TITLE: The continuity of a function defined on $\mathbb R/2\pi \mathbb Z$, and the distance for $\mathbb R/2\pi \mathbb Z.$
QUESTION [1 upvotes]: I'm handling a function $f$ from $\mathbb R/2\pi \mathbb Z$ to $\mathbb R$, $f : \mathbb R/2\pi \mathbb Z \to \mathbb R$.
And I was told that the meaning of [$f$ is continuous at  $a\in \mathbb R/2\pi \mathbb Z$] is : $$\forall \epsilon >0, \exists \delta>0 \ \mathrm{\ s.t.\ } d'(x, a)<\delta \Rightarrow |f(x)-f(a)|<\epsilon$$, where
$d'(x,y)=\min \{|a-b|\mid a\in x,b\in y \}$ for $x,y\in \mathbb R/2\pi \mathbb Z$.
But I'm wondering whether this $d'$ is metric in this situation, and whether $\{|a-b|\mid a\in x, b\in y\}$ indeed has minimum value.
Accourding to
https://en.wikipedia.org/wiki/Metric_space#Quotient_metric_spaces,

given two equivalence classes $[u]$ and
$[v]$, we define $$
     d'([u],[v]) = \inf\{d(p_1,q_1)+d(p_2,q_2)+\dotsb+d(p_{n},q_{n})\} $$ where the infimum is taken over all finite sequences $(p_1, p_2,
 \dots, p_n)$ and $(q_1, q_2, \dots, q_n)$ with $[p_1]=[u], [q_n]=[v],
 [q_i]=[p_{i+1}], i=1,2,\dots, n-1$.

In general this will only define a
pseudometric, i.e. $d'([u],[v])=0$ does not necessarily imply that
$[u]=[v]$. However for nice equivalence relations , it is a metric.

I wonder the definition $d'(x,y)=\min \{|a-b|\mid a\in x,b\in y \}$ for $x,y\in \mathbb R/2\pi \mathbb Z$ is equivalen to this.
And according to this wikipedia page, $d'$ is not metric in general, but in special case, $d'$ can be metric.
Now, I have considered whether $d'$ defined by $d'(x,y)=\min \{|a-b|\mid a\in x,b\in y \}$ for $x,y\in \mathbb R/2\pi \mathbb Z$ is metric or not.
・$d'(x,y)=0 \iff x=y$
・$d'(x,y)=d'(y,x)$
are almost obvious, but I cannot see $d'$ satisfies triangle inequality and $\min \{|a-b|\mid a\in x, b\in y\}$ really exists.
Could you explain about these things?

REPLY [2 votes]: In general, the metric $d'$ that you define doesn't necessarily satisfy the triangle inequality as explained here: Why are quotient metric spaces defined this way?
However, in this case, you have a group quotient rather than just a metric space quotient, which means that the metric $d'$ that you define is the same as the usual quotient metric.
In particular, the group structure implies that $[x] = [y]$ is equivalent to $[x - y] = [0]$, and so your metric can be written as
$$
d'([x], [y])
= \min_{k\in \mathbf{Z}} |x - y + 2k\pi|.
$$
Now, this metric does satisfy the triangle inequality.
To see this, let $x, y, z \in \mathbf{R}$ and let $m^*, n^*$ be the minimizers
\begin{align*}
m^*
&= \operatorname*{argmin}_{m \in \mathbf{Z}} |x - y + 2m\pi|, \\
n^*
&= \operatorname*{argmin}_{m \in \mathbf{Z}} |y - z + 2n\pi|,
\end{align*}
which yields
\begin{align*}
d'([x], [z])
&\leq |x - z + 2(m^* + n^*)\pi| \\
&\leq |x - y + 2 m^* \pi| + |y - z + 2 n^*\pi| \\
&= d'([x], [y]) + d'([y], [z]).
\end{align*}


TITLE: Standard deviation and field of tolerance
QUESTION [0 upvotes]: Resistor with $500\Omega$ is made with tolerance $\pm 1\Omega$.
$a)$ Determine the spoilage percentage (that falls outside of the tolerance area) if the processing gives the standard deviation $\sigma =2\Omega$.
$b)$ Increase the field of tolerance such that the spoilage percentage is $10\%$ with standard deviation $\sigma = 2\Omega$.
Normal distribution should be used.
My attempt:
$a)$
$$P(X\le 499)+P(X\ge 501)=2P(X\ge 501)$$
$$=2(1-P(X\le 501))$$
$$=2(1-P(X\le 0.5)),\Phi(0.5)=0.6915,P(X\le 0.5)=0.6915$$
$$=2(1-0.6915)$$
$$=0.6915=61.7\%$$
How to solve for $b)$?
In my book’s solution, the field of tolerance is $[496,72,503,28]$. Could someone show the procedure for finding it?

REPLY [0 votes]: You have a contratiction in your notation.
If $P(X\geq 501)$ then it is equal to $P(Z\geq 0.5)$, where $Z=\frac{X-\mu}{\sigma}=\frac{X-500}{2}$. $Z$ is normally standard distributed.
For $b)$ we use the standardized random variable as well.
$P(X\leq x)=\Phi\left(\frac{x-500}{2} \right)$. Thus
$$P(-z\leq Z \leq z)=2-2\Phi(z)$$ $$=2-2\Phi\left(\frac{x-500}{2}\right)=0.1\qquad |:2$$
$$1-1\Phi\left(\frac{x-500}{2}\right)=0.05\qquad |-0.05$$
$$0.95-1\Phi\left(\frac{x-500}{2}\right)=0\qquad $$
$$0.95=\Phi\left(\frac{x-500}{2}\right)\qquad $$
$$\Phi^{-1}(0.95)=\frac{x-500}{2}$$
$\Phi^{-1}(p)$ is inverse cumulative distribution function or quantile function of the standard normal distribution. The equation becomes
$$1.64=\frac{x-500}{2}$$
Calculate $x$. Finally the interval is $[2\mu-x,x]=[2\cdot 500-x, x]$.


TITLE: Won't Hawking radiation be sucked back into the black hole?
QUESTION [1 upvotes]: According to Stephen Hawking, black holes emit electromagnetic radiation(Hawking radiation), won't the EM radiations themselves be sucked back in by the black hole?
I also read on wiki the following:

An important difference between the black hole radiation as computed by Hawking and thermal radiation emitted from a black body is that the latter is statistical in nature, and only its average satisfies what is known as Planck's law of black body radiation, while the former fits the data better. Thus thermal radiation contains information about the body that emitted it, while Hawking radiation seems to contain no such information, and depends only on the mass, angular momentum, and charge of the black hole (the no-hair theorem). This leads to the black hole information paradox.

So if the BH does suck back in the radiation then there would be no way of finding out if it has done so because the information of Hawking radiation is not defined. To counter argue that statement, let's say that Hawking radiation is not sucked back by the BH, then does that mean that information-less systems are not affected by a BH?
Please pardon my ignorance about the topic.

REPLY [2 votes]: There is really two questions here.
One is if Hawking radiation gets absorbed (sucked in) by the BH. Generally it does not. The calculations were done to determine the output of that process (seen as two virtual particles, one sucked in and one escapes, or tunneling out; they are equivalent) at infinity. The ones that get created outside or tunnels out with enough energy (and right angular range), escapes, the other does not. The calculation determined how many escaped to infinity. So the radiation counted as emitted escapes. What does not is not counted
The second question is about the information. Neither thermal radiation nor Hawking, and both follow Plancks law, contain information. Thermal blackbody radiation is statistical and random at all times. They just mean that real emitted radiation from normal bodies (but mot radioactivity, say as an object burns) has information and we just abstract it by taking statistical averages. But then it is not exactly blackbody. Yes, blackbody is an idealization. But yes radioactive decay is truly random, and so is Hawking, and people do try to approximate true blackbody.
So, yes Hawking radiation is truly random and has no information other than the temperature; and the information paradox still needs a fully accepted resolution - the best bet is on the AdS/CFT correspondence that it somehow is on the horizon.


text:
\begin{document}

 \ \vspace{-22pt}

 \title[Interval partitions from Bertoin's study of ${\tt BES}_0(\lowercase{d})$]{Diffusions on a space of interval partitions:\\ construction from Bertoin's ${\tt BES}_0(\lowercase{d})$, $\lowercase{d}\in(0,1)$}

  \author[M.~Winkel]{Matthias Winkel}
  \address{Department of Statistics\\ University of Oxford\\ Oxford OX1 3LB\\ UK}
  \email{winkel@stats.ox.ac.uk}

\begin{abstract}
  In 1990, Bertoin constructed a measure-valued Markov process in the framework of a Bessel process of dimension between 0 and 1.
  In the present paper, we represent this process in a space of interval partitions.
  We show that this is a member of a class of interval partition diffusions introduced recently and independently by Forman, Pal, Rizzolo and Winkel
  using a completely different construction from spectrally positive stable L\'evy processes with index between 1 and 2 and with jumps marked by
  squared Bessel excursions of a corresponding dimension between $-2$ and 0.
\end{abstract}

 \keywords{Interval partition, Bessel process, measure-valued diffusion, Poisson--Dirichlet distribution, excursion theory}

 \subjclass[2010]{Primary 60J25, 60J60, 60J80; Secondary 60G18, 60G55}

\maketitle

 \ \vspace{-22pt}

\section{Introduction}
\label{sec:intro}

\noindent We define interval partitions, following Aldous \cite[Section 17]{AldousExch} and Pitman \cite[Chapter 4]{CSP}.

\begin{definition}\label{def:IP_1}
 An \emph{interval partition} is a set $\beta$ of disjoint, open subintervals of some finite real interval $[0,M]$, that cover $[0,M]$ up to a Lebesgue-null set. We write $\IPmag{\beta}$ to denote $M$. We refer to the elements of an interval partition as its \emph{blocks}. The Lebesgue measure of a block is called its \emph{mass} or \emph{size}.
\end{definition}

In this paper we construct diffusion processes in a space of interval partitions in Bertoin's \cite{Bertoin1990,Bertoin1990c} framework of
a Bessel process of dimension $d\in(0,1)$. Bertoin studied the excursions of such a Bessel process. Specifically, he first
decomposed the Bessel process \vspace{-0.1cm}
\begin{equation}\label{framework}\mathbf{R}=\mathbf{B}-(1-d)\mathbf{H}
\end{equation}
into a Brownian motion $\mathbf{B}$ and a path-continuous process
$\mathbf{H}$ with zero quadratic variation. He constructed excursions of the Markov process $(\mathbf{R},\mathbf{H})$ away from $(0,0)$, each consisting of infinitely many excursions of $\mathbf{R}$ away from $0$.

