Discussion on Lecture 06

EL-Houcine OUALI, 2022/12/02 10:06

Hello,

Thanks for this sixth lecture.

Sascha Trostorff, 2022/11/30 09:58

Dear ISEM-Team,

I have a small question concerning the definition of regular Dirichlet forms. You require that $D (Q) \cap C_{c}$ should be dense in $C_{c}$ with respect to the supremum norm and in $D (Q)$ with respect to the $Q$ -Norm in the definition. After that, you claim that regularity is equivalent to $\bar{Q |_{C_{c}}} = Q$ and it is written that one implication is obvious. How could I derive the densitiy of $C_{c} \cap D (Q)$ in $C_{c}$ with respect to $‖ \cdot ‖_{\infty}$ from $\bar{Q |_{C_{c}}} = Q$ ?

Best regards Sascha

Marcel Schmidt, 2022/12/01 14:35

Dear Sascha,

I think the formulation in the text is a bit misleading. What was meant is that $Q$ is regular if and only if $C_{c} (X) \subseteq D (Q)$ and $Q$ is the closure of $Q |_{C_{c} (X)}$ .

Best, Marcel

Sascha Trostorff, 2022/12/02 11:32

Dear Marcel,

thanks a lot for the answer.

Best regards Sascha

Peer Kunstmann, 2022/11/28 15:32

Dear virtual lecturer,

thank you very much for the nice lecture.

Some comments:

1) A typo on p. 76, proof of Lemma 4.7: In (a) it should read $\geq 0$ at the end, not $\leq 0$ .

2) On p.78, line 3, it would be nice to recall what $F$ is.

3) For the formula of $Q (φ, f)$ in line 7 (p. 78) one could mention that $φ \in C_{c} (X)$ .

4) $D (L^{(D)}) \subseteq F$ should already be mentioned in the statement of Theorem 4.9 (it is, of course, in the proof).

Best, Peer

Christian Seifert, 2022/11/29 17:18

Dear Peer,

Many thanks. I adjusted the notes accordingly.

Best, Christian

Sahiba Arora, 2022/11/24 16:49

Dear lecturers, dear all,

Thanks for the really interesting lecture. I found this lecture particularly easy to follow and enjoyed reading it.

I'm curious about the converse of Exercise 4, Dirichlet and Neumann forms: what is in between? :)

Regards, Sahiba

Marcel Schmidt, 2022/12/01 15:02, 2022/12/01 15:04

Dear Sahiba,

actually we wrote an article which characterizes all Dirichlet forms $Q$ with $D (Q^{(D)}) \subseteq D (Q) \subseteq D (Q^{(N)})$ and $Q (f) = Q^{(D)}$ for all $f \in D (Q^{(D)})$ :

Keller, Matthias; Lenz, Daniel; Schmidt, Marcel; Schwarz, Michael Boundary representation of Dirichlet forms on discrete spaces. J. Math. Pures Appl. (9) 126 (2019), 109–143.

The short (sketchy) answer is the following: If $Q$ is such a Dirichlet form, then the following holds: There exists a (not necessarily densely defined) Dirichlet form $q$ on $L^{2} (\partial X, μ)$ with $D (q) = T r D (Q) \subseteq D (q^{D N})$ such that $q - q^{D N}$ (defined on $D (q)$ ) is Markovian and

$Q (f) = Q^{(N)} (f) + q (T r f) - q^{D N} (T r f), f \in D (Q) .$

Here: $\partial X$ is the Royden boundary of the graph, $μ$ is a harmonic measure and $T r : D (Q^{(N)}) \to L^{2} (\partial X, μ)$ is a suitable trace map. Moreover, $q^{D N}$ is the Dirichlet to Neumann form on $L^{2} (\partial X, μ)$ .

Conversely, any (not necessarily densely defined) Dirichlet form on $L^{2} (\partial X, μ)$ with $D (q) \subseteq D (q^{D N})$ for which $q - q^{D N}$ is Markovian yields a form $Q_{q}$ between $Q^{(D)}$ and $Q^{(N)}$ by letting $D (Q_{q}) = {f \in D (Q^{(N)}) ∣ T r f \in D (q)}$ and

$Q_{q} (f) = Q^{(N)} (f) + q (T r f) - q^{D N} (T r f) .$

As mentioned above the map $q \mapsto Q_{q}$ is surjective but may not be injective. For example if $m \geq α$ for some $α > 0$ , then $Q^{(D)} = Q^{(N)}$ (this will be discussed in a later lecture) but the Royden boundary $\partial X$ and the set of Dirichlet forms on it may be large.

Best, Marcel

ISem26

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Discussion on Lecture 06

Discussion on Lecture 06