Continuity and Markov property – Assumptions assumption b and assumption
csection continuity In this section we prove Theorems theorem diffusion and
theorem interval partitions regular. The majority of the section is dedicated
to the proof of Theorem theorem diffusion which is completed in Section
subsection strong markov. The proof of Theorem theorem interval partitions
regular is simpler and is completed at the end of Section subsection
continuity in aggregate mass. Scale function and speed measure under
Assumptions assumption B’ and assumption c’section model assumptions We now
define a modification of Assumption assumption b which we will use to prove
intermediate results in this paper. B’ A diffusion $\mathbf{Y}$ with state
space $E\subseteq[0,\infty)$ of the form $E=[0,c]$ or $E=[0,c)$ for some
$c\leq\infty$ satisfies _Assumption 1_ if it satisfies $A1$, $A2$, $A3$, $B1$,
$B2$, $B3$. Note that this is Assumption LABEL:assumption_b to the cases where
the transformation $g$ is the identity map. We define Assumption 1 in the same
way for Assumption LABEL:assumption_c. C’ A diffusion $\mathbf{Y}$ with state
space $E\subseteq[0,\infty)$ of the form $E=[0,c]$ or $E=[0,c)$ for some
$c\leq\infty$ satisfies _Assumption 1_ if it satisfies $A1$, $A2$, $A3$, $C1$,
$C2$. We use the definition of the derivative $s^{\prime}$ of a scale function
$s$ from equation scale and speed measure and we take $b$ in equation scale
and speed measure to be $\varepsilon_{0}$ when under Assumption 1 and to be
$\varepsilon_{1}$ when under Assumption 1. We obtain upper and lower bounds
for the speed measure and scale function from the corresponding quantities in
the squared Bessel case given above. We have

$\varepsilon_{0}^{-\alpha^{+}}x^{\alpha^{+}}=\exp\left(\int_{z=x}^{\varepsilon_{0}}\frac{-\alpha^{+}}{z}dz\right)\leq
s^{\prime}(x)=\exp\left(\int_{z=x}^{\varepsilon_{0}}\frac{\mu(z)}{2z}dz\right)\leq\varepsilon_{0}^{\alpha^{-}}x^{\alpha^{-}},x\in(0,\varepsilon_{0})\text{under\ref{assumption
B'}}$ (1)

and

$\varepsilon_{1}^{\beta^{-}}x^{-\beta^{-}}=\exp\left(\int_{z=x}^{\varepsilon_{1}}\frac{\beta^{-}}{z}dz\right)\leq
s^{\prime}(x)=\exp\left(\int_{z=x}^{\varepsilon_{1}}\frac{\mu(z)}{2z}dz\right)\leq\varepsilon_{1}^{\beta^{+}}x^{-\beta^{+}},x\in(0,\varepsilon_{0})\text{under\ref{assumption
c'}}.$ (2)

For the density of the speed measure this results in

$\frac{1}{2}\varepsilon_{0}^{\alpha^{-}}x^{-1-\alpha^{-}}\leq
m(x)\leq\frac{1}{2}\varepsilon_{0}^{\alpha^{+}}x^{-1-\alpha^{+}},x\in(0,\varepsilon_{0})\text{under\ref{assumption
B'}}$ (3)

and

$\frac{1}{2}\varepsilon_{1}^{-\beta^{+}}x^{-1+\beta^{+}}\leq
m(x)\leq\frac{1}{2}\varepsilon_{1}^{-\beta^{-}}x^{-1+\beta^{-}},x\in(0,\varepsilon_{0})\text{under\ref{assumption
c'}}.$ (4)

For $s(x)$ we have

$\frac{\varepsilon_{0}^{-\alpha^{+}}}{1+\alpha^{+}}x^{1+\alpha^{+}}=\varepsilon_{0}^{-\alpha^{+}}\int_{y=0}^{x}y^{\alpha^{+}}dy\leq
s(x)\leq\varepsilon_{0}^{-\alpha^{-}}\int_{y=0}^{x}y^{\alpha^{-}}dy\leq\frac{\varepsilon_{0}^{-\alpha^{-}}}{1+\alpha^{-}}x^{1+\alpha^{-}},x\in(0,\varepsilon_{0})\text{under\ref{assumption
B'}}.$ (5)

It immediately follows that diffusions under Assumption LABEL:assumption_c
have scaffolding of bounded variation because $0$ is a regular point for these
diffusions. As for diffusions under Assumption LABEL:assumption_b, if we
further assume $A4$ then $B1$ is immediate and is not required as a condition.
We include this in the result below for completeness. Consider a diffusion
that satisfies Assumption LABEL:assumption_b and let $(\mathbf{N},\mathbf{X})$
be a scaffolding-and-spindles pair for this diffusion which is stopped at some
a.s. finite random time $T$. Then we have the aggregate health process
$M^{0}_{\mathbf{N},\mathbf{X}}(T)$ is finite a.s. Furthermore, if instead we
have a diffusion that satisfies all of Assumption LABEL:assumption_a and $B2$,
$B3$ then the diffusion also satisfies $B1$. For the first result we note that
the untransformed diffusion $\mathbf{Y}$ satisfies Assumption 1 and so we can
use the bound in equation m bounds B for its speed measure $m_{\mathbf{Y}}$.
Let $q\in(\alpha^{+},q_{0})$ and let $C_{q}$ be the Hölder constant for $g$ as
in from Assumption $B4$. Then we have

$\int_{y=0}^{\varepsilon_{0}}g(y)m_{\mathbf{Y}}(y)dy\leq\frac{1}{2}C_{q}\varepsilon_{0}^{\alpha^{+}}\int_{y=0}^{\varepsilon_{0}}y^{-1-\alpha^{+}+q}dy=\frac{C_{q}\varepsilon_{0}^{\alpha^{+}}}{2(q-\alpha^{+})}<\infty.$

For the second result we begin by noting that on $(0,\varepsilon_{0})$

$s^{\prime}(x)=\exp\left(\int_{z=x}^{\varepsilon_{0}}\frac{\mu(z)}{2z}dz\right),$

and so as $\mu$ is negative on $(0,\varepsilon_{0})$ by $B3$ we have
$s^{\prime}$ is increasing on $(0,\varepsilon_{0})$, and so $s(x)\leq
xs^{\prime}(x)$ for $x\in(0,\varepsilon_{0})$. Recall from equation scale and
speed measure that $m(x)=2/(\sigma^{2}(x)s^{\prime}(x))$. Then, as
$\sigma^{2}(x)=4x$ by $B2$ we have

$\int_{x=0}^{\varepsilon_{0}}m(x)\int_{z=0}^{x}s(z)m(z)dzdx\leq\frac{1}{2}\int_{x=0}^{\varepsilon_{0}}m(x)\int_{z=0}^{x}dzdx$

which by the bounds for $m$ given in equation m bounds B (that follow from
$B2$ and $B3$) is less than or equal to

$\frac{\varepsilon_{0}^{\alpha^{+}}}{4}\int_{x=0}^{\varepsilon_{0}}x^{-\alpha^{+}}dx$

which is finite, and so the condition in equation x^2 bound is satisfied.
Also, by the lower bound for $m(x)$ given in equation m bounds B we have

$M((0,\varepsilon_{0}))\geq\frac{\varepsilon_{0}^{\alpha^{-}}}{2}\int_{x=0}^{\varepsilon_{0}}x^{-1-\alpha^{-}}dx=\infty,$

and so the condition in equation 0 is regular is not satisfied and therefore
by Theorem LABEL:theorem_lévy_measure_condition we have that the Pitman-Yor
excursion measure satisfies equation levy measure unbounded variation as
required.

##  Continuity of local time

In our proof Theorem LABEL:theorem_diffusion we use the continuity of the
local time in space and time of the scaffolding process $\mathbf{X}$. To show
continuity of the local time, we first prove some technical lemmas and we use
them to show a necessary and sufficient condition for the continuity of local
time on Laplace exponents of SPLPs with infinite variation, from Lambert and
Simatos [lambert2015asymptotic, Lemma 2.2]. We note that necessary and
sufficient conditions for the space-time continuity of the local time of Lévy
processes more generally had been determined by Barlow [barlow1988] but the
conditions in [lambert2015asymptotic] are more accessible for us to verify in
our setting. To begin, we define the measure of the overshoot (or equivalently
of the undershoot) $\chi$ from the Lévy measure $\nu(\zeta\in\cdot)$ as

$\chi(\cdot)\coloneqq\nu_{\text{cld}}(\zeta(\hat{f}_{T^{+}_{0}})\in\cdot)=\nu_{\text{cld}}(\zeta(\check{f}_{T^{+}_{0}})\in\cdot),$

where the second equality is an immediate consequence of equation reversal
under nu clade. We also have from Lemma LABEL:lemma_excursion_overshoot-
undershoot_measure that

$\chi(dz)=\nu_{\text{cld}}(\zeta(\check{f}_{T^{+}_{0}})\in
dz)=\int_{y=0}^{\infty}\nu(\zeta\in dz+y)dy=\nu(\zeta\in(z,\infty))dz.$

Consider a diffusion that satisfies Assumption 1. Let $\nu$ be its Pitman-Yor
excursion measure and let $\chi$ be defined from $\nu$ as above. Then there
exist $\varepsilon_{2}>0$ and $k_{5}>0$ such that

$\chi((x,\infty))\geq
k_{5}x^{-\alpha^{-}}\text{forall}x\in(0,\varepsilon_{2}).$

First note that we have

$\chi((x,\infty))=\nu_{\text{cld}}(\zeta(\hat{f}_{T^{+}_{0}})>x)=k_{0}\int_{u=0}^{\infty}m(u)\mathbb{Q}^{0}_{u}(\zeta>x)du,$
(6)

where $k_{0}$ is the constant given in Lemma LABEL:lemma_excursion_overshoot-
undershoot_measure. We can bound the density of the speed measure $m$ below by
equation m bounds B. Let $u\in(0,1)$, and let $B$ be a standard Brownian
motion on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. Define the
stochastic processes $Y$, $Y^{+}$, $Y^{-}$ by the stochastic differential
equations,

$dY_{t}=u+\mu(Y_{t})dt+2\sqrt{|Y_{t}|}dB_{t};\hskip
7.11317ptdY^{+}_{t}=u-2\alpha^{+}dt+2\sqrt{|Y^{+}_{t}|}dB_{t};\hskip
7.11317ptdY^{-}_{t}=u-2\alpha^{-}dt+2\sqrt{|Y^{-}_{t}|}dB_{t},$ (7)

which are stopped when they hit $0$. We have that

$\mathbb{P}(Y\in\cdot)=\mathbb{Q}_{u}(\cdot);\hskip
7.11317pt\mathbb{P}(Y^{+}\in\cdot)={{}^{-2\alpha^{+}}}\mathbb{Q}_{u}(\cdot);\hskip
7.11317pt\mathbb{P}(Y^{-}\in\cdot)={{}^{-2\alpha^{-}}}\mathbb{Q}_{u}(\cdot),$

and furthermore that

$Y^{+}_{t}\leq Y_{t}\leq Y^{-}_{t}\text{for}t\in(0,T_{0}(Y^{+})\wedge
T_{\varepsilon_{0}}(Y^{-}))\hskip 7.11317pt\mathbb{P}\text{-a.s.},$

where the $\varepsilon_{0}$ is as in Assumption 1. Therefore, by using these
inequalities along with the marginal distributions of $Y$, $Y^{+}$, $Y^{-}$
under $\mathbb{P}$, we have that

$\split\mathbb{Q}^{0}_{u}(\zeta>x)=\mathbb{P}(T_{0}(Y)>x)&\geq\mathbb{P}\Big{(}T_{0}(Y^{+})>x,\sup_{0\leq
t\leq T_{0}(Y^{-})}\\{Y^{-}_{t}\\}<\varepsilon_{0}\Big{)}\\\
\geq{{}^{-2\alpha^{+}}}\mathbb{Q}^{0}_{u}(\zeta>x)-{{}^{-2\alpha^{-}}}\mathbb{Q}^{0}_{u}(A\geq\varepsilon_{0}).\\\
$

Göing-Jaeschke and Yor [going2003survey, Equation (13)] state $\zeta$ under
${{}^{-2\alpha}}\mathbb{Q}^{0}_{u}$ has distribution
$\text{InverseGamma}(1+\alpha,u/2)$, i.e.

${{}^{-2\alpha}}\mathbb{Q}^{0}_{u}(\zeta\in
dx)=\frac{1}{\Gamma(1+\alpha)}\Big{(}\frac{u}{2}\Big{)}^{1+\alpha}x^{-2-\alpha}\exp\Big{(}-\frac{u}{2x}\Big{)}.$
(8)

By the two-sided exit problem and equation scale function we have
${{}^{-2\alpha^{-}}}\mathbb{Q}^{0}_{u}(A\geq\varepsilon_{0})=(u/\varepsilon_{0})^{1+\alpha^{-}}$.
For $x<\varepsilon_{0}/4$, we now state an upper bound for $\chi((x,\infty))$
by using the upper bound for $\mathbb{Q}^{0}_{u}(\zeta>x)$ given above along
with the upper bound for $m$ given in equation m bounds B. We also upper bound
$\mathbb{Q}^{0}_{u}(\zeta>x)$ by
$\mathbb{Q}^{0}_{u}(\zeta\in(x,\varepsilon_{0}))$ and obtain

$\chi((x,\infty))\geq\int_{u=0}^{\varepsilon_{0}}\frac{k_{0}\varepsilon_{0}^{\alpha^{-}}}{2}u^{-1-\alpha^{-}}\Big{(}\int_{y=x}^{\varepsilon_{0}/2}\Big{(}\Big{(}\frac{1}{\Gamma(1+\alpha^{+})}\frac{u^{1+\alpha^{+}}}{y^{2+\alpha^{+}}}\exp\Big{(}-\frac{u}{2y}\Big{)}dy-\Big{(}\frac{u}{\varepsilon_{0}}\Big{)}^{1+\alpha^{-}}\Big{)}\wedge
0\Big{)}\Big{)}du$

which is greater than or equal to

$\int_{u=0}^{\varepsilon_{0}}\frac{k_{0}\varepsilon_{0}^{\alpha^{-}}}{2}u^{-1-\alpha^{-}}\int_{y=x}^{\varepsilon_{0}/2}\frac{1}{\Gamma(1+\alpha^{+})}\frac{u^{1+\alpha^{+}}}{y^{2+\alpha^{+}}}\exp\Big{(}-\frac{u}{2y}\Big{)}dydu-\int_{u=0}^{\varepsilon_{0}}\frac{k_{0}\varepsilon_{0}^{\alpha^{-}}}{2}u^{-1-\alpha^{-}}\int_{y=x}^{\varepsilon_{0}/2}\Big{(}\frac{u}{\varepsilon_{0}}\Big{)}^{1+\alpha^{-}}dydu.$

We consider the two double integrals separately. For the first double integral
we have swap the integrals to find and restrict $u\in(0,\varepsilon_{0})$ to
$u\in(0,2y)$ as $y\in(x,\varepsilon_{0}/2)$. We get a lower bound of

$\frac{k_{0}\varepsilon_{0}^{\alpha^{-}}}{2\Gamma(1+\alpha^{+})}\int_{y=x}^{\varepsilon_{0}/2}y^{-2-\alpha^{+}}\int_{u=0}^{2y}u^{\alpha^{+}-\alpha^{-}}\exp\Big{(}-\frac{u}{2y}\Big{)}dudy$

and we can bound $\exp(-u/2y)$ from below by $1/e$ to get a lower bound of

$\frac{k_{0}\varepsilon_{0}^{\alpha^{-}}}{2\Gamma(1+\alpha^{+})e}\int_{y=x}^{\varepsilon_{0}/2}y^{-2-\alpha^{+}}\int_{u=0}^{2y}u^{\alpha^{+}-\alpha^{-}}dudy=\frac{2^{\alpha^{+}-\alpha^{-}}k_{0}\varepsilon_{0}^{\alpha^{-}}}{\Gamma(1+\alpha^{+})e(1-\alpha^{+}+\alpha^{-})}\int_{y=x}^{\varepsilon_{0}/2}y^{-1-\alpha^{-}}dy$

which is equal to

$\frac{2^{\alpha^{+}-\alpha^{-}}k_{0}\varepsilon_{0}^{\alpha^{-}}}{\Gamma(1+\alpha^{+})e(1-\alpha^{+}+\alpha^{-})\alpha^{-}}\Big{(}x^{-\alpha^{-}}-\Big{(}\frac{\varepsilon_{0}}{2}\Big{)}^{-\alpha^{-}}\Big{)}.$

For the second double integral we see the upper bound given by

$\int_{u=0}^{\varepsilon_{0}}\frac{k_{0}\varepsilon_{0}^{\alpha^{-}}}{2}u^{-1-\alpha^{-}}\int_{y=0}^{\varepsilon_{0}/2}\Big{(}\frac{u}{\varepsilon_{0}}\Big{)}^{1+\alpha^{-}}dydu=\frac{k_{0}}{2\varepsilon_{0}}\int_{u=0}^{\varepsilon_{0}}\int_{y=0}^{\varepsilon_{0}/2}dydu=\frac{k_{0}\varepsilon_{0}}{4}.$

Therefore the claim in the lemma is satisfied for some $k_{5}$ and
$\varepsilon_{2}$. In the same setting as in Lemma 1, we have that, for some
positive constant $k_{6}$,

$\chi((x,\infty))\leq k_{6}(x^{-\alpha^{+}}+1)\text{forall}x\in(0,\infty),$

We use the formulation of $\chi((x,\infty))$ as given in equation chi
definition, as well as the coupled processes $Y$, $Y^{+}$ and $Y^{-}$ as in
Lemma 1. We note that

$\split\mathbb{Q}^{0}_{u}(\zeta>x)=\mathbb{P}(T_{0}(Y)>x)&=\mathbb{P}\Big{(}T_{0}(Y)>x,\sup_{0\leq
t\leq
T_{0}(Y^{-})}\\{Y^{-}_{t}\\}\geq\varepsilon_{0}\Big{)}+\mathbb{P}\Big{(}T_{0}(Y)>x,\sup_{0\leq
t\leq T_{0}(Y^{-})}\\{Y^{-}_{t}\\}<\varepsilon_{0}\Big{)}\\\
\leq\mathbb{P}(A(Y^{-})\geq\varepsilon_{0})+\mathbb{P}(\zeta(Y^{-})>x)={{}^{-2\alpha^{-}}}\mathbb{Q}_{u}^{0}(A\geq\varepsilon_{0})+{{}^{-2\alpha^{-}}}\mathbb{Q}^{0}_{u}(\zeta>x),\\\
$

where the $\varepsilon_{0}$ is as in Assumption 1. Therefore we have that

$\chi((x,\infty))\leq\frac{k_{0}\varepsilon_{0}^{\alpha^{+}}}{2}\int_{u=0}^{\varepsilon_{0}}u^{-1-\alpha^{+}}\Big{(}{{}^{-2\alpha^{-}}}\mathbb{Q}_{u}^{0}(A\geq\varepsilon_{0})+{{}^{-2\alpha^{-}}}\mathbb{Q}^{0}_{u}(\zeta>x)\Big{)}du+k_{0}\int_{u=\varepsilon_{0}}^{c}\mathbb{Q}^{0}_{u}(\zeta>x)m(u)du$

and we bound this above by

$\frac{k_{0}\varepsilon_{0}^{\alpha^{+}}}{2}\int_{u=0}^{\varepsilon_{0}}u^{-1-\alpha^{+}}\Big{(}u^{1+\alpha^{-}}+\int_{y=x}^{\infty}\frac{1}{\Gamma(1+\alpha^{-})}\frac{u^{1+\alpha^{-}}}{y^{2+\alpha^{-}}}\exp\Big{(}-\frac{u}{2y}\Big{)}dy\Big{)}du+\frac{k_{0}}{2}M(E\backslash[0,\varepsilon_{0}]).$