By extracting suitable statistics, namely the set $\{\mathbf{R}(t)\colon t\ge 0,\,\mathbf{H}(t)=y\}$, he showed \cite[Theorems II.2--II.3]{Bertoin1990c} that the
measure-valued process\vspace{-0.1cm}
  \begin{equation}\label{mvp} y\mapsto\mu^y_{[0,T]}:=\sum_{0\le t\le T\colon\mathbf{H}(t)=y,\mathbf{R}(t)\neq 0}\delta_{\mathbf{R}(t)}\vspace{-0.1cm}
  \end{equation}
is path-continuous (in the vague topology for sigma-finite point measures on $(0,\infty)$) and Markovian when $T$ is chosen suitably such
as an inverse local time $\tau_{\mathbf{R},\mathbf{H}}^0(u)$, $u\ge 0$, of $(\mathbf{R},\mathbf{H})$ at $(0,0)$. He further
showed in \cite[Theorem 4.2]{Bertoin1990} and \cite[Corollary II.4]{Bertoin1990c} that\vspace{-0.2cm}
\begin{equation}\label{eq:loctime}y\mapsto\lambda^y(T):=2\int_{(0,\infty)}x\mu_{[0,T]}^y(dx)=2\sum_{0\le t\le T\colon\mathbf{H}(t)=y}\mathbf{R}(t)
\end{equation}
is \BESQ[0], a zero-dimensional squared Bessel process. We provide a more comprehensive review of Bertoin's results in Section \ref{sec:Bertoin}.

In this paper, we represent his measure-valued process \eqref{mvp} as a diffusion in a space of interval partitions.

\begin{theorem}\label{thm:Bertoin} In the setting of \eqref{framework}--\eqref{eq:loctime}, with $T=\tau_{\mathbf{R},\mathbf{H}}^0(u)$, the interval partitions
  $$\beta^y:=\Big\{\big(\lambda^y(t-),\lambda^y(t)\big)\colon t\in[0,T],\,\mathbf{R}(t)\neq 0,\,\mathbf{H}(t)=y\Big\},\quad y\ge 0,$$
  form a diffusion process in a suitable space interval partitions.
\end{theorem}

While the interval lengths $\lambda^y(t)-\lambda^y(t-)$ of $\beta^y$ are (twice) the locations $\mathbf{R}(t)$ of atoms of $\mu_{[0,T]}^y$,
the order of the intervals is not captured by $\mu_{[0,T]}^y$. Hence, this theorem is not an immediate consequence of Bertoin's corresponding
results for $(\mu_{[0,T]}^y,\,y\ge 0)$

Indeed, we prove this theorem by identifying this diffusion process as an instance of a class of diffusion processes introduced in \cite{IPPA}, where we gave a general construction of processes in a space of interval partitions based on spectrally positive L\'evy processes
(\em scaffolding\em) whose point process of jump heights (interpreted as lifetimes of individuals) is marked by excursions (\em spindles\em, giving ``sizes'' varying during the
lifetime, one for each level crossed). Informally, the interval partition evolution, indexed by level, considers for each level $y\ge 0$
the jumps crossing that level and records for each such jump an interval whose length is the ``size'' of the individual (width of the spindle) when crossing that level,
ordered from left to right without leaving gaps. This construction and terminology is illustrated in Figure \ref{fig:skewer_1}.

\begin{figure}[t]
 \centering
 \begin{picture}(0,0)
\includegraphics[scale=0.92]{Fig_JCCP_skewer_3scale_cropped.pdf}
\end{picture}
\setlength{\unitlength}{3730sp}

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\begin{picture}(6324,2218)(304,-2699)
\put(3600,-1574){\makebox(0,0)[b]{\smash{{\SetFigFont{10}{12.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$s_j$}
}}}}
\put(4451,-736){\makebox(0,0)[rb]{\smash{{\SetFigFont{10}{12.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$(N,X)$}
}}}}
\put(5076,-2327){\rotatebox{360.0}{\makebox(0,0)[b]{\smash{{\SetFigFont{11}{13.2}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$f_j(y-X(s_j-))$}
}}}}}
\put(406,-1276){\makebox(0,0)[rb]{\smash{{\SetFigFont{11}{13.2}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$y$}
}}}}
\put(3411,-1096){\makebox(0,0)[rb]{\smash{{\SetFigFont{11}{13.2}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$X(s_j)$}
}}}}
\put(3731,-2086){\makebox(0,0)[lb]{\smash{{\SetFigFont{11}{13.2}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$X(s_j-)$}
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}}}}
\put(5775,-1000){\makebox(0,0)[b]{\smash{{\SetFigFont{10}{12.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$(f_j(z),\ z\geq 0)$}
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\put(3231,-2600){\makebox(0,0)[rb]{\smash{{\SetFigFont{11}{13.2}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$\skewer(y,N,X)$}
}}}}
\end{picture}\vspace{-0.2cm}
 \caption{Left: The sloping black lines show the scaffolding $X$. Shaded blobs decorating jumps show the corresponding spindles: points $(s_j,f_j)$ of $N$. Right: Graph of one spindle. Bottom: A skewer, with blocks shaded to correspond to spindles; not to scale.\label{fig:skewer_1}\vspace{-0.2cm}}
\end{figure}

Specifically, if $N=\sum_{i\in I}\delta_{(s_i,f_i)}$ is a point process of times $s_i\in[0,S]$ and excursions $f_i$ of excursion lengths $\zeta_i$ (spindle heights),
and $X$ is a real-valued process with jumps $\Delta X(s_i):=X(s_i)-X(s_i-)=\zeta_i$ at times $s_i$, $i\in I$, we define the interval partition $\skewer(y,N,X)$ at
level $y$, as follows.

\begin{definition}\label{def:skewer}
 For $y\in\BR$, $s\in [0,S]$, the \emph{aggregate mass} in $(N,X)$ at level $y$, up to time $s$ is
 \begin{equation}
  M_{N,X}^y(s) := \sum_{i\in I\colon s_i\le s}f_i(y - X(s_i-)).\label{eq:agg_mass_from_spindles}
 \end{equation}
 The \emph{skewer} of $(N,X)$ at level $y$, denoted by $\skewer(y,N,X)$, is defined as
 \begin{equation}
   \left\{\left(M^y_{N,X}(s-),M^y_{N,X}(s)\right)\!\colon s\in [0,S],\,M^y_{N,X}(s-) < M^y_{N,X}(s)\right\}\label{eq:skewer_def}
 \end{equation}
 and the \emph{skewer process} as $\skewerP(N,X) := \big( \skewer(y,N,X),y\!\geq\! 0\big)$.
\end{definition}

This definition is meaningful when $X$ has finitely many jumps as in Figure \ref{fig:skewer_1}, and also when $X$ has a dense set of jump times and the $f_i$ are such that $M_{N,X}$ is finite. In \cite{IPPA}, we established criteria under which $\skewerP(\bN,\bX)$ is a diffusion. Specifically, $\bN$ is a Poisson random measure ({\tt PRM}) with intensity
measure ${\tt Leb}\otimes\nu$, where $\nu$ is the Pitman--Yor excursion law \cite{PitmYor82} associated with a suitable (self-similar) $[0,\infty)$-valued diffusion,
and $\bX$ is an associated L\'evy process, suitably stopped at a time $\mathbf{S}$ when $\bX$ is zero. In
this interval partition evolution, each interval length (block) evolves independently according to the
$[0,\infty)$-valued diffusion, which we call \em block diffusion\em, while between (the infinitely many) blocks, new blocks appear at the pre-jump levels of $\bX$. The
{\tt PRM} of jumps is obtained by mapping the {\tt PRM} of spindles onto the spindle heights. Conversely, we may view the {\tt PRM} of spindles as marking the
{\tt PRM} of jumps by \em block excursions. \em See Section \ref{sec:prel} for more details.

\begin{theorem}\label{thm:diffusion_0} When the block diffusion is \BESQ[-2(1-d)], a squared Bessel process of dimension $-2(1\!-\!d)\!\in\!(-2,0)$ and
  the scaffolding L\'evy process is \Stable[2\!-\!d] stopped at an inverse local time $\tau_\mathbf{X}^0(v)$ of $\mathbf{X}$ at 0, the interval partition evolution associated via $\skewerP$ is distributed as the diffusion in Theorem
  \ref{thm:Bertoin}, for $u=2^{d-1}v$.
\end{theorem}

The remainder of this paper is organised, as follows. In Sections \ref{sec:Bertoin} and \ref{sec:prel}, we state the main results of
 \cite{Bertoin1990,Bertoin1990c} and \cite{IPPA,IPPB}, exhibiting the parallels. In Section \ref{sec:new}, we make precise the connections between
the two frameworks and deduce the theorems we have stated. In Section \ref{sec:new2}, we discuss some further observations.

\section{Bertoin's study of Bessel processes \cite{Bertoin1990,Bertoin1990c}}\label{sec:Bertoin}

\noindent Consider a Bessel process $\mathbf{R}\sim{\tt BES}_x(d)$ of dimension $d\in(0,1)$ starting from $x>0$. Let $T_\mathbf{R}(0)=\inf\{t\ge 0\colon\mathbf{R}(t)=0\}$. On $[0,T_\mathbf{R}(0))$, the Bessel process
$\mathbf{R}$ satisfies an SDE that yields
\begin{equation}\mathbf{R}(t)=x+\mathbf{B}(t)-\frac{1-d}{2}\int_0^t\frac{du}{\mathbf{R}(u)}.\label{bessde}
\end{equation}
Furthermore, this singular integral is finite as $t\uparrow T_\mathbf{R}(0)$. By time reversal, this means that this integral is also well-defined under the
excursion measure of the Bessel process. While the excursions can be stitched together to form a Bessel process that has 0 as a
reflecting boundary, the positive values of these integrals are not summable so that the representation (\ref{bessde}) fails beyond $T_\mathbf{R}(0)$. However,
(\ref{bessde}) can be extended beyond $T_\mathbf{R}(0)$ if some compensation is introduced, as follows. It is well-known that the Bessel process $\mathbf{R}$ has
jointly continuous space-time local times on $(0,\infty)^2$. To obtain a family of local times that extends continuously to $[0,\infty)^2$, it is
convenient to choose the level-$a$ local time $(L^a(t),t\ge 0)$, $a>0$, of $\mathbf{R}$ such that the occupation density of $\mathbf{R}$ is $(a^{d-1}L^a(t),a>0,t\ge 0)$. By the occupation density formula and since $L^0(t)=0$ for
$t<T_\mathbf{R}(0)$, we can write
\begin{equation}\label{eq:defH}\frac{1}{2}\int_0^t\frac{du}{\mathbf{R}(u)}=\frac{1}{2}\int_0^\infty a^{d-1}L^a(t)\frac{da}{a}=\frac{1}{2}\int_0^\infty a^{d-2}(L^a(t)-L^0(t))da=:\mathbf{H}(t)
\end{equation}
  for $t<T_\mathbf{R}(0)$.
Bertoin showed that defining $\mathbf{H}(t)$ by the right-most integral in \eqref{eq:defH} also for $t\ge T_\mathbf{R}(0)$ yields a path-continuous process $\mathbf{H}$ with unbounded variation, but
zero quadratic variation (the finiteness of $\mathbf{H}$ follows from the H\"older continuity of $L^a(t)$ in $a$).
Clearly, this process $\mathbf{H}$ is increasing on all excursion intervals of $\mathbf{R}$ away from zero, but the effect of the compensating local time at zero
is that $\mathbf{H}$ does not increase across the zero-set of $\mathbf{R}$. With this notation, we have
\begin{equation}\mathbf{R}(t)=x+\mathbf{B}(t)-(1-d)\mathbf{H}(t),\qquad \mbox{for all }t\ge 0.\label{besdec}
\end{equation}
Bertoin noted that $(\mathbf{R},\mathbf{H})$ is a Markov process and that $(0,0)$ is recurrent for this Markov process. It is instructive to
consider the excursions of $(\mathbf{R},\mathbf{H})$ away from $(0,0)$ by plotting $\mathbf{R}(t)$ against ``time'' $\mathbf{H}(t)$. Since $\mathbf{H}$
increases during each excursion of $\mathbf{R}$ away from 0, on $(\ell_s,r_s)$, say, such a plot shows a time-changed excursion of $\mathbf{R}$
starting from 0 at ``time'' $\mathbf{H}(\ell_s)$ and returning to 0 at ``time'' $\mathbf{H}(r_s)>\mathbf{H}(\ell_s)$. As $\mathbf{H}$ does not
increase across the zero-set of $\mathbf{R}$, the excursions for different $(\ell_s,r_s)$ overlap, in general, when included in the same plot.

