f\vcentcolon=\mathcal{F}^{-1}[\check{\chi}(2^{-N}\cdot)]\ast f.$

By considering their Fourier transforms, one observes

$(\Delta G_{N})\ast f=-f+[\Delta(G_{N}-G)]\ast f$ (79)

We recall operations of kernels on modelled distributions from [34, Section
5].

######  Definition C.7.

Let $\mathscr{Z}=(\Pi,\Gamma)$ be a model realizing $K$ in the sense of [34,
Definition 5.9].

  1. (a)

We set

$\mathcal{J}(x)\tau\vcentcolon=\mathcal{J}^{\mathscr{Z}}(x)\tau\vcentcolon=\sum_{\lvert
k\rvert<\lvert\tau\rvert_{+}+2}\frac{X^{k}}{k!}\big{[}D^{k}K\ast\Pi_{x}\tau(x)\big{]},\quad
x\in\mathbb{R}^{d},$

for $\tau\in\mathfrak{B}(\mathscr{T})$ and extend it linearly for
$\tau\in\mathscr{T}$.

  2. (b)

Let $\gamma\in(0,\infty)\setminus\mathbb{N}$ and
$f\in\mathcal{D}^{\gamma}(\mathscr{T},\mathscr{Z})$. We set

$\mathcal{N}f(x)\vcentcolon=\mathcal{N}^{\mathscr{Z}}_{\gamma}f(x)\vcentcolon=\sum_{\lvert
k\rvert<\gamma+2}\frac{X^{k}}{k!}D^{k}K\ast(\mathcal{R}^{\mathscr{Z}}f-\Pi_{x}f(x))(x)$
(80)

and

$\mathcal{K}f(x)\vcentcolon=\mathcal{K}^{\mathscr{Z}}_{\gamma}f(x)\vcentcolon=(\mathscr{I}+\mathcal{J}^{\mathscr{Z}}(x))f(x)+\mathcal{N}^{\mathscr{Z}}_{\gamma}f(x).$
(81)

By [34, Theorem 5.12], $\mathcal{K}$ maps
$\mathcal{D}^{\gamma}(\mathscr{T},\mathscr{Z})$ to
$\mathcal{D}^{\gamma+2}(\mathscr{T},\mathscr{Z})$ and one has
$\mathcal{R}\mathcal{K}f=K\ast\mathcal{R}f$. More precisely, one has

$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathcal{K}f\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma+2;\mathfrak{K}}\lesssim_{\mathscr{T},\gamma}(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma+2;B(\mathfrak{K},1)})^{2}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert
f\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},1)}.$
(82)

uniformly over $\mathscr{Z}\in\mathscr{M}(\mathscr{T},K)$,
$f\in\mathcal{D}(\mathscr{T},\mathscr{Z})$ and compact sets
$\mathfrak{K}\subseteq\mathbb{R}^{d}$. See [36, Theorem 5.1].

  3. (c)

For a smooth function $F$ on $\mathbb{R}^{d}$ and $\beta\in(0,\infty)$, we set

$R_{\beta}F(x)\vcentcolon=\sum_{\lvert
k\rvert<\beta}\frac{X^{k}}{k!}D^{k}F(x),\quad x\in\mathbb{R}^{d}.$

Then [34, Lemma 2.12] implies
$R_{\beta}F\in\mathcal{D}^{\beta}(\mathscr{T},\mathscr{Z})$.

######  Definition C.8.

Suppose that the model $\mathscr{Z}$ realizes $K$. For
$\gamma\in(0,\infty)\setminus\mathbb{N}$,
$f\in\mathcal{D}^{\gamma}(\mathscr{T},\mathscr{Z})$ and $N\in\mathbb{N}_{0}$,
we set

$\mathcal{G}_{N}f(x)\vcentcolon=\mathcal{G}_{N,\gamma}^{\mathscr{Z}}f(x)\vcentcolon=\mathcal{K}^{\mathscr{Z}}_{\gamma}f(x)+R_{\gamma+2}[H_{N}\ast\mathcal{R}f](x).$

Note that one has
$\mathcal{R}^{\mathscr{Z}}\mathcal{G}^{\mathscr{Z}}_{N}f=G_{N}\ast\mathcal{R}^{\mathscr{Z}}f$.
For the meaning of the parameter $N$, see Remark C.16 below.

###  C.3 Definition of modelled distributions

######  Definition C.9.

We define $\mathcal{T},\mathcal{T}_{-}\subseteq\mathscr{T}$ and
$\mathfrak{B}(\mathcal{T}),\mathfrak{B}(\mathcal{T}_{-})\subseteq\mathfrak{B}(\mathscr{T})$
as follows.

  1. (a)

For $\tau_{1},\tau_{2}\in\mathscr{T}$ we write
“$\nabla\mathscr{I}(\tau_{1})\cdot\nabla\mathscr{I}(\tau_{2})$” instead of
“$\sum_{j=1}^{d}\mathscr{I}_{j}(\tau_{1})\mathscr{I}_{j}(\tau_{2})$”.

  2. (b)

We denote by $\mathcal{T}$ the smallest subset of $\mathscr{T}$ with the
following properties:

     * •

$\Xi\in\mathcal{T}$ and

     * •

if $\tau_{1},\tau_{2}\in\mathcal{T}$, then
$\nabla\mathscr{I}(\tau_{1})\cdot\nabla\mathscr{I}(\tau_{2})\in\mathcal{T}$.

Furthermore, we associate $c(\tau)\in\mathbb{N}$ to each $\tau\in\mathcal{T}$
by setting $c(\Xi)\vcentcolon=1$ and by inductively setting for
$\tau_{1},\tau_{2}\in\mathcal{T}$

$c(\nabla\mathscr{I}(\tau_{1})\cdot\nabla\mathscr{I}(\tau_{2}))\vcentcolon=\begin{cases}2c(\tau_{1})c(\tau_{2})&\text{if
}\tau_{1}\neq\tau_{2},\\\ c(\tau_{1})c(\tau_{2})&\text{if
}\tau_{1}=\tau_{2}.\end{cases}$

  3. (c)

One defines $\mathfrak{B}(\mathcal{T})\subseteq\mathfrak{B}(\mathscr{T})$ as
the minimal subset with the following properties:

     * •

$\Xi\in\mathfrak{B}(\mathcal{T})$ and

     * •

if $\tau_{1},\tau_{2}\in\mathfrak{B}(\mathcal{T})$ and $i\in\\{1,\ldots,d\\}$,
then
$\mathscr{I}_{i}(\tau_{1})\mathscr{I}_{i}(\tau_{2})\in\mathfrak{B}(\mathcal{T})$.

  4. (d)

We set
$\mathcal{T}_{-}\vcentcolon=\\{\tau\in\mathcal{T}\nonscript\>|\nonscript\>\mathopen{}\lvert\tau\rvert_{+}<0\\}$
and
$\mathfrak{B}(\mathcal{T}_{-})\vcentcolon=\\{\tau\in\mathfrak{B}(\mathcal{T})\nonscript\>|\nonscript\>\mathopen{}\lvert\tau\rvert_{+}<0\\}$.

######  Remark C.10.

After Lemma 1.12, we have discussed the algorithm to obtain the tree expansion
($\tau_{1},\tau_{2},\tau_{3},\ldots$) for $W$. The constant $c(\tau)$
represents the coefficient of the tree $\tau$ in that expansion.

######  Definition C.11.

Given a model $\mathscr{Z}$ realizing $K$, we associate
$\tau^{\mathcal{K}}=\tau^{\mathcal{K},\mathscr{Z}}\in\mathcal{D}^{\gamma_{\tau}}(\mathscr{T},\mathscr{Z})$
to each $\tau\in\mathcal{T}_{-}$ by setting $\Xi^{\mathcal{K}}\vcentcolon=\Xi$
and by inductively setting

$\gamma_{\tau}\vcentcolon=\min\\{\gamma_{\tau_{1}}+1+\lvert\tau_{2}\rvert_{+},\gamma_{\tau_{2}}+1+\lvert\tau_{1}\rvert_{+}\\},\quad\tau^{\mathcal{K}}\vcentcolon=\sum_{i=1}^{d}\mathscr{D}_{i}[\mathcal{K}\tau_{1}^{\mathcal{K}}]\star\mathscr{D}_{i}[\mathcal{K}\tau_{2}^{\mathcal{K}}]$

for $\tau=\nabla\mathscr{I}(\tau_{1})\cdot\nabla\mathscr{I}(\tau_{2})$. The
exponent $\gamma_{\Xi}$ is choosen so that $\gamma_{\tau}>2$ for every
$\tau\in\mathcal{T}_{-}$.

######  Remark C.12.

Thanks to Proposition C.2 and [34, Theorem 5.12], indeed one has
$\tau^{\mathcal{K},\mathscr{Z}}\in\mathcal{D}^{\gamma_{\tau}}$. Furthermore,
for $\tau=\nabla\mathscr{I}(\tau_{1})\cdot\nabla\mathscr{I}(\tau_{2})$ and a
compact set $\mathfrak{K}$, one has

$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\tau^{\mathcal{K},\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma_{\tau};\mathfrak{K}}\lesssim_{\mathscr{T}}(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma_{\tau_{1}}+\gamma_{\tau_{2}}+2;B(\mathfrak{K},1)})^{6}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\tau_{1}^{\mathcal{K},\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma_{\tau_{1}};B(\mathfrak{K},1)}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\tau_{2}^{\mathcal{K},\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma_{\tau_{2}};B(\mathfrak{K},1)}.$

Therefore, there exist a constant $\gamma,C\in(0,\infty)$ and integers
$k,l\in\mathbb{N}$, which depend only on $\mathscr{T}$, such that

$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\tau^{\mathcal{K},\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;\mathfrak{K}}\leq
C(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)})^{k}$
(83)

uniformly over $\tau\in\mathcal{T}_{-}$,
$\mathscr{Z}\in\mathscr{M}(\mathscr{T},\mathscr{Z})$ and compact sets
$\mathfrak{K}\subseteq\mathbb{R}^{d}$.