We deal with the two parts of the integral separately. We have

$\frac{k_{0}\varepsilon_{0}^{\alpha^{+}}}{2}\int_{u=0}^{\varepsilon_{0}}u^{-1-\alpha^{+}}\int_{y=x}^{\infty}\frac{1}{\Gamma(1+\alpha^{-})}\frac{u^{1+\alpha^{-}}}{y^{2+\alpha^{-}}}\exp\Big{(}-\frac{u}{2y}\Big{)}dydu\leq\frac{k_{0}\varepsilon_{0}^{\alpha^{+}}}{2\Gamma(1+\alpha^{-})}\int_{y=x}^{\infty}y^{-2-\alpha^{-}}\int_{u=0}^{\varepsilon_{0}}u^{\alpha^{-}-\alpha^{+}}dudy$

which equals

$\frac{k_{0}\varepsilon_{0}^{1+\alpha^{-}}}{2\Gamma(1+\alpha^{-})(1+\alpha^{-}-\alpha^{+})}\int_{y=x}^{\infty}y^{-1-\alpha^{+}}dy=\frac{k_{0}\varepsilon_{0}^{1+\alpha^{-}}}{2\Gamma(1+\alpha^{-})(1+\alpha^{-}-\alpha^{+})\alpha^{+}}x^{-\alpha^{+}}.$

For the second part of the integral we have that

$\frac{k_{0}\varepsilon_{0}^{\alpha^{+}}}{2}\int_{u=0}^{\varepsilon_{0}}u^{\alpha^{-}-\alpha^{+}}du=\frac{k_{0}\varepsilon_{0}^{1+\alpha^{-}}}{2(1+\alpha^{-}-\alpha^{+})},$

and so

$\chi((x,\infty))\leq\frac{k_{0}\varepsilon_{0}^{1+\alpha^{-}}}{2\Gamma(1+\alpha^{-})(1+\alpha^{-}-\alpha^{+})\alpha^{+}}x^{-\alpha^{+}}+\frac{k_{0}\varepsilon_{0}^{1+\alpha^{-}}}{2(1+\alpha^{-}-\alpha^{+})}+\frac{k_{0}}{2}M(E\backslash[0,\varepsilon_{0}])$

and the result follows. Consider a diffusion that satisfies Assumption 1, and
let $\psi$ be the Laplace exponent for the scaffolding process $\mathbf{X}$,
see equation laplace exponent. Then we have for $\lambda\in(0,\infty)$ that

$\psi(\lambda)=\int_{x=0}^{\infty}\lambda^{2}e^{-\lambda
x}\chi((x,\infty))dx.$ (9)

This follows by integration-by-parts applied twice to
$\psi(\lambda)=\int_{x=0}^{\infty}(e^{-\lambda x}-1+\lambda x)\Lambda(dx)$ and
we require the upper bound for $\chi((x,\infty))$ in Lemma 1 to do this. We
now use a necessary and sufficient condition from Lambert
[lambert2015asymptotic] to determine that the local time is a.s. jointly
continuous in time and space. Consider a diffusion under Assumption
LABEL:assumption_b and let $(\mathbf{N},\mathbf{X})$ be a scaffolding-and-
spindles construction. Then the local time $(\ell^{u}_{t}:t,u\in\mathbb{R})$
is jointly continuous a.s. By Lambert [lambert2015asymptotic, Lemma 2.2], as
$\mathbf{X}$ is a SPLP that satisfies equation levy measure unbounded
variation, a necessary and sufficient condition for the local time to be
jointly continuous a.s. is

$\int^{\infty-}\frac{d\lambda}{\psi(\lambda)\sqrt{\log(\lambda)}}<\infty.$

Now, with the lower bound for $\chi((x,\infty))$ from Lemma 1 we have that

$\psi(\lambda)\geq k_{5}\int_{x=0}^{\delta}\lambda^{2}e^{-\lambda
x}x^{-\alpha^{-}}dx.$

We let $\lambda>1/\delta$ and consider the Taylor expansion of $e^{-\lambda
x}$ to obtain that

$\psi(\lambda)\geq
k_{5}\sum_{n=0}^{\infty}\frac{(-1)^{n}\lambda^{n+2}}{n!}\int_{x=0}^{1/\lambda}x^{n-\alpha^{-}}dx=k_{5}\Big{(}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!(n+1-\alpha^{-})}\Big{)}\lambda^{1+\alpha^{-}}$

and therefore

$\int_{\lambda=1/\delta}^{\infty}\frac{d\lambda}{\psi(\lambda)\sqrt{\log(\lambda)}}\leq\Big{(}k_{5}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!(n+1-\alpha^{-})}\Big{)}^{-1}\int_{\lambda=1/\delta}^{\infty}\frac{d\lambda}{\lambda^{1+\alpha^{-}}\sqrt{\log(\lambda)}}<\infty.$

##  Some further calculations

In this section we give upper bounds of certain probabilities and expectations
for diffusions with a state space $E=[0,\infty)$ and we use these results in
our proof of Theorem LABEL:theorem_diffusion. In Section 1 we prove results
for a general such diffusions conditioned to not hit $0$ and we reference the
literature on quasi-stationary distributions, Méléard and Villemonais
[meleard2012quasi]. In Section 1 we consider squared Bessel cases of negative
dimension.

###  Expectation for a diffusion conditioned not to hit level $0$

In this section we calculate conditional moments for the deviation of a
diffusion with state space $E\subseteq[0,\infty)$. As before, for $s>0$,
$\mathbb{Q}_{s}$ denotes the law of the diffusion started from $s$ and
$\mathbb{Q}^{0}_{s}$ denotes the law of the diffusion started from $s$ and
stopped at $0$. Consider a diffusion that satisfies $A1$ of Assumption
LABEL:assumption_a. Then for $r$, $s>0$, $0<a<c$ and $0<t\leq r$ we have

$\mathbb{Q}^{0}_{s}(Y_{t}\in E\backslash[0,a))\leq\mathbb{Q}^{0}_{s}(Y_{t}\in
E\backslash[0,a)|\zeta>r).$

Let $t\in(0,r)$. We first note that

$b\coloneqq\mathbb{Q}^{0}_{s}(\zeta>r|Y_{t}\in[0,a))\leq\mathbb{Q}^{0}_{a}(\zeta>r-t)\leq\mathbb{Q}^{0}_{s}(\zeta>r|Y_{t}\in
E\backslash[0,a))=:c,$

by the Markov property and almost-sure continuity of the sample paths. Let
$p=\mathbb{Q}^{0}_{s}(Y_{t}\in E\backslash[0,a))$. We then have that

$\mathbb{Q}^{0}_{s}(Y_{t}\in
E\backslash[0,a))\mathbb{Q}^{0}_{s}(\zeta>r)=p(cp+b(1-p))\leq
cp=\mathbb{Q}^{0}_{s}(Y_{t}\in[a,\infty),\zeta>r)$

which implies that

$\mathbb{Q}^{0}_{s}(Y_{t}\in
E\backslash[0,a))\leq\mathbb{Q}^{0}_{s}(Y_{t}\in[a,\infty)|\zeta>r).$

Taking $t\uparrow r$ we have by a.s. continuity of the sample paths that

$\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,a))\leq\mathbb{Q}^{0}_{s}(Y_{r}\in[a,\infty)|\zeta>r).$

In the same setting as Lemma 1, for $r$, $s$, $u>0$ and $0<a<c$ we have

$\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,a)|\zeta>r)\leq\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,a)|\zeta>r+u).$

In a very similar fashion to the previous lemma we first note that

$b_{u}\coloneqq\mathbb{Q}^{0}_{s}(\zeta>r+u|Y_{r}\in(0,a))\leq\mathbb{Q}^{0}_{a}(\zeta>u)\leq\mathbb{Q}^{0}_{s}(\zeta>r+u|Y_{r}\in
E\backslash[0,a))=:c_{u},$

and let
$p\coloneqq\mathbb{Q}^{0}_{s}(Y_{r}\in(a,\infty))\leq\mathbb{Q}^{0}_{s}(\zeta>y)=:q$.
Then we have that

$\mathbb{Q}^{0}_{s}(\zeta>r+u,Y_{r}\in E\backslash[0,a))=c_{u}p.$

Furthermore note that

$\mathbb{Q}^{0}_{s}(\zeta>r+u|\zeta>r)=c_{u}\frac{p}{q}+b_{u}\frac{q-p}{q}\leq
c_{u},\hskip 7.11317pt\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,a)|\zeta>r)=\frac{p}{q},$

and therefore

$\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,a),\zeta>r+u|\zeta>r)=\frac{pc}{q}\geq\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,a)|\zeta>r)\mathbb{Q}^{0}_{s}(\zeta>r+u|\zeta>r),$

and the claim follows. This final lemma, Lemma 1, uses the following
convergence result for the $\uparrow$-diffusion from [meleard2012quasi,
Theorem 27]. In the same setting as the previous lemmas, for $r$, $s$, $a>0$
we have

$\lim_{u\rightarrow\infty}\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,a)|\zeta>r+u)=\mathbb{Q}^{\uparrow}_{s}(Y_{r}\in(a,\infty)E\backslash[0,a)).$

Let $F:[0,\infty)\rightarrow\mathbb{R}$ be a continuous increasing function
with derivative $F^{\prime}$ Lebesgue-almost everywhere. In the same setting
as the previous lemmas, for $r$, $s>0$ we have

$\mathbb{E}_{\mathbb{Q}^{0}_{s}}(F(Y_{r}))\leq\mathbb{E}_{\mathbb{Q}^{0}_{s}}(F(Y_{r})|\zeta>r)\leq\mathbb{E}_{\mathbb{Q}^{\uparrow}_{s}}(F(Y_{r})).$

We first note that

$\mathbb{E}_{\mathbb{Q}^{0}_{s}}(F(Y_{r}))=\int_{z=0}^{\infty}\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,z))F^{\prime}(z)dz,$

and similarly for $\mathbb{E}_{\mathbb{Q}^{0}_{s}}(F(Y_{r})|\zeta>r)$ and
$\mathbb{E}_{\mathbb{Q}^{\uparrow}_{s}}(F(Y_{r}))$. By Lemmas 1, 1 and 1 we
have that for any $a\in(0,\infty)$

$\mathbb{Q}^{0}_{s}(Y_{r}\in E\backslash[0,a))\leq\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,a)|\zeta>r)\leq\lim_{u\rightarrow\infty}\mathbb{Q}^{0}_{s}(Y_{r}\in
E\backslash[0,a)|\zeta>r+u)=\mathbb{Q}^{\uparrow}_{s}(Y_{r}\in
E\backslash[0,a)),$

and the result follows.

###  Certain calculations concerning squared Bessel processes with negative
dimension

In this section we work with squared Bessel processes with dimension
$-2\alpha$, where $\alpha\in(0,1)$. We make a slight abuse of notation and
write ${{}^{-2\alpha}}\mathbb{Q}^{0}_{s}$ in place of
$\mathbb{E}_{{{}^{-2\alpha}}\mathbb{Q}^{0}_{s}}$ for readability. Let $r$,
$q>0$, $0<\alpha,s<1$ and $n\in\mathbb{N}$. Then we have

${{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn}|\zeta>r)\leq{{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn})+{{}^{4+2\alpha}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn}).$

We note that the functions

$|z-s|^{qn}\mathbbm{1}_{z\geq s};\hskip
14.22636pt-|z-s|^{qn}\mathbbm{1}_{z\leq s},$

are increasing and Lebesgue almost everywhere differentiable. Therefore, by
Lemma 1 we have that

$\split{{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn}|\zeta>r)&\leq{{}^{-2\alpha}}\mathbb{Q}^{\uparrow}_{s}(|Y_{r}-s|^{qn}\mathbbm{1}_{z\geq
s}|\zeta>r)-{{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(-|Y_{r}-s|^{qn}\mathbbm{1}_{z\leq
s}|\zeta>r)\\\
\leq{{}^{4+2\alpha}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn})+{{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn}).\\\
$

We establish upper and lower bounds for the probability that the lifetime
$\zeta$ is greater than $r$ under ${{}^{-2\alpha}}\mathbb{Q}^{0}_{s}$. For
$r$, $s>0$, $0<\alpha<1$, we have

${{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(\zeta\leq
r)\leq\frac{\Gamma(1+\alpha+v)2^{v}}{\Gamma(1+\alpha)}\frac{r^{v}}{s^{v}};\text{for}v>0\hskip
14.22636pt{{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(\zeta>r)\leq\frac{\Gamma(1+\alpha-v)}{\Gamma(1+\alpha)2^{v}}\frac{s^{v}}{r^{v}}\text{for}0<v<1+\alpha.$
(10)

For the first inequality let $v>0$ and we apply the Markov inequality and
equation zeta distribution to see that

${{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(\zeta\leq
r)={{}^{-2\alpha}}\mathbb{Q}^{0}_{s}\left(\frac{1}{\zeta^{v}}\geq\frac{1}{r^{v}}\right)\leq{{}^{-2\alpha}}\mathbb{Q}^{0}_{s}\left(\frac{1}{\zeta^{v}}\right)r^{v}=r^{v}\int_{x=0}^{\infty}\frac{s^{1+\alpha}}{2^{1+\alpha}\Gamma(1+\alpha)}\frac{1}{x^{2+\alpha+v}}\exp\left(-\frac{s}{2x}\right)dx$

and we integrate out the probability density function of an
$\text{InverseGamma}(1+\alpha+v,s/2)$ to obtain

$\frac{\Gamma(1+\alpha+v)2^{v}}{\Gamma(1+\alpha)}\frac{r^{v}}{s^{v}}\int_{x=0}^{\infty}\frac{s^{1+\alpha+v}}{2^{1+\alpha+v}\Gamma(1+\alpha+v)}\frac{1}{x^{2+\alpha+v}}\exp\left(-\frac{s}{2x}\right)dx=\frac{\Gamma(1+\alpha+v)2^{v}}{\Gamma(1+\alpha)}\frac{r^{v}}{s^{v}}.$

For the second inequality let $0<v<1+\alpha$, we use the same ideas:

${{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(\zeta>r)\leq\frac{1}{r^{v}}{{}^{-2\alpha}}\mathbb{Q}^{0}_{s}(\zeta^{v})=\frac{1}{r^{v}}\int_{x=0}^{\infty}\frac{s^{1+\alpha}}{2^{1+\alpha}\Gamma(1+\alpha)}\frac{1}{x^{2+\alpha-v}}\exp\left(-\frac{s}{2x}\right)dx=\frac{\Gamma(1+\alpha-v)}{\Gamma(1+\alpha)2^{v}}\frac{s^{v}}{r^{v}},$

where we have integrated out the probability distribution function of an
$\text{InverseGamma}(1+\alpha-v,s/2)$. We also establish a further lemma with
some useful probability bounds for squared Bessel processes. Let
$\gamma\in(-\infty,0)\cup[2,\infty)$, and $d>0$. Then there exists a constant
$C=C(\gamma,d)$ such that for any $r$, $s>0$, we have

${{}^{\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{d})\leq Cr^{d/2}(r^{d/2}\vee
s^{d/2}).$

We use [forman2018uniform, Equation (15)] to note that for $d\geq 2$ and
$\gamma\in\mathbb{R}$ that we have

$\split\left({{}^{\gamma}}\mathbb{Q}_{s}(|Y_{r}-s|^{d})\right)^{(1/d)}&\leq|\gamma|r+2\sqrt{(d-1)(|s|r+r^{2}(|\gamma|+(d-2)))}\\\
\leq\Big{(}|\gamma|+2\sqrt{(d-1)(1+(|\gamma|+(d-2)))}\Big{)}r^{1/2}\Big{(}r^{1/2}\vee
s^{1/2}\Big{)}$ (11)

Squared Bessel processes with negative dimension never return to the positive
half-line after the first hitting time of $0$ a.s. Also, squared Bessel
processes with dimension greater than or equal to $2$ that start positive
never hit $0$ a.s. Therefore, for $\gamma\in(-\infty,0)\cup[2,\infty)$ we have
that

$\left({{}^{\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{d})\right)^{(1/d)}\leq\Big{(}|\gamma|+2\sqrt{(d-1)(1+(|\gamma|+(d-2)))}\Big{)}r^{1/2}\Big{(}r^{1/2}\vee
s^{1/2}\Big{)}\text{for}d\geq 2,\gamma\in(-\infty,0)\cup[2,\infty)$

by equation squared bessel bound. Now let $d<2$. Then $4/d>2$ and so we can
apply Jensen’s inequality to obtain that

${{}^{\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{d})\leq({{}^{\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{4}))^{d/4}\leq
C(\gamma,4)^{d/4}r^{d/2}\Big{(}r^{d/2}\vee
s^{d/2}\Big{)}\text{for}0<d<2,\gamma\in(-\infty,0)\cup[2,\infty).$

Let $n\in\mathbb{N}$, $0<\alpha,\gamma<1$, $0<\kappa<1-\alpha$, and
$\alpha+\kappa<q<1$. Then there exists a constant
$C=C(n,q,\alpha,\gamma,\kappa)>0$ such that for $0<r\leq 1$ we have

$\int_{s=0}^{1}{{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn})s^{-1-\alpha}ds\leq
Cr^{n((q-\alpha-\kappa)\wedge q/2)}.$

We can apply Lemma 1 as

${{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}^{q}-s^{q}|^{n})\leq{{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn})\text{for}0<q<1,n\geq
1.$

We split the integral. We first consider the expectation on the event
$\\{\zeta\leq r\\}$ and use the upper bound for
${{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(\zeta\leq r)$ in equation lifetime upper
bound with $v=qn-\alpha-\kappa$:

$\split\int_{s=0}^{1}{{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn}:\zeta\leq
r)s^{-1-\alpha}ds&\leq\frac{\Gamma(1+\gamma+qn-\alpha-\kappa)2^{qn-\alpha-\kappa}}{\Gamma(1+\alpha)}\int_{s=0}^{1}s^{-1-\alpha}s^{qn}\frac{r^{qn-\alpha-\kappa}}{s^{qn-\alpha-\kappa}}ds\\\
=\frac{\Gamma(1+\gamma+qn-\alpha-\kappa)2^{qn-\alpha-\kappa}}{\Gamma(1+\alpha)\kappa}r^{qn-\alpha-\kappa}.\\\
$

We now consider the integral on the event $\\{\zeta>r\\}$ on the range
$0<s<r$. By Lemmas 1 and 1 we have that

${{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn}|\zeta>r)\leq(C(-2\alpha,qn)+C(4+2\alpha,qn))r^{qn/2}(s^{qn/2}\vee
r^{qn/2}),$

where $C(-2\alpha,qn)$ and $C(4+2\alpha,qn)$ are from Lemma 1. Let
$K=(C(-2\alpha,qn)+C(4+2\alpha,qn))$. We use equation lifetime upper bound
with $v=\alpha+\kappa$ to bound above
${{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(\zeta>r)$:

$\split\int_{s=0}^{r\wedge
1}{{}^{-2\gamma}}\mathbb{Q}^{0}_{s}[|Y_{r}-s|^{qn}:\zeta>r]s^{-1-\alpha}ds&\leq\frac{K\Gamma(1+\gamma-\alpha-\kappa)}{\Gamma(1+\gamma)2^{\alpha+\kappa}}\int_{s=0}^{r\wedge
1}\frac{s^{\alpha+\kappa}}{r^{\alpha+\kappa}}s^{-1-\alpha}r^{qn}ds\\\
\leq\frac{K\Gamma(1+\gamma-\alpha-\kappa)}{\Gamma(1+\gamma)2^{\alpha+\kappa}}r^{qn-\alpha-\kappa}\int_{s=0}^{1}s^{-1+\kappa}ds\\\
=\frac{K\Gamma(1+\gamma-\alpha-\kappa)}{\Gamma(1+\gamma)2^{\alpha+\kappa}\kappa}r^{qn-\alpha}.\\\
$