Since $\mathbf{H}$ is increasing when $\mathbf{R}$ is away from 0, and can only decrease across the zero-set of $\mathbf{R}$, the excursions of
$(\mathbf{R},\mathbf{H})$ away from $(0,0)$ typically consist of many excursions of $\mathbf{R}$. Specifically, each excursion of
$(\mathbf{R},\mathbf{H})$ can be decomposed into three parts: first, at the ``beginning'', there is an escape from $(0,0)$ towards the
left by an accumulation of short $\mathbf{R}$-excursions until, in the ``middle'', one $\mathbf{R}$-excursion takes $(\mathbf{R},\mathbf{H})$
across to positive $\mathbf{H}$-values and, at the ``end'', there is a final approach back to $(0,0)$ from the right by an accumulation of short $\mathbf{R}$-excursions.\medskip

\noindent The main objects of interest in Bertoin's work \cite{Bertoin1990,Bertoin1990c} are
\begin{itemize}\item the excursions away from $(0,0)$ of $(\mathbf{R},\mathbf{H})$, and associated quantities,
  \item the excursions away from 0 of $\widetilde{\mathbf{R}}\!:=\!2\mathbf{R}\circ T_\mathbf{H}^+$, where $T_\mathbf{H}^+(y)\!=\!\inf\{t\!\ge\! 0\colon\!\mathbf{H}(t)\!>\!y\}$,\vspace{0.1cm}
  \item local time processes $(\lambda^y(t),y\ge 0)$ and $(\lambda^{-y}(t),y\ge 0)$ of $\mathbf{H}$, up to time $t\ge 0$,\vspace{0.1cm}
  \item measure-valued processes $y\!\mapsto\!\mu_{[0,T]}^y\!=\!\sum_{0\le t\le T\colon\mathbf{H}(t)=y,\mathbf{R}(t)\neq 0}\delta_{\mathbf{R}(t)}$ for some $T\!>\!0$.
\end{itemize}

Specifically, some of the main results of \cite{Bertoin1990} are the following. We use Bertoin's numbering for
ease of reference.

\begin{enumerate}
  \item[2.4] The inverse local time $\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}$ of $(\mathbf{R},\mathbf{H})$ at $(0,0)$ is stable with index $(1-d)/2$.
  \item[3.1] The It\^o excursion process of $(\mathbf{R},\mathbf{H})$ is a \PRM.\vspace{0.1cm}
  \item[3.2] Under the excursion measure, excursions of $(\mathbf{R},\mathbf{H})$ are time-reversible.\vspace{0.1cm}
  \item[3.3]\begin{enumerate}
                    \item[(i)] A.s., all excursions of $(\mathbf{R},\mathbf{H})$ away from $(0,0)$ start into $[0,\infty)\!\times\!(-\infty,0)$,
                                                                         cross $(0,\infty)\times\{0\}$ at a unique time $U$
                                                                         and finish from $[0,\infty)\times(0,\infty)$.
    	         \item[(ii)] The \PRM\ has points at excursions whose value of $\mathbf{R}$ when the excursion is crossing the line $\mathbf{H}=0$
    	         							    has (sigma-finite) law $((1-d)/\Gamma(d))x^{d-2}dx$, and
    	         \item[(iii)] points at excursions with an $\mathbf{H}$-infimum below $-y$ occur at rate $y^{d-1}$.\vspace{0.1cm}
    	       \end{enumerate}
  \item[3.4] Mid-excursion Markov property: conditionally given an $\mathbf{R}$-value of $\mathbf{R}(U)=x$ at the crossing time $U$ of the line $\mathbf{H}=0$, the
    post-$U$ part of the excursion and the time-reversed pre-$U$ part are independent and distributed as the process $(\mathbf{R},\mathbf{H})$ starting from $(x,0)$ and
    stopped when hitting $(0,0)$.\vspace{0.1cm}
  \item[4.1] In the setting of result 3.4, conditionally given $\mathbf{R}(U)\!=\!x$, the above-0 and below-0 local
    time processes  $(\lambda^y(t),y\ge 0)$ and $(\lambda^{-y}(t),y\ge 0)$ of $\mathbf{H}$ during an excursion of $(\mathbf{R},\mathbf{H})$ are two independent $\BESQ_{2x}(0)$.
  \item[4.2] The level-0 local time $\lambda^0$ of $\mathbf{H}$ time-changed by the inverse local time $\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}$ is a
    stable subordinator of index $1-d$. Given $\lambda^0(\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u))\!=\!x$, the processes $(\lambda^y(\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u)),y\!\ge\! 0)$ and
    $(\lambda^{-y}(\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u)),y\!\ge\! 0)$ are two independent $\BESQ_x(0)$.
\end{enumerate}
The main additional results of \cite{Bertoin1990c} are the following.

\begin{enumerate}
  \item[I.5] Under the It\^o excursion measure of $\widetilde{\mathbf{R}}$, excursions
    \begin{enumerate}\item[(i)] start positive with initial values at rate $(2^{1-d}(1\!-\!d)/\Gamma(d))x^{d-2}dx$, and
                                   \item[(ii)] when starting from $x$ evolve as $\BESQ_x(-2(1-d))$.\vspace{0.1cm}
    \end{enumerate}
  \item[I.6] The semi-group of $\widetilde{\mathbf{R}}$ is characterised by its Laplace transforms, for $\gamma\ge 0$,
    $\mathbb{E}_x(\exp(-\gamma\widetilde{\mathbf{R}}(y)))=\exp(-x/2y)\left((1+2\gamma y)^{1-d}\exp(x/(2+4\gamma y))-(2\gamma y)^{1-d}\right)$.
  \item[II.1] The measure-valued process $y\!\mapsto\!\mu^y_{[0,T_{\mathbf{H}}(-1)]}$ for $T_{\mathbf{H}}(-1)\!:=\!\inf\{t\!\ge\! 0\colon\mathbf{H}(t)\!=\!-1\}$
    admits a continuous version in the space $\mathcal{N}((0,\infty))$ of point measures that are finite on $(\varepsilon,\infty)$ for all $\varepsilon>0$,
    equipped with the topology of vague convergence.
  \item[II.2]
    \begin{enumerate}
      \item[(i)] The process $(\mu^y_{[0,T_{\mathbf{H}}(-1)]},y\ge 0)$ is Markovian.
      \item[(ii)] Its semi-group $\kappa_y^{\mathcal{N}}$, $y\!\ge\! 0$, acts on functions $f_\varphi(\sum_{i\in I}n_i\delta_{x_i})\!=\!\prod_{i\in I}(\varphi(x_i))^{n_i}$ for
        continuous $\varphi\colon(0,\infty)\rightarrow[0,1]$ as $\kappa_y^{\mathcal{N}}f_\varphi=f_{\varphi_y}$, where $\varphi_y(x)$ is given by
        $e^{-x/y}\!+\!\int_0^\infty\varphi(a)p_y(x,da)\big/\big(1+y^{1-d}\!\int_0^\infty((1\!-\!d)/\Gamma(d))s^{d-2}(1\!-\!\varphi(s))e^{-s/y}ds\big)$, where
        $\int_0^\infty e^{-\gamma a}p_y(x,da)=(1+\gamma y)^{1-d}\big(e^{-\gamma x/(1+\gamma y)}-e^{-x/y}\big)$, for all $\gamma\ge 0$.
      \item[(iii)] The process $(\mu^{-1+y}_{[0,T_{\mathbf{H}}(-1)]},0\!\le\! y\!\le\! 1)$ is Markovian with semi-group $\widetilde{\kappa}_y^{\mathcal{N}}$, $y\!\ge\! 0$, given by
        $\widetilde{\kappa}_y^{\mathcal{N}}f_\varphi=f_{\varphi_y}\big/\big(1+y^{1-d}\!\int_0^\infty((1\!-\!d)/\Gamma(d))s^{d-2}(1\!-\!\varphi(s))e^{-s/y}ds\big)$.
    \end{enumerate}
  \item[II.3] Given $\mu^0_{[0,\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(1)]}$, the processes $(\mu^y_{[0,\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(1)]},y\!\ge\! 0)$ and $(\mu^{-y}_{[0,\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(1)]},y\!\ge\! 0)$ are conditionally independent and have the semi-group $(\kappa_y^\mathcal{N},y\!\ge\! 0)$, of II.2(ii).
  \item[II.4]
    \begin{enumerate}
      \item[(i)] The process $(\lambda^{-1+y}(T_\mathbf{H}(-1)),0\!\le\! y\!\le\! 1)$ is a ${\tt BESQ}_0(2-2d)$.
      \item[(ii)] Given $\lambda^0(T_\mathbf{H}(-1))=x$, the process $(\lambda^y(T_\mathbf{H}(-1)),y\!\ge\! 0)$ is a ${\tt BESQ}_x(0)$.
      \item[(iii)]  Given $\lambda^0(\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(1))=x$, the process $(\lambda^y(\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(1)),y\!\ge\! 0)$ is a ${\tt BESQ}_x(0)$. \vspace{-0.1cm}
    \end{enumerate}
\end{enumerate}

\section{Skewer processes of marked L\'evy processes \cite{IPPA,IPPB}\vspace{-0.1cm}}
\label{sec:prel}