######  Definition C.13.

Let $F\in C_{c}^{\infty}(\mathbb{R})$ be such that $F(x)=-e^{2x}$ if $\lvert
x\rvert\leq 2$. Given $N\in\mathbb{N}$ and a model $\mathscr{Z}$ realizing
$K$, we set

$\bm{X}\vcentcolon=\bm{X}^{\mathscr{Z}}\vcentcolon=\sum_{\tau\in\mathcal{T}_{-}}c(\tau)\tau^{\mathcal{K},\mathscr{Z}},$
$\bm{W}_{N}\vcentcolon=\bm{W}^{\mathscr{Z}}_{N}\vcentcolon=\mathfrak{p}_{<2}\mathcal{G}_{N,2}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}}$

and

$\displaystyle\bm{Y}_{N}\vcentcolon=$ $\displaystyle\bm{Y}_{N}^{\mathscr{Z}}$
$\displaystyle\vcentcolon=$
$\displaystyle\mathfrak{p}_{<\delta}\Big{[}F^{\star}(\bm{W}_{N}^{\mathscr{Z}})\star\Big{\\{}\sum_{\begin{subarray}{c}\tau_{1},\tau_{2}\in\mathcal{T}_{-},\\\
\lvert\tau_{1}\rvert_{+}+\lvert\tau_{2}\rvert_{+}>-2\end{subarray}}\sum_{i=1}^{d}c(\tau_{1})c(\tau_{2})\mathscr{D}_{i}[\mathcal{K}^{\mathscr{Z}}\tau^{\mathcal{K},\mathscr{Z}}_{1}]\star\mathscr{D}_{i}[\mathcal{K}^{\mathscr{Z}}\tau^{\mathcal{K},\mathscr{Z}}_{2}]$
$\displaystyle+2\sum_{i=1}^{d}\mathscr{D}_{i}[\mathcal{K}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}}]\star
R_{2}[\partial_{i}\\{H_{N}\ast(\mathcal{R}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}})\\}]\Big{\\}}\Big{]}.$

######  Proposition C.14.

Suppose that a model $\mathscr{Z}$ realizes $K$. Let $N\in\mathbb{N}$. Then,
one has
$\bm{W}_{N}^{\mathscr{Z}}\in\mathcal{D}_{0}^{2}(\mathscr{T},\mathscr{Z})$ and
$\bm{Y}_{N}^{\mathscr{Z}}\in\mathcal{D}^{\delta}_{-1+\delta}(\mathscr{T},\mathscr{Z})$.
More precisely, there exist constants $\gamma,C\in(0,\infty)$ and integers
$k,l\in\mathbb{N}$ such that the following estimates hold uniformly over
$N\in\mathbb{N}$,
$\mathscr{Z},\overline{\mathscr{Z}}\in\mathscr{M}(\mathscr{T},K)$ and convex
compact sets $\mathfrak{K}\subseteq\mathbb{R}^{d}$:

$\displaystyle\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{X}^{\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{2;\mathfrak{K}}$
$\displaystyle\leq
C(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)})^{k},$
$\displaystyle\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{W}^{\mathscr{Z}}_{N}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{2;\mathfrak{K}}$
$\displaystyle\leq
C\\{(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)})^{k}+\lVert
H_{N}\ast(\mathcal{R}^{\mathscr{Z}}X^{\mathscr{Z}})\rVert_{C^{2}(\mathfrak{K})}\\},$
$\displaystyle\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{Y}_{N}^{\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\delta;\mathfrak{K}}$
$\displaystyle\leq
C(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)}+\lVert
H_{N}\ast(\mathcal{R}^{\mathscr{Z}}X^{\mathscr{Z}})\rVert_{C^{2}(\mathfrak{K})})^{k},$

and furthermore

$\displaystyle\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{X}^{\mathscr{Z}};\bm{X}^{\overline{\mathscr{Z}}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{2;\mathfrak{K}}$
$\displaystyle\leq
C(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\overline{\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)})^{k}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z};\overline{\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)},$
$\displaystyle\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{W}^{\mathscr{Z}}_{N};\bm{W}^{\overline{\mathscr{Z}}}_{N}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{2;\mathfrak{K}}$
$\displaystyle\leq
C(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)}+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\overline{\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)})^{k}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z};\overline{\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)}$
$\displaystyle\qquad+\lVert
H_{N}\ast(\mathcal{R}^{\mathscr{Z}}X^{\mathscr{Z}}-\mathcal{R}^{\overline{\mathscr{Z}}}X^{\overline{\mathscr{Z}}})\rVert_{C^{2}(\mathfrak{K})}$
$\displaystyle\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{Y}_{N}^{\mathscr{Z}};\bm{Y}_{N}^{\overline{\mathscr{Z}}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\delta;\mathfrak{K}}$
$\displaystyle\leq
C\Big{(}1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)}+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\overline{\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)}$
$\displaystyle\qquad+\lVert
H_{N}\ast(\mathcal{R}^{\mathscr{Z}}X^{\mathscr{Z}})\rVert_{C^{2}(\mathfrak{K})}+\lVert
H_{N}\ast(\mathcal{R}^{\overline{\mathscr{Z}}}X^{\overline{\mathscr{Z}}})\rVert_{C^{2}(\mathfrak{K})}\Big{)}^{k}$
$\displaystyle\qquad\qquad\times(\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z};\overline{\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(\mathfrak{K},l)}+\lVert
H_{N}\ast(\mathcal{R}^{\mathscr{Z}}X^{\mathscr{Z}}-\mathcal{R}^{\overline{\mathscr{Z}}}X^{\overline{\mathscr{Z}}})\rVert_{C^{2}(\mathfrak{K})}).$

###### Proof.

The estimate for $\bm{X}^{\mathscr{Z}}$ follows from (83). As for the estimate
of $\bm{W}_{N}^{\mathscr{Z}}$, the Schauder estimate (82) gives the estimate
for $\mathcal{K}\bm{X}^{\mathscr{Z}}$. The estimate for
$R_{2}[H_{N}\ast(\mathcal{R}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}})]$ follows from
the estimate

$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert
R_{2}[H_{N}\ast(\mathcal{R}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}})]\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{2;\mathfrak{K}}\leq\sum_{m:\lvert
m\rvert\leq
2}\lVert\partial^{m}[H_{N}\ast(\mathcal{R}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}})]\rVert_{L^{\infty}(\mathfrak{K})},$

where the convexity of $\mathfrak{K}$ is used. The estimate for
$\bm{Y}_{N}^{\mathscr{Z}}$ follows from Proposition C.2, the estimate (78) and
the Schauder estimate (82).

For the estimates of the differences, let us just mention that for differences
there exist analogue estimates to those in Proposition C.2, the estimate (78)
and (82), see [34, Proposition 4.10], [37, Proposition 3.11] and [34, Theorem
5.12] respectively. Using them, we can prove the last three inequalities of
the differences similarly. ∎

######  Definition C.15.

Given a model $\mathscr{Z}$ realizing $K$ and $N\in\mathbb{N}$, we set

$X\vcentcolon=X^{\mathscr{Z}}\vcentcolon=\mathcal{R}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}},\quad
W_{N}\vcentcolon=W^{\mathscr{Z}}_{N}\vcentcolon=\mathcal{R}^{\mathscr{Z}}\bm{W}^{\mathscr{Z}}_{N}$

and

$Y_{N}\vcentcolon=Y^{\mathscr{Z}}_{N}\vcentcolon=\mathcal{R}^{\mathscr{Z}}\bm{Y}^{\mathscr{Z}}_{N}+F(W_{N}^{\mathscr{Z}})\Big{\\{}\lvert\nabla[H_{N}\ast(X^{\mathscr{Z}})]\rvert^{2}+[\Delta(G_{N}-G)]\ast
X^{\mathscr{Z}}\Big{\\}}.$

######  Remark C.16.

The parameter $N$ will be used to ensure $W_{N}$ is bounded on a given bounded
domain. Therefore, $N$ will be random and will depend on the domain. The idea
of introducing such parameter is also used in [57]. As noted in Definition
C.8, one has $W_{N}=G_{N}\ast X$.

######  Lemma C.17.

Let $\varepsilon\in(0,1)$. To simplify notation, we write
$X^{\mathrm{can}}\vcentcolon=X^{\mathscr{Z}^{\mathrm{can},\varepsilon}}$ here
for instance. Then, one has the following identity:

$\lvert\nabla W_{N}^{\mathrm{can}}\rvert^{2}+\Delta
W_{N}^{\mathrm{can}}=-\xi_{\varepsilon}\\\
+\sum_{\begin{subarray}{c}\tau_{1},\tau_{2}\in\mathcal{T}_{-},\\\
\lvert\tau_{1}\rvert_{+}+\lvert\tau_{2}\rvert_{+}>-2\end{subarray}}c(\tau_{1})c(\tau_{2})\nabla(K\ast\mathcal{R}^{\mathrm{can}}\tau_{1}^{\mathcal{K},\mathrm{can}})\cdot\nabla(K\ast\mathcal{R}^{\mathrm{can}}\tau_{2}^{\mathcal{K},\mathrm{can}})\\\
+2\nabla[K\ast
X^{\mathrm{can}}]\cdot\nabla[H_{N}\ast(X^{\mathrm{can}})]+\lvert\nabla[H_{N}\ast(X^{\mathrm{can}})]\rvert^{2}+[\Delta(G_{N}-G)]\ast
X^{\mathrm{can}}$

###### Proof.

One has $W_{N}^{\mathrm{can}}=K\ast X^{\mathrm{can}}+H_{N}\ast
X^{\mathrm{can}}$ and

$\lvert\nabla
W_{N}\rvert^{2}=\sum_{\tau_{1},\tau_{2}\in\mathcal{T}_{-}}c(\tau_{1})c(\tau_{2})\nabla[K\ast\mathcal{R}^{\mathrm{can}}\tau_{1}^{\mathcal{K},\mathrm{can}}]\cdot\nabla[K\ast\mathcal{R}^{\mathrm{can}}\tau_{2}^{\mathcal{K},\mathrm{can}}]\\\
+2\nabla[K\ast X^{\mathrm{can}}]\cdot\nabla[H_{N}\ast
X^{\mathrm{can}}]+\lvert\nabla H_{N}\ast X^{\mathrm{can}}\rvert^{2}$

Furthermore,

$\Delta
W_{N}=-\sum_{\tau\in\mathcal{T}_{-}}c(\tau)\mathcal{R}^{\mathrm{can}}\tau^{\mathcal{K},\mathrm{can}}+[\Delta(G_{N}-G)]\ast
X^{\mathrm{can}}.$

Now it remains to observe

$\sum_{\tau_{1},\tau_{2}\in\mathcal{T}_{-}}c(\tau_{1})c(\tau_{2})\nabla[K\ast\mathcal{R}^{\mathrm{can}}\tau_{1}^{\mathcal{K},\mathrm{can}}]\cdot\nabla[K\ast\mathcal{R}^{\mathrm{can}}\tau_{2}^{\mathcal{K},\mathrm{can}}]-\sum_{\tau\in\mathcal{T}_{-}}c(\tau)\mathcal{R}^{\mathrm{can}}\tau^{\mathcal{K},\mathrm{can}}\\\
=-\xi_{\varepsilon}+\sum_{\begin{subarray}{c}\tau_{1},\tau_{2}\in\mathcal{T}_{-},\\\
\lvert\tau_{1}\rvert_{+}+\lvert\tau_{2}\rvert_{+}>-2\end{subarray}}c(\tau_{1})c(\tau_{2})\nabla(K\ast\mathcal{R}^{\mathrm{can}}\tau_{1}^{\mathcal{K},\mathrm{can}})\cdot\nabla(K\ast\mathcal{R}^{\mathrm{can}}\tau_{2}^{\mathcal{K},\mathrm{can}}).\qed$

###  C.4 BPHZ renormalization for $\bm{X}$

The goal of this section is to show
$X^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}=X^{\mathscr{Z}^{\mathrm{can},\varepsilon}}-c_{\varepsilon}$
(Proposition C.25). To this end, our first goal is to obtain the basis
expansion for modelled distributions
$\tau^{\mathcal{K},\mathscr{Z}}\in\mathcal{T}_{-}$, which will be given in
Lemma C.20.