We note that

$r^{qn-\alpha-\kappa}\leq r^{n((q-\alpha-\kappa)\wedge q/2)}\text{for}0<r<1.$

Finally, we consider the integral on the event $\\{\zeta>r\\}$ over the range
$r\wedge 1\leq s\leq 1$. We split this calculation into two parts to deal with
the cases $q/2\geq q-\alpha-\kappa$ and $q/2<q-\alpha-\kappa$ separately. In
the case where $q/2\geq q-\alpha-\kappa$ we use the bound in equation lifetime
upper bound for ${{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(\zeta>r)$ with
$0<v=\alpha+\kappa-q/2<1+\alpha$:

$\split\int_{s=r\wedge
1}^{1}{{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn}:\zeta>r)s^{-1-\alpha}ds&\leq\frac{K\Gamma(1+\gamma+q/2-\alpha-\kappa)}{\Gamma(1+\alpha)2^{\alpha+\kappa-q/2}}\int_{s=r\wedge
1}^{1}\frac{s^{\alpha+\kappa-q/2}}{r^{\alpha+\kappa-q/2}}s^{-1-\alpha}r^{qn/2}s^{qn/2}ds\\\
\leq\frac{K\Gamma(1+\gamma+q/2-\alpha-\kappa)}{\Gamma(1+\alpha)2^{\alpha+\kappa-q/2}}r^{qn/2+q/2-\alpha-\kappa}\int_{s=0}^{1}s^{-1+\kappa}ds\\\
\leq\frac{K\Gamma(1+\gamma+q/2-\alpha-\kappa)}{\Gamma(1+\alpha)2^{\alpha+\kappa-q/2}\kappa}r^{n(q-\alpha-\kappa)}\text{for}q/2\geq
q-\alpha-\kappa.\\\ $

For the case where $q/2<q-\alpha-\kappa$ we do not use equation lifetime upper
bound to bound ${{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(\zeta>r)$ (we instead use
the trivial bound of $1$):

$\split\int_{s=r\wedge
1}^{1}{{}^{-2\gamma}}\mathbb{Q}^{0}_{s}(|Y_{r}-s|^{qn}:\zeta>r)s^{-1-\alpha}ds&\leq
K\int_{s=r\wedge 1}^{1}s^{-1-\alpha}r^{qn/2}s^{qn/2}ds\\\ =K\int_{s=r\wedge
1}^{1}s^{-1-\alpha+qn/2}dsr^{qn/2}\\\ \leq
Kr^{qn/2}\int_{s=0}^{1}s^{-1+\kappa}ds=\frac{K}{\kappa}r^{qn/2}\text{for}q/2<q-\alpha-\kappa.\\\
$

##  Continuity of the skewer process

In this section we prove the continuity results for Theorems
LABEL:theorem_diffusion and LABEL:theorem_interval_partitions_regular. We
begin with some definitions. Consider a diffusion under Assumption 1 with
Pitman-Yor excursion measure $\nu$, and a transformation $g:E\rightarrow g(E)$
(where $E=[0,c)$ or $E=[0,c]$ for some $c\leq\infty$) that satisfies the
conditions stated in Assumption LABEL:assumption_b. Without loss of generality
we can take the $\varepsilon_{0}$ in Assumption LABEL:assumption_b to be less
than $1$ and we do this here. As before, let $\mathbf{N}$ be a stopped PRM
with intensity $\text{Leb}\otimes\nu$, where the stopping is at a positive
random variable $T$. Finally, for $M$, $\varepsilon_{0}>0$ we define
$\mathbf{N}_{M,\varepsilon_{0}}$ by

$\mathbf{N}_{M,\varepsilon_{0}}\coloneqq\mathbbm{1}_{\sup_{u\in\mathbb{R}}\ell^{u}(T)<M}\sum_{\substack{(t,f)\text{atomof}\mathbf{N}:\\\
A(f)<\varepsilon_{0}}}\delta_{(t,f)}.$ (12)

Let $x$, $y\in\mathbb{R}$, $z\in\\{x,y\\}$, and let $k\in\mathbb{N}$. Then we
have that

$\split&\mathbf{E}\Bigg{(}\Bigg{(}\sum_{\substack{(t,f)\text{atomof}\mathbf{N}_{M,\varepsilon_{0}}:\\\
\mathbf{X}_{t-}<z<\mathbf{X}_{t},\ell^{z}(t)>0}}\Big{|}g\big{(}f(y-\mathbf{X}_{t-})\big{)}-g\big{(}f(x-\mathbf{X}_{t-})\big{)}\Big{|}\Bigg{)}^{k}\Bigg{)}\\\
\leq C^{k}_{q}\sum_{\substack{\left(n_{1},\dots,n_{a}\right)\\\
\text{composition}\\\
\text{of}k}}(k_{0}M)^{a}\prod_{i=1}^{a}\int_{s=0}^{\varepsilon_{0}}m(s)\Big{(}\hskip
0.28436pt{{}^{-2\alpha^{+}}}\mathbb{Q}^{0}_{s}(|Y_{|x-y|}-s|^{qn_{i}})+{{}^{-2\alpha^{-}}}\mathbb{Q}^{0}_{s}(|Y_{|x-y|}-s|^{qn_{i}})\Big{)}ds,$
(13)

where $C_{q}$ is the Hölder constant for $g$ as in Assumption
LABEL:assumption_b and $k_{0}$ is as in Proposition
LABEL:proposition_crossing_width_density. The bound is trivial for $x=y$ so
assume $x\neq y$. First recall from Assumption LABEL:assumption_b that there
exists $C_{q}>0$ such that $|g(a)-g(b)|\leq C_{q}|a-b|^{q}$ for $0<a$,
$b<\varepsilon_{0}$. We can bound the term on the left-hand-side of equation
deviations bound by

$C_{q}^{k}\mathbf{E}\Bigg{(}\Bigg{(}\sum_{\substack{(t,f)\text{atomof}\mathbf{N}:A(f)<\varepsilon_{0}\\\
0<\ell^{z}(t)<M,\mathbf{X}_{t-}<z<\mathbf{X}_{t}}}\Big{|}f(y-\mathbf{X}_{t-})-f(x-\mathbf{X}_{t-})\Big{|}^{q}\Bigg{)}^{k}\Bigg{)}.$
(14)

As $T^{z}$ is a stopping time for $\mathbf{X}$, we have by Lemma
LABEL:lemma_joint_probability_undershoot_overshoot that the process
$\xi(\mathbf{N}|^{\leftarrow}_{[T^{z},\infty)})$ can be split into excursions
about level $0$, and all these excursions a.s. include a single jump across
level $0$. We have a.s. a one-to-one correspondence between crossing across a
level and excursions about the level. Therefore the sum in equation bound by
tau M less than or equal to a sum indexed by the Poisson point process of
crossing widths at level $z$ and is summing functionals of the clades or anti-
clades associated with each of these crossing widths. Specifically, let
$\mathbf{N}^{\prime}$ be the unstopped Poisson random measure, then

$\mathbf{J}\coloneqq\sum_{\substack{t:0<\ell^{z}(t)<M,\\\
A(f)<\varepsilon_{0}}}\delta_{(\ell^{z}(t),\mathbf{N}^{\prime}|^{\leftarrow}_{(\ell^{z}(t)-,\ell^{z}(t))})}$

is distributed as a Poisson random measure with intensity

$\text{Leb}|_{[0,M]}\otimes
k_{0}\int_{s=0}^{\varepsilon_{0}}m(s)\nu_{\text{cld}}(\mathbbm{1}_{A(f_{T^{+}_{0}})<\varepsilon_{0}}N\in\cdot|f_{T^{+}_{0}}(-\xi_{N}(T^{+}_{0}-))=s)ds,$

where $k_{0}$ is as in Proposition LABEL:proposition_crossing_width_density
and $\varepsilon_{0}$ is as in Assumption LABEL:assumption_b. Let
$w\in\\{x,y\\}\backslash\\{z\\}$. Then, we define

$F(N)\coloneqq|f_{T^{+}_{0}}(w-z-\xi_{N}(T^{+}_{0}-))-f_{T^{+}_{0}}(-\xi_{N}(T^{+}_{0}-))|^{q}\text{for}N\in\mathcal{N}^{\text{sp}}_{\pm\text{cld}}$

and therefore the expectation in equation bound by tau M is bounded above by

$\mathbf{E}\Bigg{(}\Bigg{(}\sum_{\substack{(s,N)\text{atom}\\\
\text{of}\mathbf{J}}}F(N)\Bigg{)}^{k}\Bigg{)}=\sum_{\substack{\left(n_{1},\dots,n_{a}\right)\\\
\text{composition}\\\
\text{of}k}}\mathbf{E}\bigg{(}\sum_{\substack{(s_{1},N_{1}),\dots,(s_{a},N_{a})\\\
\text{distinctatomsof}\mathbf{J}}}\prod_{i=1}^{a}F(N_{i})^{n_{i}}\bigg{)}.$

We apply Campbell’s formula for Poisson random measures, see Kingman
[kingman1992poisson, Equation (3.14)], to obtain, for a composition
$(n_{1},\dots,n_{a})$ of $k$, that the expectation in the right-hand-side of
the above equation equals

$(k_{0}M)^{a}\prod_{i=1}^{a}\int_{s=0}^{\varepsilon_{0}}m(s)\nu_{\text{cld}}\Big{(}|f_{T^{+}_{0}}(w-z-\xi_{N}(T^{+}_{0}-))-f_{T^{+}_{0}}(-\xi_{N}(T^{+}_{0}-))|^{qn_{i}}\mathbbm{1}_{A(f_{T^{+}_{0}})<\varepsilon_{0}}\Big{|}f_{T^{+}_{0}}(-\xi_{N}(T^{+}_{0}-))=s\Big{)}ds.$

Now, let $\Tilde{\mathbf{Y}}$ be a diffusion with infinitesimal parameters

$\sigma^{2}_{\Tilde{\mathbf{Y}}}(x)=4x;\hskip
14.22636pt\mu_{\Tilde{\mathbf{Y}}}(x)=\left\\{\matrix{}{0.25cm}\mu(x)\text{for}x\in(0,\varepsilon_{0}];\\\
\mu(\varepsilon_{0})\text{for}x>\varepsilon_{0},\\\ \right.$ (15)

and let $\Tilde{\nu}_{\text{cld}}$ be the bi-clade excursion measure for
$\Tilde{\mathbf{Y}}$. Then we have for any measurable set
$B\subseteq\mathcal{C}$ of continuous paths we have for
$C\coloneqq\\{N\in\mathcal{N}^{\text{sp}}_{\pm\text{cld}}:f_{T^{+}_{0}}\in
B\\}$ that

$\nu_{\text{cld}}\Big{(}\\{A(f_{T^{+}_{0}})<\varepsilon_{0}\\}\cap
C\Big{|}f_{T^{+}_{0}}(-\xi_{N}(T^{+}_{0}-))=s\Big{)}=\Tilde{\nu}_{\text{cld}}\Big{(}\\{A(f_{T^{+}_{0}})<\varepsilon_{0}\\}\cap
C\Big{|}f_{T^{+}_{0}}(-\xi_{N}(T^{+}_{0}-)))=s\Big{)}$

because the two excursion measures differ only on excursions with amplitude
greater than $\varepsilon_{0}$. Indeed, you can construct a Poisson random
measure $\mathbf{N}$ with bi-clade measure $\nu_{\text{cld}}$ by first
constructing a Poisson random measure $\Tilde{\mathbf{N}}$ with bi-clade
measure $\Tilde{\nu}_{\text{cld}}$, removing the finitely many bi-clades with
spindles with amplitude greater than $\varepsilon_{0}$ (up to a finite time
$T$) and adding finitely many bi-clades with spindles with amplitude greater
than $\varepsilon_{0}$ since such bi-clades have finite intensity under the
bi-clade measures. In Proposition LABEL:proposition_disintegration we
established that the bi-clade measure about a fixed level disintegrates with
respect to the crossing width. Let $n\in\mathbb{N}$. The terms in the sum in
equation bound by tau M are the functionals of the clade (if
$z=\min\\{x,y\\}$) or the anti-clade (if $z>\min\\{x,y\\}$) of excursions
about level $z$. Therefore we have an upper bound given by

$\Tilde{\nu}_{\text{cld}}\Big{(}|f_{T^{+}_{0}}(w-z-\xi_{N}(T^{+}_{0}-))-f_{T^{+}_{0}}(-\xi_{N}(T^{+}_{0}-))|^{qn}\Big{|}f_{T^{+}_{0}}(-\xi_{N}(T^{+}_{0}-)))=s\Big{)}=\Tilde{\mathbb{Q}}^{0}_{s}(|Y_{|x-y|}-s|^{qn}),$

and as $-2\alpha^{+}\leq\mu_{\Tilde{\mathbf{Y}}}(x)\leq-2\alpha^{-}$ we have,
through a coupling argument, that

$\Tilde{\mathbb{Q}}^{0}_{s}(|Y_{|x-y|}-s|^{qn})\leq{{}^{-2\alpha^{+}}}\mathbb{Q}^{0}_{s}(|Y_{|x-y|}-s|^{qn})+{{}^{-2\alpha^{-}}}\mathbb{Q}^{0}_{s}(|Y_{|x-y|}-s|^{qn}),$
(16)

which gives the bound stated in this lemma.

[width=]figure3newfinal.pdf

Figure : Left: The skewer process at level $x$ constructed from the PRM
$\mathbf{N}$ and scaffolding $\mathbf{X}$. Right: The skewer process at level
$x$ constructed from the reduced PRM $\mathbf{N}_{M,\varepsilon_{0}}$ from
equation n with m and epsilon and scaffolding $\mathbf{X}$.

The next lemma deals with the first spindle to cross level $z>0$. In this
case, we do not have a complete excursion about level $z$, and moreover the
distribution of the crossing width is not clear. However, it is sufficient to
know that some probability density function $p_{z},z\in[0,\infty)$ gives the
distribution for the crossing width where the anti-clade has infimum less than
or equal to $-z$, i.e. the distribution of

$\nu_{\text{cld}}(f_{T^{+}_{0}}(-\xi_{N}(T^{+}_{0}-)))\in\cdot|\inf\xi_{N}\leq-z).$

In the setting of Lemma 1 we have that

$\split&\mathbf{E}\Bigg{(}\Bigg{(}\sum_{\substack{(t,f)\text{atomof}\mathbf{N}:\\\
t<T,\ell^{z}(t)=0,\\\
A(f)<\varepsilon_{0},\mathbf{X}_{t-}<z<\mathbf{X}_{t}}}\Big{|}g\big{(}f(y-\mathbf{X}_{t-})\big{)}-g\big{(}f(x-\mathbf{X}_{t-})\big{)}\Big{|}\Bigg{)}^{k}\Bigg{)}\\\
\leq C_{q}^{k}\int_{s=0}^{\varepsilon_{0}}p_{z}(s)\Big{(}\hskip
0.28436pt{{}^{-2\alpha^{+}}}\mathbb{Q}^{0}_{s}(|Y_{|x-y|}-s|^{qk})+{{}^{-2\alpha^{-}}}\mathbb{Q}^{0}_{s}(|Y_{|x-y|}-s|^{qk})\Big{)}ds\\\
$ (17)

where $p_{z}$ is the probability density function on $[0,\infty)$ that gives
the distribution for $\nu_{\text{cld}}(m^{0}\in\cdot|\zeta^{+}(N^{-})>z)$
(i.e. the distribution for the crossing width under $\nu_{\text{cld}}$
conditional on the anti-clade having depth at least $-z$) and where $C_{q}$ is
the Hölder constant given in Assumption LABEL:assumption_b. We use the same
method as in the previous lemma. The sum on the left-hand-side consists of the
first spindle to cross level $z$ a.s. if it exists and has amplitude less than
$\varepsilon_{0}$. We can then apply Lemma LABEL:lemma_mid-spindle with
stopping time $T^{\geq z}$. This tells us that the spindle above level $z$
evolves diffusively. We compare this to the diffusion $\Tilde{\mathbf{Y}}$ to
upper bound this expectation in the same way as in Lemma 1. Consider a
diffusion that satisfies Assumption LABEL:assumption_b. Let
$(\mathbf{N},\mathbf{X})$ be a scaffolding-and-spindles construction for this
diffusion on some probability space $(\Omega,\mathcal{A},\mathbf{P})$. Let $T$
be any $(0,\infty)$-valued $\mathbf{P}$-random variable. Then there exists a
continuous IP-evolution that is a modification of
$(\text{skewer}(y,\mathbf{N},\mathbf{X}),y\in\mathbb{R})$ which takes values
on $(\mathcal{I}_{H}(g(E)),d^{\prime}_{H})$. First assume $g(E)$ is closed and
$x$, $y\geq 0$. Let $\kappa>0$ be small enough so that $\alpha^{+}+\kappa<q$.
We first show that there exists $A\geq 0$ such that

$\mathbf{E}\Bigg{(}\Bigg{(}\sum_{\substack{(t,f)\text{atomof}\mathbf{N}_{M,\varepsilon_{0}}:\\\
\mathbf{X}_{t-}<x<\mathbf{X}_{t}\\\
\text{or}\mathbf{X}_{t-}<y<\mathbf{X}_{t}}}\Big{|}g\big{(}f(y-\mathbf{X}_{t-})\big{)}-g\big{(}f(x-\mathbf{X}_{t-})\big{)}\Big{|}\Bigg{)}^{k}\Bigg{)}\leq
A|x-y|^{k(q/2\wedge(q-\alpha^{+}-\kappa))}.$ (18)

First we upper bound sum in the equation above by

$\Bigg{(}\sum_{\substack{(t,f)\text{atomof}\\\
\mathbf{N}_{M,\varepsilon_{0}}:t<T,\\\ \mathbf{X}_{t-}<x<\mathbf{X}_{t},\\\
\ell^{x}(t)=0}}+\sum_{\substack{(t,f)\text{atomof}\\\
\mathbf{N}_{M,\varepsilon_{0}}:t<T,\\\ \mathbf{X}_{t-}<y<\mathbf{X}_{t},\\\
\ell^{y}(t)=0}}+\sum_{\substack{(t,f)\text{atomof}\\\
\mathbf{N}_{M,\varepsilon_{0}}:t<T,\\\ \mathbf{X}_{t-}<x<\mathbf{X}_{t},\\\
\ell^{x}(t)>0}}+\sum_{\substack{(t,f)\text{atomof}\\\
\mathbf{N}_{M,\varepsilon_{0}}:t<T,\\\ \mathbf{X}_{t-}<y<\mathbf{X}_{t},\\\
\ell^{y}(t)>0}}\Bigg{)}\Big{|}g\big{(}f(y-\mathbf{X}_{t-})\big{)}-g\big{(}f(x-\mathbf{X}_{t-})\big{)}\Big{|}.$

Now note that for any $a$, $b$, $c$, $d\geq 0$ we have that $(a+b+c+d)^{k}\leq
4^{k-1}(a^{k}+b^{k}+c^{k}+d^{k})$ and therefore we can deal with each of the
sums in the term above individually. We apply Lemmas 1 and 1 and the upper
bound for $m$ given in equation m bounds B to obtain an upper bound for the
third sum in the equation above:

