Discussion on Lecture 10

Peer Kunstmann, 2023/01/30 14:43, 2023/01/30 15:00

Dear lecturers,

I am a bit late … Nevertheless, two remarks on the proof of Lemma 8.5:

- On p. 121, line -3, the left hand side should be divided by $m (x)$ .

- On p. 122, line 4, it should read $b (x, y) (u (x) f (x) - u (y) f (y))$ .

In Corollary 8.8 (p. 123) it should read $\dots \geq Q_{u} (φ / u) \geq 0,$ since Theorem 8.7 yields $= Q_{u} (φ / u)$ in line 6 of the proof (of Corollary 8.8). Moreover, this is what is written in the proof of Theorem 8.10 (p. 124, line 17).

Best, Peer

Christian Seifert, 2023/01/30 16:38

Dear Peer,

Many thanks for spotting the typos!

Best, Christian

Patrizio Bifulco, 2023/01/27 09:40

Dear lecturers,

many thanks for this very interesting lecture!

In Bonus Exercise 1, I believe you want the graph to be connected and $u$ non-trivial, right?

Otherwise, in the case that the underlying graph is not connected, one can consider a graph with three vertices $x, y, z$ and one edge connecting two of them, say $x$ and $y$ (and no potential, i.e. $c \equiv 0$ ). Then for $u$ given by $u (x) = u (y) = 1$ and $u (z) = 0$ and $w = 1 - u$ we get $(L - w) u = 0$ but also $0 = D e g (z) < w (z) = 1$ .

In the case that $u \equiv 0$ , one can consider a path graph with two vertices and standard weights and just choose $w : X \to R$ larger than $1$ in one of the vertices.

Best wishes, Patrizio

Christian Seifert, 2023/01/30 16:33

Dear Patrizio,

Many thanks; your are right soncerning the two assumptions.

Best, Christian

Joachim Hofmann, 2023/01/18 11:57

Dear all,

first of all thanks for the new lecture to the organisers. I have the following problem: In the last part of proof of Theorem 8.1 respectively in Remark 8.2 it is not clear to me why there is no non-trivial supersolution for $λ = \frac{deg (x)}{m (x)}$ in the case where the infimum is not attained.

And does non-trivial also mean non-constant? Because if we allow constants I think I found the following counter-example: Let $X := N$ , $b (x, y) := | x - y |$ , if $| x - y | = 1$ and $0$ otherwise, $m (x) := x$ as well as $c = 0$ . Then $inf_{x \in X} \frac{deg (x)}{m (x)} = 0$ is not attained but for $λ = 0$ any positive constant function $u$ satisfies $(L - λ) u = 0$ , i.e. $u$ is a supersolution.

Best, Joachim.

Marcel Schmidt, 2023/01/25 11:12, 2023/01/25 11:15

Dear Joachim,

you are correct, there is a mistake in the text. By a non-trvial supersolution we mean a supersolution $f \neq 0$ . In this sense constant functions are non-trivial.

The following is true: In the proof of Theorem 8.1 we show that if there is a nontrivial supersolution to $λ$ , then $\deg (x) / m (x) > λ$ for every $x \in X$ (this uses connectedness of the graph).

At the end of the proof of Theorem 8.1 one needs to justify that for any path $x_{0}, \dots, x_{n}$ from $x$ to $x^{*}$ the constant

$C_{x, x^{*}} (λ) = \prod_{j = 0}^{n - 1} \frac{\deg (x_{j}) - λ m (x_{j})}{b (x_{j}, x_{j + 1})}$

is positive for 'relevant' $λ$ , i.e., for $λ$ with a non-trivial supersolution. But this is guaranteed by the observation $\deg (x) / m (x) > λ$ for every $x \in X$ if there exists a non-trivial supersolution to $λ$ .

My argument also shows that if $λ > inf_{x \in X} \deg (x) / m (x)$ , then there is no non-trivial supersolution to $λ$ . Hence, Remark 8.2 should be adjusted.

Best Marcel

Anna Muranova, 2023/01/17 20:03

Dear lecturers,

thank you for the lecture!

1. I suspect that in Lemma 8.5 it should be $L_{u} = L_{b_{u}, u^{2} w m, u^{2} m}$ instead $L_{u} = L_{b_{u}, u^{2} w, u^{2} m}$ (i.e. one $m$ is missing on the rhs), isn't it? Then in the first equation in the proof there should be additionally $m (x)$ on the l.h.s.

2. Is the fact that $(L - λ)^{- 1} 1_{o}$ in the very beginning of the proof of Theorem 8.10 obvious or mentioned somewher before?

3 Shoudn't at the end of the proof of Theorem 9.7 be $w (x) := \frac{1}{v (x)} L v (x)$ (without $m (x)$ )?

Best, Anna

With the best wishes, Anna

Joachim Hofmann, 2023/01/18 11:33

Dear Anna,

1. I think you are right about this one.

2. Do you mean that $(L - λ)^{- 1} 1_{o}$ is non-trivial and positive? That would be because the graph is assumed to be connected so that the resolvent $(L - λ)^{- 1}$ is positivity improving.

3. I was asking myself the same question. The funtcion $w$ defined in the proof of Theorem 9.7 is the function $w$ that we need for (ii) in Theorem 9.7 and not the function to which we apply Theorem 8.7. For the application of Theorem 8.7 we need the funtion $\tilde{w} (x) := \frac{1}{v (x)} L v (x)$ , as you mentioned yourself.

Best, Joachim

Marcel Schmidt, 2023/01/25 11:50, 2023/01/25 12:01

Dear Anna and Joachim,

I think Joachim's answers are correct.

Concerning 3: The usage of $w$ in the proof of Theorem 9.7 and Theorem 8.7 is indeed not consistent but this is just a matter of scaling. Since $m$ is strictly positive, in the formulation of Theorem 9.7 one could replace (ii) by the following:

(ii)' There exists a non-trivial $w \geq 0$ such that

$Q (φ) \geq \sum_{x \in X} φ (x)^{2} w (x) m (x)$

for all $φ \in C_{c} (X)$ .

With this convention $w$ in the proof of Theorem~9.7 and in the Theorem on the ground state transform are compatible.

Personally I prefer (ii)' because (ii) somehow treats the counting measure as the canonical underlying measure on a discrete space, whereas (ii)' refers to the measure $m$ , which is also utilized when defining the Green function. Even though recurrence/transience does not depend on the choice of the measure, (ii)' is more in line with the philosophy that $Q_{b, c}$ on $ℓ^{2} (X, m)$ is a toy model for regular Dirichlet forms, where in general one does not have a distinguished measure on the underlying space.

Best Marcel

ISem26

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Discussion on Lecture 10

Discussion on Lecture 10