######  Lemma C.18.

For every $\tau_{1},\tau_{2}\in\mathcal{T}_{-}$ with
$\lvert\tau_{1}\rvert_{+},\lvert\tau_{2}\rvert_{+}<-1$ and
$i,j\in\\{1,\ldots,d\\}$, one has

$\Delta^{\circ}_{+}[\mathscr{I}_{i}(\tau_{1})]=\mathscr{I}_{i}(\tau_{1})\otimes\bm{1}_{+},\quad\Delta^{\circ}_{+}[\mathscr{I}_{i}(\tau_{1})\mathscr{I}_{j}(\tau_{2})]=[\mathscr{I}_{i}(\tau_{1})\mathscr{I}_{j}(\tau_{2})]\otimes\bm{1}_{+}.$

In particular, the constant map
$x\mapsto\mathscr{I}_{i}(\tau_{1})\mathscr{I}_{j}(\tau_{2})$ belongs to
$\mathcal{D}^{\infty}_{\lvert\tau_{1}\rvert+\lvert\tau_{2}\rvert+2}(\mathscr{T},\mathscr{Z})$
for any model $\mathscr{Z}=(\Pi,\Gamma)$ and

$\mathcal{R}[\mathscr{I}_{i}(\tau_{1})\mathscr{I}_{j}(\tau_{2})]=\Pi_{x}[\mathscr{I}_{i}(\tau_{1})\mathscr{I}_{j}(\tau_{2})],$

where the right-hand side is independent of $x$.

###### Proof.

In view of the recursive formula [13, Proposition 4.17], one can prove the
claim by induction on $\lvert\cdot\rvert_{+}$. Indeed, suppose one is going to
prove $\Delta^{\circ}_{+}\tau=\tau\otimes\bm{1}_{+}$, where
$\tau=\mathscr{I}_{i}(\tau_{1})\mathscr{I}_{j}(\tau_{2})$ and
$\Delta^{\circ}_{+}\tau_{k}=\tau_{k}\otimes\bm{1}_{+}$. By Lemma B.29,
$\Delta^{\circ}_{+}\tau=\Delta^{\circ}_{+}[\mathscr{I}_{i}(\tau_{1})]\Delta^{\circ}_{+}[\mathscr{I}_{j}(\tau_{2})]$.
Therefore, it suffices to show
$\Delta^{\circ}_{+}[\mathscr{I}_{i}(\tau_{1})]=[\mathscr{I}_{i}(\tau_{1})]\otimes\bm{1}_{+}$.
By [13, Proposition 4.17], one has

$\Delta^{\circ}_{+}\mathscr{I}_{i}(\tau_{1})=(\mathscr{I}_{i}\otimes\operatorname{Id})\Delta\tau_{1}+\sum_{k:\lvert\tau\rvert_{+}+1-\lvert
k\rvert>0}\frac{X^{k}}{k!}\otimes\hat{\mathscr{I}}_{e_{i}+k}(\tau_{1}).$

It remains to observe that
$(\mathscr{I}_{i}\otimes\operatorname{Id})\Delta\tau_{1}=[\mathscr{I}_{i}(\tau_{1})]\otimes\bm{1}_{+}$
by hypothesis of the induction and that the set over which $k$ ranges is
empty. ∎

######  Definition C.19.

We use some notations from Section B.1. Let $\tau\in\mathfrak{B}(\mathcal{T})$
and let $e$ be an edge of $\tau$ with $\mathfrak{t}(e)=\mathscr{I}$. By
removing the edge $e$, we obtain a decorated forest with two connected
components. We denote by

$\operatorname{Remove}(\tau;e)$

the component containing the root of $\tau$, with decoration inherited from
$\tau$. For instance,

$\operatorname{Remove}(\leavevmode\hbox to62.35pt{\vbox
to48.12pt{\pgfpicture\makeatletter\hbox{\hskip 31.17409pt\lower-2.72133pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.2pt}\pgfsys@invoke{ }
{{}}{{{{}}}}{}{}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
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where  represents the noise (6). We set

$\displaystyle\operatorname{Remove}(\mathfrak{B}(\mathcal{T}))$
$\displaystyle\vcentcolon=\\{\operatorname{Remove}(\tau;e)\nonscript\>|\nonscript\>\mathopen{}\tau\in\mathfrak{B}(\mathcal{T}),e\in
E_{\tau}\text{ with }\mathfrak{t}(e)=\mathscr{I}\\},$
$\displaystyle\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T}))$
$\displaystyle\vcentcolon=\\{(T,0)^{\mathfrak{n},0}_{\mathfrak{e}}\nonscript\>|\nonscript\>\mathopen{}(T,0)^{0,0}_{\mathfrak{e}}\in\operatorname{Remove}(\mathfrak{B}(\mathcal{T}))\\}.$

######  Lemma C.20.

Suppose $\mathscr{Z}=(\Pi,\Gamma)$ is a model realizing $K$. Then, one has a
claim for $\tau\in\mathcal{T}_{-}$ as follows.

  1. (a)

If $\tau=\Xi$ or
$\tau=\nabla\mathscr{I}(\tau_{1})\cdot\nabla\mathscr{I}(\tau_{2})$ with
$\lvert\tau_{1}\rvert_{+},\lvert\tau_{2}\rvert_{+}<-1$, then
$\tau^{\mathcal{K},\mathscr{Z}}=\tau$.

  2. (b)

If $\tau=\nabla\mathscr{I}(\tau_{1})\cdot\nabla\mathscr{I}(\tau_{2})$ with
$\lvert\tau_{1}\rvert_{+}>-1$ and $\lvert\tau_{2}\rvert_{+}<-1$, then one has
the expansion

$\tau^{\mathcal{K},\mathscr{Z}}(x)=\tau+\sum_{\sigma\in\mathfrak{V}(\tau)}a_{\tau,\sigma}^{\mathscr{Z}}(x)\sigma,$
(84)

with the following properties:

     * •

$\mathfrak{V}(\tau)$ is a finite subset of
$\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T}))$ that is
independent of $\mathscr{Z}$,

     * •

one has

$\displaystyle a_{\tau,\sigma}^{\mathscr{Z}}(x)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}j\in\\{1,\ldots,d\\},n\in\mathbb{N}_{0},\rho\in\mathcal{T}_{-},\\\
l_{1},\ldots,l_{n}\in\mathbb{N}_{0}^{d},\sigma_{1},\ldots,\sigma_{n}\in\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T}))\\\
\lvert\sigma_{k}\rvert_{+}+2-l_{k}>0,-1<\lvert\rho\rvert_{+}<\lvert\tau\rvert_{+}\end{subarray},}c_{\tau,\sigma,\rho}^{l_{1},\ldots,l_{n},\sigma_{1},\ldots,\sigma_{n}}(\mathcal{P})[\partial_{j}K\ast\Pi_{x}\rho^{\mathcal{K},\mathscr{Z}}(x)]\prod_{k=1}^{n}[\partial^{l_{k}}K\ast\Pi_{x}\sigma_{k}](x)$
$\displaystyle+$
$\displaystyle\sum_{\begin{subarray}{c}n\in\mathbb{N}_{0},\rho\in\mathcal{T}_{-},\\\
l,l_{1},\ldots,l_{n}\in\mathbb{N}_{0}^{d},\sigma_{1},\ldots,\sigma_{n}\in\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T}))\\\
\lvert\sigma_{k}\rvert_{+}+2-l_{k}>0,-1<\lvert\rho\rvert_{+}<\lvert\tau\rvert_{+}\end{subarray},}c_{\tau,\sigma,\rho,l}^{l_{1},\ldots,l_{n},\sigma_{1},\ldots,\sigma_{n}}(\mathcal{R})[\partial^{l}K\ast(\mathcal{R}^{\mathscr{Z}}\rho^{\mathcal{K},\mathscr{Z}}-\Pi_{x}\rho^{\mathcal{K},\mathscr{Z}}(x))](x)$
$\displaystyle\hskip
227.62204pt\times\prod_{k=1}^{n}[\partial^{l_{k}}K\ast\Pi_{x}\sigma_{k}](x),$

where the sum is actually finite and the constants

$c_{\tau,\sigma,\rho}^{l_{1},\ldots,l_{n},\sigma_{1},\ldots,\sigma_{n}}(\mathcal{P})\quad\text{and}\quad
c_{\tau,\sigma,\rho,l}^{l_{1},\ldots,l_{n},\sigma_{1},\ldots,\sigma_{n}}(\mathcal{R})$

are independent of $\mathscr{Z}$.

###### Proof.

To see the claim (a), if $\lvert\tau\rvert_{+}<-1$, thanks to Lemma C.18, the
identity (81) becomes

$\mathcal{K}\tau=\mathscr{I}\tau+(K\ast\mathcal{R}\tau)(x)\bm{1}$

and hence $\mathscr{D}_{i}\mathcal{K}\tau=\mathscr{I}_{i}\tau$. The claim (b)
seems complicated but can be proven easily by induction. Suppose that one has
$\tau=\nabla\mathscr{I}(\tau_{1})\cdot\nabla\mathscr{I}(\tau_{2})$ such that
$\tau_{1}$ has the expansions of the form (84) and
$\tau_{2}^{\mathcal{K}}=\tau_{2}$. Furthermore, one has
$-1<\lvert\tau_{1}\rvert_{+}<0$ since $\lvert\tau\rvert_{+}<0$. Therefore, one
has

$\tau_{1}^{\mathcal{K}}=\tau_{1}+\sum_{\sigma\in\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T}))}a_{\sigma}\sigma,\quad\tau_{2}^{\mathcal{K}}=\tau_{2}$
(85)

where $a_{\sigma}$ has the desired property. By the definition (81) of
$\mathcal{K}$ , one has

$\mathscr{D}_{i}\mathcal{K}\tau^{\mathcal{K}}_{1}(x)=\mathscr{I}_{i}\tau_{1}+\sum_{\sigma\in\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T}))}a_{\sigma}(x)\mathscr{I}_{i}(\sigma)+[\partial_{i}K\ast\Pi_{x}\tau_{1}](x)\bm{1}\\\
+\sum_{\begin{subarray}{c}\sigma\in\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T})),l\in\mathbb{N}_{0}^{d}\\\
\lvert\sigma\rvert+1-|l|>0\end{subarray}}a_{\sigma}(x)[\partial^{\bm{e}_{i}+l}K\ast\Pi_{x}\sigma](x)\frac{X^{l}}{l!}+\sum_{\lvert
l\rvert<\gamma_{\tau_{1}}+1}[\partial^{\bm{e}_{i}+l}K\ast(\mathcal{R}\tau_{1}^{\mathcal{K}}-\Pi_{x}\tau_{1}^{\mathcal{K}})](x)\frac{X^{l}}{l!},$

where $\gamma_{\tau_{1}}$ is chosen so that
$\tau_{1}^{\mathcal{K}}\in\mathcal{D}^{\gamma_{\tau_{1}}}(\mathscr{T},\mathscr{Z})$,
see Remark C.12. Since
$\mathscr{D}_{i}\mathcal{K}\tau_{2}^{\mathcal{K}}=\mathscr{I}_{i}\tau_{2}$ as
shown in the part (a), one has

$\mathscr{I}_{i}(\sigma)\mathscr{I}_{i}(\tau_{2}),X^{l}\mathscr{I}_{i}(\tau_{2})\in\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T})).$

Since $\lvert\tau_{1}\rvert_{+}<\lvert\tau\rvert_{+}$, we complete the
induction. ∎

We recall an explicit formula of the BPHZ realization.