$\mathbf{E}\Bigg{(}\Bigg{(}\sum_{\substack{(t,f)\text{atomof}\mathbf{N}_{M,\varepsilon_{0}}:\\\
t<T,\mathbf{X}_{t-}<x<\mathbf{X}_{t},\ell^{x}(t)>0}}\Big{|}g\big{(}f(y-\mathbf{X}_{t-})\big{)}-g\big{(}f(x-\mathbf{X}_{t-})\big{)}\Big{|}\Bigg{)}^{k}\Bigg{)}\leq
A^{\prime}|x-y|^{k(q/2\wedge(q-\alpha^{+}-\kappa))},$

for some $A^{\prime}\geq 0$ and similarly for the fourth sum. We deal with the
first two sums in the same way, using Lemma 1 and therefore obtain the upper
bound given in equation sum of differences x and y. We now consider the
$\mathbf{P}$-random times

$S_{0}\coloneqq 0;\hskip
14.22636ptS_{n}\coloneqq\inf\\{t>S_{n-1}:(t,f)\text{atomof}\mathbf{N},\hskip
7.11317ptA(f)\geq\varepsilon_{0}\\}\text{for}n\geq 1.$ (19)

As $\mathbf{N}$ is a stopped PRM stopped at random time $T$ there exists
$P\in\mathbb{N}$ such that $S_{P-1}\leq T<S_{P}$. For $1\leq n\leq P$ we
define the random measures

$\mathbf{N}_{M,\varepsilon_{0},n}\coloneqq\sum_{\substack{(t,f)\text{atomof}\mathbf{N}_{M,\varepsilon_{0}}:\\\
t\in[0,T]\cap(S_{n},S_{n+1})}}\delta_{(t,f)}\text{for}n\geq 0$

so that
$\mathbf{N}_{M,\varepsilon_{0}}=\sum_{n=0}^{P}\mathbf{N}_{M,\varepsilon_{0},n}$.
Let $n\in\mathbb{N}$. Consider the correspondence between
$\text{skewer}(x,\mathbf{N}_{M,\varepsilon_{0},n},\mathbf{X})$ and
$\text{skewer}(y,\mathbf{N}_{M,\varepsilon_{0},n},\mathbf{X})$ given by
pairing all the intervals generated by spindles that cross both levels $x$ and
$y$. This is a correspondence as we only have finitely many spindles cross
level $x$ and level $y$ $\mathbf{P}$-a.s and has distortion

$\sum_{\substack{(t,f)\text{atom}\\\
\text{of}\mathbf{N}_{M,\varepsilon_{0},n}:\\\ \mathbf{X}_{t-}<x\leq
y<\mathbf{X}_{t}}}\Big{|}g\big{(}f(y-\mathbf{X}_{t-})\big{)}-g\big{(}f(x-\mathbf{X}_{t-})\big{)}\Big{|}+\max\Big{\\{}\sum_{\substack{(t,f)\text{atom}\\\
\text{of}\mathbf{N}_{M,\varepsilon_{0},n}:\\\
x\leq\mathbf{X}_{t-}<y<\mathbf{X}_{t}}}g\big{(}f(y-\mathbf{X}_{t-})\big{)},\sum_{\substack{(t,f)\text{atom}\\\
\text{of}\mathbf{N}_{M,\varepsilon_{0}}:\\\
\mathbf{X}_{t-}<x<\mathbf{X}_{t}\leq
y}}g\big{(}f(x-\mathbf{X}_{t-})\big{)}\Big{\\}}.$

This sum is less than or equal to the sum in equation sum of differences x and
y. Therefore we have
$\overline{\text{skewer}}(\mathbf{N}_{M,\varepsilon_{0},n},\mathbf{X})$
satisfies the Kolmogorov-Čentsov continuity criterion and so there exists a
continuous modification $(\gamma^{y}_{M,n},y\geq 0)$. We can take $M$ to
infinity as by Proposition 1 we have that $\sup_{u\in\mathbb{R}}\ell^{u}(T)$
is finite $\mathbf{P}$-a.s. and so we have a continuous modification
$(\gamma_{n}^{y},y\geq 0)$ of
$\overline{\text{skewer}}(\mathbf{N}|_{(S_{n},S_{n+1})},\mathbf{X})$. From
here we can construct a continuous modification of
$\overline{\text{skewer}}(\mathbf{N},\mathbf{X})$. For $n\in\mathbb{N}$ let
$f_{n}$ be the $n^{\text{th}}$ spindle of $\mathbf{N}$ with amplitude greater
than or equal to $\varepsilon_{0}$, i.e. so that $\delta_{(S_{n},f_{n})}$ is
an atom of $\mathbf{N}$. Then we define the IP-evolution $(\gamma^{y},y\geq
0)$ by

$\gamma^{y}\coloneqq\bigstar_{n=1}^{P}\big{(}\gamma^{y}_{n}\star\\{(0,f_{n}(y-\mathbf{X}_{S_{n}-}))\\}\big{)}.$
(20)

Note there are only finitely many spindles with amplitude greater than or
equal to $\varepsilon_{0}$ on $[0,T]$ $\mathbf{P}$-a.s. Therefore we have that
$(\gamma^{y},y\geq 0)$ is the concatenation of finitely many
$d^{\prime}_{H}$-continuous IP-evolutions (the IP-evolutions
$(\\{(0,f_{n}(y-\mathbf{X}_{S_{n}}))\\},y\geq 0)$ are clearly
$d^{\prime}_{H}$-continuous) and therefore $(\gamma^{y},y\geq 0)$ is
$d^{\prime}_{H}$-continuous. Finally we note that concatenating modificiations
gives modifications and so $(\gamma^{y},y\geq 0)$ is a modification of
$\overline{\text{skewer}}(\mathbf{N},\mathbf{X})$ on
$(\mathcal{I}_{H},d^{\prime}_{H})$. The subspace
$(\mathcal{I}_{H}(g(E)),d^{\prime}_{H})$ is closed and so this modification
exists on the appropriate subspace. To finish the proof assume $g(E)$ is open.
This only occurs if $g(c)$ is finite and $g(E)=[0,g(c))$. In this case however
we still obtain our result as we do not take a modification of the diffusions
with amplitude greater than $\varepsilon_{0}$ and as each of these diffusions
with amplitude greater than $\varepsilon_{0}$ do not hit $g(c)$ then
$(\gamma^{y},y\geq 0)$ does take values on the subspace
$(\mathcal{I}_{H}(g(E)),d^{\prime}_{H})$. Finally, assume $x$, $y\leq 0$. Then
this case follows ultimately by equation reversal under nu clade. If $x\leq
0\leq y$ then we have the result by the previous two cases and the triangle
inequality. We conclude this section with the analogous result for diffusions
under Assumption LABEL:assumption_c. Let $\mathbf{Z}$ be a diffusion that
satisfies Assumption LABEL:assumption_c, and let $(\mathbf{N},\mathbf{X})$ be
a scaffolding-and-spindles pair stopped at an a.s. finite stopping time. Then
$(\overline{\text{skewer}}(\mathbf{N},\mathbf{X})$ is a.s. continuous and
takes values on $(\mathcal{I}_{H}(g(E)),d^{\prime}_{H})$. Let $\mathbf{Y}$ be
an untransformed diffusion that satisfies Assumption 1 and $g$ be a
transformation that satisfies $C3$ of Assumption LABEL:assumption_c such that
$\mathbf{Z}=g(\mathbf{Y})$. By the bounds in equation m bounds C we have that
$M((0,\varepsilon_{0}))$ is finite. Therefore by Theorem
LABEL:theorem_lévy_measure_condition the Pitman-Yor excursion measure for
$\mathbf{Y}$ (equivalently $\mathbf{Z}$) satisfies equation levy measure
bounded variation. For the diffusion $\mathbf{Z}$ let
$(\mathbf{N},\mathbf{X})$ be a scaffolding-and-spindles pair stopped at an
a.s. finite stopping time. To show continuity of the IP-evolution we first
show that the sum of the amplitudes of all of the spindles is finite. From
Theorem LABEL:definition_Williams_decomposition we know that the intensity of
spindles of $\mathbf{N}$ with amplitude greater than $x$ is $1/s(x)$.
Therefore, the amplitudes of the spindles is a Poisson point process with
intensity $s^{\prime}(x)/s(x)^{2}$ which, when stopped at an a.s. finite
random time, is summable iff is finite . With the substitution $y=s(x)$ we
have that it is equal to

$\int_{x=0}^{b}g(x)\frac{s^{\prime}(x)}{s(x)^{2}}dx=\int_{y=0}^{s(b)}\frac{g(s^{-1}(y))}{y^{2}}dy<\infty$
(21)

for some (equivalently all) $b\in(0,c)$ (we have included the integral
obtained with the substitution $y=s(x)$ here as we use it below). By the bound
in equation s’ bounds of $s^{\prime}$ we have

$\frac{\varepsilon_{1}^{\beta^{-}}}{1-\beta^{-}}x^{1-\beta^{-}}\leq
s(x)\text{for}x\in(0,\varepsilon_{1}).$

Therefore for
$y=\big{(}\varepsilon_{1}^{-\beta^{-}}(1-\beta^{-})\big{)}^{1/(1-\beta^{-})}x^{1/(1-\beta^{-})}$
we have

$x=\frac{\varepsilon_{1}^{\beta^{-}}}{1-\beta^{-}}y^{(1-\beta^{-})}\leq
s(y)\text{for}y\in(0,\varepsilon_{1}).$

This means that

$s^{-1}(x)\leq\big{(}\varepsilon_{1}^{-\beta^{-}}(1-\beta^{-})\big{)}^{1/(1-\beta^{-})}x^{1/(1-\beta^{-})}\hskip
14.22636pt\text{for}x<s(\varepsilon_{1})$

and so the second integral in equation needs to be summable is less than or
equal to

$\int_{y=0}^{s(b)}\frac{g(\big{(}\varepsilon_{1}^{-\beta^{-}}(1-\beta^{-})\big{)}^{1/(1-\beta^{-})}y^{1/(1-\beta^{-})})}{y^{2}}dy\hskip
14.22636pt\text{providedthat}b<\varepsilon_{1}.$

To put this into the form of $C3$ requires a linear transformation $z=ay$ for
$a=\big{(}\varepsilon_{1}^{-\beta^{-}}(1-\beta^{-})\big{)}$. Therefore, the
integral in equation needs to be summable is finite by $C3$ and the
summability of the amplitudes follows. From here we can show that
$\overline{\text{skewer}}(\mathbf{N},\mathbf{X})$ is continuous a.s. with
respect to the metric $d^{\prime}_{H}$. Let $I$ be an index set for the
spindles of $\mathbf{N}$, $\varepsilon>0$ and $C=\sum_{i\in I}A(f_{i})<\infty$
be the sum of amplitudes Let $(t_{1},f_{1}),\dots,(t_{K},f_{K})$ be atoms of
$\mathbf{N}$ such that $\sum_{i=1}^{K}A(f_{i})>C-\varepsilon/2$, and let
$\Tilde{\mathbf{N}}$ be the Poisson random measure $\mathbf{N}$ with the atoms
$(t_{1},f_{1}),\dots,(t_{K},f_{K})$ removed. The spindles $f_{i}$ are
continuous functions with finite lifetimes so are uniformly continuous. Let
$\delta>0$ such that for any $1\leq i\leq K$ and any $s$, $t$ with
$|s-t|<\delta$ we have $|f_{i}(s)-f_{i}(t)|<\varepsilon/2K$. Let $x$,
$y\in\mathbb{R}$ with $|x-y|<\delta$. Then, for the interval partitions
$\text{skewer}(x,\mathbf{N},\mathbf{X})$ and
$\text{skewer}(y,\mathbf{N},\mathbf{X})$ we can define the correspondence of
size at most $K$ that matches every interval generated by spindle $f_{i}$ in
$\text{skewer}(x,\mathbf{N},\mathbf{X})$ to the corresponding interval in
$\text{skewer}(y,\mathbf{N},\mathbf{X})$. The remaining intervals in
$\text{skewer}(x,\mathbf{N},\mathbf{X})$ or
$\text{skewer}(y,\mathbf{N},\mathbf{X})$ have sum less than $\varepsilon/2$
and therefore the distortion is equal to

$\sum_{j=1}^{K}|f_{i}(x-\mathbf{X}_{t_{i}-})-f_{i}(y-\mathbf{X}_{t_{i}-})|+\max\Big{\\{}\sum_{(t,f)\text{atomof}\Tilde{\mathbf{N}}}f(x-\mathbf{X}_{t-}),\sum_{(t,f)\text{atomof}\Tilde{\mathbf{N}}}f(y-\mathbf{X}_{t-})\Big{\\}}\leq
K\frac{\varepsilon}{2K}+\frac{\varepsilon}{2}$

and therefore
$d^{\prime}_{H}(\text{skewer}(x,\mathbf{N},\mathbf{X}),\text{skewer}(y,\mathbf{N},\mathbf{X}))\leq\varepsilon$.
As in the previous result the process takes values on the subspace
$(\mathcal{I}_{H}(g(E)),d^{\prime}_{H})$ as required.

##  Simple Markov property and proof of Theorem
LABEL:theorem_interval_partitions_regular

For a measurable map $g:E\rightarrow g(E)$ (where as before $E=[0,c)$ or
$E=[0,c]$ for some $c\leq\infty$) we define
$g^{*}:\mathcal{I}_{H}(E)\rightarrow g^{*}(\mathcal{I}_{H}(E))$ in the
following way: for an interval partition $\beta=(U_{i},i\in I)$ such that
$\sum_{i\in I}g(\text{Leb}(U_{i}))$ is finite we define $g^{*}(\beta)$ to be
the interval partition $\gamma\coloneqq(V_{i},i\in I)$ where we have
$\text{Leb}(V_{i})=g(\text{Leb}(U_{i}))$ and the order of the intervals in
$\beta$ and $\gamma$ is consistent; when $\sum_{i\in I}g(\text{Leb}(U_{i}))$
is infinite we take $g^{*}(\beta)\coloneqq\emptyset$, the empty interval
partition. Note that,
$g^{*}(\mathcal{I}_{H}(E))\subseteq\mathcal{I}_{H}(g(E))$. Also, as $g$ is an
injection, $g^{*}$ restricted to
$\\{\beta\in\mathcal{I}_{H}(E):\sum_{U\in\beta}g(\text{Leb}(U))<\infty\\}$ has
an inverse on $\\{\mathcal{I}\hskip 3.55658pt|\hskip
3.55658pt\mathcal{I}\subset g^{*}(\mathcal{I}_{H})\\}$ and we have that
$(g^{*})^{-1}(\mathcal{I})=(g^{-1})^{*}(\mathcal{I})$. We also note that
$g^{*}$ is a measurable function. To see this, first define the functions
$(g_{n},n\in\mathbb{N})$ by $g_{n}(x)=g(x)$ if $x>1/n$ and $g_{n}(x)=x$
otherwise and observe that $g^{*}$ is the limit of the sequence of the
functions $(g_{n}^{*},n\in\mathbb{N})$. The measurability of $g^{*}$ then
follows by the fact that for all $n\in\mathbb{N}$ the functions $g_{n}^{*}$
are continuous. The proof of continuity of the functions
$(g_{n}^{*},n\in\mathbb{N})$ follows from the continuity of $g$ and that for
any interval partition $\beta$ the interval partition differs from its image
$g_{n}^{*}(\beta)$ on only finitely many interval widths (see also Lemma 1).
The next step is to develop a simple Markov property. In Corollary
LABEL:corollary_assumption_b'_starting_interval_partition we have that for an
interval partition $\beta\in\mathcal{I}_{H}(E)$ and a diffusion that satisfies
Assumption 1, we can construct a scaffolding-and-spindles pair
$(\mathbf{N}_{\beta},\mathbf{X}_{\beta})$ as in Definition
LABEL:definition_starting_interval_partition. We state a corollary here for
diffusions under Assumption LABEL:assumption_b. Consider a diffusion
$\mathbf{Z}$ that satisfies Assumption LABEL:assumption_b and let $\gamma\in
g^{*}(\mathcal{I}_{H}(E))$. Consider a scaffolding-and-spindles pair started
from $\gamma$, denoted as $(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ as in
Definition LABEL:definition_starting_interval_partition. Then we have that
$\text{len}(\mathbf{N}_{\gamma})=\sum_{U\in\gamma}\text{len}(\mathbf{N}_{U})$
is finite a.s. Let $\mathbf{Y}$ be the untransformed diffusion that satisfies
Assumption 1, let $\gamma\in g^{*}(\mathcal{I}_{H}(E))$ and then let
$\beta\in\mathcal{I}_{H}(E)$ such that $g^{*}(\beta)=\gamma$. As mentioned
above, by Corollary LABEL:corollary_assumption_b'_starting_interval_partition
we can construct a scaffolding-and-spindles pair
$(\mathbf{N}_{\beta}^{\mathbf{Y}},\mathbf{X}_{\beta}^{\mathbf{Y}})$ for
$\mathbf{Y}$ started from $\beta$ and $\text{len}(\mathbf{N}_{\beta})$ is
finite a.s. Therefore we can define a scaffolding-and-spindles pair for
diffusion $\mathbf{Z}$ and started from $\gamma$ by