\noindent Let $\alpha\!\in\!(0,1)$
and $\bX$ a spectrally positive \Stable[1\!+\!\alpha]-process with Laplace exponent $\psi(c)=c^{1+\alpha}/2^\alpha\Gamma(1\!+\!\alpha)$. We call $\bX$ \em scaffolding \em and proceed to decorate it. Specifically, consider the \PRM\ $\sum_{i\in\mathbf{I}}\delta_{(\mathbf{s}_i,\Delta\mathbf{X}(\mathbf{s}_i))}$ of its jumps. For each jump $\Delta\mathbf{X}(\mathbf{s}_i)$, consider an independent $\BESQ(-2\alpha)$ excursion (\em spindle\em) $\mathbf{f}_i$
of length $\zeta(\mathbf{f}_i)=\Delta\mathbf{X}(\mathbf{s}_i)$. These excursions were studied by Pitman and Yor \cite{PitmYor82}, who also noted, in their Remark (5.8) on pp. 453f., that when
conditioned on their length, they are $\BESQ(4+2\alpha)$ bridges from 0 to 0. By standard marking of \PRM s, $\bN:=\sum_{i\in\mathbf{I}}\delta_{(\mathbf{s}_i,\mathbf{f}_i)}$ is itself a \PRM\ on the space $[0,\infty)\times\Exc$,
where $\Exc$ is the space of (continuous) excursion paths.
This is illustrated in a simplified way in Figure \ref{fig:skewer_1}. The intensity measure ${\tt Leb}\otimes\nu$ of $\mathbf{N}$ is the Pitman--Yor excursion measure of
\cite{PitmYor82}, which can be described by entrance laws and a further evolution as unconditioned $\BESQ(-2\alpha)$ processes.

Recall that the skewer of Definition \ref{def:skewer} extracts from $N=\sum_{i\in I}\delta_{(s_i,f_i)}$
all level-$y$ spindle masses $f_i(y-X(s_i))$, where $y\in(X(s_i-),X(s_i))$, $i\in I$, and builds the interval partition that has these as interval lengths in the order given by
the $s_i$, $i\in I$. The set $\mathcal{I}_H$ of all interval partitions can be equipped with a distance $d_H$ that applies the Hausdorff metric to the set of points not
covered by the intervals.

Since $\bX$ is spectrally positive, its (c\`adl\`ag) excursions away from 0 (or any other level $y$) start negative, jump across zero and end positive.
Applied to $(\bN,\bX)$, the skewer at level $y$ extracts one block from each excursion of $\bX$ away from $y$.
In \cite{IPPA}, we denote the \PRM\ of excursions of $\bX$ away from $y$ by $\bG^y$ and enhance the excursion theory of $\bX$ to include $\bN$:
each excursion $e_{[\ell,r]}\!:=\!(-y\!+\!\bX|_{[\ell,r]}(\ell\!+\!s),s\!\in\![0,r\!-\!\ell])$ of $\bX$ has its jumps marked by spindles.
We denote  by $\bF^y$ the associated random measure whose points are pairs of $e_{[\ell,r]}$ and the restriction $\bN|_{[\ell,r]\times\Exc}$ shifted to $[0,r-\ell]\times\Exc$. In each excursion with spindle marks, the \em central spindle \em crossing 0 can be viewed as the ``middle'' of
three parts, separating the spindles of the ``beginning'' where $\bX$ is negative from the spindles of the ``end'' where $\bX$ is positive.

We refer to excursions of $(\bX,\bN)$ as \em bi-clades\em,
to the negative part of such an excursion including the central spindle up to level 0 as an \em anti-clade\em,
and to the remainder as a \em clade\em.
To start an interval-partition-valued process from any interval partition $\beta$, we consider \em clades starting from ${\tt Leb}(V)$\em, $V\!\in\!\beta$, as follows.
For each \pagebreak interval $V\!\in\!\beta$ independently, consider $\ff_V\!\sim\!\BESQ_{{\tt Leb}(V)}(-2\alpha)$ and an independent $(\bX,\bN)$ stopped at $S_\mathbf{X}(-\zeta(\ff_V))\!:=\!\inf\{s\!\ge\! 0\colon\bX(s)\!=\!-\zeta(\ff_V)\}$, for the length $\zeta(\ff_V)$ of $\ff_V$, then form the clade
$(\bX_V,\mathbf{N}_V)\!:=\!(\zeta(\ff_V)\!+\!\bX|_{[0,S_\mathbf{X}(-\zeta(\ff_V))]},\delta_{(0,\ff_V)}\!+\!\bN|_{[0,S_\mathbf{X}(-\zeta(\ff_V))]\times\Exc})$.
We stitch together all excursions $\bX_V$ in the left-to-right order of $V\!\in\!\beta$ to form a scaffolding $\mathbf{X}_\beta$,
similarly build $\mathbf{N}_\beta$ from $\mathbf{N}_V$, $V\!\in\!\beta$, and consider $\skewerP(\mathbf{N}_\beta,\mathbf{X}_\beta)$.

\medskip

\noindent Some of the main objects of interest are

\begin{itemize}\item the pair $(\bX,\bN)$ of the \Stable[1+\alpha] scaffolding $\bX$ and the \PRM\ $\bN$ of spindles,
  \item the random point measures $\bF^y$, $y\ge 0$, of bi-clades of $(\bX,\bN)$,
  \item the \em type-1 evolution \em $(\beta^y,y\!\ge\! 0):=\skewerP(\bN_\beta,\bX_\beta)$, extracting intervals from the spindles in
    jumps of $\mathbf{X}_\beta$ crossing level $y$, for any initial interval partition $\beta$.
  \item the total mass process $(\IPmag{\beta^y},\,y\ge 0)$.
\end{itemize}

\noindent Some of the main results of \cite{IPPA} are the following, in the numbering of \cite{IPPA}.

\begin{enumerate}
  \item[1.3] The interval-partition-valued process $y\mapsto\beta^y$ admits a continuous version.
  \item[1.4] Type-1 evolutions $y\mapsto\beta^y$ are path-continuous Hunt processes: they can be started from any interval partition in a Lusin state space $(\mathcal{I}_H,d_H)$, are
    continuous in the initial condition and satisfy the strong Markov property.
  \item[3.2] The level-$y$ aggregate mass process $s\mapsto M_{\bN,\bX}^y(s)$ of \eqref{eq:agg_mass_from_spindles}, time-changed by the inverse local time $\tau^y_\bX$ of $\bX$ at level $y$ is a stable
    subordinator of index $\alpha$.
  \item[4.9] $\bF^y$ is a \PRM, whose intensity measure we call \em bi-clade excursion measure\em.
  \item[4.11] Bi-clades are space/time-reversible in the sense that reversing scaffolding time and block diffusion time in spindles yields
    the same bi-clade excursion measure.
  \item[4.15] Mid-bi-clade Markov property: conditionally given a spindle mass $f_R(\mathbf{X}_{R-})=x$ of
    the spindle $(R,f_R)$ in $\mathbf{N}$ at the time $R$ when the scaffolding $\mathbf{X}$ crosses the line $\mathbf{X}=0$, the clade part (post-$R$) and the time-reversed anti-clade part (pre-$R$) are independent and
    distributed as clades starting from $x$.
  \item[5.5] The skewer processes of $(\mathbf{N},\mathbf{X})$ stopped at stopping times including $\tau_\mathbf{X}^0(u)$ and $S_\mathbf{X}(-u)$ for $u\ge 0$
    are type-1 evolutions.
\end{enumerate}

\noindent In \cite{IPPB}, we further prove the following.
\begin{enumerate}
  \item[1.2] The type-1 semi-group $\kappa_y^\mathcal{I}$, $y\!\ge\!0$, is as follows. Independently for each block $V\!\in\!\beta^0$ of size $b\!=\!{\tt Leb}(V)$, there is a contribution to
     level $y$ with probability $1\!-\!e^{-b/2y}$.
     Such a contribution consists of a left-most interval with Laplace transform $(1\!+\!\gamma/r)^\alpha(e^{br^2/(r+\gamma)}\!-\!1)/(e^{br}\!-\!1)$,
     where $r\!=\!1/2y$,
     concatenated with a scaled ${\tt PDIP}(\alpha,\alpha)$, the interval partition formed by the excursion intervals of a ${\tt BES}(2-2\alpha)$-bridge,
     scaled by an independent ${\tt Gamma}(\alpha,1/2y)$-distributed random factor. The contributions are
     concatenated in the order of $V\in\beta^0$ to give the distribution under the time-$y$ transition kernel starting from $\beta^0$.
  \item[1.3] The kernels $\widetilde{\kappa}_y^\mathcal{I}$ obtained by concatenating
     a ${\tt PDIP}(\alpha,\alpha)$  scaled by an independent ${\tt Gamma}(\alpha,1/2y)$ with the interval partition from $\kappa_y^{\mathcal{I}}$, $y\!\ge\!0$,
     also form the semi-group of a path-continuous Hunt process, called \em type-0 evolution\em.
  \item[1.4]\begin{enumerate}
                    \item[(i)] The total mass process $(\IPmag{\beta^y},y\!\ge\! 0)$ of a type-1 evolution is $\BESQ_{\IPmag{\beta^0}}(0)$,
                      for all initial $\beta^0\in\mathcal{I}_H$, hence including the case of a single clade.
                   \item[(ii)] The total mass processes of type-0 evolutions are $\BESQ_{\IPmag{\beta^0}}(2\alpha)$.
                 \end{enumerate}
  \item[3.2] \begin{enumerate}\item[(i)] The \PRM\ $\mathbf{F}^y$ has points at bi-clades whose value of the central spindle mass when crossing $\mathbf{X}=0$
                                                                  has (sigma-finite) law $(\alpha/\Gamma(1-\alpha)) x^{-\alpha-1}dx$, and
                                                  \item[(ii)] points at bi-clades with $\mathbf{X}$-supremum above $y$ at rate $2^{-\alpha}y^{-\alpha}$.
                   \end{enumerate}
  \item[3.10] $(\skewer(y,\mathbf{N}|_{[0,S_\mathbf{X}(-u)]\times\Exc},u\!+\!\mathbf{X}|_{[0,S_\mathbf{X}(-u)]}),0\!\le\!y\!\le\! u)$ is a type-0 evolution.
\end{enumerate}

\section{Construction of $(\mathbf{X},\mathbf{N})$ from $(\mathbf{R},\mathbf{H})$ and vice versa}\label{sec:new}