######  Definition C.21 ([13, Theorem 6.18]).

Let $\hat{\mathscr{T}}_{-}$ be the free algebra generated by $\mathscr{T}$
under the forest product. (In fact, recalling $H^{R}_{1}$ from Definition
B.22, we have $\hat{\mathscr{T}}_{-}=H^{R}_{1}$.) We define the algebra
homomorphism $g_{\varepsilon}^{-}:\hat{\mathscr{T}}_{-}\to\mathbb{R}$
characterized by

$g_{\varepsilon}^{-}(\mathfrak{i}_{\circ}\tau)\vcentcolon=\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau(0)],$

where $\mathfrak{i}_{\circ}:\mathscr{T}\to\hat{\mathscr{T}}_{-}$ is the
natural injection. Then, we have

$\mathbf{\Pi}^{\mathrm{BPHZ},\varepsilon}=(g_{\varepsilon}^{-}\hat{\mathscr{A}}_{-}\otimes\mathbf{\Pi}^{\mathrm{can},\varepsilon}\Delta^{\circ}_{-}).$
(86)

In view of the identity (86) and Lemma C.20, we need to understand
$(g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\otimes\mathbf{\Pi}^{\mathrm{can},\varepsilon})\Delta_{-}^{\circ}\tau$
for $\tau\in\mathcal{T}_{-}$ and
$\tau\in\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T}))$. As
one can easily guess from the definition of $g^{-}_{\varepsilon}$, it is
necessary to estimate
$\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau(0)]$ for such $\tau$.
The following simple lemma is a consequence of the symmetry of the noise
$\xi$.

######  Lemma C.22.

For $\tau\in\operatorname{Remove}(\mathfrak{B}(\mathcal{T}))$, one has
$\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau(0)]=0$.

###### Proof.

Let
$\tau=(T,0)^{0,0}_{\mathfrak{e}}\in\operatorname{Remove}(\mathfrak{B}(\mathcal{T}))$.
Let $\mathbf{\Pi}^{\text{minus}}$ be the canonical realization for
$\xi_{\varepsilon}(-\cdot)$. Since
$\xi\overset{\operatorname{d}}{=}\xi(-\cdot)$, one has
$\mathbf{\Pi}^{\text{minus}}\sigma\overset{\operatorname{d}}{=}\mathbf{\Pi}^{\mathrm{can},\varepsilon}\sigma$
for every $\sigma\in\mathscr{T}$. If we set

$n(T)\vcentcolon=\\#\\{e\in
E_{T}\nonscript\>|\nonscript\>\mathopen{}\mathfrak{t}(e)=\mathscr{I}\\},$

by using the identity

$\partial_{i}K\ast[f(-\cdot)]=-[\partial_{i}K\ast f](-\cdot),$

where the fact $K=K(-\cdot)$ is used, one has
$\mathbf{\Pi}^{\text{minus}}\tau=(-1)^{n(T)}\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau$.
However, since $\tau\in\operatorname{Remove}(\mathfrak{B}(\mathcal{T}))$,
$n(T)$ is odd. Therefore, one has

$\mathbf{\Pi}^{\text{minus}}\tau\overset{\operatorname{d}}{=}\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau\quad\text{and}\quad\mathbf{\Pi}^{\text{minus}}\tau=-\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau,$

and concludes $\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau(0)]=0$.
∎

######  Lemma C.23.

For
$\tau=(F,\hat{F})^{\mathfrak{n},\mathfrak{o}}_{\mathfrak{e}}\in\mathfrak{B}(\mathcal{T})\cup\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T}))$
and $x\in\mathbb{R}^{d}$, one has

$\Delta^{\circ}_{-}\tau=\tau\otimes\bm{1}+\bm{1}_{-}\otimes\tau+\ker(g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\otimes\Pi^{\mathrm{can},\varepsilon}_{x})\cap\ker(g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\otimes\mathbf{\Pi}^{\mathrm{can},\varepsilon}).$

###### Proof.

Recall from Definition B.2-(a) that edges are oriented. We call an edge
$e=(a,b)$ a leaf if $b$ is not followed by any edge. We call a node $a$ of $F$
true if there exists an edge $e=(a,b)$ such that
$\mathfrak{t}(e)=\mathscr{I}$. We denote by $N^{\text{true}}$ the set of all
true nodes of $F$. For a subforest $G$ of $F$, we set

$N_{G}^{j}\vcentcolon=\\{a\in N_{G}\cap
N^{\text{true}}\nonscript\>|\nonscript\>\mathopen{}\text{ there exist exactly
$j$ outgoing edges in $G$ at $a$}\\}.$

Recalling the coproduct formula (73), one has

$\Delta^{\circ}_{-}\tau=\tau\otimes\mathscr{R}_{\lvert\tau\rvert_{+}}\bm{1}+\bm{1}_{-}\otimes\tau\\\
+\sum_{G\subseteq
F,G\neq\varnothing}\sum_{\mathfrak{n}_{G}\neq\mathfrak{n},\varepsilon^{F}_{G}}\frac{1}{\varepsilon^{F}_{G}!}\binom{\mathfrak{n}}{\mathfrak{n}_{G}}(G,0)^{\mathfrak{n}_{G}+\pi\varepsilon^{F}_{G},0}_{\mathfrak{e}}\otimes\mathscr{K}(F,\mathbbm{1}_{G})^{\mathfrak{n}-\mathfrak{n}_{G},\pi(\varepsilon^{F}_{A}-\mathfrak{e}\mathbbm{1}_{G})}_{\mathfrak{e}\mathbbm{1}_{E_{F}\setminus
E_{G}}+\varepsilon^{F}_{G}},$

where $\mathscr{R}_{\alpha}$ is defined in Definition 3.8. However, note that
$\mathbf{\Pi}^{\mathrm{can},\varepsilon}\mathscr{R}_{\alpha}\bm{1}=\mathbf{\Pi}^{\mathrm{can},\varepsilon}\bm{1}$.
We fix $G\neq\varnothing$, $\mathfrak{n}_{G}\neq\mathfrak{n}$ and
$\varepsilon^{F}_{G}$ and set

$\tau_{1}\vcentcolon=(G,0)^{\mathfrak{n}_{G}+\pi\varepsilon^{F}_{G},0}_{\mathfrak{e}},\quad\tau_{2}\vcentcolon=\mathscr{K}(F,\mathbbm{1}_{G})^{\mathfrak{n}-\mathfrak{n}_{G},\pi(\varepsilon^{F}_{A}-\mathfrak{e}\mathbbm{1}_{G})}_{\mathfrak{e}\mathbbm{1}_{E_{F}\setminus
E_{G}}+\varepsilon^{F}_{G}}$

We will prove
$(g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\otimes\Pi^{\mathrm{can},\varepsilon}_{x})(\tau_{1}\otimes\tau_{2})=0$
by considering various cases, which will complete the proof. When a case is
studied, we exclude all cases considered before.

  1. (a)

Suppose that $G\neq F$ and that a connected component $T$ of $G$ satisfies
$N_{T}^{0}=\varnothing$ and $N_{T}^{1}=N_{F}^{1}\cap N_{G}$. Then, the forest
$\tau_{2}$ contains a leaf $(a,\rho_{T})$ of edge type $\mathscr{I}$ and hence
$\Pi^{\mathrm{can},\varepsilon}_{x}\tau_{2}=\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau_{2}=0$.

  2. (b)

Suppose $G$ contains a leaf of edge type $\mathscr{I}$. Then, in view of the
recursive formula (76), this is also the case for each forest appearing in
$\hat{\mathscr{A}}_{-}\tau_{1}$ and hence
$g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\tau_{1}=0$.

  3. (c)

Suppose $N_{G}^{0}\neq\varnothing$. If the case 2 is excluded, then a
connected component of $\tau_{1}$ is of the form
$\bullet^{\mathfrak{n}_{1},0}$ and hence $\tau_{1}=0$ (as an element of
$\mathscr{T}_{-}$).

  4. (d)

Suppose $\tau_{1}$ contains a connected component
$\tau_{3}=(T,0)^{\mathfrak{n},0}_{\mathfrak{e}}$ such that $\\#N^{1}_{T}\geq
2$. Let $a\in N^{1}_{T}$.

     * •

If $a$ is the root of $T$, then $\tau_{3}=\mathscr{I}_{i}(\tau_{4})$ and hence
$\tau_{1}=0$ (as an element of $\mathscr{T}_{-}$).

     * •

If $a$ is not the root of $T$, one can merge two consecutive edges $(a_{1},a)$
and $(a,a_{2})$ into a single edge $(a_{1},a_{2})$ to obtain a new tree
$\tau_{5}\in\mathfrak{T}_{\circ}$ with
$\lvert\tau_{3}\rvert_{-}=\lvert\tau_{5}\rvert_{-}+1$. Since
$\lvert\sigma\rvert_{-}\geq-2+\delta$ for every
$\sigma\in\mathfrak{T}_{\circ}$, if $\\#(N_{T}^{1}\setminus\\{\rho_{T}\\})\geq
2$, then $\lvert\tau_{3}\rvert_{-}>0$ and hence $\tau_{1}=0$ (as an element of
$\mathscr{T}_{-}$).

  5. (e)

Suppose that $\tau_{1}$ contains a connected component
$\tau_{6}=(T_{6},0)^{\mathfrak{n}_{6},0}_{\mathfrak{e}}$ such that
$N_{T_{6}}^{0}=N_{T_{6}}^{1}=\varnothing$. Then, $T_{1}=T_{6}=F$ and
$\tau_{1}\in\mathfrak{B}(\mathcal{T})$. However, this implies
$\mathfrak{n}=\mathfrak{n}_{G}=0$, which is excluded.