$\mathbf{N}_{\gamma}^{\mathbf{Z}}=\sum_{(t,f)\text{atomof}\mathbf{N}_{\beta}^{\mathbf{Y}}}\delta_{(t,g(f))},$

where for a function $(f(x),x\in\mathbb{R})$ we write $g(f)$ to be the
function $(g(f(x)),x\in\mathbb{R})$. Then define
$\mathbf{X}_{\gamma}^{\mathbf{Z}}=\xi(\mathbf{N}_{\gamma}^{\mathbf{Z}})$ which
is equal to $\mathbf{X}_{\beta}^{\mathbf{Y}}$ as the spatial transformation of
the spindles does not change the scaffolding. We now state a continuity result
for an IP-evolution with a fixed initial state. For
$N\in\mathcal{N}^{\text{sp}}_{\text{fin}}$, let
$\zeta^{+}(N)\coloneqq\sup_{t\in\mathbb{R}}\xi_{N}(t)$. Consider a diffusion
$\mathbf{Z}$ that satisfies Assumption LABEL:assumption_b and let $\gamma\in
g^{*}(\mathcal{I}_{H}(E))$. Let $(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ be
a scaffolding-and-spindles pair $\gamma$ as in Definition
LABEL:definition_starting_interval_partition. Then there exists a modification
of $\overline{\text{skewer}}(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ that we
call $(\gamma^{y},y\geq 0)$ that is continuous for $y\in(0,\infty)$.
Furthermore, for $y\in(0,\infty)$ fixed, we have $\gamma^{y}\in
g^{*}(\mathcal{I}_{H}(E))$ a.s. Let $n\in\mathbb{N}$ and $\gamma\in
g^{*}(\mathcal{I}_{H}(E))$. From Corollary 1 we can construct
$(\mathbf{N}_{U},U\in\gamma)$ as in Definition
LABEL:definition_starting_interval_partition. Recall that
$\mathbf{N}_{U}\coloneqq\delta_{(0,f_{U})}+\mathbf{N}|_{[0,T^{-\zeta(f_{U})}]}$
where $\mathbf{N}$ is a Poisson random measure. The process
$(\text{skewer}(y,\mathbf{N}|_{[0,T^{-\zeta(f_{U})}]},\xi(\mathbf{N}|_{[0,T^{-\zeta(f_{U})}]})),y\in\mathbb{R})$
has a continuous modification by Theorem 1 and it follows that
$\overline{\text{skewer}}(\mathbf{N}_{U},\mathbf{X}_{U})$ has a continuous
modification. Also, from the fact that
$\sum_{U\in\gamma}\text{len}(\mathbf{N}_{U})$ is finite a.s., from Corollary
1, we have that for only finitely many $U\in\gamma$ that
$\zeta^{+}(\mathbf{N}_{U})>1/n$. Therefore we can conclude that there exists a
continuous modification of
$\overline{\text{skewer}}(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ on
$(1/n,\infty)$. As this is true for all $n\in\mathbb{N}$, we have a continuous
modification on $(0,\infty)$. It remains to show that for $y\in(0,\infty)$
fixed, we have $\gamma^{y}\in g^{*}(\mathcal{I}_{H}(E))$ a.s. Let $y>0$. For
each $U\in\gamma$ we have that $\mathbf{N}_{U}$ is constructed by an initial
incomplete spindle $f_{U}$ and then a stopped Poisson random measure
$\mathbf{N}$ with stopping time $T^{-\zeta(f_{U})}$. We claim that
$\text{skewer}(y,\mathbf{N}_{U},\mathbf{X}_{U})\in g^{*}(\mathcal{I}_{H}(E))$
a.s. By the spatial homogeneity of the scaffolding-and-spindles pair and the
independence of $f_{U}$ and $\mathbf{N}$ it is sufficient to show that
$\text{skewer}(0,\mathbf{N},\mathbf{X})\in g^{*}(\mathcal{I}_{H}(E))$ a.s. We
have that $\text{skewer}(0,\mathbf{N},\mathbf{X})$ is in $\mathcal{I}_{H}(E)$
by the first part of Proposition 1 and we can conclude that it is in fact in
$g^{*}(\mathcal{I}_{H}(E))$ by again using the first part of Proposition 1
with the identity transformation $g(x)=x$ (this tells us the untransformed
spindles produce an interval partition and therefore the transformed spindles
produce an interval partition in $g^{*}(\mathcal{I}_{H}(E))$ a.s.). We
concatenate the interval partitions
$\text{skewer}(y,\mathbf{N}_{U},\mathbf{X}_{U})$ for the finitely many of the
$U\in\gamma$ with $\zeta^{+}(\mathbf{N}_{U})>1/n$ to form
$\text{skewer}(y,\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ and conclude that
$\text{skewer}(y,\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})\in
g^{*}(\mathcal{I}_{H}(E))$ a.s. and therefore the modification $\gamma^{y}\in
g^{*}(\mathcal{I}_{H}(E))$ a.s. We define the filtration
$(\mathcal{F}_{\mathcal{I}}^{y},y\geq 0)$ to be the right-continuous
filtration generated by $(\gamma^{y},y\in(0,\infty))$ on
$\Omega\coloneqq\mathcal{C}((0,\infty),\mathcal{I}_{H})$, the class of all
paths in $\mathcal{I}_{H}$ continuous on $(0,\infty)$. We define a filtered
probability space
$(\Omega,\mathcal{F}_{\mathcal{I}},(\mathcal{F}_{\mathcal{I}}^{y},y\geq
0),\mathbb{P}_{\gamma})$ where
$\mathcal{F}_{\mathcal{I}}\coloneqq\bigcup_{y\geq
0}\mathcal{F}_{\mathcal{I}}^{y}$, and $\mathbb{P}_{\gamma}(\cdot)$ is the
probability measure

$\mathbb{P}_{\gamma}(\cdot)=\mathbf{P}((\gamma^{y},y\geq 0)\in\cdot)$

where $(\gamma^{y},y\geq 0)$ is the modification of
$\overline{\text{skewer}}(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ that is
continuous on $(0,\infty)$ that is given in the lemma above. [Simple Markov
property under Assumption LABEL:assumption_b] Consider a diffusion under
Assumption LABEL:assumption_b. Let $\mu$ be a probability distribution on
interval partitions that is supported on $g^{*}(\mathcal{I}_{H}(E))$, and
define the probability measure $\mathbb{P}_{\mu}(\cdot)=\int_{\gamma\in
g^{*}(\mathcal{I}_{H}(E))}\mathbb{P}_{\gamma}(\cdot)\mu(d\gamma)$. For any
$y\in(0,\infty)$ let $\theta_{y}$ be the shift operator, and let
$\eta,f:\mathcal{C}((0,\infty),\mathcal{I}_{H})\rightarrow[0,\infty)$ be
measurable where $\eta$ is measurable with respect to
$\mathcal{F}_{\mathcal{I}}^{y}$. Then we have that

$\mathbb{P}_{\mu}\big{[}\eta
f\circ\theta_{y}\big{]}=\mathbb{P}_{\mu}\big{[}\eta\mathbb{P}_{\gamma^{y}}[f]\big{]}.$

This proof uses the same arguments as the analogous result [construction,
Corollary 6.13]; specifically, by Lemma 1 we have that the sample paths a.s.
are in $\Omega$ and the simple Markov property for the skewer process on a
larger probability space is given in Proposition
LABEL:proposition_simple_markov_skewer. We now show the analogous result for
diffusions under Assumption LABEL:assumption_c and prove Theorem
LABEL:theorem_interval_partitions_regular. We let
$\mathcal{I}_{\text{fin}}\subset\mathcal{I}_{H}$ be the interval partitions
that consist of finitely many intervals and
$\mathcal{I}_{\text{fin}}(g(E))=\mathcal{I}_{\text{fin}}\cap\mathcal{I}_{H}(g(E))$.
For any interval partition $\gamma\in\mathcal{I}_{\text{fin}}(g(E))$ we define
$\mathbb{P}_{\gamma}$ as before and note that the scaffolding-and-spindles
pair $(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ can always be constructed for
such initial interval partitions $\gamma$. Let
$\Omega_{0}\coloneqq\mathcal{C}([0,\infty),\mathcal{I}_{H})$ be the space of
continuous paths on $\mathcal{I}_{H}$ indexed by $[0,\infty)$. [Proof of
Theorem LABEL:theorem_interval_partitions_regular] Let $\mathbf{Z}$ be a
diffusion under Assumption LABEL:assumption_c, and let
$\gamma\in\mathcal{I}_{\text{fin}}(g(E))$. Let
$(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ be a scaffolding-and-spindles pair
started at $\gamma$ as defined in Definition
LABEL:definition_starting_interval_partition. For $y\geq 0$ let
$\gamma^{y}=\text{skewer}(y,\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$. Then
$(\gamma^{y},y\geq 0)$ is continuous by the same reasoning as in Proposition
1; for each $y$ we have only concatenated finitely many interval partitions.
In this setting let $(\mathcal{F}_{\mathcal{I}}^{y},y\geq 0)$ be the right-
continuous filtration generated by $(\gamma^{y},y\geq 0)$ on the smaller space
$\Omega_{0}$, rather than on $\Omega$ as we had done previously. The simple
Markov property for the skewer process from Proposition
LABEL:proposition_simple_markov_skewer applies in this setting as for fixed
$y>0$ the interval partition $\gamma^{y}\in\mathcal{I}_{\text{fin}}(g(E))$
a.s. To see this, note that for each $U\in\gamma$ the SPLP $\mathbf{X}_{U}$ is
of bounded variation and so only crosses a fixed level finitely many times
a.s., [bertoin1996levy, Corollary VII.5], and therefore $\gamma^{y}$ consists
of finitely many intervals as it is the concatenation of finitely many
interval partitions that each consist of finitely many intervals. To conclude
the proof for the setting described in Theorem
LABEL:theorem_interval_partitions_regular we note that, for $n\in\mathbb{N}$,
if we stop the PRM $\mathbf{N}$ of a scaffolding-and-spindles pair
$(\mathbf{N},\mathbf{X})$ at the $n^{\text{th}}$ hitting time of $0$ then
$\text{skewer}(0,\mathbf{N},\mathbf{X})\in\mathcal{I}_{\text{fin}}(g(E))$ a.s.
We used $\Omega_{0}\subset\Omega$ under Assumption LABEL:assumption_c whereas
under Assumption LABEL:assumption_b we were required to use $\Omega$ as we did
not have a proof of continuity at time $0$. However, if we restrict the class
of initial interval partitions then we can use $\Omega_{0}$ to state the
simple Markov property under Assumption LABEL:assumption_b, and we do this in
the following subsection. This subsection is not necessary to prove Theorem
LABEL:theorem_diffusion but we include it to show the class of initial
interval partitions for which we have a proof of continuity at time $0$.

###  Continuity at time $0$ under Assumption LABEL:assumption_b

To prove the results of this section we define certain classes of interval
partitons. For any $q>0$ and any class of interval partitions
$\mathcal{I}\subseteq\mathcal{I}_{H}$ we define

$\mathcal{I}^{(q)}\coloneqq\\{\beta=(U_{i},i\in I)\in\mathcal{I}:\sum_{i\in
I}\text{Leb}(U_{i})^{q}<\infty\\}.$

Note that for $q\geq 1$ we have that $\mathcal{I}^{(q)}=\mathcal{I}$. We now
state a corollary of Theorem 1. Consider a diffusion that satisfies Assumption
1 with a stopped scaffolding-and-spindles pair $(\mathbf{N},\mathbf{X})$ on
some probability space $(\Omega,\mathcal{A},\mathbf{P})$, and let
$q>\alpha^{+}$. Then, for $(\text{skewer}(y,\mathbf{N},\mathbf{X}),y\geq 0)$,
there exists a modification $(\beta^{y},y\geq 0)$ which is continuous on
$(\mathcal{I}^{(q)}_{H}(E),d^{\prime}_{H})$. We first note that for
$q\in(0,1)$ we can apply Theorem 1 with the diffusion $\mathbf{Y}$ with the
transformation $h(x)=x^{q}$, to produce the continuous IP-evolution
$(\phi^{y},y\geq 0)$. Also, note that the map $(h^{-1})^{*}$ is continuous on
the space of interval partitions $\mathcal{I}_{H}$ (this ultimately follows
because the map $h^{-1}$ is Lipschitz on $[0,1]$, see also Lemma 1). Therefore
$((h^{-1})^{*}(\phi^{y}),y\geq 0)$ is $\mathbf{P}$-a.s. continuous in
$\mathcal{I}_{H}$. Now, $(\phi^{y}),y\geq 0)$ is a modification of
$h^{*}(\overline{\text{skewer}}(\mathbf{N},\mathbf{X}))$ and therefore
$((h^{-1})^{*}(\phi^{y}),y\geq 0)$ is a modification of
$\overline{\text{skewer}}(\mathbf{N},\mathbf{X})$. Also, $(\beta^{y},y\geq 0)$
is a modification of $\overline{\text{skewer}}(\mathbf{N},\mathbf{X})$ and so
is also a modification of $((h^{-1})^{*}(\phi^{y}),y\geq 0)$. As both
$(\beta^{y},y\geq 0)$ and $((h^{-1})^{*}(\phi^{y}),y\geq 0)$ are both
continuous processes they are therefore indistinguishable from one another,
and therefore $(\beta^{y},y\geq 0)$ is a continuous process in
$\mathcal{I}_{H}^{(q)}(E)$ a.s. As in the previous sections, we consider a
diffusion $\mathbf{Z}$ under Assumption LABEL:assumption_b, which is equal in
distribution to $g(\mathbf{Y})$ where $\mathbf{Y}$ is a diffusion under
Assumption 1 and $g:E\rightarrow g(E)$ is a continuous bijection as in
Assumption LABEL:assumption_b. Let $q_{0}$, $\alpha^{+}$, $\alpha^{-}$ be the
associated constants and let $q\in(\alpha^{+},q_{0})$. Finally, let
$m_{\mathbf{Y}}$ and $m_{\mathbf{Z}}$ be the speed measure densities of
$\mathbf{Y}$ and $\mathbf{Z}$. We now state a corollary of Lemma
LABEL:lemma_diversity_is_local_time and Lemma 1. Consider a stopped
scaffolding-and-spindles pair $(\mathbf{N},\mathbf{X})$ for the diffusion
$\mathbf{Z}$ which satisfies Assumption LABEL:assumption_b, and let
$(\gamma^{y},y\geq 0)$ be a continuous modification of
$\overline{\text{skewer}}(\mathbf{N},\mathbf{X})$. Then for any fixed $y\geq
0$ we have that

$\gamma^{y}\in
g^{*}(\mathcal{I}^{(q)}_{m_{\mathbf{Y}}}(E))=\Big{\\{}\gamma=(V_{i},i\in I)\in
g^{*}(\mathcal{I}_{m_{\mathbf{Y}}}(E)):\sum_{i\in
I}g^{-1}(\text{Leb}(V_{i}))^{q}<\infty\Big{\\}}\hskip
7.11317pt\mathbf{P}-\text{a.s.}$

Let $\Tilde{\mathbf{N}}$ be the Poisson random measure of untransformed
spindles. The $m_{\mathbf{Y}}$-diversity of
$\overline{\text{skewer}}(\Tilde{\mathbf{N}},\mathbf{X})$ at any fixed level
has already been established in Lemma LABEL:lemma_diversity_is_local_time.
This shows that $\text{skewer}(y,\mathbf{N},\mathbf{X})\in
g^{*}(\mathcal{I}_{m_{\mathbf{Y}}}(E))$ a.s. and so $\gamma^{y}\in
g^{*}(\mathcal{I}_{m_{\mathbf{Y}}}(E))$. It remains to show that
$\gamma^{y}\in g^{*}(\mathcal{I}^{(q)}_{H}(E))$. To do this we note that the
process $(\gamma^{y},y\geq 0)$ is a modification of
$g^{*}(\overline{\text{skewer}}(\Tilde{\mathbf{N}},\mathbf{X}))$. Therefore
$((g^{-1})^{*}(\gamma^{y}),y\geq 0)$ is a modification of
$\overline{\text{skewer}}(\Tilde{\mathbf{N}},\mathbf{X})$. By Lemma 1
$\overline{\text{skewer}}(\Tilde{\mathbf{N}},\mathbf{X})$ has a continuous
modification that takes values in $\mathcal{I}^{(q)}_{H}(E)$, and the result
follows as for fixed $y\geq 0$ we have that
$(g^{-1})^{*}(\gamma^{y})=\text{skewer}(y,\Tilde{\mathbf{N}},\mathbf{X})$ a.s.
The following lemma concerns the initial spindles of the IP-evolution, i.e.
the spindles that are not equal to $0$ at time $0$. Consider a diffusion that
satisfies Assumption 1. Let $(a_{i},i\in I)$ be a potentially infinite
sequence of positive numbers such that $\sum_{i\in I}a_{i}^{q\wedge
1}<\infty$. Consider a family of independent diffusions $(f_{i},i\in I)$
distributed as independent diffusions started from $(a_{i},i\in I)$ stopped at
$0$ on some common probability space $(\Omega,\mathcal{A},\mathbf{P})$, i.e.
$f_{i}\sim\mathbb{Q}^{0}_{a_{i}}$. Then we have that $\sum_{i\in
I}A(f_{i})^{q}<\infty$ $\mathbf{P}$-a.s. By the coupling given in equation
coupling of diffusions we have for $0<a<y<\varepsilon_{0}$ that
$\mathbb{Q}^{0}_{a}(A>y)\leq{{}^{-2\alpha^{-}}}\mathbb{Q}^{0}_{a}(A>y)$ which
in turn is equal to $(a/y)^{1+\alpha^{-}}$ (by equation scale function
property and as $x^{1+\alpha^{-}}$ for $x\in[0,\infty)$ is a scale function
for a squared Bessel process with dimension $-2\alpha^{-}$). Therefore we have

$\sum_{i\in
I:a_{i}<\varepsilon_{0}}\mathbf{P}(A(f_{i})>\varepsilon_{0})\leq\varepsilon_{0}^{-1-\alpha^{-}}\sum_{i\in
I}a_{i}^{1+\alpha^{-}}<\infty,$

and so by the Borel-Cantelli lemma and by the fact that only finitely many
$a_{i}$ are such that $a_{i}\geq\varepsilon_{0}$, we have that a.s. only
finitely many of the incomplete spindles $f_{i}$ will have amplitude greater
than $\varepsilon_{0}$. Furthermore, again with the upper bound for
$\mathbb{Q}^{0}_{a}(A>y)$ stated above, we have

$\mathbf{E}(A(f_{i})\mathbbm{1}_{A(f_{i})<\varepsilon_{0}})=\int_{y=0}^{\varepsilon_{0}}\mathbf{P}(y\leq
A(f_{i})<\varepsilon)dy.$

This is less than or equal to

$\int_{y=0}^{\varepsilon_{0}}\mathbb{Q}_{a_{i}}^{0}(A>y)dy=\int_{y=0}^{\varepsilon_{0}}\Big{(}(a_{i}/y)^{1+\alpha^{-}}\wedge
1\Big{)}dy=\Big{(}\frac{1}{\alpha^{-}}+1\Big{)}a_{i}-\frac{a_{i}^{1+\alpha^{-}}}{\alpha^{-}\varepsilon_{0}^{\alpha^{-}}}\leq\Big{(}\frac{1}{\alpha^{-}}+1\Big{)}a_{i}.$

Therefore, when $q<1$, we have that

$\mathbf{E}(A(f_{i})^{q}\mathbbm{1}_{A(f_{i})<\varepsilon_{0}})\leq\Big{(}\mathbf{E}(A(f_{i})\mathbbm{1}_{A(f_{i})<\varepsilon_{0}})\Big{)}^{q}\leq
C_{q}\Big{(}\frac{1}{\alpha^{-}}+1\Big{)}^{q}a_{i}^{q}$

where $C_{q}$ is the Hölder constant from Assumption LABEL:assumption_b.
Therefore we have that $\sum_{i\in
I}\mathbf{E}(A(f_{i})^{q}:A(f_{i})<\varepsilon_{0})$ is finite, and so
$\sum_{i\in I:A(f_{i})<\varepsilon_{0}}A(f_{i})^{q}$ is finite
$\mathbf{P}$-a.s. In the next lemma we give a result that compares certain
interval partitions with $\text{skewer}(0,\mathbf{N},\mathbf{X})$ for a
particular stopped scaffolding-and-spindles pairs $(\mathbf{N},\mathbf{X})$,
which we then use in a coupling argument in the proposition after to prove
continuity of a IP-evolutions with certain initial interval partitions.
[Interval partition domination] Consider a diffusion $\mathbf{Z}$ that
satisfies Assumption LABEL:assumption_b and let
$\gamma\in\overline{\mathcal{I}}_{m_{\mathbf{Z}}}(g(E))$. Recall the notation
$\mathcal{D}$ and $\overline{\mathcal{D}}$ from Definition LABEL:definition_m-
diversity and let $(\mathbf{N},\mathbf{X})$ be a scaffolding-and-spindles pair
stopped at
$\tau^{0}(\overline{\mathcal{D}}^{m_{\mathbf{Z}}}_{\gamma}(\infty)+1)$ (the
inverse local time of $\mathbf{X}$ at level $0$), and let
$\Tilde{\gamma}\coloneqq\text{skewer}(0,\mathbf{N},\mathbf{X})$. Firstly we
have that

$\mathcal{D}_{\Tilde{\gamma}}^{m_{\mathbf{Z}}}(\infty)=\overline{\mathcal{D}}_{\gamma}^{m_{\mathbf{Z}}}(\infty)+1\text{a.s.}$