\noindent Let $\tau_\mathbf{R}^0(s)\!=\!\inf\{L^0(t)\!>\!s\}$, $s\!\ge\! 0$, be the inverse local time of $\mathbf{R}$ at $0$,
and $\mathbf{K}$ the \PRM\ of excursions of $\mathbf{R}$ away from $0$. For each excursion interval
$(\tau^0_\mathbf{R}(s-),\tau^0_\mathbf{R}(s))=(\ell,r)$ of $\mathbf{R}$, we decompose the Bessel excursion $(\mathbf{R}(\ell+t),0\le t\le r-\ell)$ in $\mathbf{K}$ as in
$\mathbf{R}=\mathbf{B}-(1-d)\mathbf{H}$ in \eqref{besdec}, and we define the associated occupation density local time process $\lambda_s:=(\lambda^y_s,y\ge 0)$ of
$(\mathbf{H}(\ell+t)-\mathbf{H}(\ell),0\le t\le r-\ell)$.

\begin{proposition}\label{prop:Bertoin} The random measure $\sum_{s\ge 0\colon\tau^0_\mathbf{R}(s-)<\tau^0_\mathbf{R}(s)}\delta_{\lambda_s}$ is
  a $\PRM\big({\tt Leb}\otimes\overline{\nu}\big)$, where $\overline{\nu}$ is a Pitman--Yor excursion measure associated with \BESQ[-2(1-d)], the process $\mathbf{H}\circ\tau^0_\mathbf{R}$ is a spectrally
  positive stable process of index $2\!-\!d$. The pair has the same distribution as $(\mathbf{N},\mathbf{X})$ in Section \ref{sec:prel}, with $\alpha=1-d$,
  up to a linear time-change.
\end{proposition}
\begin{proof} Since $t\mapsto\mathbf{H}(t)$ is differentiable almost everywhere, with derivative $\mathbf{H}^\prime(t)=1/2\mathbf{R}(t)$,
  its local time at level $\mathbf{H}(t)$ increases by a jump of $2\mathbf{R}(t)$ at time $t$. Specifically, during each excursion interval $(\tau^0_\mathbf{R}(s-),\tau^0_\mathbf{R}(s))=(\ell,r)$ of $\mathbf{R}$ away from 0,
  we get
  \begin{equation}\label{eq:excloctime}\lambda_s^{\mathbf{H}(\ell+t)-\mathbf{H}(\ell)}=2\mathbf{R}(\ell+t),\quad 0\le t\le r-\ell,
  \end{equation}
  i.e.\ the local times of $\mathbf{H}$ during excursions are continuous time-changes of the excursions of $2\mathbf{R}$.
  Hence, the jump sizes of $\mathbf{H}\circ\tau^0_\mathbf{R}$ are
  $$\mathbf{H}(\tau^0_\mathbf{R}(s))-\mathbf{H}(\tau^0_\mathbf{R}(s-))=\inf\{y>0\colon\lambda^y_s=0\}=\sup\{y\ge 0\colon\lambda^y_s>0\},\ s\ge 0.$$
  Bertoin \cite[Proof of Lemma 3.2]{Bertoin1990} showed that $\mathbf{H}\circ\tau^0_\mathbf{R}$ is a spectrally positive stable process of index
  $2\!-\!d$. Furthermore, by standard mapping of \PRM s, $\sum_{s\ge 0\colon\tau^0_\mathbf{R}(s-)<\tau^0_\mathbf{R}(s)}\!\delta_{\lambda_s}$ is a \PRM.
  We will identify its intensity measure as a \BESQ[-2(1-d)] excursion measure by \cite[(3.1) First description]{PitmYor82}.

  Specifically, this description requires us to check three points.
  (i) Neither excursion measure charges the zero excursion.
  (ii) Whether the hitting time $T_x$ of level $x$ by the excursion of $\mathbf{R}$ is finite or not is not affected by the time-change,
  and the associated rate under either excursion measure is proportional to $1/s(x)$
  where $s(x)=x^{2-d}$ is the common scale function of ${\tt BES}(d)$ and \BESQ[-2(1-d)], see e.g. \cite[(3.5) Examples]{PitmYor82}.
  (iii) We will show that the pre-$T_x$ and post-$T_x$ processes are, as required.
  The pre-$T_x$ part of the excursion of $\mathbf{R}$ is a ${\tt BES}_0(d)$ conditioned to stay positive,
  i.e. a ${\tt BES}_0(4-d)$ (see again \cite[(3.5) Examples]{PitmYor82}).
  The time-change relation \eqref{eq:excloctime} transforms this into a $\BESQ_0(4+2(1-d))$, by \cite[Proposition XI.(1.11)]{RevuzYor},
  which is a $\BESQ_0(-2(1-d))$ conditioned to stay positive, as required.
  Bertoin \cite[bottom of p.\ 117]{Bertoin1990c} noted the corresponding time-change relation
  for ${\tt BES}(d)$ and \BESQ[-2(1-d)] starting from $y$ stopped when hitting 0. This identifies the post-$T_x$ parts of the
  excursions and completes the proof.
\end{proof}

\begin{proof}[Proof of Theorems \ref{thm:Bertoin} and \ref{thm:diffusion_0}] This will follow from Proposition \ref{prop:Bertoin} because applying $\skewer(y,\cdot)$ to the scaffolding-and-spindles pair of the proposition yields an interval partition with blocks
\begin{equation}\label{corrr}
  \lambda_s^{y-\mathbf{H}(\tau^0_\mathbf{R}(s-))}=2\mathbf{R}(t)=\lambda^y(t)\!-\!\lambda^y(t-)\quad\mbox{if }y\!=\!\mathbf{H}(t)\mbox{ and }t\!\in\!(\tau^0_\mathbf{R}(s-),\tau^0_\mathbf{R}(s)),
\end{equation}
  by \eqref{eq:excloctime} and \eqref{eq:loctime}. Since $\skewerP(y,\mathbf{N},\mathbf{X})$ is unaffected by (linear) changes of scaffolding time of
  $\mathbf{X}$ and $\mathbf{N}$, the process of Theorem \ref{thm:Bertoin} can be constructed as claimed in Theorem \ref{thm:diffusion_0}, when stopped at a time
  that corresponds to an inverse local time $\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u)$ of $(\mathbf{R},\mathbf{H})$ at $(0,0)$ and that after time change by $\tau^0_\mathbf{R}$, is an inverse local time
  of the ${\tt Stable}(2-d)$ process $\mathbf{H}\circ\tau^0_\mathbf{R}$.

Let us work out the constant $c$ for which stopping $(\mathbf{X},\mathbf{N})$ at $\tau_{\mathbf{X}}^0(cu)$ yields the same initial distribution for the skewer process
as the stopped scaffolding-and-spindles pair constructed from $(\mathbf{R},\mathbf{H})$ stopped at $\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u)$.
We do this using the parts of \cite[Lemma 3.3]{Bertoin1990} and \cite[Proposition 3.2]{IPPB} that we recalled in Sections \ref{sec:Bertoin} and \ref{sec:prel} here.
Specifically, the statistics of excursions of $(\mathbf{R},\mathbf{H})$ of $\mathbf{H}$-infima
directly transfer to $\mathbf{H}\circ\tau_{\mathbf{R}}^0$-infima that correspond to $\mathbf{X}$-infima, which, by bi-clade reversibility (see 4.11 above) or the
mid-bi-clade Markov property (see 4.15 above) have the same rates as $\mathbf{X}$-suprema in a bi-clade.
But the rates of $\mathbf{H}$-infima and $\mathbf{X}$-suprema differ by the constant $c=2^\alpha=2^{1-d}$,
hence $\mathbf{X}$ needs to run longer than $\mathbf{H}\circ\tau_{\mathbf{R}}^0$, by a factor of $c$,
to achieve the same number of excursions exceeding any given level $y$.

  Finally, we note that the skewer process associated with $(\mathbf{N}|_{[0,\tau_{\mathbf{X}}^0(v)]\times\Exc},\mathbf{X}|_{[0,\tau_{\mathbf{X}}^0(v)]})$ is a diffusion by \cite[Theorem 1.4]{IPPA}, again as recalled in Section \ref{sec:prel} here.
\end{proof}

A similar argument to work out $c$ can be based on the values of $\mathbf{R}$ when crossing $\mathbf{H}=0$ and the mass of the central spindle of $\mathbf{N}$ when crossing $\mathbf{X}=0$. Note, however, that these also differ by a factor of 2, by \eqref{eq:excloctime}.

\begin{corollary}\label{cor:spindles} In Bertoin's setting, under the It\^o excursion measure of $\mathbf{R}$, the local time process of $\mathbf{H}$ has as its law a
  Pitman--Yor excursion measure of \BESQ[-\!2(1\!-\!d)].
\end{corollary}

Proposition \ref{prop:Bertoin} makes precise the sense
in which the framework of a single Bessel process $\mathbf{R}\sim{\tt BES}_0(d)$ of \cite{Bertoin1990,Bertoin1990c},
via $(\mathbf{R},\mathbf{H})$,
yields the scaffolding-and-spindles framework $(\mathbf{N},\mathbf{X})$ of \cite{IPPA,IPPB}.
The main step in the proof is time-changing the excursions of $\mathbf{R}$ away from 0 to form ${\tt BESQ}(-2(1-d))$ spindles.
Let us invert this time-change and construct $\mathbf{R}$ from the spindles of $\mathbf{N}$.
To this end, recall our notation $\nu$ for the Pitman--Yor excursion measure of ${\tt BESQ}(-2\alpha)$ of Section \ref{sec:prel}.