  6. (f)

Therefore, it remains to consider the case where every connected component
$\tau_{7}=(T_{7},0)^{\mathfrak{n}_{7},0}_{\mathfrak{e}}$ of $\tau_{1}$
satisfies $\\#N_{T_{7}}^{1}=1$ and $N^{0}_{T_{7}}=\varnothing$ and all leaves
of $\tau_{7}$ are of type $\Xi$, namely
$\tau_{7}\in\operatorname{Remove}^{\mathfrak{n}}(\mathfrak{B}(\mathcal{T}))$.
If $\mathfrak{n}_{7}\neq 0$ on $N_{T_{7}}$, then $\lvert\tau_{7}\rvert_{-}>0$.
Thus, we suppose $\mathfrak{n}_{7}=0$. We will show
$g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\tau_{7}=0$, which implies
$g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\tau_{1}=0$ since the character
$g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}$ is multiplicative. To apply the
recursive formula (76), consider the expansion

$\hat{\Delta}_{-}\tau_{7}-\tau_{7}\otimes\bm{1}_{-}=\bm{1}\otimes\tau_{7}+\sum_{\tau_{8}}c_{\tau_{8}}\tau_{8}\otimes\tau_{9}.$

Then, one has

$g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\tau_{7}=-\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau_{7}(0)]-\sum_{\tau_{8}}c_{\tau_{8}}\times\big{(}g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\tau_{8}\big{)}\times\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau_{9}(0)].$

By the same reasoning as before, one can suppose that every component
$\tau_{10}=(T_{10},0)^{0,0}_{\mathfrak{e}}$ of $\tau_{8}$ belongs to
$\operatorname{Remove}(\mathfrak{B}(\mathcal{T}))$. However, since $T_{10}$
has a strictly smaller number of edges than $T_{7}$ does, one can assume
$g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\tau_{8}=0$ by induction. Therefore,
it remains to show
$\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau_{7}(0)]=0$. But this
was shown in Lemma C.22. ∎

######  Corollary C.24.

If $\tau\in\operatorname{Remove}(\mathfrak{B}(\mathcal{T}))$, then
$g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\tau=0$. If $\tau\in\mathcal{T}_{-}$,
then

$g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\tau=-\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau(0)].$

###### Proof.

The claim for $\tau\in\operatorname{Remove}(\mathfrak{B}(\mathcal{T}))$ is
proved in the proof of Lemma C.23, see the case 6. If
$\tau\in\mathcal{T}_{-}$, by Lemma C.23 one has

$\mathbf{\Pi}^{\mathrm{BPHZ},\varepsilon}\tau=\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau+g^{-}_{\varepsilon}\hat{\mathscr{A}}\tau.$

However, since $\lvert\tau\rvert_{-}<0$, one has
$\mathbb{E}[\mathbf{\Pi}^{\mathrm{BPHZ},\varepsilon}\tau(0)]=0$ by definition,
which completes the proof. ∎

######  Proposition C.25.

For $\tau\in\mathcal{T}_{-}$, one has

$\displaystyle\Pi_{x}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}\tau^{\mathcal{K},\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}(x)$
$\displaystyle=\Pi_{x}^{\mathscr{Z}^{\mathrm{can},\varepsilon}}\tau^{\mathcal{K},\mathscr{Z}^{\mathrm{can},\varepsilon}}(x)-\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau(0)],\quad
x\in\mathbb{R}^{d},$ (87)
$\displaystyle\mathcal{R}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}\tau^{\mathcal{K},\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}$
$\displaystyle=\mathcal{R}^{\mathscr{Z}^{\mathrm{can},\varepsilon}}\tau^{\mathcal{K},\mathscr{Z}^{\mathrm{can},\varepsilon}}-\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau(0)].$

In particular,

$X^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}=X^{\mathscr{Z}^{\mathrm{can},\varepsilon}}-c_{\varepsilon}.$

where

$c_{\varepsilon}\vcentcolon=\sum_{\tau\in\mathcal{T}_{-}}c(\tau)\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau(0)].$
(88)

###### Proof.

To simplify notation, we write
$\mathcal{R}^{\mathrm{BPHZ}}\vcentcolon=\mathcal{R}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}$
here, for instance. Since

$\mathcal{R}^{\\#}\tau^{\mathcal{K},\\#}(x)=[\Pi^{\\#}_{x}\tau^{\mathcal{K},\\#}(x)](x),\quad\\#\in\\{\mathrm{can},\mathrm{BPHZ}\\},$

it suffices to prove  (87). By Lemma C.20, one has the expansion

$\tau^{\mathcal{K},\mathrm{BPHZ}}(x)=\tau+\sum_{\sigma}a_{\tau,\sigma}^{\mathrm{BPHZ}}(x)\sigma.$

In the expression of $a_{\tau,\sigma}^{\mathrm{BPHZ}}$ given in Lemma C.20,
every $\rho$ in the sum satisfies $\lvert\rho\rvert_{+}<\lvert\tau\rvert_{+}$.
Therefore, one can assume
$a_{\sigma}^{\mathrm{BPHZ}}=a_{\sigma}^{\mathrm{can}}$ by induction. By Lemma
C.23 and Corollary C.24,

$\Delta^{\circ}_{-}\tau^{\mathcal{K},\mathrm{BPHZ}}(x)=\tau\otimes\bm{1}+\bm{1}_{-}\otimes\tau+\sum_{\sigma}a_{\sigma}^{\mathrm{can}}(x)\bm{1}_{-}\otimes\sigma+\ker(g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\otimes\Pi^{\mathrm{can}}_{x}).$

Furthermore, by [13, Theorem 6.16], one has

$\Pi_{x}^{\mathrm{BPHZ}}=(g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\otimes\Pi^{\mathrm{can}}_{x})\Delta^{\circ}_{-}.$

Therefore,

$\displaystyle\Pi^{\mathrm{BPHZ}}_{x}\tau^{\mathcal{K},\mathrm{BPHZ}}(x)$
$\displaystyle=g^{-}_{\varepsilon}\hat{\mathscr{A}}_{-}\tau+\Pi^{\mathrm{can}}_{x}\tau+\sum_{\sigma}a_{\sigma}^{\mathrm{can}}(x)\Pi^{\mathrm{can}}_{x}\sigma$
$\displaystyle=-\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\tau(0)]+\Pi^{\mathrm{can}}_{x}\tau^{\mathcal{K},\mathrm{can}}(x),$

where we applied Corollary C.24 to get the last equality. ∎

###  C.5 BPHZ renormalization for $\bm{Y}_{N}$

The goal of this section is to compare
$\bm{Y}_{N}^{\mathscr{Z}^{\mathrm{can},\varepsilon}}$ and
$\bm{Y}_{N}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}$, as we did for $\bm{X}$
in the previous section. Again, we need to obtain the basis expansion for
$\bm{Y}_{N}$.

######  Lemma C.26.

Let $\tau_{1},\tau_{2}\in\mathcal{T}_{-}$, $i\in\\{1,\ldots,d\\}$ and
$N\in\mathbb{N}$. Let $\mathscr{Z}$ be a model realizing $K$. Assume
$\lvert\tau_{1}\rvert_{+}+\lvert\tau_{2}\rvert_{+}>-2$. Then, for
$x\in\mathbb{R}^{d}$, one has

$\mathfrak{p}_{<\delta}\big{\\{}F(\bm{W}_{N}^{\mathscr{Z}})(x)\star\mathscr{D}_{i}[\mathcal{K}\tau_{1}^{\mathcal{K},\mathscr{Z}}](x)\star\mathscr{D}_{i}[\mathcal{K}\tau_{2}^{\mathcal{K},\mathscr{Z}}](x)\big{\\}}\\\
=\mathfrak{p}_{<\delta}\Big{\\{}\sum_{k\in\mathbb{N}_{0}}\frac{D^{k}F(W_{N}^{\mathscr{Z}}(x))}{k!}\Big{(}\sum_{\tau\in\mathcal{T}_{-}}\mathscr{I}\tau\Big{)}^{\star
k}\star\mathscr{D}_{i}[\mathcal{K}\tau_{1}^{\mathcal{K},\mathscr{Z}}](x)\star\mathscr{D}_{i}[\mathcal{K}\tau_{2}^{\mathcal{K},\mathscr{Z}}](x)\Big{\\}}$

and

$\mathfrak{p}_{<\delta}\big{\\{}F(\bm{W}_{N}^{\mathscr{Z}})(x)\star\mathscr{D}_{i}[\mathcal{K}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}}](x)\star
R_{2}[\partial_{i}\\{H_{N}\ast(\mathcal{R}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}})\\}](x)\big{\\}}\\\
=\mathfrak{p}_{<\delta}\Big{\\{}\sum_{k\in\mathbb{N}_{0}}\frac{D^{k}F(W_{N}^{\mathscr{Z}}(x))}{k!}\partial_{i}[H_{N}\ast(X^{\mathscr{Z}})](x)\Big{(}\sum_{\tau\in\mathcal{T}_{-}}\mathscr{I}\tau\Big{)}^{\star
k}\star\mathscr{D}_{i}[\mathcal{K}^{\mathscr{Z}}\bm{X}^{\mathscr{Z}}](x)\Big{\\}}.$

###### Proof.

By Lemma C.20, one has

$\bm{W}_{N}^{\mathscr{Z}}(x)=\sum_{\tau\in\mathcal{T}_{-}}\mathscr{I}\tau+W_{N}^{\mathscr{Z}}(x)\bm{1}+\bm{W}_{N}^{\mathscr{Z},+}(x),$

where $\bm{W}_{N}^{\mathscr{Z},+}(x)\in\oplus_{\alpha\geq
1}\mathscr{T}_{\alpha}$. Recalling Definition C.3, one has

$F(\bm{W}_{N}^{\mathscr{Z}})(x)=\sum_{k\in\mathbb{N}_{0}}\frac{D^{k}F(W_{N}^{\mathscr{Z}}(x))}{k!}\Big{(}\sum_{\tau\in\mathcal{T}_{-}}\mathscr{I}\tau+\bm{W}_{N}^{\mathscr{Z},+}(x)\Big{)}^{\star
k}.$

Since Lemma C.20 implies that

$\mathscr{D}_{i}[\mathcal{K}\tau_{1}^{\mathcal{K},\mathscr{Z}}](x)\star\mathscr{D}_{i}[\mathcal{K}\tau_{2}^{\mathcal{K},\mathscr{Z}}](x)$

is $\oplus_{\alpha\geq-1+\delta}\mathscr{T}_{\alpha}$-valued, one can ignore
the contribution from $\bm{W}_{N}^{\mathscr{Z},+}(x)$ when the projection
$\mathfrak{p}_{<\delta}$ is applied. This observation proves the claimed
identities. ∎