Also, with positive probability we have that there exists a matching between
$\gamma$ and $\Tilde{\gamma}$ such that every interval of $\gamma$ is matched
with a larger interval of $\Tilde{\gamma}$ (but that does not necessarily
respect the total-order of the intervals). We say that $\Tilde{\gamma}$
dominates $\gamma$ by matching in this case. Finally we have that
$g^{*}(\overline{\mathcal{I}}_{m_{\mathbf{Y}}}(E))=\overline{\mathcal{I}}_{m_{\mathbf{Z}}}(g(E))$
and also
$g^{*}(\mathcal{I}_{m_{\mathbf{Y}}}(E))=\mathcal{I}_{m_{\mathbf{Z}}}(g(E))$.
This first part of this lemma generalises [construction, Lemma 6.8] to
interval partitions with finite $m_{\mathbf{Z}}$-diversity and the proof is
the same. The result further generalises to any interval partition with a
finite limsup-$m_{\mathbf{Z}}$-diversity as intervals can be added to an
interval partition with finite limsup-$m_{\mathbf{Z}}$-diversity to produce an
interval partition with a finite $m_{\mathbf{Z}}$-diversity and, therefore,
the domination result carries over. For the second part first note that if a
pair of interval partitions $\hat{\beta}$, $\hat{\gamma}\neq\emptyset$ satisfy
$g^{*}(\hat{\beta})=\hat{\gamma}$ then
$\hat{\beta}\in\mathcal{I}_{m_{\mathbf{Y}}}(E)$ if and only if
$\hat{\gamma}\in\mathcal{I}_{m_{\mathbf{Z}}}(g(E))$ and also
$\mathcal{D}_{\hat{\beta}}^{m_{\mathbf{Y}}}(\infty)=\mathcal{D}_{\hat{\gamma}}^{m_{\mathbf{Z}}}(\infty)$
(and similarly for limsup diversities). Now, for $\Tilde{\gamma}$ defined
above there exists $\Tilde{\beta}$ such that
$g^{*}(\Tilde{\beta})=\Tilde{\gamma}$ by the aggregate health summability in
the transformed and untransformed cases shown in Proposition 1. Therefore we
have for the $\gamma\in\overline{\mathcal{I}}_{m_{\mathbf{Z}}}(g(E))$ defined
above that $\gamma\in g^{*}(\overline{\mathcal{I}}_{m_{\mathbf{Y}}}(E))$. This
can be seen as
$\sum_{V\in\gamma}g^{-1}(\text{Leb}(V))\leq\norm{\Tilde{\beta}}$. The
generality of $\gamma$ here means that
$\overline{\mathcal{I}}_{m_{\mathbf{Z}}}(g(E))\subseteq
g^{*}(\overline{\mathcal{I}}_{m_{\mathbf{Y}}}(E))$. The result follows as it
is immediate that
$g^{*}(\overline{\mathcal{I}}_{m_{\mathbf{Y}}}(E))\subseteq\overline{\mathcal{I}}_{m_{\mathbf{Z}}}(g(E))$.
The other case
$g^{*}(\mathcal{I}_{m_{\mathbf{Y}}}(E))=\mathcal{I}_{m_{\mathbf{Z}}}(g(E))$
follows by the same reasoning. Let $\gamma=(V_{i},i\in I)\in
g^{*}(\overline{\mathcal{I}}^{(q)}_{m_{\mathbf{Y}}}(E))$ and let
$(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ be a scaffolding-and-spindles pair
for $\gamma$ as in Definition LABEL:definition_starting_interval_partition.
Then there exists a modification of
$\overline{\text{skewer}}(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ that is
continuous on $[0,\infty)$. As in the proof of Theorem 1 first assume that
$g(E)$ is closed. Let $\gamma\in
g^{*}(\overline{\mathcal{I}}_{m_{\mathbf{Y}}}(E))=\overline{\mathcal{I}}_{m_{\mathbf{Z}}}(g(E))$
and let $\Tilde{\gamma}$ be a random interval partition such that
$\mathcal{D}_{\Tilde{\gamma}}^{m_{\mathbf{Z}}}(\infty)=\overline{\mathcal{D}}_{\gamma}^{m_{\mathbf{Z}}}(\infty)+1$
and that dominates $\gamma$ as in Lemma . We describe a coupling for a
scaffolding-and-spindles pair $(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ and
another stopped scaffolding-and-spindles pair $(\mathbf{N},\mathbf{X})$. Let
$(\Tilde{V}_{i},i\in\Tilde{I})$ be the intervals of $\Tilde{\gamma}$ and let
$I\subset\Tilde{I}$ so that $(V_{i},i\in I)$ denotes the matching intervals of
$\gamma$. For each $i\in I$, we consider an independent Brownian motion
$(B^{i}_{y},y\geq 0)$ and we define $f_{\Tilde{V}_{i}}$ and $f_{V_{i}}$ to be
strong solutions to the stochastic differential equations

$df_{\Tilde{V}_{i}}(y)=\text{Leb}(\Tilde{V}_{i})+\mu_{\mathbf{Z}}(f_{\Tilde{V}_{i}}(y))dy+\sigma_{\mathbf{Z}}(f_{\Tilde{V}_{i}}(y))dB^{i}_{y};\hskip
14.22636ptdf_{V_{i}}(y)=\text{Leb}(V_{i})+\mu_{\mathbf{Z}}(f_{V_{i}}(y))dy+\sigma_{\mathbf{Z}}(f_{V_{i}}(y))dB^{i}_{y},$

and we stop $f_{\Tilde{V}_{i}}$ and $f_{V_{i}}$ when they hit $0$. With this
construction we have that $(f_{\Tilde{V}_{i}},i\in I)$ are independent
diffusions with the same dynamics and with initial states $(\Tilde{V}_{i},i\in
I)$, and similarly for $(f_{V_{i}},i\in I)$. We also have for each $i\in I$
that $\zeta(f_{V_{i}})\leq\zeta(f_{\Tilde{V}_{i}})$ and for each $y>0$ that
$f_{V_{i}}(y)\leq f_{\Tilde{V}_{i}}(y)$. We then define a pair of point
measures $(\mathbf{N},\mathbf{N}_{\gamma})$. For $z>0$, let
$\mathbb{Q}^{0}_{z}$ denote the law of the diffusion $\mathbf{Z}$ started from
$z$, and let $\nu$ be the Pitman-Yor excursion measure of $\mathbf{Z}$.
Consider independent Poisson random measures $(\mathbf{N}_{i},i\in I)$ with
intensity $\text{Leb}\otimes\nu$ and scaffolding
$\xi(\mathbf{N}_{i})=\mathbf{X}_{i}$ for $i\in I$; we use this to define

$\mathbf{N}_{\Tilde{V}_{i}}\coloneqq\delta(0,f_{\Tilde{V}_{i}})+\mathbf{N}_{i}|_{[0,\Tilde{T}_{i}]}\hskip
7.11317pt\text{where}\hskip
7.11317pt\Tilde{T}_{i}\coloneqq\inf\\{t>0:\mathbf{X}_{i}(t)=-\zeta(f_{\Tilde{V}_{i}})\\}.$

We further define $\mathbf{N}_{\gamma}\coloneqq\bigstar_{i\in
I}\mathbf{N}_{V_{i}}$ where $\mathbf{N}_{V_{i}}$ is given by

$\mathbf{N}_{V_{i}}\coloneqq\delta(0,f_{V_{i}})+\mathbf{N}_{i}|^{\leftarrow}_{(T_{i},\Tilde{T_{i}}]}\hskip
7.11317pt\text{with}\hskip 7.11317ptT_{i}\coloneqq\inf\\{t\geq
0:\mathbf{X}_{i}(t)=\zeta(f_{V_{i}})-\zeta(f_{\Tilde{V}_{i}})\\}$ (22)

and $\Tilde{T_{i}}$ is as stated above. Let $T=\sum_{i\in I}T_{i}$ and note
$T$ is a $\mathbf{P}$-a.s. finite random variable. Finally let
$\mathbf{N}\coloneqq\bigstar_{i\in I}\mathbf{N}_{\Tilde{V}_{i}}$ and
$\mathbf{N}_{\gamma}\coloneqq\bigstar_{i\in I}\mathbf{N}_{V_{i}}$. The rest of
the proof uses the same ideas as the proof of Theorem 1. Let $q$, $\kappa>0$
such that $\alpha^{+}+\kappa<q<q_{0}$ where $\alpha^{+}$, $q_{0}$ are as in
Assumption LABEL:assumption_b and let
$\rho\coloneqq(q-\alpha^{+}-\kappa)\wedge q/2$. We recall
$\mathbf{N}_{M,\varepsilon_{0}}$ from equation n with m and epsilon and define
$\mathbf{N}_{\gamma,M,\varepsilon_{0}}$:

$\mathbf{N}_{M,\varepsilon_{0}}\coloneqq\mathbbm{1}_{\sup_{u\in\mathbb{R}}\ell^{u}(T)<M}\sum_{\substack{(t,f)\text{atomof}\\\
\mathbf{N}:A(f)<\varepsilon_{0}}}\delta_{(t,f)};\hskip
7.11317pt\mathbf{N}_{\gamma,M,\varepsilon_{0}}\coloneqq\mathbbm{1}_{\sup_{u\in\mathbb{R}}\ell^{u}(T)<M}\sum_{\substack{(t,f)\text{atomof}\mathbf{N}_{\gamma}:\\\
A(f)<\varepsilon_{0},f(0)=0}}\delta_{(t,f)}$

where $\ell^{u}$ for $u\in\mathbb{R}$ denotes the local time of
$\xi(\mathbf{N})$. We note that the set of atoms of
$\mathbf{N}_{\gamma,M,\varepsilon_{0}}$ is a subset of the set of atoms of
$\mathbf{N}_{M,\varepsilon_{0}}$ and so we can use the bound in equation sum
of differences x and y as $\mathbf{N}$ is a Poisson random measure stopped at
$\tau^{0}(\overline{\mathcal{D}}^{m_{\mathbf{Z}}}_{\gamma}(\infty)+1)$. The
constant in front of the bound in equation sum of differences x and y will
change from $A$ to $A/p$ where
$p\coloneqq\mathbf{P}(\Tilde{\gamma}\text{dominates}\gamma)>0$ where we
defined domination in Lemma  (note that the fact that the domination of
$\gamma$ by $\Tilde{\gamma}$ does not respect the order of the intervals in
the interval partition does not impact the upper bound for the sum). At this
point we need to incorporate the initial spindles and the spindles with
amplitude greater than $\varepsilon_{0}$. In the proof of Theorem 1, we only
had the spindles with amplitude greater than $\varepsilon_{0}$ to add in, and
as there were only finitely many of these we could concatenate finitely many
continuous IP-evolutions to finish the proof. In this setting however as we
have potentially infinitely many initial spindles we need to take care to
incorporate these. We adapt the proof of the Kolmogorov-Čentsov continuity
criterion from [revuz2013continuous, Theorem 1.2.1]. We define

$D_{m}\coloneqq\\{i2^{-m}:i\in\mathbb{N}\cap[0,2^{m})\\}\text{for}m\in\mathbb{N};\hskip
14.22636ptD\coloneqq\bigcup_{m\in\mathbb{N}}D_{m}$

and

$M_{\rho}\coloneqq\sup_{x,y\in D,x\neq
y}|x-y|^{-\rho}\Big{\\{}\sum_{\substack{(t,f)\text{atomof}\mathbf{N}_{M,\varepsilon_{0}}:,\\\
\mathbf{X}_{t-}<x<\mathbf{X}_{t}\text{or}\\\
\mathbf{X}_{t-}<y<\mathbf{X}_{t}}}\Big{|}g\big{(}f(y-\mathbf{X}_{t-})\big{)}-g\big{(}f(x-\mathbf{X}_{t-})\big{)}\Big{|}\Big{\\}}.$

Then it follows from the proof in [revuz2013continuous, Theorem 1.2.1] that
$M_{\rho}$ is finite $\mathbf{P}$-a.s. as we have shown the bound in equation
sum of differences x and y with $\mathbf{N}_{\gamma,M,\varepsilon_{0}}$ in
place of $\mathbf{N}_{M,\varepsilon_{0}}$. Let
$\mathbf{N}_{\gamma,M}\coloneqq\mathbbm{1}_{\sup_{u\in\mathbb{R}}\ell^{u}(T)<M}\mathbf{N}_{\gamma}$.
From this point we show that
$\overline{\text{skewer}}(\mathbf{N}_{\gamma,M},\mathbf{X}_{\gamma})$ is
$d^{\prime}_{H}$-continuous on $D$. Let $\varepsilon>0$, let $K\in\mathbb{N}$,
$i_{1},\dots,i_{K}\in I$ such that $\sum_{i\in I:i\neq
i_{1},\dots,i_{K}}A(f_{V_{i}})<\varepsilon/4$, and let
$K^{\prime}\in\mathbb{N}$ and
$(t_{1},f_{1}),\dots,(t_{K^{\prime}},f_{K^{\prime}})$ be the atoms of
$\mathbf{N}_{\gamma}$ with amplitude greater than or equal to
$\varepsilon_{0}$. There exists $\delta_{1}>0$ such that for $x$, $y\in D$
with $|x-y|<\delta_{1}$ we have

$\sum_{j=1}^{K}\Big{|}g(f_{V_{j}}(y))-g(f_{V_{j}}(x))\Big{|}<\frac{\varepsilon}{4};\hskip
14.22636pt\sum_{j=1}^{K^{\prime}}\Big{|}g\big{(}f_{j}(y-\mathbf{X}_{t_{j}-})\big{)}-g\big{(}f_{j}(x-\mathbf{X}_{t_{j}-})\big{)}\Big{|}<\frac{\varepsilon}{4}.$

Furthermore, for $x$, $y\in D$ with
$|x-y|<\delta_{2}\coloneqq(\varepsilon/4M_{\rho})^{1/\rho}$ we have

$\sum_{\substack{(t,f)\text{atomof}\mathbf{N}_{M,\varepsilon_{0}}:,\\\
\mathbf{X}_{t-}<x<\mathbf{X}_{t}\text{or}\\\
\mathbf{X}_{t-}<y<\mathbf{X}_{t}}}\Big{|}g\big{(}f(y-\mathbf{X}_{t-})\big{)}-g\big{(}f(x-\mathbf{X}_{t-})\big{)}\Big{|}<\frac{\varepsilon}{4}$

Let $\delta=\min\\{\delta_{1},\delta_{2}\\}$. Then for all $x$, $y\in D$ with
$|x-y|<\delta$ the sum of $|g(f(y-\mathbf{X}_{t-}))-g(f(x-\mathbf{X}_{t-}))|$
over all atoms $(t,f)$ in $\mathbf{N}_{\gamma,M}$ is less than $\varepsilon$
$\mathbf{P}$-a.s. This is an upper bound for the distortion between
$\text{skewer}(x,\mathbf{N}_{\gamma,M},\mathbf{X}_{\gamma})$ and
$\text{skewer}(y,\mathbf{N}_{\gamma,M},\mathbf{X}_{\gamma})$ for the
correspondence that matches intervals with each other if they both are
generated by the same spindle. This is a correspondence as we have only
finitely many spindles with lifetime greater than $|x-y|$; it is the same
correspondence used in Theorem 1. We can take $M$ to infinity at this stage as
by Proposition 1 the local time of $\mathbf{X}$ is continuous and therefore
bounded. Therefore we have that
$\overline{\text{skewer}}(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ is
continuous on $D$ $\mathbf{P}$-a.s. To finish, we can define for
$y\in[0,\infty)$

$\gamma^{y}\coloneqq\lim_{\substack{s\rightarrow y\\\ s\in
D}}\text{skewer}(s,\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$

and $(\gamma^{y},y\geq 0)$ is a continuous modification of
$\overline{\text{skewer}}(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$. This
modification is in the appropriate subspace
$(\mathcal{I}_{H}(g(E)),d^{\prime}_{H})$ as this subspace is closed. Now
assume $g(E)$ is open. To complete the argument in this case we avoid taking
modifications of interval partition evolutions with intervals great than
$\varepsilon_{0}$. As in Theorem 1 there are only finitely many spindles with
amplitude greater than $\varepsilon_{0}$ and so we can consider stopping times
in equation stopping times Sn with $\mathbf{N}_{\gamma}$ in place of
$\mathbf{N}$. Denoting these stopping times by
$(S_{n}^{\gamma},n\in\mathbb{N})$ we can obtain finitely many skewer processes
$(\overline{\text{skewer}}(\mathbf{N}_{\gamma}|_{(S_{n}^{\gamma},S_{n+1}^{\gamma})}),n\leq
P)$ for some $P\in\mathbb{N}$ and concatenate these with the skewer processes
generated by the individual spindles with amplitude larger than
$\varepsilon_{0}$ as in equation concatenate. The previous proposition shows
path continuity at $0$ for certain initial states but not for all possibles
ones. This is a partial response to the claim just after [construction,
Corollary 6.19] that states We believe that $d_{H}^{\prime}$-path-continuity
extends to time $0$ which is in reference to their branching interval
partition evolutions generated from self-similar diffusions. As their self-
similar diffusions satisfy our Assumption LABEL:assumption_b we have confirmed
that this belief is correct for at least some initial states. [Simple Markov
property under Assumption LABEL:assumption_b on $\Omega_{0}$] Consider a
diffusion under Assumption LABEL:assumption_b. Let $\gamma\in
g^{*}(\overline{\mathcal{I}}^{(q)}_{m_{\mathbf{Y}}}(E))$ and let
$(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ be a scaffolding-and-spindles pair
with initial state
$\text{skewer}(0,\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})=\gamma$ a.s. as in
Definition LABEL:definition_starting_interval_partition. Let
$(\gamma^{y},y\geq 0)$ be a modification of
$\overline{\text{skewer}}(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ that is
continuous on $[0,\infty)$ which exists by Proposition 1. Recall the
filtration $(\mathcal{F}_{\mathcal{I}}^{y},y\geq 0)$ from Corollary . We
define $(\Omega_{0},\mathcal{F},(\mathcal{F}_{\mathcal{I}}^{y},y\geq
0),\mathbb{P}_{\gamma})$, a filtered probability space, where $\Omega_{0}$ is
the set of all paths over $\mathcal{I}_{H}$ that are continuous on
$[0,\infty)$ and with initial states in
$g^{*}(\overline{\mathcal{I}}^{(q)}_{m_{\mathbf{Y}}})$,
$\mathcal{F}\coloneqq\bigcup_{y\geq 0}\mathcal{F}_{\mathcal{I}}^{y}$, and also
$\mathbb{P}_{\gamma}(\cdot)\coloneqq\mathbf{P}((\gamma^{y},y\geq 0)\in\cdot)$.
Finally, let $\mu$ be a probability distribution on interval partitions that
is supported on $g^{*}(\overline{\mathcal{I}}^{(q)}_{m_{\mathbf{Y}}}(E))$, and
define the probability measure $\mathbb{P}_{\mu}(\cdot)=\int_{\gamma\in
g^{*}(\mathcal{I}_{H}(E))}\mathbb{P}_{\gamma}(\cdot)\mu(d\gamma)$. Then
$(\gamma^{y},y\geq 0)$ is a simple Markov process on
$(\Omega_{0},\mathcal{F},(\mathcal{F}_{\mathcal{I}}^{y},y\geq
0),\mathbb{P}_{\mu})$. This proof uses the same arguments as the analogous
result [construction, Corollary 6.13]. In particular, we have continuity of
the sample paths from Proposition 1 with initial states in
$g^{*}(\overline{\mathcal{I}}^{(q)}_{m_{\mathbf{Y}}}(E))$ as well as the
simple Markov property for the skewer process given in Proposition
LABEL:proposition_simple_markov_skewer. Furthermore, at each level we have,
for $y\in(0,\infty)$, that the $\text{skewer}(y,\mathbf{N},\mathbf{X})\in
g^{*}(\mathcal{I}^{(q)}_{m_{\mathbf{Y}}}(E))\subset
g^{*}(\overline{\mathcal{I}}^{(q)}_{m_{\mathbf{Y}}}(E))$ a.s. by Corollary 1.
Therefore, for any fixed $y\in(0,\infty)$, the continuous modification takes
values in $g^{*}(\overline{\mathcal{I}}^{(q)}_{m_{\mathbf{Y}}}(E))$ a.s. as
required.