\begin{proposition}\label{prop:reverse} For $\mathbf{N}=\sum_{i\in\mathbf{I}}\delta_{(\mathbf{s}_i,\mathbf{f}_i)}\sim{\tt PRM}({\tt Leb}\otimes\nu)$,
  set $\zeta_i:=\int_0^{\zeta(\mathbf{f}_i)}\mathbf{f}_i(y)dy$ and
  $$\mathbf{e}_i(t):=\frac{1}{2}\mathbf{f}_i\left(\inf\left\{z\ge 0\colon\int_0^z\mathbf{f}_i(y)dy>t\right\}\right),\quad
      t\in\left[0,\zeta_i\right),\quad\mbox{and}\quad\mathbf{e}_i(\zeta_i)=0.
  $$
  Then $\sum_{i\in\mathbf{I}}\delta_{(\mathbf{s}_i,\mathbf{e}_i)}$ has the same distribution as the It\^o excursion process $\mathbf{K}$ of
  $\mathbf{R}\sim{\tt BES}_0(d)$, up to a linear time change, with $d=1-\alpha$.
  In particular, the $\mathbf{e}_i$ can be stitched together in the order of the $\mathbf{s}_i$, $i\in\mathbf{I}$, to yield a process $\overline{\mathbf{R}}\sim{\tt BES}_0(1-\alpha)$.
\end{proposition}
\begin{proof} This follows from Proposition \ref{prop:Bertoin}. Specifically, mapping $\mathbf{f}_i$ to $\mathbf{e}_i$ is elementary
  since all $\mathbf{f}_i$ are continuous with compact support a.s.. In present notation, we can write \eqref{eq:excloctime} as
  $$\mathbf{f}_i\left(\int_0^t\frac{du}{\mathbf{e}_i(u)}\right)=2\mathbf{e}_i(t),\quad t\in[0,\zeta_i].$$
  This is a.s.\ well-defined for all $\mathbf{e}_i$, $i\in\mathbf{I}$, so the time-changes relating $\mathbf{f}_i$ and $\mathbf{e}_i$ are bijective, and hence the
  associated \PRM s are bijectively related by standard mapping of \PRM s. In particular, we deduce the claimed distributional identities up to a linear time change.
  The construction of Markov processes from excursions has been well-studied \cite{Salisbury1986a}.
  Note that a linear time change of the \PRM\ has no effect on the ${\tt BES}_0(1-\alpha)$-excursions themselves. Specifically, we define
  $\overline{\tau}(s)=\sum_{i\in\mathbf{I}\colon\mathbf{s}_i\le s}\zeta_i$, $s\ge 0$, and $\overline{\mathbf{R}}(\overline{\tau}(\mathbf{s}_i-)+t)=\mathbf{e}_i(t)$,
  $0\le t\le\zeta_i$, also setting $\overline{\mathbf{R}}(t)=0$ for $t\not\in\bigcup_{i\in\mathbf{I}}[\overline{\tau}(\mathbf{s}_i-),\overline{\tau}(\mathbf{s}_i)]$ and
  obtain $\overline{\mathbf{R}}\sim{\tt BES}_0(1\!-\!\alpha)$, and this is the same process as if we replace $\mathbf{s}_i$ by $a\mathbf{s}_i$, $i\in\mathbf{I}$,
  throughout, $a\!>\!0$. The process $\overline{\tau}$ is an inverse local time of $\overline{\mathbf{R}}$ at 0, and replacing $\mathbf{s}_i$ by $a\mathbf{s}_i$ corresponds to
  a different choice of local time.
\end{proof}

\begin{corollary} For $(\mathbf{X},\mathbf{N})$ as in Section \ref{sec:prel} and notation $\overline{\mathbf{R}}$ as in Proposition \ref{prop:reverse}, with
  $\overline{\tau}(s)=\sum_{i\in\mathbf{I}\colon\mathbf{s}_i\le s}\zeta_i$, $s\ge 0$, define $\overline{\mathbf{H}}$ on the range of $\overline{\tau}$ as
  $\overline{\mathbf{H}}(\overline{\tau}(s)):=\mathbf{X}(s)$, $s\ge 0$, and outside the range of $\overline{\tau}$ as
  $$\overline{\mathbf{H}}\left(\overline{\tau}(\mathbf{s}_i-)+\frac{1}{2}\int_0^z\mathbf{f}_i(y)dy\right):=\mathbf{X}(\mathbf{s}_i-)+z,\quad 0\le z<\Delta\mathbf{X}(\mathbf{s}_i)=\zeta(\mathbf{f}_i).$$
  Then the pair $(\overline{\mathbf{R}},\overline{\mathbf{H}})$ has the same distribution as $(\mathbf{R},\mathbf{H})$ of Section \ref{sec:Bertoin}.
\end{corollary}
\begin{proof} Since $\mathbf{H}$ is determined by $\mathbf{R}$ via \eqref{eq:defH} and $\overline{\mathbf{R}}\stackrel{d}=\mathbf{R}$, it suffices to show that $\overline{\mathbf{H}}$ relates to
  $\overline{\mathbf{R}}$ in the same way. Indeed, we have $\overline{\mathbf{H}}\circ\overline{\tau}=\mathbf{X}$, by construction, and $\mathbf{X}(s)$ is the
  compensated limit of its jumps $\Delta\mathbf{X}(\mathbf{s}_i)\!=\!\zeta(\mathbf{f}_i)$ for $i\!\in\!\mathbf{I}$ with $\mathbf{s}_i\!\le\! s$. But
  $$\zeta(\mathbf{f}_i)=\int_0^{\zeta(\mathbf{e}_i)}\frac{du}{\mathbf{e}_i(u)}=\int_0^\infty a^{d-2}L^a_i(\infty)da,$$
  where $(a^{d-1}L^a_i(\infty),a\ge 0)$ is the continuous version of the total occupation density local time of $\mathbf{e}_i$ at level $a$. This entails that the right-most equality of
  \eqref{eq:defH} holds for $t=\overline{\tau}(s)$, when $(\mathbf{R},\mathbf{H})$ is replaced by $(\overline{\mathbf{R}},\overline{\mathbf{H}})$. Since these
  limits exist almost surely uniformly for $s$ in compact intervals, they also hold at $t=\overline{\tau}(\mathbf{s}_i-)$, $i\in\mathbf{I}$.
  Beyond the range of $\overline{\tau}$, we have, for each $i\in\mathbf{I}$,
  $$\overline{\mathbf{H}}\left(\overline{\tau}(\mathbf{s}_i-)+\frac{1}{2}\int_0^z\mathbf{f}_i(y)dy\right)=\overline{\mathbf{H}}(\overline{\tau}(\mathbf{s}_i-))+z,
      \quad 0\le z\le\zeta(\mathbf{f}_i).$$
  But according to the bijective time change relationships transforming $\mathbf{f}_i$ into $\mathbf{e}_i$ noted in the proof of Proposition \ref{prop:reverse}, we have
  $$z=\int_0^t\frac{du}{\mathbf{e}_i(u)}\quad\mbox{if and only if}\quad t=\frac{1}{2}\int_0^z\mathbf{f}_i(y)dy.$$
  Hence, we obtain
  $$\overline{\mathbf{H}}(\overline{\tau}(\mathbf{s}_i-)+t)=\overline{\mathbf{H}}(\overline{\tau}(\mathbf{s}_i-))+\int_0^t\frac{du}{\mathbf{e}_i(u)},
      \quad 0\le t\le\zeta(\mathbf{e}_i),$$
  and this completes the proof.
\end{proof}

\section{Further consequences of the connection between \cite{Bertoin1990,Bertoin1990c} and \cite{IPPA,IPPB}}\label{sec:new2}

\begin{table}[b]
  \begin{center}
   \begin{tabular}{|c||c|c|c|c|c|c||c|c|c|c|}
    \hline
     &&&&&&&&&&\\[-0.3cm]
    \cite{IPPA}$\|$\cite{IPPB}                          &1.3 & 1.4 & 4.9 & 4.11 & 4.15&5.5& 1.2 &$\!$1.2, 1.3, 3.10$\!$&    1.4            &3.2\\[0.1cm]
    \hline
    &&&&&&&&&&\\[-0.3cm]
    \cite{Bertoin1990,Bertoin1990c} &II.1& II.2 &3.1 & 3.2 & 3.4 &   II.2& II.3&     II.2            &$\!$4.1, 4.2, II.4$\!$&3.3\\[0.1cm]
    \hline
  \end{tabular}
  \bigskip

  \caption{Each column lists pairs of results (or groups of results) from \cite{IPPA} or \cite{IPPB}, and from \cite{Bertoin1990,Bertoin1990c} that are analogues of each other.\label{table}}
  \end{center}
\end{table}

In the light of the results of Section \ref{sec:new}, the results of \cite{Bertoin1990,Bertoin1990c} and \cite{IPPA,IPPB} are closely related. Indeed, many results of \cite{Bertoin1990,Bertoin1990c} can now be deduced from \cite{IPPA,IPPB}, and the approach of \cite{Bertoin1990,Bertoin1990c} could be refined to handle the
additional order structure needed for the interval partitions of \cite{IPPA,IPPB}. Table \ref{table} pairs the analogous results, which will mostly have
been evident already from the formulations in Sections \ref{sec:Bertoin} and \ref{sec:prel}.

One may note, however, that these results differ in detail, not just because $(\beta^y,y\!\ge\!0)$ and $(\mu^y_{[0,T]},y\!\ge\! 0)$ have different state spaces. Specifically,
\cite[Theorem 1.4]{IPPA} and \cite[Theorem 1.3]{IPPB} establish interval-partition-valued processes as path-continuous Hunt processes that are continuous in the
initial condition, while \cite[Theorem II.2]{Bertoin1990c} does not push beyond the simple Markov property. On the other hand,
\cite[Proposition 2.4]{Bertoin1990} and \cite[Proposition 3.2]{IPPA} find stable inverse local times of different indices, but fundamentally play the same role, since they
provide the time parameterisations for the \PRM s of excursions of $(\mathbf{R},\mathbf{H})$ and of bi-clades, respectively.

The observation of Corollary \ref{cor:spindles}, that $\mathbf{H}$-local time processes in $\mathbf{R}$-excursions are $\BESQ(-2(1-d))$, is related to
\cite[Theorem I.5 or (0.3)]{Bertoin1990c}, which notes $\BESQ(-2(1-d))$ evolution of time-changed $\mathbf{R}$-excursions after they exceed previous
$\mathbf{H}$-suprema. In the context of \cite{IPPA,IPPB}, the corresponding result is a consequence of the construction from $\BESQ(-2\alpha)$ spindles
(and the Markov property). But \cite[Theorem I.5]{Bertoin1990c} goes further and yields the following result when translated into the framework of \cite{IPPA,IPPB}.

\begin{corollary} For each $y\ge 0$, let $T_\mathbf{X}^+(y)=\inf\{s\ge 0\colon\mathbf{X}(s)>y\}$ and denote by
  $\mathbf{L}(y):=\mathbf{f}_{T_\mathbf{X}^+(y)}(y-\mathbf{X}(T_\mathbf{X}^+(y)-))$ the value of the left-most spindle $\mathbf{f}_{T_\mathbf{X}^+(y)}$
  that crosses level $y$. Then $(\mathbf{L}(y),y\!\ge\! 0)$ is a Markov process whose excursions away from 0 start with a jump of intensity
  $(2^{\alpha}\alpha/\Gamma(1-\alpha))x^{-1-\alpha}dx$ and then evolve as $\BESQ_x(-2\alpha)$.
\end{corollary}

Similarly, \cite[Theorem I.6]{Bertoin1990c} then yields the semi-group of $\mathbf{L}$.