######  Lemma C.27.

Let $N\in\mathbb{N}$. Then, one has

$\mathcal{R}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}\bm{Y}_{N}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}=F(W_{N}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}})\\\
\times\Big{\\{}\sum_{\begin{subarray}{c}\tau_{1},\tau_{2}\in\mathcal{T}_{-},\lvert\tau_{1}\rvert_{+}+\lvert\tau_{2}\rvert_{+}>-2\end{subarray}}c(\tau_{1})c(\tau_{2})\nabla(K\ast\mathcal{R}^{\mathscr{Z}^{\mathrm{can},\varepsilon}}\tau_{1}^{\mathcal{K},\mathscr{Z}^{\mathrm{can},\varepsilon}})\cdot\nabla(K\ast\mathcal{R}^{\mathscr{Z}^{\mathrm{can},\varepsilon}}\tau_{2}^{\mathcal{K},\mathscr{Z}^{\mathrm{can},\varepsilon}})\\\
+2\nabla[K\ast
X^{\mathscr{Z}^{\mathrm{can},\varepsilon}}]\cdot\nabla[H_{N}\ast
X^{\mathrm{can},\varepsilon}]\Big{\\}}$

###### Proof.

To simplify notation, we write
$\Pi_{x}^{\mathrm{BPHZ},\varepsilon}\vcentcolon=\Pi_{x}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}$
here, for instance. One has

$\mathcal{R}^{\mathrm{BPHZ},\varepsilon}\bm{Y}_{N}^{\mathrm{BPHZ},\varepsilon}(x)=[\Pi_{x}^{\mathrm{BPHZ},\varepsilon}\bm{Y}_{N}^{\mathrm{BPHZ},\varepsilon}(x)](x).$

In view of Lemma C.20, Proposition C.25 and Lemma C.26, it suffices to show

$\displaystyle\Pi^{\mathrm{BPHZ},\varepsilon}_{x}[\mathscr{I}(\tau_{1})\cdots\mathscr{I}(\tau_{n})\mathscr{I}_{i}(\tau_{n+1})]$
$\displaystyle=\Pi^{\mathrm{can},\varepsilon}_{x}[\mathscr{I}(\tau_{1})\cdots\mathscr{I}(\tau_{n})\mathscr{I}_{i}(\tau_{n+1})],$
$\displaystyle\Pi^{\mathrm{BPHZ},\varepsilon}_{x}[\mathscr{I}(\tau_{1})\cdots\mathscr{I}(\tau_{n})\mathscr{I}_{i}(\tau_{n+1})\mathscr{I}_{i}(\tau_{n+2})]$
$\displaystyle=\Pi^{\mathrm{can},\varepsilon}_{x}[\mathscr{I}(\tau_{1})\cdots\mathscr{I}(\tau_{n})\mathscr{I}_{i}(\tau_{n+1})\mathscr{I}_{i}(\tau_{n+2})],$
(89)

for $\tau_{1},\ldots,\tau_{n},\tau_{n+1}\in\mathcal{T}_{-}$ and
$\tau_{n+2}\in\operatorname{Remove}(\mathfrak{B}(\mathcal{T}))$. We only prove
the second identity of (89). We set

$\bm{\tau}\vcentcolon=(F,0)^{0,0}_{\mathfrak{e}}\vcentcolon=\mathscr{I}(\tau_{1})\cdots\mathscr{I}(\tau_{n})\mathscr{I}_{i}(\tau_{n+1})\mathscr{I}_{i}(\tau_{n+2}),\quad(F_{j},0)^{0,0}_{\mathfrak{e}}\vcentcolon=\tau_{j}.$

The proof of (89) follows the argument in the proof of Lemma C.23. We claim

$\Delta^{\circ}_{-}\bm{\tau}=\bm{1}_{-}\otimes\bm{\tau}+\sum_{J\subseteq\\{1,\ldots,n\\}}\big{[}\mathscr{I}_{i}(\tau_{n+1})\prod_{j\in
J}\mathscr{I}(\tau_{j})\big{]}\otimes\big{[}\mathscr{I}_{i}(\tau_{n+2})\prod_{j\notin
J}\mathscr{I}(\tau_{j})\big{]}\\\
+\sum_{J\subseteq\\{1,\ldots,n\\}}\big{[}\mathscr{I}_{i}(\tau_{n+1})\mathscr{I}_{i}(\tau_{n+2})\prod_{j\in
J}\mathscr{I}(\tau_{j})\big{]}\otimes\prod_{j\notin J}\mathscr{I}(\tau_{j}).$
(90)

Indeed, let $\sigma\otimes\sigma^{\prime}$ be a basis appearing in the
coproduct formula (73) for $\Delta^{\circ}_{-}\bm{\tau}$. If we set
$(G,0)^{\mathfrak{n},0}_{\mathfrak{e}}\vcentcolon=\sigma$ and
$\sigma_{k}\vcentcolon=(G\cap F_{j},0)^{\mathfrak{n},0}_{\mathfrak{e}}$, by
repeating the argument in the proof of Lemma C.23, the forest $\sigma_{k}$ is
either $\varnothing$, $\tau_{k}$ or $\operatorname{Remove}(\rho_{k};e_{k})$
for some $\rho_{k}$ and $e_{k}$.

  * •

If $\sigma_{k}=\varnothing$, then $\sigma=0$ in $\mathscr{T}_{-}$ unless
$(\rho_{\bm{\tau}},\rho_{\tau_{k}})\notin E_{\sigma}$.

  * •

If $\sigma_{k}=\tau_{k}$, then $\sigma^{\prime}$ has a leaf of type
$\mathscr{I}$ unless $(\rho_{\bm{\tau}},\rho_{\tau_{k}})\in E_{\sigma}$.

  * •

If $\sigma_{k}=\operatorname{Remove}(\rho_{k};e_{k})$, then
$\lvert\sigma\rvert_{+}>0$ and hence $\sigma=0$ in $\mathscr{T}_{-}$.

Therefore, the claimed identity (90) is established. It remains to show

$g_{\varepsilon}^{-}\hat{\mathscr{A}}_{-}\big{[}\mathscr{I}_{i}(\tau_{n+1})\prod_{j\in
J}\mathscr{I}(\tau_{j})\big{]}=0,\quad
g_{\varepsilon}^{-}\hat{\mathscr{A}}_{-}\big{[}\mathscr{I}_{i}(\tau_{n+1})\mathscr{I}_{i}(\tau_{n+2})\prod_{j\in
J}\mathscr{I}(\tau_{j})\big{]}=0.$ (91)

Without loss of generality, we can suppose $J=\\{1,\ldots,n\\}$. The proof is
based on induction. We only consider the first identity of (91). As for the
case $n=0$, the first identity of (91) is shown in Lemma C.22. Similarly to
(90), one can show

$\hat{\Delta}_{-}\bm{\tau}=\bm{1}_{-}\otimes\bm{\tau}+\sum_{J\subseteq\\{1,\ldots,n\\}}\big{[}\mathscr{I}_{i}(\tau_{n+1})\prod_{j\in
J}\mathscr{I}(\tau_{j})\big{]}\otimes\prod_{j\notin J}\mathscr{I}(\tau_{j})$

In view of the recursive formula (76) and the hypothesis of the induction, it
remains to show

$\mathbb{E}[\mathbf{\Pi}^{\mathrm{can},\varepsilon}\bm{\tau}(0)]=0.$

However, this can be proved as in Lemma C.22, since $\bm{\tau}$ has an odd
number of edges $e$ such that $\mathfrak{t}(e)=\mathscr{I}$ and
$\lvert\mathfrak{e}(e)\rvert=1$. ∎

######  Proposition C.28.

Let $U$ be a bounded domain.

Suppose that $M$ and $\varepsilon$ are random variables (depending on $U$)
with values in $\mathbb{N}_{0}$ and $(0,\infty)$, respectively, such that
$\lvert W_{M}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}\rvert\leq 2$ on $U$
and $\lVert
W_{M}^{\operatorname{AH},\varepsilon}-W_{M}^{\operatorname{AH}}\rVert_{L^{\infty}(U)}\leq
1$ almost everywhere.

Then,

$\lvert\nabla W_{N}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}\rvert^{2}+\Delta
W_{N}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}+e^{-2W_{N}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}}Y_{N}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}=-\xi_{\varepsilon}+c_{\varepsilon}\quad\text{on
}U,$ (92)

where the constant $c_{\varepsilon}$ is defined in (88).

###### Proof.

By Proposition C.25, one has
$W_{N}^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}=W_{N}^{\mathscr{Z}^{\mathrm{can},\varepsilon}}$.
Therefore, by Lemma C.17 and Lemma C.27, the left-hand side of (92) is equal
to

$-\xi_{\varepsilon}-[\Delta(G_{N}-G)]\ast
c_{\varepsilon}=-\xi_{\varepsilon}+c_{\varepsilon}.\qed$

###  C.6 Stochastic estimates and Besov regularity

Proposition C.14 gives pathwise estimates for the modelled distributions
$\bm{X}$, $\bm{W}_{N}$ and $\bm{Y}_{N}$. Here we give stochastic estimates for
$X$ and $Y_{N}$ in suitable Besov spaces. To this end, we will need a wavelet
characterization of weighted Besov spaces.

######  Theorem C.29 ([54], [71, Theorem 1.61]).

For any $k\in\mathbb{N}$, there exist
$\psi_{\mathfrak{f}},\psi_{\mathfrak{m}}\in C_{c}^{k}(\mathbb{R})$ with the
following properties.

  * •

For $n\in\mathbb{N}_{0}$, if we denote by $V_{n}$ the subspace of
$L^{2}(\mathbb{R})$ spanned by

$\\{\psi_{\mathfrak{f}}(2^{n}\cdot-m)\nonscript\>|\nonscript\>\mathopen{}m\in\mathbb{Z}\\},$

then the inclusions $V_{0}\subseteq V_{1}\subseteq\cdots\subseteq
V_{n}\subseteq V_{n+1}\subseteq\cdots$ hold and $L^{2}(\mathbb{R})$ is the
closure of $\cup_{n\in\mathbb{N}_{0}}V_{n}$.

  * •

The set

$\\{\psi_{\mathfrak{f}}(\cdot-m)\nonscript\>|\nonscript\>\mathopen{}m\in\mathbb{Z}\\}\cup\\{\psi_{\mathfrak{m}}(\cdot-m)\nonscript\>|\nonscript\>\mathopen{}m\in\mathbb{Z}\\}$

forms an orthonormal basis of $V_{1}$. Therefore, the set

$\\{\psi_{\mathfrak{f}}(\cdot-m)\nonscript\>|\nonscript\>\mathopen{}m\in\mathbb{Z}\\}\cup\\{2^{\frac{n}{2}}\psi_{\mathfrak{m}}(2^{n}\cdot-m)\nonscript\>|\nonscript\>\mathopen{}n\in\mathbb{N}_{0},m\in\mathbb{Z}\\}$

forms an orthonormal basis of $L^{2}(\mathbb{R})$.

