Dirichlet forms

Let $F_{1}=\left\{ f \in C^{1}[0, \infty ): f\right.$ has compact support $\}$ and $F_{2}=F_{1} \cap \left\{ f \in C^{1}[0, \infty )\right.$ : $f$ has compact support, $\left.f^{\prime }(0)=0\right\}$. Show that the closures of $F_{1}$ and $F_{2}$ with respect to the norm $\left(\int \left(|f(x)|^{2}+\left|f^{\prime }(x)\right|^{2}\right) d x\right)^{1 / 2}$ are the same.

If $H^{1}$ is the completion of $C_{K}^{1}$, the $C^{1}$ functions on $\mathbb {R}^{d}$ with compact support, relative to the norm given by ( 38.11 ), show $H^{1}$ is a Hilbert space.

Show that the resolvent operator $R_{\lambda }$ defined in Theorem 38.1 is a symmetric operator, that is, if $f, g \in \mathcal{B}$, then $\left\langle R_{\lambda } f, g \right\rangle = \left\langle f, R_{\lambda } g \right\rangle$.

Show that if the resolvent operator $R_{\lambda }$ is a symmetric operator, then the transition operators $P_{t}$ are also symmetric: if $f, g \in \mathcal{B}$, then $\left\langle P_{t} f, g \right\rangle = \left\langle f, P_{t} g \right\rangle$.

To do the next few exercises, you will have to know some functional analysis, specifically, the spectral theorem for self-adjoint operators. See Lax (2002).

Let $\mathcal{E}$ be a Dirichlet form with domain $\mathcal{D}(\mathcal{E})$ and let $\mathcal{L}$ be the infinitesimal generator of the semigroup $P_{t}$ that corresponds to $\mathcal{L}$. Let $E(d \lambda )$ be a spectral resolution of the identity for $-\mathcal{L}$. (The operator $\mathcal{L}$ is a negative operator, so $-\mathcal{L}$ is a positive one.) Then a consequence of the spectral theorem is that

and

Also

Show that if $f, g \in \mathcal{D}$, then

Hint: First prove it for $f=R_{a} h$. Write

To extend this to all $f$ in the domain of $\mathcal{E}$, use the fact that $\mathcal{E}$ is closed.

If $\mathcal{L}$ is the infinitesimal generator of the semigroup associated with the Dirichlet form $\mathcal{E}$, show that $\mathcal{D}(\sqrt{-\mathcal{L}})=\mathcal{D}(\mathcal{E})$.

Show that if $f \in \mathcal{D}(\mathcal{E})$, then $a R_{a} f$ converges to $f$ with respect to the norm induced by $\mathcal{E}_{1}$.

Show that if $b>0$, then $\left\{ R_{b} f: f \in L^{2}\right\}$ is a dense subset of $\mathcal{D}(\mathcal{E})$ with respect to the norm induced by $\mathcal{E}_{1}$.

Show that $\left\{ P_{t} f: f \in L^{2}, t>0\right\}$ is a dense subset of $\mathcal{D}(\mathcal{E})$ with respect to the norm induced by $\mathcal{E}_{1}$.

This exercise shows how to approximate $\mathcal{E}$ by forms whose domain is all of $\mathcal{B}$. Let

Show that if $f \in \mathcal{D}(\mathcal{E})$, then $\mathcal{E}^{(t)}(f, f)$ increases to $\mathcal{E}(f, f)$. Show that if $f, g \in \mathcal{D}(\mathcal{E})$, then $\mathcal{E}^{(t)}(f, g)$ converges to $\mathcal{E}(f, g)$.

Show that if $u \in \mathcal{D}(\mathcal{E})$, then $|u| \in \mathcal{D}(\mathcal{E})$ and $\mathcal{E}(|u|,|u|) \leq \mathcal{E}(u, u)$.

Hint: Use Exercise 38.10 .

Use Exericse 38.11 to show that if $u \in \mathcal{D}(\mathcal{E})$, then $\mathcal{E}\left(u^{+}, u^{-}\right) \leq 0$.

Suppose $\left\{ P_{t}\right\}$ are the transition probabilities corresponding to a Dirichlet form $\mathcal{E}$. Suppose there exist functions $p_{t}(x, y)$ such that for each $t$,

for almost every $x$. Prove that for almost every pair $(x, y)$ with respect to the product measure $m \times m, p_{t}(x, y)=p_{t}(y, x)$.

Let $f \in L^{2}(m)$ and define the functional

for $u$ in the domain of $\mathcal{E}$. Prove that $\psi$ is minimized by $u=R_{\lambda } f$, and that this function is the unique minimizer.

Let $P_{t}$ be the semigroup associated with a Dirichlet form and define

This is a continuation of the previous exercise. If $f$ is a function on the state space, we say that $g$ is a normal contraction of $f$ if $|g(x)| \leq |f(x)|$ for all $x$ and $|g(x)-g(y)| \leq |f(x)-f(y)|$ for all $x$ and $y$. As an example, note that if $g(x)=-1 \vee (f(x) \wedge 1)$, then $g$ is a normal contraction of $f$. Prove that if $f \in \mathcal{D}(\mathcal{E})$, where $\mathcal{E}$ is a Dirichlet form and $g$ is a normal contraction of $f$, then for each $t>0$,