Less immediate are the consequences of some further results of \cite{IPPB}, which we have not stated in Section \ref{sec:prel}, about what we call pseudo-stationarity
of type-0 and type-1 evolutions, and the passage to normalised interval-partition evolutions on the subspace $\mathcal{I}_{H,1}$ of interval partitions via suitable
time-change.
We observe in the context of Bertoin \cite[Theorem II.2]{Bertoin1990c} that the marginal distributions of
$\mu_{[0,T_\mathbf{H}(-1)]}^{-1+y}$, $y\in[0,1]$, a process starting from the zero measure $0\in\mathcal{N}((0,\infty))$, can be read from
\begin{equation}\label{pseudostat1}
  \widetilde{\kappa}_y^\mathcal{N}f_\varphi(0)=\frac{y^{d-1}}{y^{d-1}\!+\!\int_0^\infty(1\!-\!\varphi(s))\Pi_y(ds)}=\mathbb{E}\Bigg(\!f_\varphi\Bigg(\sum_{j\in\mathbf{J}\colon 0\le\mathbf{r}_j\le E_y}\!\delta_{\Delta\sigma_y(\mathbf{r}_j)}\!\Bigg)\!\Bigg),
\end{equation}
where $\Pi_y(ds)\!=\!((1\!-\!d)/\Gamma(d))s^{d-2}e^{-s/y}ds$ is the L\'evy measure of a subordinator $(\sigma_y(r),r\!\ge\! 0)$ with \PRM\
$\sum_{j\in\mathbf{J}}\delta_{(\mathbf{r}_j,\Delta\sigma_y(\mathbf{r}_j))}$ of its jumps, and $E_y\sim{\tt Exp}(y^{d-1})$ is an independent random variable. Now \cite[Proposition 21]{PitmYorPDAT} showed, for $y=1$, that the decreasing rearrangement
$(\sigma_y(E_y))^{-1}(\Delta\sigma_y(t),0\le t\le E_y)^\downarrow$ of normalised jump sizes of $\sigma_y|_{[0,E_y]}$ has Poisson--Dirichlet
distribution ${\tt PD}(1-d,1-d)$, and is independent of $\sigma_y(E_y)\sim{\tt Gamma}(1-d,1/y)$, and a simple change of variables extends this to all $y>0$. As ${\tt PD}(1-d,1-d)$ is preserved at all times $y>0$ (while ${\tt Gamma}(1-d,1/y)$ depends on $y>0$), we call this behaviour
\em pseudo-stationarity\em, cf. \cite[Theorem 1.5]{IPPB}.

Furthermore, it is well-known that adding an independent $\mathbf{R}(0)\sim{\tt Gamma}(d,1/y)$ variable to the jumps of
$\sigma_y|_{[0,E_y]}$, we obtain $\mathbf{R}(0)+\sigma_y(E_y)\sim{\tt Exp}(1/y)$ independent of
$(\mathbf{R}(0)+\sigma_y(E_y))^{-1}(\mathbf{R}(0);\Delta\sigma_y(t),0\le t\le E_y)^\downarrow\sim{\tt PD}(1-d,0)$, and there is
pseudo-stationarity in the following sense.

\begin{theorem}\label{pseudo} Let $\mathbf{R}$ be a ${\tt BES}(d)$ starting from $\mathbf{R}(0)\sim{\tt Gamma}(d,\rho)$ and let
  $\mathbf{Q}$ be a ${\tt BESQ}(0)$ starting from $\mathbf{Q}(0)\sim{\tt Exp}(\rho/2)$ independent of
  $(\mathbf{x}_i,i\ge 1)\sim{\tt PD}(1-d,0)$. Then $\mu_{[0,T_{\mathbf{H}}(-1/\rho)]}^y$ has the same distribution as
  $\sum_{i\ge 1}\delta_{\mathbf{x}_i\mathbf{Q}(y)/2}$, for each fixed $y>0$.
\end{theorem}
\begin{proof} Let $\mathbf{f}_0:=\lambda_0=(\lambda_0^y,y\ge0)$ be the local time process of $(\mathbf{H}(t),0\le t\le T_{\mathbf{R}}(0))$.
  Then $\mathbf{f}_0(y)=\widetilde{\mathbf{R}}(y)$ for all $0\le y<\mathbf{H}(T_\mathbf{R}(0))=\zeta(\mathbf{f}_0)$,
  and by \cite[Theorem I.5]{Bertoin1990c}, $\mathbf{f}_0$ is a ${\tt BESQ}(-2(1-d))$,
  starting from $2\mathbf{R}(0)\sim{\tt Gamma}(d,\rho/2)$.
  Proceeding as in Proposition \ref{prop:Bertoin}, where $\mathbf{R}(0)=0$, we obtain here,
  after the linear time-change noted in that proposition, a point measure $\mathbf{N}_0:=\delta_{(0,\mathbf{f}_0)}+\mathbf{N}$, in which
  $\mathbf{N}\sim{\tt PRM}({\tt Leb}\otimes\nu)$ is independent of $\mathbf{f}_0\sim{\tt BESQ}_{2\mathbf{R}(0)}(-2(1-d))$ and scaffolding
  $\mathbf{X}_0:=\zeta(\mathbf{f}_0)+\mathbf{X}$.

  By the strong Markov property of $(\mathbf{R},\mathbf{H})$ at $T_\mathbf{H}(0)$, and by \eqref{pseudostat1} with $y=1/\rho$, we find that
  $\mu_{[0,T_{\mathbf{H}}(-1/\rho)]}^0$ has the claimed initial distribution.
  Similarly, but now based on the strong Markov property of $(\mathbf{N}_0,\mathbf{X}_0)$ at $S_{\mathbf{X}_0}(0)$
  and on \cite[Theorem 1.3]{IPPB}, $(\mathbf{N}_0,\mathbf{X}_0)$ correspondingly
  stopped at $S_{\mathbf{X}_0}(-1/\rho)$, has a skewer process $(\beta^y,y\!\ge\! 0)$ starting from a ${\tt PDIP}(\alpha,0)$
  scaled by the independent ${\tt Exp}(\rho/2)$. By \cite[Theorem 1.5]{IPPB}, $(\beta^y,y\!\ge\!0)$ is pseudo-stationary
  with $\beta^y$ distributed as a ${\tt PDIP}(\alpha,0)$ scaled by an independent $\mathbf{Q}(y)$, where $\mathbf{Q}$ is a ${\tt BESQ}(0)$
  starting from $\mathbf{Q}(0)\sim{\tt Exp}(\rho/2)$.

  But as $(\mathbf{N}_0,\mathbf{X}_0)$ has been constructed from $(\mathbf{R},\mathbf{H})$ as Proposition \ref{prop:Bertoin} did for the proof of Theorems \ref{thm:Bertoin}
  and \ref{thm:diffusion_0}, we read from \eqref{corrr} the coupling
  \begin{equation}\label{coupling}\mu_{[0,T_\mathbf{H}(-1/\rho)]}^y=\phi(\beta^y),\qquad\mbox{where }\phi(\beta)=\sum_{V\in\beta}\delta_{{\tt Leb}(V)/2},
  \end{equation}
  so the distribution of $\mu_{[0,T_\mathbf{H}(-1/\rho)]}^y$ follows from the distribution of the ranked sequence of interval lengths of the pseudo-stationary $\beta^y$, which are
  ${\tt PD}(\alpha,0)$ scaled by independent $\mathbf{Q}(y)$, as required.
\end{proof}

In the light of this coupling \eqref{coupling}, \cite[Theorem 1.6]{IPPB} has the following corollary. Let
$\mathcal{N}_1((0,\infty)):=\big\{\sum_{i\in I}\delta_{x_i}\in\mathcal{N}((0,\infty))\colon\sum_{i\in I}x_i=1\big\}$ and consider the map
$\mu=\sum_{i\in I}\delta_{x_i}\mapsto\overline{\mu}:=\sum_{i\in I}\delta_{x_i/\sum_{j\in I}x_j}$ from $\mathcal{N}((0,\infty))\setminus\{0\}$ to $\mathcal{N}_1((0,\infty))$.

\begin{corollary} Let $\mathbf{R}$ be as in Theorem \ref{pseudo}, set $T:=T_\mathbf{H}(-1)$ and consider the time-change
  $$\varrho(u):=\inf\left\{y\ge 0\colon\int_0^y\frac{dz}{\lambda^z(T)}>u\right\},\qquad u\ge 0.$$
  Then the process $\Big(\overline{\mu}_{[0,T]}^{\varrho(u)}, u\!\ge\! 0\Big)$, obtained from $\big(\mu_{[0,T]}^y,y\!\ge\!0\big)$ by
  first time-changing by $\varrho$ and then mapping under $\mu\mapsto\overline{\mu}$, is a stationary Markov process whose invariant distribution
  is the distribution of $\sum_{i\ge 1}\delta_{\mathbf{x}_i}$ for $\big(\mathbf{x}_i,i\ge 1\big)\sim{\tt PD}(\alpha,0)$.
\end{corollary}
\begin{proof} By \cite[Theorem 1.6]{IPPB}, the corresponding interval-partition-valued process is a stationary Markov process, whose invariant
  distribution is ${\tt PDIP}(\alpha,0)$. Specifically, for $\beta\!\in\!\mathcal{I}_H\!\setminus\!\{\emptyset\}$, let
  $\overline{\beta}:=\{(a/\|\beta\|,b/\|\beta\|)\colon(a,b)\!\in\!\beta\}
    \in\mathcal{I}_{H,1}:=\{\gamma\!\in\!\mathcal{I}_H\colon\|\gamma\|\!=\!1\}$.
  Recall that with the coupling \eqref{coupling}, for $\rho\!=\!1$, we have $\lambda^y(T)=\|\beta^y\|$ for all $y\ge 0$. Then
  $$\overline{\mu}^{\varrho(u)}_{[0,T]}=\sum_{V\in\beta^{\varrho(u)}}\delta_{{\tt Leb}(V)/\|\beta^{\varrho(u)}\|}
                                                         =\sum_{V\in\overline{\beta}^u}\delta_{{\tt Leb}(V)}.$$
  This yields that each $\overline{\mu}^{\varrho(u)}_{[0,T]}$ has the claimed distribution. The Markov property will follow from Dynkin's
  criterion for when a function of a Markov process is a Markov process. Specifically, any two $\beta,\gamma\!\in\!\mathcal{I}_H$ with
  $\phi(\beta)\!=\!\phi(\gamma)$ have intervals of the same lengths, so
  there is a bijection $\eta\colon\gamma\rightarrow\beta$ such that ${\tt Leb}(\eta(V))={\tt Leb}(V)$ for all $V\!\in\!\gamma$. We construct
  coupled type-1 evolutions $(\beta^y,y\!\ge\!0)$ starting from $\beta$ and $(\gamma^y,y\!\ge\!0)$ starting from $\gamma$ by building
  $(\mathbf{X}_\beta,\mathbf{N}_\beta)$ from $(\mathbf{X}_V,\mathbf{N}_V)$, $V\!\in\!\beta$, as in Section \ref{sec:prel}, and then building
  $(\mathbf{X}_\gamma,\mathbf{N}_\gamma)$ by stitching together the same $(\mathbf{X}_{\eta(V)},\mathbf{N}_{\eta(V)})$, $V\!\in\!\gamma$,
  in the order given by $\gamma$. Then $\phi(\beta^y)\!=\!\phi(\gamma^y)$ and $\phi(\overline{\beta}^u)\!=\!\phi(\overline{\gamma}^u)$.
  In particular, the distributions of $\phi(\overline{\beta}^u)$ and $\phi(\overline{\gamma}^u)$ coincide, as required for Dynkin's criterion.
\end{proof}

