Dear all,

We can be more ambitious with the beautiful and surprising Bonus Exercise 1 (Effective resistance II).

Once we prove that $\rho :(x,y)\mapsto W_{\text{eff}}(x,y)$ is a distance. We can prove that ${\rho }_{\alpha }:(x,y)\mapsto W_{\text{eff}}^{\alpha }(x,y)$ is a distance for all $\alpha \in (0,1]$.

For this, we need these typical exercises in Functional Analysis Courses:

1. For $\alpha \in (0,1]$ and $x,y\ge 0$ the function $\phi :x\mapsto x^{\alpha }$ is not decreasing and $(x+y{)}^{\alpha }\le x^{\alpha }+y^{\alpha }.$

2. If d is a metric, then $(\varphi \circ d)$ is also a metric provided that $\varphi$ is a non-decreasing mapping such that $\varphi (s+r)\le \varphi \left(s\right)+\varphi \left(r\right).$

Best Regards,

Paco

Hello,

Regarding exercise 2a) a question came up. Is “finite measure space” a typo for “finite measure set space” or are we talking about a possibly infinite set $X$?

Regards Ragon

Dear all,

is the graph in Exercise 2.4 assumed to be connected? Otherwise, I won't see, why $W_{eff}(x,y)<\infty$ for $x,y\in X$. Indeed, if we for instance assume that $X$ consists of two elements and $b=0$, then $Q\left(h\right)=0$ for each $h\in C\left(X\right)$ and hence, $W_{eff}(x,y)=\infty$ for the two distinct points in $X$.

Best regards Sascha

Hello, Thank you very much for this scond lecture.

Dear all,

I do not really understand the definition of the heat kernel (1.28). How does the definition $e^{-tL}f\left(x\right)$ fit to the map $p:[0,\infty )\times X\times X\to [0,\infty )$? $e^{-tL}f\left(x\right)$ looks to me like a map with domain $[0,\infty )\times C\left(X\right)\times X$ instead of $[0,\infty )\times X\times X$ What am I missing? Furthermore, how is $p_t$ defined?

Best, Kai

Dear all,

In the proof of (i)⇒(ii) in Prop. 1.24 you need to know that the Laplacian $L_{b,c,m}$ has only non-negative Eigenvalues to apply the Laplace transform. If $\lambda <0$, then $L_{b,c,m}-\lambda I=L_{b,c-\lambda ,m}$ is bijective by Lemma 1.30 because $c-\lambda >0$. Thus $\lambda$ is no Eigenvalue of $L_{b,c,m}$. Alternatively you could apply the maximum principle to see this.

Best, Brian

Dear ISEM-Team,

I wonder why you prove Lemma 1.30, since this is an immediate consequence of Exercise 1.3.

Best regards

Sascha

Dear all,

I have a couple of small questions:

1. Why in Proposition 1.24 (i) the positivity for one t is equal to positivity for all?

2. Concerning Definition 1.28 (Heat kernel), I am afraid I don't quite understand it. Is $p_t(x,y)$ unique? Why it exists? Do I understamd correctly, that it follows from linear algebra? Why $e^{-tL}$ is bijective?

Best, Anna

Dear all,

thank you very much for lecture number 2.

Some minor points:

• p18, middle: The if-clause after `desired equivalence' seems to be a bit awkward (at least to me). The word `satisfies' suggests that $u_0=1$ is a consequence of (i). One rather chooses $u$ to be given by $u_t:=e^{-tL}1$. Then, by (i), the derivative ${\partial }_t{u}_t$ is non-negative.
• p18, middle (personal taste) Even though, I think I know what `non-increasing' means I feel way more comfortable saying `decreasing' (and understanding that this also allows for equality signs)
• p18, middle: I find `the inequality above' not specific enough. More so as (at least I think this is the argument) one uses positivity preserving for the first and the second inequality and only the third is derived from the inequality above. I would rather argue that positivity preserving implies

$$0\le e^{-tL}f\le e^{-tL}1,$$ and from $e^{-tL}1\le 1$, it follows that the semi-group satisfies (iii).