##  Strong Markov property and proof of Theorem LABEL:theorem_diffusion

In this section we assume that the diffusion satisfies Strong Assumption
LABEL:assumption_b, that is Assumption LABEL:assumption_b along with the
assumption that either $\liminf_{x\rightarrow 0}g(x)/x>0$ or
$\limsup_{x\rightarrow 0}g(x)/x<\infty$. Note that the identity transformation
$g(x)=x$ satisfies these conditions, and therefore Strong Assumption
LABEL:assumption_b is more restrictive than Assumption LABEL:assumption_b but
more general than Assumption 1. Before we begin the proof we give a class of
transformations $g$ that do not satisfy $B5$ of Strong Assumption
LABEL:assumption_b to show that this really is a restriction. [Transformation
$g$ that does not satisfy $B5$] Consider a function
$g:[0,1/32]\rightarrow[0,1/32]$ such that for all $n\geq 3$ we have:

$g(2^{-2^{n}})=2^{-2^{n}};\hskip 7.11317ptg(2^{-2^{n}-1})=2^{-2^{n}-n};\hskip
7.11317ptg(2^{-2^{n+1}+1})=2^{-2^{n+1}+n},$

and where for all $n\geq 3$ $x\in[2^{-2^{n+1}},2^{-2^{n+1}+1}]$,
$[2^{-2^{n+1}+1},2^{-2^{n}-1}]$, $[2^{-2^{n}-1},2^{-2^{n}}]$ that $g$ is
smooth and strictly increasing and $g(0)=0$. Then $g$ is a smooth continuous
strictly increasing bijection and moreover

$\liminf_{x\downarrow
0}\frac{g(x)}{x}\leq\lim_{n\rightarrow\infty}\frac{g(2^{-2^{n}-1})}{2^{-2^{n}-1}}=\lim_{n\rightarrow\infty}2^{-n+1}=0;\hskip
7.11317pt\limsup_{x\downarrow
0}\frac{g(x)}{x}\geq\lim_{n\rightarrow\infty}\frac{g(2^{-2^{n+1}+1})}{2^{-2^{n+1}+1}}=\lim_{n\rightarrow\infty}2^{n-1}=\infty.$

We state key results about the transformations $g^{*}$ and $(g^{-1})^{*}$
which apply under Strong Assumption LABEL:assumption_b. Let $g:E\rightarrow
g(E)$ be a strictly increasing continuous function with $g(0)=0$ where
$E\subseteq[0,\infty)$ is of the form $E=[0,c)$ or $E=[0,c]$ for some $c>0$
and $g(E)\subseteq[0,\infty)$. If $\limsup_{x\rightarrow 0}g(x)/x<\infty$ then
the map $g^{*}:\mathcal{I}_{H}(E)\rightarrow\mathcal{I}_{H}(g(E))$ is
injective and continuous. If instead $\liminf_{x\rightarrow 0}g(x)/x>0$ then
the map $(g^{-1})^{*}:\mathcal{I}_{H}(g(E))\rightarrow\mathcal{I}_{H}(E)$ is
injective and continuous. First suppose that $\limsup_{x\rightarrow
0}g(x)/x<\infty$. Let $\varepsilon>0$, let $\beta\in\mathcal{I}_{H}(E)$ and
let $(a_{j})_{j\in\mathbb{N}}$ be the interval lengths of $\beta$ in a non-
decreasing order and let $(U_{j},j\in\mathbb{N})$ be the corresponding
intervals of $\beta$. If $g(c)$ is finite let $D=(a_{1}+g(c))/2$ and if $g(c)$
is infinite let $D=a_{1}+1$. Let $C\coloneqq\sup_{x\in(0,D]}g(x)/x$. We note
that $\sum_{j\in\mathbb{N}}g(a_{j})\leq C\sum_{j\in\mathbb{N}}a_{j}$ which is
finite so $g^{*}(\beta)$ is a non-trivial interval partition. The fact that
$g^{*}$ is injective then follows immediately from $g$ being injective. Let
$K\in\mathbb{N}$ be large enough so that $\sum_{j>K}a_{j}<\varepsilon/4C$. As
$g$ is continuous, then $g$ is uniformly continuous on $[0,D]$, and therefore
let $\Tilde{\varepsilon}>0$ be small enough so that for $x$, $y\in[0,D]$ with
$|x-y|<\Tilde{\varepsilon}$ we have $|g(x)-g(y)|<\varepsilon/2K$. Let
$\delta\coloneqq\frac{1}{2}\min\\{\Tilde{\varepsilon},a_{K}/2,\varepsilon/4C,D-a_{1}\\}$.
Let $\gamma\in\mathcal{I}_{H}(E)$ such that
$d^{\prime}_{H}(\beta,\gamma)<\delta$. Consider a correspondence
$(\Tilde{U}_{j},\Tilde{V}_{j})_{1\leq j\leq\Tilde{K}}$ from $\beta$ to
$\gamma$ with distortion less than $\delta$. Since $\delta<a_{K}$, we must
have that $\Tilde{K}\geq K$ and $\\{U_{j}\\}_{1\leq j\leq
K}\subseteq\\{\Tilde{U}_{j}\\}_{1\leq j\leq\Tilde{K}}$. Let $(V_{j})_{1\leq
j\leq K}$ denote the intervals of $\gamma$ paired in the correspondence with
the respective $U_{j}$, and let $b_{j}\coloneqq\text{Leb}(V_{j})$. Consider
the correspondence of $g^{*}(\beta)$ and $g^{*}(\gamma)$ that uses the
corresponding interval pairs to $(U_{j},V_{j})_{1\leq j\leq K}$. This has
distortion

$\sum_{j=1}^{K}|g(a_{j})-g(b_{j})|+\max\Big{\\{}\norm{g^{*}(\beta)}-\sum_{j=1}^{K}g(a_{j}),\norm{g^{*}(\gamma)}-\sum_{j=1}^{K}g(b_{j})\Big{\\}}.$

First, let $1\leq j\leq K$. Then $|a_{j}-b_{j}|<\tilde{\varepsilon}$ and
therefore we have $|g(a_{j})-g(b_{j})|<\varepsilon/2K$. Also note that
$\norm{\beta}-\sum_{j=1}^{K}a_{j}<\varepsilon/4C$, and therefore we have

$\norm{\gamma}-\sum_{j=1}^{K}b_{j}\leq
d^{\prime}_{H}(\beta,\gamma)+\norm{\beta}-\sum_{j=1}^{K}a_{j}<\frac{\varepsilon}{2C}.$

From this we can conclude that

$\max\Big{\\{}\norm{g^{*}(\beta)}-\sum_{j=1}^{K}g(a_{j}),\norm{g^{*}(\gamma)}-\sum_{j=1}^{K}g(b_{j})\Big{\\}}<\frac{\varepsilon}{2}$

and the first result follows. To prove the second result, we show that if
$\liminf_{x\rightarrow 0}g(x)/x>0$ then

$\limsup_{x\rightarrow 0}\frac{g^{-1}(x)}{x}\leq\Big{(}\liminf_{y\rightarrow
0}\frac{g(y)}{y}\Big{)}^{-1}<\infty$

and the result then follows as previously. To see this let $(x_{n},n\geq 1)$
be a positive sequence that converges to $0$. Then $(g^{-1}(x_{n}),n\geq 1)$
is also a sequence that converges to $0$. Therefore we have that

$\liminf_{n\rightarrow\infty}\frac{x_{n}}{g^{-1}(x_{n})}=\liminf_{n\rightarrow\infty}\frac{g(g^{-1}(x_{n}))}{g^{-1}(x_{n})}\geq\liminf_{y\rightarrow
0}\frac{g(y)}{y}>0.$

Therefore $\limsup_{n\rightarrow\infty}g^{-1}(x_{n})/x_{n}\leq
1/(\liminf_{y\rightarrow 0}g(y)/y)<\infty$ and the second result follows in
the same way as the first result. In fact the conditions in Lemma 1 are
necessary as well as sufficient in the sense that if $\limsup_{x\rightarrow
0}g(x)/x=\infty$ then the map $g^{*}$ is not injective and if
$\liminf_{x\rightarrow 0}g(x)/x=0$ then the map $(g^{-1})^{*}$ is not
injective (they both map certain interval partitions to $\emptyset$ as the
transformed intervals are not summable). We now give certain bounds for the
diffusion under Strong Assumption LABEL:assumption_b which we will use to
prove continuity in the initial state in Proposition . Consider a diffusion
$\mathbf{Z}$ that satisfies Strong Assumption LABEL:assumption_b with
untransformed diffusion $\mathbf{Y}$ and spatial transformation $g$ such that
$\liminf_{x\rightarrow 0}g(x)/x>0$, and let $z>0$ and let $\varepsilon>0$.
Then there exists $k_{7}=k_{7}(\varepsilon,z)>0$ such that for any interval
partition $\gamma\in\mathcal{I}_{H}(g(E))$ such that $\norm{\gamma}<k_{7}$ we
have that

$\mathbf{P}(\zeta^{+}(\mathbf{X}_{\gamma})<z)>1-\varepsilon,$

where $\mathbf{X}_{\gamma}$ is constructed as in Definition
LABEL:definition_starting_interval_partition and
$\zeta^{+}(X)\coloneqq\sup_{t\in[0,\text{len}(X)]}X(t)$ for $X\in\mathcal{D}$.
Let $\varepsilon$, $z>0$. The injectivity of $(g^{-1})^{*}$ from Lemma 1 leads
to the surjectivity of $g^{*}$ and so
$g^{*}(\mathcal{I}_{H}(E))=\mathcal{I}_{H}(g(E))$. Also we note that the
untransformed diffusion satisfies Corollary
LABEL:corollary_assumption_b'_starting_interval_partition. Therefore we can
create a scaffolding-and-spindles pair
$(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ with an initial interval partition
$\gamma\in\mathcal{I}_{H}(g(E))$ as in Definition
LABEL:definition_starting_interval_partition where we construct the
scaffolding by adding $\zeta(f_{V_{i}})$ jumps for each interval $V_{i}$ of
the initial interval partition $\gamma$. We will compare this to an SPLP with
a single initial jump $P=\sum_{U\in\gamma}\zeta(f_{U})$ at the beginning of
the process which we run until it hits level $0$ and we will denote this
process as $\mathbf{X}_{P}$. By a natural coupling we have
$\mathbf{X}_{P}(t)\geq\mathbf{X}_{\gamma}(t)$ for all $t\geq 0$ and therefore
the event $\\{\zeta^{+}(\mathbf{X}_{\gamma})>z\\}$ is a subset of by
$\\{\zeta^{+}(\mathbf{X}_{P})>z\\}$. We construct $\mathbf{X}_{P}$ by
considering a Lévy process started at $\sum_{U\in\beta}\zeta(f_{U})$ such that
$\mathbf{X}_{P}-\sum_{U\in\beta}\zeta(f_{U})$ has the same distribution as an
SPLP $\mathbf{X}$ with Lévy measure $\nu(\zeta\in\cdot)$. Let $W$ be the scale
function for the Lévy process $\mathbf{X}$. This function is continuous and
increasing with $W(0)=0$, [bertoin1996levy, Theorem VII.8], and therefore
there exists $x>0$ such that $W(x)/W(z)\leq\varepsilon/2$. Also note that for
the empty interval partition $\emptyset$ we have
$d^{\prime}_{H}(\gamma,\emptyset)=\norm{\gamma}$. Recall that we can apply
Proposition LABEL:proposition_starting_interval_partition to the untransformed
diffusion by Corollary
LABEL:corollary_assumption_b'_starting_interval_partition. By Lemma 1
$(g^{-1})^{*}$ is continuous at $\emptyset$ and therefore there exists
$k_{7}>0$ such that for all $\gamma\in\mathcal{I}_{H}(g(E))$ with
$\norm{\gamma}<k_{7}$ we have $\norm{(g^{-1})^{*}(\gamma)}<\varepsilon
x/2(1+k_{3})$ where $k_{3}$ is constant from Proposition
LABEL:proposition_starting_interval_partition. Recall $\varepsilon^{\prime}$
from the same proposition and take $k_{7}<g(\varepsilon^{\prime})$. Then, by
equation sum of starting diffusion lifetimes, we therefore have that

$\mathbf{E}\Big{(}\sum_{i\in I}\zeta(f_{V_{i}})\Big{)}\leq(1+k_{3})\sum_{i\in
I}g^{-1}(\text{Leb}(V_{i}))=(1+k_{3})\norm{(g^{-1})^{*}(\gamma)}.$

Therefore, for $\gamma\in\mathcal{I}_{H}(g(E))$ such that $||\gamma||<k_{7}$,
we have that

$\mathbf{P}\big{(}\zeta^{+}(\mathbf{X}_{\gamma})>z\big{)}\leq\mathbf{P}\big{(}\zeta^{+}(\mathbf{X}_{P})>z\big{)}\leq\mathbf{P}\Big{(}\sum_{i\in
I}\zeta(f_{V_{i}})>x\Big{)}+\frac{W(x)}{W(z)}\leq\varepsilon.$

In the following lemma, for $x\in g(E)$, let $\mathbb{Q}^{0}_{x}(\cdot)$
denote the law of the diffusion under Assumption LABEL:assumption_b. Consider
a diffusion under Assumption LABEL:assumption_b, let $D\in g(E)$ and let
$d_{0}$, $d_{1}>0$. Then there exists $d_{2}>0$ such that for all $a$,
$b\in[0,D]$ with $a<b<a+d_{2}$ we have

$\mathbb{Q}^{0}_{b}(T_{a}<d_{1})>1-d_{0}.$

This result uses the Markov inequality. We first note that

$\mathbb{Q}^{0}_{b}(T_{a}>d_{1})\leq\frac{\mathbb{E}_{\mathbb{Q}^{0}_{b}}(T_{a})}{d_{1}}$

and recall from Corollary
LABEL:corollary_assumption_b'_starting_interval_partition that the function
$r(x):=\mathbb{E}_{\mathbb{Q}^{0}_{g^{-1}(x)}}(\zeta)$ has a derivative that
is bounded above near $0$ (we have the transformation $g^{-1}$ here as the
corollary we use only applies to untransformed diffusions). Therefore, setting
$C:=\sup_{x\in(0,g^{-1}(D))}r^{\prime}(x)$ we have for all $0\leq a<b\leq D$

$\mathbb{E}_{\mathbb{Q}^{0}_{b}}(T_{a})=r(g^{-1}(b))-r(g^{-1}(a))\leq
C|g^{-1}(b)-g^{-1}(a)|.$

Now, $g^{-1}$ is uniformly continuous on $[0,D]$ and so there exists $d_{2}>0$
such that for $a$, $b\in[0,D]$ with $b>a$ and $b-a<d_{2}$ we have
$g^{-1}(b)-g^{-1}(a)<d_{0}d_{1}/C$ and the result follows. [Continuity in the
initial state for $\overline{\text{skewer}}(\mathbf{N}_{U},\mathbf{X}_{U})$]
Let $d_{3}$, $d_{4}$, $z>0$, $D\in g(E)$ and let $a\in[0,D]$. Then there
exists $d_{5}>0$ such that for $b\in[0,D]$ with $|a-b|<d_{5}$ we have the
following. Let $U=(0,a)$ and $V=(0,b)$. Then, on some probability space
$(\Omega,\mathcal{A},\mathbf{P})$, there exists a coupled pair of scaffolding
and spindles $(\mathbf{N}_{U},\mathbf{X}_{U})$,
$(\mathbf{N}_{V},\mathbf{X}_{V})$ started from $U$ and $V$ as in Definition
LABEL:definition_starting_interval_partition on which we have

$\mathbf{P}(d^{\prime}_{H}(\text{skewer}(z,\mathbf{N}_{U},\mathbf{X}_{U}),\text{skewer}(z,\mathbf{N}_{V},\mathbf{X}_{V}))>d_{3})<d_{4}.$

Let $d_{3}$, $d_{4}$, $z>0$ and let $a\leq D\in g(E)$. Let $d_{6}>0$ be small
enough that $W(z-2d_{6})/W(z)>1-d_{4}/5$ where $W$ is the scale function for
the SPLP from Lemma 1. Let $d_{7}=d_{2}/2$ where $d_{2}$ is the constant
obtained from Lemma 1 when applied with $d_{0}=d_{4}/5$ and $d_{1}=d_{6}$. Let
$f$ be a $0$-diffusion started at $(a+d_{7})\wedge D$ and note that

$\mathbf{P}\Big{(}G_{0}\coloneqq\Big{\\{}T_{(a-d_{7})\vee
0}(f)<d_{6}\Big{\\}}\Big{)}>1-\frac{d_{4}}{5}$

where $T_{(a-d_{7})\vee 0}(f)$ is the hitting time of $(a-d_{7})\vee 0$ by
$f$. The diffusion $f$ has a finite lifetime and so is uniformly continuous
and therefore there exists $d_{8}>0$ such that

$\mathbf{P}\Big{(}G_{1}\coloneqq\Big{\\{}|f(x)-f(y)|<\frac{d_{3}}{2}\hskip
7.11317pt\forall
x,y\in\mathbb{R}:|x-y|<d_{8}\Big{\\}}\Big{)}>1-\frac{d_{4}}{5}.$

Now consider an independent Poisson random measure $\mathbf{N}$ with
scaffolding $\mathbf{X}$ which is stopped at $T_{-\zeta(f)}$. First note that

$\mathbf{P}\Big{(}G_{2}\coloneqq\Big{\\{}\sup_{t\in[T_{-\zeta(f)+2d_{6}},T_{-\zeta(f)}]}\mathbf{X}_{t}\leq
z-\zeta(f)\Big{\\}}\Big{)}\leq\frac{W(z-2d_{6})}{W(z)}>1-\frac{d_{4}}{5}$

by the independence of $f$ and $\mathbf{N}$ and the strong Markov property of
$\mathbf{X}$ and where we follow convention that the supremum over the empty
set is $-\infty$. Furthermore note that
$(\text{skewer}(y,\mathbf{N},\mathbf{X}),y\in\mathbb{R})$ has a uniformly
continuous modification $(\gamma^{y},y\in\mathbb{R})$ by Theorem 1. Let
$d_{9}>0$ be small enough so that

$\mathbf{P}\Big{(}G_{3}\coloneqq\Big{\\{}d^{\prime}_{H}(\gamma^{x},\gamma^{y})<d_{3}/2\text{forall}x,y\in\mathbb{R}:|x-y|<d_{9}\Big{\\}}\Big{)}>1-\frac{d_{4}}{5}.$

Now, let $d_{10}=d_{2}/2$ where $d_{2}$ is now the constant obtained from
Lemma 1 when applied with $d_{1}=d_{8}\wedge d_{9}$ and $d_{0}=d_{4}/5$ and
let $d_{5}\coloneqq d_{7}\wedge d_{10}$. Then we have

$\mathbf{P}\Big{(}G_{4}\coloneqq\Big{\\{}T_{(a-d_{5})\vee
0}(f)-T_{(a+d_{5})\wedge D}(f)<d_{8}\wedge
d_{9}\Big{\\}}\Big{)}>1-\frac{d_{4}}{5}.$

Now let $b\in[0,D]$ with $|a-b|<d_{5}$. Let $f_{U}(\cdot)\coloneqq
f(T_{a}+\cdot)$ and let $f_{V}(\cdot)\coloneqq f(T_{b}+\cdot)$. We define