  * •

One has $\int_{\mathbb{R}}x^{l}\psi_{\mathfrak{m}}(x)\,\mathrm{d}x=0$ for
every $l\in\\{1,2,\ldots,k\\}$.

One can build an orthonormal basis of $L^{2}(\mathbb{R}^{d})$ as follows.

######  Proposition C.30 ([71, Proposition 1.53]).

Let $k\in\mathbb{N}$ and let $\psi_{\mathfrak{f}},\psi_{\mathfrak{m}}\in
C_{c}^{k}(\mathbb{R}^{d})$ be as in Theorem C.29. For $n\in\mathbb{N}_{0}$, we
define the sets of $d$-tuples by

$\mathfrak{G}^{n}\vcentcolon=\begin{cases}\\{(\mathfrak{f},\ldots,\mathfrak{f})\\}&\text{if
}n=0,\\\
\\{(G_{1},\ldots,G_{d})\in\\{\mathfrak{f},\mathfrak{m}\\}^{d}\nonscript\>|\nonscript\>\mathopen{}\exists
j\text{ s.t. }G_{j}=\mathfrak{m}\\}&\text{if }n\geq 1.\end{cases}$

For $n\in\mathbb{N}_{0}$, $G\in\mathfrak{G}^{n}$, $m\in\mathbb{Z}^{d}$ and
$x\in\mathbb{R}^{d}$, we set

$\Psi_{m}^{n,G}(x)\vcentcolon=2^{\frac{d\max\\{n-1,0\\}}{2}}\prod_{j=1}^{d}\psi_{G_{j}}(2^{\max\\{n-1,0\\}}x_{j}-m_{j}).$
(93)

The set
$\\{\Psi_{m}^{n,G}\nonscript\>|\nonscript\>\mathopen{}n\in\mathbb{N}_{0},G\in\mathfrak{G}^{n},m\in\mathbb{Z}^{d}\\}$
forms an orthonormal basis of $L^{2}(\mathbb{R}^{d})$.

With the expansion by the basis
$\\{\Psi_{m}^{n,G}\nonscript\>|\nonscript\>\mathopen{}n\in\mathbb{N}_{0},G\in\mathfrak{G}^{n},m\in\mathbb{Z}^{d}\\}$,
one can give a wavelet characterization of weighted Besov spaces.

######  Proposition C.31 ([71, Theorem 6.15]).

Let $p,q\in[1,\infty]$, $r\in\mathbb{R}$ and $\sigma\in(0,\infty)$. Suppose

$k>\max\Big{\\{}r,\frac{2d}{p}+\frac{d}{2}-r\Big{\\}}$

and let
$\\{\Psi_{m}^{n,G}\nonscript\>|\nonscript\>\mathopen{}n\in\mathbb{N}_{0},G\in\mathfrak{G}^{n},m\in\mathbb{Z}^{d}\\}$
be as in Proposition C.30. Then, there exists a constant $C\in(0,\infty)$ such
that for every $f\in B_{p,q}^{r,\sigma}(\mathbb{R}^{d})$ one has

$C^{-1}\lVert f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}\\\
\leq\Big{\lVert}\Big{(}2^{n(r-d/p)}\Big{(}\sum_{G\in\mathfrak{G}^{n},m\in\mathbb{Z}^{d}}w_{\sigma}(2^{-n}m)^{p}\lvert
2^{nd/2}\langle
f,\Psi^{n,G}_{m}\rangle\rvert^{p}\Big{)}^{1/p}\Big{)}_{n\in\mathbb{N}_{0}}\Big{\rVert}_{l^{q}(\mathbb{N}_{0})}\\\
\leq C\lVert f\rVert_{B_{p,q}^{r,\sigma}(\mathbb{R}^{d})}.$

We fix $k\in\mathbb{N}$ such that $k>\frac{5d}{2}+2$, and we consider the
orthonormal basis $\\{\Psi^{n,G}_{m}\\}$ given by (93). We set
$\Psi\vcentcolon=\Psi^{0,(\mathfrak{f},\ldots,\mathfrak{f})}_{0}$.

######  Definition C.32.

Let
$\mathscr{Z}=(\Pi,\Gamma),\overline{\mathscr{Z}}=(\overline{\Pi},\overline{\Gamma})\in\mathscr{M}(\mathscr{T},K)$.
Given a compact set $\mathfrak{K}\subseteq\mathbb{R}^{d}$, we set

$\displaystyle\llbracket\mathscr{Z}\rrbracket_{\mathfrak{K}}$
$\displaystyle\vcentcolon=\sup_{\tau=(T,0)^{\mathfrak{n},0}_{\mathfrak{e}}\in\mathfrak{B}(\mathscr{T})\cap\mathscr{T}_{<0}}\sup_{n\in\mathbb{N}}\sup_{x\in\mathfrak{K}\cap
2^{-n}\mathbb{Z}^{d}}2^{n\lvert\tau\rvert_{+}}\lvert\langle\Pi_{x}\tau,2^{nd}\Psi(2^{n}(\cdot-x))\rangle_{\mathbb{R}^{d}}\rvert,$
$\displaystyle\llbracket\mathscr{Z};\overline{\mathscr{Z}}\rrbracket_{\mathfrak{K}}$
$\displaystyle\vcentcolon=\sup_{\tau=(T,0)^{\mathfrak{n},0}_{\mathfrak{e}}\in\mathfrak{B}(\mathscr{T})\cap\mathscr{T}_{<0}}\sup_{n\in\mathbb{N}}\sup_{x\in\mathfrak{K}\cap
2^{-n}\mathbb{Z}^{d}}2^{n\lvert\tau\rvert_{+}}\lvert\langle\Pi_{x}\tau-\overline{\Pi}_{x}\tau,2^{nd}\Psi(2^{n}(\cdot-x))\rangle_{\mathbb{R}^{d}}\rvert.$

######  Lemma C.33.

For each $\gamma\in\mathbb{R}$, there exist a constant $C\in(0,\infty)$ and an
integer $k\in\mathbb{N}$ such that the following estimates hold uniformly over
$\mathscr{Z},\overline{\mathscr{Z}}\in\mathscr{M}(\mathscr{T},K)$ and compact
sets $\mathfrak{K}\subseteq\mathbb{R}^{d}$:

$\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;\mathfrak{K}}\leq
C(1+\llbracket\mathscr{Z}\rrbracket_{\mathfrak{K}})^{k},\quad\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z};\overline{\mathscr{Z}}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;\mathfrak{K}}\leq
C(1+\llbracket\mathscr{Z}\rrbracket_{\mathfrak{K}})^{k}(\llbracket\mathscr{Z};\overline{\mathscr{Z}}\rrbracket_{\mathfrak{K}}+\llbracket\mathscr{Z};\overline{\mathscr{Z}}\rrbracket_{\mathfrak{K}}^{k}).$

###### Proof.

Using the recursive formula [13, Proposition 4.17], one can prove the claim as
in [48, Lemma 2.3]. ∎

######  Lemma C.34.

Let $L\in[1,\infty)$ and set $Q_{L}\vcentcolon=[-L,L]^{d}$. Let $p\in
2\mathbb{N}$. Under Assumption 3.10, if $p\delta^{\prime}>d+1$, one has

$\mathbb{E}[\llbracket\mathscr{Z}^{\mathrm{BPHZ}}\rrbracket_{Q_{L}}^{p}]\leq
C_{p}^{\mathrm{BPHZ}}L^{d},\quad\mathbb{E}[\llbracket\mathscr{Z}^{\mathrm{BPHZ}};\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}\rrbracket_{Q_{L}}^{p}]\leq\bm{\varepsilon}^{\mathrm{BPHZ}}_{p}(\varepsilon)L^{d}.$

###### Proof.

The proof is essentially the repetition of [48, Lemma 4.11]. Set

$\mathfrak{B}_{0}(\mathscr{T})\vcentcolon=\\{\tau=(T,0)^{\mathfrak{n},0}_{\mathfrak{e}}\in\mathfrak{B}(\mathscr{T})\nonscript\>|\nonscript\>\mathopen{}\lvert\tau\rvert_{+}<0\\}.$

If we write
$\Psi^{\lambda}_{x}\vcentcolon=\lambda^{-d}\Psi(\lambda^{-1}(\cdot-x))$, one
has

$\displaystyle\mathbb{E}[\llbracket\mathscr{Z}^{\mathrm{BPHZ}}\rrbracket_{Q_{L}}^{p}]$
$\displaystyle=\mathbb{E}[\sup_{\tau\in\mathfrak{B}_{0}(\mathscr{T})}\sup_{n\in\mathbb{N}}\sup_{x\in
Q_{L}\cap
2^{-n}\mathbb{Z}^{d}}2^{n\lvert\tau\rvert_{+}p}\lvert\langle\Pi_{x}\tau,\Psi_{x}^{2^{-n}}\rangle_{\mathbb{R}^{d}}\rvert^{p}]$
$\displaystyle\lesssim\sum_{\tau\in\mathfrak{B}_{0}(\mathscr{T})}\sum_{n\in\mathbb{N}}2^{nd}L^{d}2^{n\lvert\tau\rvert_{+}p}\mathbb{E}[\lvert\langle\Pi_{0}\tau,\Psi_{0}^{2^{-n}}\rangle_{\mathbb{R}^{d}}\rvert^{p}],$

where the stationarity of the noise $\xi$ and the estimate $\\#(Q_{L}\cap
2^{-n}\mathbb{Z}^{d})\lesssim 2^{nd}L^{d}$ are used. By Assumption 3.10,

$\mathbb{E}[\lvert\langle\Pi_{0}\tau,\Psi_{0}^{2^{-n}}\rangle_{\mathbb{R}^{d}}\rvert^{p}]\lesssim_{\psi_{\mathfrak{f}}}C_{p}^{\mathrm{BPHZ}}2^{-np(\lvert\tau\rvert_{+}+\delta^{\prime})}.$

Therefore,

$\mathbb{E}[\llbracket\mathscr{Z}^{\mathrm{BPHZ}}\rrbracket_{Q_{L}}^{p}]\lesssim
C^{\mathrm{BPHZ}}_{p}L^{d}\lvert\mathfrak{B}_{0}(\mathscr{T})\rvert(2^{p\delta^{\prime}-d}-1)^{-1}.$

The estimate for the second claimed inequality is similar. ∎

######  Lemma C.35.

Let $\mathfrak{K}\subseteq\mathbb{R}^{d}$ be a compact set and
$\sigma\in(0,\infty)$. Then, there exists a constant $C\in(0,\infty)$ such
that for all $N\in\mathbb{N}$

$\lVert H_{N}\ast X\rVert_{C^{2}(\mathfrak{K})}\leq C2^{3N}\lVert
X\rVert_{\mathcal{C}^{-2,\sigma}(\mathbb{R}^{d})}.$

###### Proof.

Let $\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$ be such that $\phi\equiv 1$ on
$\mathfrak{K}$. By Lemma A.4, one has

$\lVert H_{N}\ast X\rVert_{C^{2}(\mathfrak{K})}\lesssim\lVert\phi(H_{N}\ast
X)\rVert_{\mathcal{C}^{2}(\mathbb{R}^{d})}\lesssim_{\sigma}\lVert H_{N}\ast
X\rVert_{\mathcal{C}^{2,\sigma}(\mathbb{R}^{d})}.$

It remains to apply Corollary A.10. ∎

Recall from Definition B.6 that we have, for instance,
$\lvert\Xi\rvert_{+}=-2+\delta+\kappa$ for some
$\kappa\in(0,\delta^{\prime})$.