Finally, we rewrite $\mathbf{R}(t)=\mathbf{B}(t)-(1\!-\!d) \mathbf{H}(t)$ as a decomposition of Brownian motion
$\mathbf{B}(t)=\mathbf{R}(t)+(1-d)\mathbf{H}(t)$. In Proposition \ref{prop:Bertoin}, we time-changed $\mathbf{H}$ by the inverse local time $\tau_\mathbf{R}^0$. But then $(1\!-\!d)\mathbf{H}(\tau^0_\mathbf{R}(s))=\mathbf{B}(\tau_\mathbf{R}^0(s))$ is a time-changed Brownian motion. Also,
during each jump $\mathbf{H}(\tau^0_\mathbf{R}(s))-\mathbf{H}(\tau_\mathbf{R}^0(s-))$, we can write
$$\;\mathbf{B}(\tau^0_\mathbf{R}(s-)\!+\!t)-\mathbf{B}(\tau^0_\mathbf{R}(s-))
							=\mathbf{R}(\tau^0_\mathbf{R}(s-)\!+\!t)+(1\!-\!d)\!\left(\mathbf{H}(\tau_\mathbf{R}^0(s-)\!+\!t)\!-\!\mathbf{H}(\tau_\mathbf{R}^0(s-))\right)\!,\;$$
which is the part of the Brownian motion from which the corresponding excursion of $\mathbf{R}$ away from 0 is built. From that excursion, we built
the corresponding (increasing) stretch of $(1-d)\mathbf{H}$, whose local time is a ${\tt BESQ}(-2(1-d))$ excursion of length
$\mathbf{H}(\tau_\mathbf{R}^0(s))-\mathbf{H}(\tau_\mathbf{R}^0(s-))$, by Corollary \ref{cor:spindles}. Note that this part of the Brownian motion is
positive (relative to its starting level) and indeed it stays above the increasing stretch of $(1-d)\mathbf{H}$ since $\mathbf{R}(t)>0$
during $(\tau_\mathbf{R}^0(s-),\tau_\mathbf{R}^0(s))$.

At $\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u)$, when $\mathbf{R}$ and $\mathbf{H}$ both vanish, we also have
$\mathbf{B}(\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u))=0$. Bertoin \cite[Lemma 3.2]{Bertoin1990} noted that
$\widetilde{\mathbf{R}}(t)=\mathbf{R}(\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u)-t)$ and $\widetilde{\mathbf{H}}(t)=-\mathbf{H}(\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u)-t)$ give $\widetilde{\mathbf{R}}\stackrel{d}{=}\mathbf{R}$ and $\widetilde{\mathbf{H}}\stackrel{d}{=}\mathbf{H}$ on $[0,\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u)]$. This yields\vspace{-0.1cm}
$$-\widetilde{\mathbf{R}}(t)=\widetilde{\mathbf{B}}(t)-(1-d)\widetilde{\mathbf{H}}(t),\qquad t\in[0,\tau_{\mathbf{R},\mathbf{H}}^{(0,0)}(u)],$$
with a minus sign on the left-hand side, so the reversibility is rather subtle. It would be interesting to understand more fully the behaviour of
$\mathbf{B}$ on intervals
$[\tau_\mathbf{R}^0(s-),\tau_\mathbf{R}^0(s)]$. E.g., what are the local times of $\mathbf{B}$ on
$[\tau_\mathbf{R}^0(s-),\tau_\mathbf{R}^0(s)]$, $s\ge 0$?

\bibliographystyle{abbrv}
\bibliography{AldousDiffusion}

\section*{Acknowledgements.} The author would like to thank Jean Bertoin for pointing out parallels between his work \cite{Bertoin1990,Bertoin1990c} and the author's joint work \cite{IPPA,IPPB}, and for several fruitful discussions that led to this project.
The author would also like to thank his co-authors Noah Forman and Douglas Rizzolo for some feedback on a draft and for further discussions.

\end{document}

config: arxiv


TITLE: evaluate the integral: $\int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)dx}$
QUESTION [0 upvotes]: $$\int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)dx}$$
I know this is a simple problem, but I don't have the answer for it and I just want to make sure that I'm correct!

REPLY [3 votes]: $$ \int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)dy} = y(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3) + \text{const}. $$
So I'm assuming your integral is actually:
$$ \int (3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3) \color{red}{dx}$$
By trig identities $$\csc(x) \cot(x) = \frac{\cos(x)}{\sin^2(x)}.$$
Now if we use $u = \sin(x)$ then $du = \cos(x)dx,$ and $\sin^2(x) =u^2.$ So
$$ \int \frac{\cos(x)}{\sin^2(x)} dx = \int \frac{1}{u^2} du .$$
Can you take it from here?

Edit: the complete integral is then
$$ \int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)\ \text{d}x} = -\csc(x) - \frac{5}{8}x^8 + 4 \log(|x|) + 3x + \text{const}. $$

REPLY [1 votes]: $$\forall -1\neq n\in\Bbb R\,\,,\,\int x^n\,dx=\frac{x^{n+1}}{n+1}+K\,\,,\,\int x^{-1}\,dx=\log|x|+K$$
$$\int\csc x\cot x\,dx=\int\frac{\cos x\,dx}{\sin^2 x}=\int\frac{d(\sin x)}{(\sin x)^2}=-\frac{1}{\sin x}+K\,$$


TITLE: For any two almost complex structures on infinite-dimensional space: Do they give isomorphic vector spaces? Are they similar?
QUESTION [1 upvotes]: I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying backwards as much as possible, but I have been stuck on the concepts of almost complex structures and complexification. I have studied several books and articles on the matter including ones by Keith Conrad, Jordan Bell, Gregory W. Moore, Steven Roman, Suetin, Kostrikin and Mainin, Gauthier
I have several questions on the concepts of almost complex structures and complexification. Here are some:
Assumptions: Let $W = (W,\text{Add}_W: W^2 \to W,s_W: \mathbb R \times W \to W)$ be an $\mathbb R$-vector space, which may be infinite-dimensional. Suppose $W$ has an almost complex structure $H \in Aut_{\mathbb R}(W)$. $H$ uniquely corresponds to the $\mathbb C$-vector space $(W,H)$, where scalar multiplication is given by the complex structure $s_W^{H}: \mathbb C \times W \to W$, $s_W^{H}(a+ib,v) := s_W(a,v) + s_W(b,H(v))$. Note that $s_W^{H}$ agrees with the original real scalar multiplication $s_W$.
What I know:

A1. For $W$ finite-dimensional and for any other almost complex structure $J \in Aut_{\mathbb R}(W)$, we have that $(W,H)$ and $(W,J)$ are $\mathbb C$-isomorphic but not necessarily by the identity map $id_W$ on $W$.
A2. For $W$ finite-dimensional and for any other almost complex structure $J \in Aut_{\mathbb R}(W)$, we have that $H$ and $J$ are similar, i.e. $H \circ S = S \circ J$, for some $S \in Aut_{\mathbb R}(W)$
A3. (A2) may be proven using (A1). See here and here. (I'm not sure if Moore (Section 9.1) uses (A1)).

Questions:

Is (A1) true if $W$ were instead infinite-dimensional? (You may use axiom of choice.)
Is (A2) true if $W$ were instead infinite-dimensional? (You may use axiom of choice.)

What I've tried:

If yes to Question 1, then I think yes to Question 2 because I think we can still make the same argument as in the answers in the links in (A3). If no to Question 1, I think we don't necessarily have no to Question 2.

Related question: Different almost complex structures: $\mathbb C$-isomorphism for $(W,K)$ and $(W,H)$

REPLY [2 votes]: Yes, and I don't know of any proof in the finite-dimensional case that does not also work in the infinite-dimensional case with trivial modifications.  Given an almost complex structure $J$ on $W$, you can think of $W$ as a $\mathbb{C}$-vector space via $J$.  Pick a basis $B$ for this $\mathbb{C}$-vector space.  Then $B\cup J(B)$ is a basis for $W$ as an $\mathbb{R}$-vector space.  So, $\dim_\mathbb{R} W=2\cdot \dim_\mathbb{C} W$.  In particular, this means that $\dim_\mathbb{C} W$ is uniquely determined by $\dim_\mathbb{R} W$ (it is half the dimension when it is finite, or the same dimension when it is infinite).  So any other almost complex structure $H$ makes $W$ a $\mathbb{C}$-vector space of the same dimension as $J$ does, so they are isomorphic.
Note that question 2 is equivalent to question 1.  Indeed, $S:W\to W$ witnesses that $J$ and $H$ are similar iff $S$ is an isomorphism $(W,J)\to (W,H)$ of complex vector spaces (this is trivial from the definitions).


TITLE: What is the right embedding between $C^{\infty}_c (\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$?
QUESTION [0 upvotes]: There exists an embedding relation “$\subset$” or “$\supset$” between $C^{\infty}_c (\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$?
Could anyone give me an example? Thank you in advance!

REPLY [1 votes]: A smooth compactly supported function has weak derivatives of all order that are in every $L^p$ space. In particular,
$$ C^\infty_c (\mathbb R^n)\subset H^1(\mathbb R^n).$$
In fact more is true, $C^\infty_c(\mathbb R^n)$ is dense in $H^1(\mathbb R^n)$. A $C^1_c$ function that is not smooth shows that $H^1\neq C^\infty_c$.
"bump functions" are your standard example of a function in $C^\infty_c$, and there's lots of discussion on them already on MSE. For example : Where can we find examples of smooth functions with compact support? Is there a book?. This example is related to the smooth but non-analytic function in this wikipedia page.
These examples are for 1 variable but it is trivial to modify them for higher dimensions by e.g. forcing them to be radially symmetric, or multiplying toegether one for each coordinate.


TITLE: Exercise #26 from "Abel's Theorem in Problems and Solutions" by Alekseev
QUESTION [4 upvotes]: Exercise:
Prove, that for every element $a$ of the group:
$(a^m)^{-1}=(a^{-1})^m$, where $m$ is natural.
On the #23 was proven that
$$(a_1 \centerdot \ldots \centerdot a_n)^{-1} = a_n^{-1} \centerdot \ldots \centerdot a_1^{-1}$$
and I believe that #26 directly follows from this, when $a_i=a$ for all $i$ from $1$ to $n$. Would you be so kind to tell me am I right?  (Author gives another, not so simple answer.)
Thanks.

REPLY [1 votes]: You are correct, assuming that $m$ is a natural number (if $m=0$ it doesn't follow from #26 but its not hard to see why its true in this case).
You can also prove this easily from the definition, just recall that $a,a^{-1}$ commute beetwen them