$\Tilde{\mathbf{N}}_{U}\coloneqq\mathbf{N}|_{[0,T_{-\zeta(f_{U})}(\mathbf{N})]};\hskip
7.11317pt\mathbf{N}_{U}\coloneqq\delta_{(0,f_{U})}+\Tilde{\mathbf{N}}_{U};\hskip
7.11317pt\Tilde{\mathbf{N}}_{V}\coloneqq\mathbf{N}|_{[0,T_{-\zeta(f_{V})}(\mathbf{N})]};\hskip
7.11317pt\mathbf{N}_{V}\coloneqq\delta_{(0,f_{V})}+\Tilde{\mathbf{N}}_{V}.$

Note that

$\mathbf{P}\Big{(}G_{5}\coloneqq\Big{\\{}\gamma^{z-\zeta(f_{U})}=\text{skewer}(z-\zeta(f_{U}),\mathbf{N},\mathbf{X}),\hskip
7.11317pt\gamma^{z-\zeta(f_{V})}=\text{skewer}(z-\zeta(f_{V}),\mathbf{N},\mathbf{X})\Big{\\}}\Big{)}=1$

by the spatial homogeneity of the scaffolding-and-spindles pair and the
independence of $f_{U}$ and $\mathbf{N}$ and of $f_{V}$ and $\mathbf{N}$.
Therefore we have

$\text{skewer}(z,\Tilde{\mathbf{N}}_{U},\mathbf{X}_{U})=\gamma^{z-\zeta(f_{U})},\hskip
7.11317pt\text{skewer}(z,\Tilde{\mathbf{N}}_{V},\mathbf{X}_{V})=\gamma^{z-\zeta(f_{V})}\text{on}G_{0}\cap
G_{2}\cap G_{5}.$

Furthermore we have that

$|f_{U}(z)-f_{V}(z)|<d_{3}/2\text{on}G_{1}\cap G_{4};\hskip
7.11317pt\text{and}\hskip
7.11317ptd^{\prime}_{H}(\gamma^{z-\zeta(f_{U})},\gamma^{z-\zeta(f_{V})})<d_{3}/2\text{on}G_{3}\cap
G_{4}.$

Therefore, as

$d^{\prime}_{H}(\text{skewer}(z,\mathbf{N}_{U},\mathbf{X}_{U}),\text{skewer}(z,\mathbf{N}_{V},\mathbf{X}_{V}))\leq|f_{U}(z)-f_{V}(z)|+d^{\prime}_{H}(\text{skewer}(z,\Tilde{\mathbf{N}}_{U},\mathbf{X}_{U}),\text{skewer}(z,\Tilde{\mathbf{N}}_{V},\mathbf{X}_{V}))$

then it follows that

$d^{\prime}_{H}(\text{skewer}(z,\mathbf{N}_{U},\mathbf{X}_{U}),\text{skewer}(z,\mathbf{N}_{V},\mathbf{X}_{V}))<d_{3}\text{on}G\coloneqq
G_{0}\cap G_{1}\cap G_{2}\cap G_{3}\cap G_{4}\cap G_{5},$

and as $\mathbf{P}(G)>1-d_{4}$ the result follows. [Continuity in the initial
state] Consider a diffusion under Strong Assumption LABEL:assumption_b for
which $\liminf_{x\rightarrow 0}g(x)/x>0$. Let $z>0$ and let
$f:\mathcal{I}_{H}\rightarrow[0,\infty)$ be a bounded and continuous function.
Then the map $\gamma\mapsto\mathbb{E}_{\gamma}(f(\gamma^{z}))$ is continuous
on $(\mathcal{I}_{H}(g(E)),d^{\prime}_{H})$. This was shown for self-similar
diffusions in [construction, Proposition 6.15]. We adapt the proof to
generalise to our setting. As in Lemma 1 we are under the assumption
$\liminf_{x\rightarrow 0}g(x)/x>0$ and so
$g^{*}(\mathcal{I}_{H}(E))=\mathcal{I}_{H}(g(E))$. Let
$\gamma\in\mathcal{I}_{H}(g(E))$, $\varepsilon$, $z>0$, and let
$(\Omega,\mathcal{F}_{\mathcal{I}},\mathbb{P}_{\gamma})$ be the probability
space as defined in Section 1, where
$\Omega\coloneqq\mathcal{C}((0,\infty),\mathcal{I}_{H})$,
$\mathbb{P}_{\gamma}$ is the probability measure for $(\gamma^{y},y\geq 0)$
and $\mathcal{F}_{\mathcal{I}}$ is the natural filtration for
$(\gamma^{y},y\geq 0)$. As $f$ is continuous at $\gamma^{z}$ we can define a
$(0,\infty)$-valued random variable $A_{\varepsilon}(\gamma^{z})$ on
$(\Omega,\mathcal{F}_{\mathcal{I}},\mathbb{P}_{\gamma})$ for which we have
that for $\beta\in\mathcal{I}_{H}(g(E))$ we have
$d^{\prime}_{H}(\gamma^{z},\beta)<A_{\varepsilon}(\gamma^{z})$ implies that
$|f(\gamma^{z})-f(\beta)|<\varepsilon/3$. Let
$M\coloneqq\sup_{\beta\in\mathcal{I}_{H}(g(E))}f(\beta)$. Then there is
$\Tilde{\varepsilon}>0$ such that
$\mathbb{P}_{\gamma}(A_{\varepsilon}(\gamma^{z})<\Tilde{\varepsilon})<\varepsilon/3M$.
We take $\Hat{\gamma}\in\mathcal{I}_{H}(g(E))$ and consider a probability
space $(\Tilde{\Omega},\mathcal{A},\mathbf{P})$ for a pair of processes
$(\gamma^{y},y\geq 0)$ and $(\Hat{\gamma}^{y},y\geq 0)$ which have marginal
distributions $\mathbb{P}_{\gamma}$ and $\mathbb{P}_{\Hat{\gamma}}$
respectively; we will later specify an explicit coupling. By Lemma 1 note that
$g^{*}$ is surjective so as the untransformed diffusion satisfies Corollary
LABEL:corollary_assumption_b'_starting_interval_partition it is possible to
construct IP-evolutions $(\gamma^{y},y\geq 0)$ and $(\hat{\gamma}^{y},y\geq
0)$ with initial states $\gamma$ and $\hat{\gamma}$. We first note that

$\Big{|}\mathbb{E}_{\gamma}(f(\gamma^{z}))-\mathbb{E}_{\Hat{\gamma}}(f(\Hat{\gamma}^{z}))\Big{|}\leq\mathbf{E}\Big{(}|f(\gamma^{z})-f(\Hat{\gamma}^{z})|\Big{)}.$

Then we can further bound the expectation above by

$\mathbf{E}\Big{(}|f(\gamma^{z})-f(\Hat{\gamma}^{z})|\Big{(}\mathbbm{1}_{d^{\prime}_{H}(\gamma^{z},\Hat{\gamma}^{z})<\Tilde{\varepsilon},A_{\varepsilon}(\gamma^{z})\geq\Tilde{\varepsilon}}+\mathbbm{1}_{d^{\prime}_{H}(\gamma^{z},\Hat{\gamma}^{z})\geq\Tilde{\varepsilon}}+\mathbbm{1}_{A_{\varepsilon}(\gamma^{z})<\Tilde{\varepsilon}}\Big{)}\Big{)}.$

The first of the three terms is less than or equal to $\varepsilon/3$; the
second is less than or equal to
$M\mathbf{P}(d^{\prime}_{H}(\gamma^{z},\Hat{\gamma}^{z})\geq\Tilde{\varepsilon})$;
and the third is less than or equal to $\varepsilon/3$. Therefore it suffices
to show that there is some $\delta>0$ such that for
$\Hat{\gamma}\in\mathcal{I}_{H}(g(E))$ with
$d^{\prime}_{H}(\gamma,\Hat{\gamma})<\delta$ we have a coupled pair of
evolutions $(\gamma^{y},y\geq 0)$ and $(\Hat{\gamma}^{y},y\geq 0)$ starting
from these two initial states, with

$\mathbf{P}(d^{\prime}_{H}(\gamma^{z},\Hat{\gamma}^{z})>\Tilde{\varepsilon})<\frac{\varepsilon}{3M}.$
(23)

To this end, let $U_{1}$, $U_{2}$, $\dots$ denote the blocks of $\gamma$,
listed in non-increasing order by mass and let
$a_{j}\coloneqq\text{Leb}(U_{j})$. Let $k_{7}(\varepsilon/9M,z)$ be the
constant from Lemma 1 and let $K$ be large enough that
$\sum_{j>K}a_{j}<\frac{1}{2}k_{7}$. If $a_{1}<g(c)$ then take $D\in g(E)$ such
that $a_{1}<D<g(c)$ and if $a_{1}=g(c)$ then take $D=a_{1}=g(c)$. For $1\leq
j\leq K$ let $\delta_{j}=d_{5}$ as in Lemma  with
$d_{3}=\Tilde{\varepsilon}/K$, $d_{4}=\varepsilon/9MK$, $D$ as above and
$a=a_{j}$. Now let

$\delta<\left\\{\matrix{\min}\\{\delta_{1},\dots,\delta_{K},a_{K},\frac{k_{7}}{2},D-a_{1}\\}\text{if}a_{1}<D<g(c),\\\
\min\\{\delta_{1},\dots,\delta_{K},a_{K},\frac{k_{7}}{2}\\}\text{ifinstead}a_{1}=D=g(c).\\\
\right.$

Let $\Hat{\gamma}\in\mathcal{I}_{H}(g(E))$ with
$d^{\prime}_{H}(\gamma,\Hat{\gamma})<\delta$. By definition of
$d^{\prime}_{H}$, there exists a correspondence
$(\Tilde{U}_{j},\Tilde{V}_{j})_{1\leq j\leq\Tilde{K}}$ from $\gamma$ to
$\Hat{\gamma}$ with distortion less than $\delta$. Since $\delta<a_{K}$, we
get $\Tilde{K}\geq K$ and $\\{U_{j}\\}_{1\leq j\leq
K}\subset\\{\Tilde{U}_{j}\\}_{1\leq j\leq\Tilde{K}}$. Let $(V_{j})_{1\leq
j\leq K}$ denote the terms paired with the respective $U_{j}$ in the
correspondence. For $1\leq j\leq K$, let $b_{j}\coloneqq\text{Leb}(V_{j})$.
Let $(V_{j},j>K)$ be the remaining intervals of $\Hat{\gamma}$. Note that
necessarily we have $|a_{j}-b_{j}|<\delta$ for $1\leq j\leq K$. Consider
coupled scaffolding and spindles pairs
$(\mathbf{N}_{U_{j}},\mathbf{X}_{U_{j}})$,
$(\mathbf{N}_{V_{j}},\mathbf{X}_{V_{j}})$ independently for each $1\leq j\leq
K$ as in Lemma . For $j>K$ we consider independent scaffolding and spindles
$(\mathbf{N}_{U_{j}},\mathbf{X}_{U_{j}})$ and
$(\mathbf{N}_{V_{j}},\mathbf{X}_{V_{j}})$. We construct coupled scaffolding
and spindles $(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ and
$(\mathbf{N}_{\Hat{\gamma}},\mathbf{X}_{\Hat{\gamma}})$ constructed from
coupled scaffolding and spindles $(\mathbf{N}_{U_{j}},\mathbf{X}_{U_{j}})$ and
$(\mathbf{N}_{V_{j}},\mathbf{X}_{V_{j}})$ for $j\in\mathbb{N}$ as in
Definition LABEL:definition_starting_interval_partition. Let
$(\gamma^{y},y\geq 0)$ and $(\Hat{\gamma}^{y},y\geq 0)$ be the modifications
of $\overline{\text{skewer}}(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ and
$\overline{\text{skewer}}(\mathbf{N}_{\Hat{\gamma}},\mathbf{X}_{\Hat{\gamma}})$
that are continuous on $(0,\infty)$ which are shown to exist in Lemma 1. First
we make the obvious note that

$\mathbf{P}\Big{(}F_{0}\coloneqq\Big{\\{}\gamma^{z}=\text{skewer}(z,\mathbf{N}_{\gamma},\mathbf{X}_{\gamma});\hskip
7.11317pt\Hat{\gamma}^{z}=\text{skewer}(z,\mathbf{N}_{\Hat{\gamma}},\mathbf{X}_{\Hat{\gamma}})\Big{\\}}\Big{)}=1$

as $(\gamma^{y},y\geq 0)$ and $(\Hat{\gamma}^{y},y\geq 0)$ are modifcations of
the skewer processes. Also, for $1\leq j\leq K$, we have
$|a_{j}-b_{j}|<\delta_{j}$ and so

$\mathbf{P}\Big{(}F_{1}^{j}\coloneqq\Big{\\{}d^{\prime}_{H}(\text{skewer}(z,\mathbf{N}_{U_{j}},\mathbf{X}_{U_{j}}),\text{skewer}(z,\mathbf{N}_{V_{j}},\mathbf{X}_{V_{j}}))<\frac{\Tilde{\varepsilon}}{K}\Big{\\}}\Big{)}>1-\frac{\varepsilon}{9MK}\text{for}1\leq
j\leq K.$

Furthermore we have

$\mathbf{P}\Big{(}F_{2}\coloneqq\Big{\\{}\forall
j>K,\zeta^{+}(\mathbf{N}_{U_{j}})<z\Big{\\}}\Big{)}\geq
1-\frac{\varepsilon}{9M}\text{byLemma\ref{lemma all under z}.}$

Also we have that

$||\Hat{\gamma}||-\sum_{j=1}^{K}b_{j}\leq
d^{\prime}_{H}(\gamma,\Hat{\gamma})+||\gamma||-\sum_{j=1}^{K}a_{j}<\delta+\frac{1}{2}k_{7}<k_{7}.$

Therefore

$\mathbf{P}\Big{(}F_{3}\coloneqq\Big{\\{}\zeta^{+}(\mathbf{N}_{V_{j}})<z,\forall
j>K\Big{\\}}\Big{)}\geq 1-\frac{\varepsilon}{9M}\text{againbyLemma\ref{lemma
all under z}.}$

Pulling all these together we conclude

$\mathbf{P}\Big{(}F\coloneqq
F_{0}\cap\Big{(}\bigcap_{j=1}^{K}F_{1}^{j}\Big{)}\cap F_{2}\cap
F_{3}\Big{)}\geq 1-\frac{\varepsilon}{3M},$

and we now show that
$d^{\prime}_{H}(\gamma^{z},\Hat{\gamma}^{z})<\Tilde{\varepsilon}$ on $F$ to
finish the proof. First we note that on $F_{2}$ $\gamma^{z}$ consists only of
intervals from spindles from $\mathbf{N}_{U_{j}}$ for $1\leq j\leq K$ and
similarly for $\Hat{\gamma}^{z}$ on $F_{3}$ and therefore

$d^{\prime}_{H}(\gamma^{z},\Hat{\gamma}^{z})\leq\sum_{j=1}^{K}d^{\prime}_{H}(\text{skewer}(z,\mathbf{N}_{U_{j}},\mathbf{X}_{U_{j}}),\hskip
3.55658pt\text{skewer}(z,\mathbf{N}_{V_{j}},\mathbf{X}_{V_{j}}))\text{on}F_{0}\cap
F_{2}\cap F_{3},$

and so $d^{\prime}_{H}(\gamma^{z},\Hat{\gamma}^{z})\leq\Tilde{\varepsilon}$ on
$F$ and the result follows. [Strong Markov property] Consider a diffusion
under Strong Assumption LABEL:assumption_b and then let $\gamma\in
g^{*}(\mathcal{I}_{H}(E))$. Let $(\mathbf{N}_{\gamma},\mathbf{X}_{\gamma})$ be
a scaffolding-and-spindles pair started from $\gamma$ as in Definition
LABEL:definition_starting_interval_partition and let $(\gamma^{y},y\geq 0)$ be
a modification that is continuous on $(0,\infty)$ as in Lemma 1. Recall the
appropriate filtered probability space a filtered probability space
$(\Omega,\mathcal{F}_{\mathcal{I}},(\mathcal{F}_{\mathcal{I}}^{y},y\geq
0),\mathbb{P}_{\gamma})$ as used in Corollary . Then we have that
$(\gamma^{y},y\geq 0)$ is a strong Markov process on this filtered probability
space. We first consider a diffusion that satisfies Strong Assumption
LABEL:assumption_b and we begin with the case where $\liminf_{x\rightarrow
0}g(x)/x>0$. As in [construction, Proposition 6.17] we extend the result in
Proposition  to the claim of continuity of the process

$\gamma\mapsto\mathbb{E}_{\gamma}\Big{(}\prod_{i=1}^{m}f_{i}(\gamma^{y_{i}})\Big{)},$

where $f_{1},\dots,f_{m}:\mathcal{I}_{H}\rightarrow[0,\infty)$ are bounded and
continuous. This can be shown by induction on $m$ using the simple Markov
property from Corollary . Then we show the strong Markov property from this by
the method given by Kallenberg [KallenbergOlav2021Fomp, Theorem 17.17] where
the continuity in the initial state stated above then works in place of the
Feller property that is used in their result, as in [construction, Proposition
6.17]. Finally we consider the case where $\limsup_{x\rightarrow
0}g(x)/x<\infty$. We first note that Proposition  applies to the identity
transformation $g(x)=x$. Also we have the map
$g^{*}:\mathcal{I}_{H}(E)\rightarrow g^{*}(\mathcal{I}_{H}(E))$ is continuous
and bijective by Lemma 1 which means that the Strong Markov property in the
untransformed case leads to the strong Markov property in the transformed
case. We can now prove Theorem LABEL:theorem_diffusion. [Proof of Theorem
LABEL:theorem_diffusion] First consider the case where we have a diffusion
under Assumption LABEL:assumption_b and we have a scaffolding-and-spindles
pair $(\mathbf{N},\mathbf{X})$ for this diffusion stopped at an a.s. finite
random time $T$. Then there exists a continuous modification of
$\overline{\text{skewer}}(\mathbf{N},\mathbf{X})$ by Theorem 1 and the
furthermore $\mathbf{X}$ has a local time that is continuous in time and space
by Proposition 1. Now, instead consider a diffusion under Assumption 1 and let
$s>0$. Let $(\Tilde{\mathbf{N}},\Tilde{\mathbf{X}})$ be a scaffolding-and-
spindles pair for this untransformed diffusion that is stopped at
$\tau^{0}(s)$. Then by Corollary
LABEL:corollary_assumption_b'_starting_interval_partition the diffusion
satisfies equation starting IP condition. Therefore, as the diffusion also
satisfies $B1$ we can apply Proposition
LABEL:proposition_clade_poisson_random_measure as $\tau^{0}(s)$ satisfies the
conditions of the proposition. This gives a regular conditional distribution
of the clade component for level $0$ given
$\beta^{0}\coloneqq\text{skewer}(0,\Tilde{\mathbf{N}},\Tilde{\mathbf{X}})$ and
by Proposition LABEL:proposition_simple_markov_skewer we have a simple Markov
property for $\overline{\text{skewer}}(\Tilde{\mathbf{N}},\Tilde{\mathbf{X}})$
on the probability space $(\Omega,\mathcal{A},\mathbf{P})$ for
$\Tilde{\mathbf{N}}$. We also have aggregate health summability at level $0$
of the transformed diffusion by Proposition 1 and Theorem
LABEL:theorem_aggregate_mass_summability. Therefore by Theorem  we have a
strong Markov property for a modification of the transformed skewer process on
the space of paths continuous on $(0,\infty)$ which we denote as
$(\Omega,\mathcal{F},\mathbb{P}_{\mu})$ where $\mu$ is the probability
distribution on $g^{*}(\mathcal{I}_{H}(E))$ of $g^{*}(\beta^{0})$.