######  Proposition C.36.

Under Assumption 3.10, there exist a deterministic integer
$k=k(\delta_{-})\in\mathbb{N}$ such that for all $\sigma\in(0,\infty)$, $p\in
2\mathbb{N}$ with $p>(d+1)/\min\\{\delta^{\prime}-\kappa,\sigma\\}$ and
$N\in\mathbb{N}$ we have the following:

$\displaystyle\mathbb{E}[\lVert
X^{\mathscr{Z}^{\mathrm{BPHZ}}}\rVert_{B_{p,p}^{-2+\delta+\kappa/2,\sigma}(\mathbb{R}^{d})}^{p}]$
$\displaystyle\lesssim_{\delta,\delta^{\prime},\kappa,\sigma,p}C_{kp}^{\mathrm{BPHZ}},$
$\displaystyle\mathbb{E}[\lVert
Y^{\mathscr{Z}^{\mathrm{BPHZ}}}_{N}\rVert_{B_{p,p}^{-1+\delta+\kappa/2,\sigma}(\mathbb{R}^{d})}^{p}]$
$\displaystyle\lesssim_{\delta,\delta^{\prime},\kappa,\sigma,p}C_{kp}^{\mathrm{BPHZ}}2^{kpN}$

and

$\displaystyle\mathbb{E}[\lVert
X^{\mathscr{Z}^{\mathrm{BPHZ}}}-X^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}\rVert_{B_{p,p}^{-2+\delta+\kappa/2,\sigma}(\mathbb{R}^{d})}^{p}]$
$\displaystyle\lesssim_{\delta,\delta^{\prime},\kappa,\sigma,p}C_{kp}^{\mathrm{BPHZ}}[\bm{\varepsilon}^{\mathrm{BPHZ}}_{kp}(\varepsilon)+\bm{\varepsilon}^{\mathrm{BPHZ}}_{p}(\varepsilon)],$
$\displaystyle\mathbb{E}[\lVert
Y^{\mathscr{Z}^{\mathrm{BPHZ}}}_{N}-Y^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}_{N}\rVert_{B_{p,p}^{-1+\delta+\kappa/2,\sigma}(\mathbb{R}^{d})}^{p}]$
$\displaystyle\lesssim_{\delta,\delta^{\prime},\kappa/2,\sigma,p}C_{kp}^{\mathrm{BPHZ}}2^{kpN}[\bm{\varepsilon}^{\mathrm{BPHZ}}_{kp}(\varepsilon)+\bm{\varepsilon}^{\mathrm{BPHZ}}_{p}(\varepsilon)].$

###### Proof.

Set $\mathscr{Z}\vcentcolon=\mathscr{Z}^{\mathrm{BPHZ}}$. In the proof, we
drop superscripts for $\mathrm{BPHZ}$. Natural numbers $k,l,\gamma$ depend
only on $\mathscr{T}$ and they vary from line to line. We will not write down
the dependence on $\mathscr{T},\delta,\delta_{-},p,\sigma$. Recall the
notation $\Psi^{n,G}_{m}$ from (93).

Suppose we are given a modelled distribution
$f\in\mathcal{D}^{\gamma}_{\alpha}(\mathscr{T},\mathscr{Z})$ with
$\alpha<0<\gamma$. We decompose

$\langle\mathcal{R}f,2^{nd/2}\Psi^{n,G}_{m}\rangle_{\mathbb{R}^{d}}\\\
=\langle\mathcal{R}f-\Pi_{2^{-n}m}f(2^{-n}m),2^{nd/2}\Psi^{n,G}_{m}\rangle_{\mathbb{R}^{d}}+\langle\Pi_{2^{-n}m}f(2^{-n}m),2^{nd/2}\Psi^{n,G}_{m}\rangle_{\mathbb{R}^{d}}.$

Using (77), the first term is bounded by a constant times

$2^{-n\gamma}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert
f\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(2^{-n}m,l)}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(2^{-n}m,l)}.$

To estimate the second term, consider the basis expansion

$f(x)=\sum_{\sigma}a_{\sigma}(x)\sigma.$

One has $\lvert
a_{\sigma}(2^{-n}m)\rvert\leq\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert
f\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(2^{-n}m,l)}$ and

$\lvert\langle\Pi_{2^{-n}m}\sigma,2^{nd/2}\Psi^{n,G}_{m}\rangle_{\mathbb{R}^{d}}\rvert\lesssim
2^{-n\alpha}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(2^{-n}m,l)}.$

Therefore,

$\lvert\langle\mathcal{R}f,2^{nd/2}\Psi^{n,G}_{m}\rangle_{\mathbb{R}^{d}}\rvert\lesssim
2^{-n\alpha}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert
f\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(2^{-n}m,l)}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(2^{-n}m,l)}.$
(94)

Applying the estimate (94) to $\bm{X}$ and $\bm{Y}_{N}$, by Proposition C.31,
we get

$\lVert X\rVert_{B_{p,p}^{-2+\delta+\kappa/2,\sigma}(\mathbb{R}^{d})}^{p}\\\
\lesssim\sum_{n\in\mathbb{N}_{0}}2^{-n(d+\kappa/2)}\sum_{G\in
G^{n},m\in\mathbb{Z}^{d}}w_{\sigma}(2^{-n}m)^{p}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{X}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{2;B(2^{-n}m,l)}^{p}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{2;B(2^{-n}m,l)}^{p},$
$\lVert
Y_{N}\rVert_{B_{p,p}^{-1+\delta+\kappa/2,\sigma}(\mathbb{R}^{d})}^{p}\\\
\lesssim\sum_{n\in\mathbb{N}_{0}}2^{-n(d+\kappa/2)}\sum_{G\in
G^{n},m\in\mathbb{Z}^{d}}w_{\sigma}(2^{-n}m)^{p}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\bm{Y}_{N}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\delta;B(2^{-n}m,l)}^{p}\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\delta;B(2^{-n}m,l)}^{p}.$

To estimate $\lVert
X\rVert_{B_{p,p}^{-2+\delta+\kappa/2,\sigma}(\mathbb{R}^{d})}$, we use Lemma
C.14 and stationarity to obtain

$\mathbb{E}[\lVert
X\rVert_{B_{p,p}^{-2+\delta+\kappa/2,\sigma}(\mathbb{R}^{d})}^{p}]\lesssim\sum_{n\in\mathbb{N}_{0}}2^{-n(d+(\delta-\delta_{-}))}\sum_{G\in
G^{n},m\in\mathbb{Z}^{d}}w_{\sigma}(2^{-n}m)^{p}\mathbb{E}[(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(0,l)})^{kp}].$

Since

$\sum_{m\in\mathbb{Z}^{d}}w_{\sigma}(2^{-n}m)^{p}\lesssim\int_{\mathbb{R}^{d}}(1+\lvert
2^{-n}x\rvert^{2})^{-\frac{p\sigma}{2}}dx=2^{nd}\lVert
w_{\sigma}\rVert_{L^{p}(\mathbb{R}^{d})}^{p},$

and by Lemma C.33 and by Lemma C.34

$\mathbb{E}[(1+\lvert\kern-1.07639pt\lvert\kern-1.07639pt\lvert\mathscr{Z}\rvert\kern-1.07639pt\rvert\kern-1.07639pt\rvert_{\gamma;B(0,l)})^{kp}]\lesssim
C_{k^{\prime}p}^{\mathrm{BPHZ}}$

for some $k^{\prime}\in\mathbb{N}$, we conclude

$\mathbb{E}[\lVert
X\rVert_{B_{p,p}^{-2+\delta+\kappa/2,\sigma}(\mathbb{R}^{d})}^{p}]\lesssim
C_{kp}^{\mathrm{BPHZ}}.$

The estimate of $Y_{N}$ is similar by using Lemma C.35. The estimates of the
differences can be proved similarly by using [34, (3.4)]. ∎

######  Corollary C.37.

Under Assumption 3.10, let $\sigma\in(0,\infty)$, $p\in[1,\infty)$ and
$N\in\mathbb{N}$. Then, as $\varepsilon\downarrow 0$,
$(X^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}})_{\varepsilon\in(0,1)}$
converges in $L^{p}(\mathbb{P})$ to $X^{\mathscr{Z}^{\mathrm{BPHZ}}}$ in
$\mathcal{C}^{-2+\delta,\sigma}(\mathbb{R}^{d})$, and
$(Y^{\mathscr{Z}^{\mathrm{BPHZ},\varepsilon}}_{N})_{\varepsilon\in(0,1)}$
converges in $L^{p}(\mathbb{P})$ to $Y_{N}^{\mathscr{Z}^{\mathrm{BPHZ}}}$ in
$\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})$. Furtheremore, there exists a
deterministic $k=k(\delta)\in\mathbb{N}$, independent of $\sigma$ and $N$,
such that

$\sup_{N\in\mathbb{N}}2^{-kN}\lVert
Y_{N}^{\mathscr{Z}^{\mathrm{BPHZ}}}\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}\in
L^{p}(\mathbb{P}).$ (95)

###### Proof.

The claim on the convergence follows from Proposition C.36 and by applying
Besov embeddings. To show (95), let $q\in 2\mathbb{N}$ be such that
$d/q<\kappa/2$ and $q>p$. By Proposition C.36 and the Besov embedding, for
some $k^{\prime}\in\mathbb{N}$,

$\mathbb{E}[\lVert
Y_{N}^{\mathscr{Z}^{\mathrm{BPHZ}}}\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}^{q}]\lesssim_{q,\delta,\sigma}\mathbb{E}[\lVert
Y_{N}^{\mathscr{Z}^{\mathrm{BPHZ}}}\rVert_{B^{-1+\delta+d/q,\sigma}_{q,q}(\mathbb{R}^{d})}^{q}]\lesssim_{q,\delta,\kappa,\sigma}2^{qk^{\prime}N}.$

Therefore, if $k>k^{\prime}$,

$\sum_{N\in\mathbb{N}}2^{-kqN}\mathbb{E}[\lVert
Y_{N}^{\mathscr{Z}^{\mathrm{BPHZ}}}\rVert_{\mathcal{C}^{-1+\delta,\sigma}(\mathbb{R}^{d})}^{q}]<\infty.\qed$

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