathbb{P}_x\text{-a.s.}$
(iv) X is quasi left-continuous, i.e. for all stopping times T and (T_n)_n such that T_n ↑ T a.s.
$\lim_{n \to \infty} X_{T_n} = X_T, \quad \mathbb{P}_x\text{-a.s. on } \{T < \infty\}.$

Remark

Note that quasi left-continuity does not necessarily imply left-continuity, because the set
$A = \left\{ \lim_{n \to \infty} X_{s_n} = X_t \right\}$

might depend on the choice of sequence (s_n)_n, s_n ↑ t.
In more generality, one might consider situations where (ℙ_x(X_t ∈ ·))_x,t are sub-probability measures (i.e. all the axioms of probability measures are satisfied but ℙ_x(X_t ∈ Ω) ≤ 1). In that case we can perform a one-point compactification of X by introducing a cemetery state ∂ ∉ X and redefine ℙ_x to be a probability measure on X ∪ {∂}.

The following theorem can then be proved, see the book [FOT], Theorem 4.2.1.

Theorem (Fukushima) Assume that ℰ is regular, then there exists a Hunt process ((X_t)_t≥0, (ℙ_x)_x∈X) such that for μ-a.e. x ∈ X, A ∈ ℬ(X) and t ≥ 0,

$\mathbb{P}_x (X_t \in A)=p_t(x,A)$

where p_t(x, ·) are the heat kernel measures associated to the Dirichlet form (ℰ, 𝒟(ℰ)).

Intrinsic metric

Definition. The Dirichlet form ℰ is called strongly local if for any two functions f,g ∈ ℱ with compact supports such that f is constant in a neighborhood of the support of g, we have ℰ(f,g) = 0.

With respect to ℰ we can define the following intrinsic metric d_ℰ on X by

$d_{\mathcal{E}}(x,y)=\sup\{u(x)-u(y)\, :\, u\in\mathcal{F}\cap C_0(X)\text{ and } d\Gamma(u,u)\le d\mu\}.$

Here the condition dΓ(u,u) ≤ dμ means that Γ(u,u) is absolutely continuous with respect to μ with Radon-Nikodym derivative bounded by 1.

The term “intrinsic metric” is potentially misleading because in general there is no reason why d_ℰ is a metric on X (it could be infinite for a given pair of points x,y or zero for some distinct pair of points).

Definition. A strongly local regular Dirichlet space is called strictly local if d_ℰ is a metric on X and the topology induced by d_ℰ coincides with the topology on X.

Example (Riemannian manifolds) Let (M,g) be a complete n-dimensional Riemannian manifold with Riemannian volume measure μ. We consider the standard Dirichlet form ℰ on M, which is obtained by closing the bilinear form

$\mathcal{E}(f,g)=\int_\mathbb{M} \langle \nabla f ,\nabla g \rangle d\mu, \quad f,g \in C_0^\infty(\mathbb M).$

Then ℰ is a strictly local Dirichlet form such that

$d_\mathcal{E} (x,y)=d_g(x,y).$

Example (Carnot groups) Let G be a Carnot group with sub-Laplacian

$L=\sum_{i=1}^d V_i^2$

and Dirichlet form

$\mathcal{E}(f)=\int_{\mathbb G} \sum_{i=1} (V_if)^2 d\mu.$

Then ℰ is a strictly local Dirichlet form such that

$d_\mathcal{E} (x,y)=d_{CC} (x,y)$

where d_CC is the so-called Carnot-Carathéodory distance which is defined as follows.

An absolutely continuous curve γ : [0,T] → G is said to be subunit for the operator L if for every smooth function f : G → ℝ we have

$\left| \frac{d}{dt} f ( \gamma(t) ) \right| \le \sqrt{ (\Gamma f) (\gamma(t)) }.$

We then define the subunit length of γ as ℓ_s(γ) = T.

Given x,y ∈ G, we indicate with

S(x,y) = {γ : [0,T] → G | γ is subunit for Γ, γ(0) = x, γ(T) = y}.

It is a consequence of the Chow-Rashevskii theorem that

S(x,y) ≠ ∅, for every x, y ∈ G.

One defines then

$d_{CC}(x,y) = \inf\{ \ell_s(\gamma) \mid \gamma \in S(x,y)\}$

Example Consider on the Sierpinski gasket the standard Dirichlet form ℰ. Then ℰ is regular, but unless f is constant, for f ∈ ℱ, dΓ(f) is singular with respect to the Hausdorff measure μ, see [KM19]. As a consequence ℰ is not strictly local.