• p18, second to last line of this proof, I think there is a 1 missing in the limit expression.
• p19, -4: For $L1=0$ it might make sense to refer to Lemma 1.6 again.
• p20, Proposition 1.24: I would have wished for a clearer argument why 'one' can be replaced by 'all'. The obvious argument for (i) and (ii) using the semi-group property and the resolvent identity (in addition to positivity) allows me to deduce the same only for eventually all large $t$ and $\alpha$, respectively.
• p20, middle: I think the term `$U$ is (or is not) connected to $X\setminus U$' is not defined in the lecture notes. One should rather write “As there is no path from any $x\in U$ to any $y\in X\setminus U$, the operator …” (or something similar).
• p26,just above Claim 2: For me it was helpful to remind myself that $y$ not belonging to $B$ means it belongs to $A$. Then, as $b(x_{j-1},y)>0$ by definition of path, $\{x_0,\dots ,x_j=y\}$ is a connected subset of $A$ and since $\{x_0,\dots ,x_{j-1}\}\subseteq Z$ it follows that $x_j\in Z$ by the connectedness of $Z$.

Cheers,

Marcus

Dear Lecturers,

thank you for the second lecture!

I am wodering, what would be the object corressponding to the positivity preserving semigroup (and all equivalent conditions e.g. $Q\left(|f|\right)\le Q\left(f\right)$), i.e. analogue of the Theorem 1.20 for the first Beurling–Deny criterion? Is it also some kind of weighted graph, which is simply not discussed here?

Dear all,

Thanks a lot to the virtual lecturers for the very interesting lecture!

Here are a few comments:

(1) Proving the implication “(i) ⇒ (iii)” in Theorem 1.18 by means of the Lie-Trotter product formula is (at least in the finite-dimensional case) more complicated than necessary; here's a very simple argument: since all off-diagonal entries of $-L$ are $\ge 0$, there exists a number $c\ge 0$ such that $-L+cI\ge 0$ (where $I$ denotes the identity matrix). Hence, $e^{-tL}=e^{-tc}{e}^{t(-L+cI)}\ge 0$, where the inequality in the end follows from the series representation of the matrix exponential function.

(2) The first part of the proof of Theorem 1.26 consist of showing that, if $u$ is an eigenvector for ${\lambda }_0$, then so is $|u|$. The argument involves a preliminary claim about the form $Q$, which in turn relies on the spectral theorem. Here's a simple alternative argument that works for every positivity-improving semigroup $(e^{-tL}{)}_{t\ge 0}$ in finite dimensions (without appealing to forms or self-adjointness):

We may assume that ${\lambda }_0=0$. Then, for all $t>0$, one has $|u|=|e^{-tL}u|\le e^{-tL}|u|$. Assume for a contradiction that, for some $t>0$, the inequality $|u|\le e^{-tL}|u|$ is not an equality. By the positivity improving property one thus gets $|u|\le e^{-tL}|u|<e^{-2tL}|u|$, so $(1+\epsilon )|u|\le e^{-2tL}|u|$ for some $\epsilon >0$. So the spectral radius of $e^{-2tL}$ is at least $1+\epsilon$, which is a contradiction since the smallest spectral value of $L$ is $0$.

(3) I was wondering why the property “$0\le f\le 1\Rightarrow 0\le e^{-tL}f\le 1$” is called Markov property, since from a probabilistic point of view it rather looks like a sub-Markov property (as some mass might get lost).

(4) Historical-terminological nitpick: It seems a bit unconventional to call Theorem 1.26 a Perron-Frobenius theorem, as the result considers the positivity improving case only; this makes it a “Perron theorem” rather than a “Perron-Frobenius theorem” (historically, Perron first analyzed the spectrum of positivity improving matrices, and Frobenius later adapted the result to the case of positivity preserving matrices - thus, theorems for the positivity improving case are typically associated with the name Perron only; compare for instance this nice survey article).

Best wishes, Jochen

Hello Thank you for the course