If furthermore one has a sequence of models $Π_n → Π$ with associated subspaces $B_n ⊂ D^{γ,w}(V)$ and reconstruction operators $\hat{R}_n$, as well as a sequence $F_n ∈ B_n$ such that $||F_n; F||_{γ,w} → 0$ and $||\hat{R}_n F_n - \hat{R}F||_η → 0$, then the sequence of modelled distributions $G_n$ constructed in the first part of the statement converges to $G$ in the sense that $||G_n; G||_{\tilde{γ},\tilde{w}} → 0$.

*Proof.* The proof is virtually identical to that of Corollary B.4. The only difference is that we use [9, Lem. 4.12] to define maps $K_k : B → D^{\tilde{γ}_k, \tilde{w}_k}$ (with $\tilde{γ}_k$ and $\tilde{w}_k$ defined like $\tilde{γ}$ and $\tilde{w}$, but with $β$ replaced by $β + |k|_S$) such that $RK_k F = K_k \hat{R} F$. (Since $η + β > -2$ by assumption, the assumptions of that lemma are satisfied and $RK_k F$ is always well-defined as a distribution on the whole space-time.) □

## Appendix C: Reconstruction Theorem

In this section, we present a version of the reconstruction theorem that allows to bypass to some extent the condition $ν > -1$ appearing in [9]. This appendix was written in collaboration with Máté Gerencsér. In this section we assume that $P_0 = \{0\} × R^d$, $P_1 = R × ∂D$, and $P = P_0 ∪ P_1$, and we write $|x|_{P_i}$ for the parabolic distance between $x$ and $P_i$. Generic points $x, y$, etc are space-time points.

Throughout this section, we fix a regularity structure $(T, G)$ containing the polynomial structure and such that the product with elements of the polynomial structure $\bar{T}$ is well-defined in $T$. We also only consider models such that $Γ_{x,y}$ acts on $\bar{T}$ by translations by $y-x$. Let us recall from [11, Sec 8] that, given any regularity structure $(T, G)$, a model $(Π, Γ)$ can alternatively be described by a linear map $Π: T → D'$, together with a continuous map $F: R^d → G$ such that, setting $Π_x = Π F_x$, $Γ_{xy} = (F_x)^{-1} F_y$, the analytic bounds for models are satisfied. The main assumption we impose on our models in the present setting is the following:

**Assumption C.1.** Setting $T_< = ⊕_{α≤1} T_α$, there exist linear maps $Π^+, Π^-: T_< → D'$ with $Π^+ + Π^- = Π ↕ T_<$ and such that

* $(Π^+τ)(ψ) = 0$ for all $τ ∈ T_<$ and all $ψ$ supported in $R × D^c$ and $(Π^-τ)(ψ) = 0$ for all $ψ$ supported in $R × D$;

* Setting $Π_x^+ = Π^+ F_x$, $Π_x^- := Π^- F_x$ the pairs $(Π^+, Γ)$ and $(Π^-, Γ)$ are models on $(T_<, G)$ in the sense of [11]. (But they are not admissible in general!)

We will always write $α$ for the lowest degree appearing in our ambient regularity structure $T$. We also fix $γ > 0$ as well as exponents $η$ and $σ$ on which we make the following assumption:

**Assumption C.2.** The exponents satisfy the condition

$$0 > σ > -1 \geq \alpha \geq η > -2. \qquad (C.1)$$

We also use the shorthand $w = (η, σ, η)$ similarly to [9] (except that we make the simplifying assumption that the “corner exponent” coincides with $η$ which is not essential but simplifies our argument). One crucial ingredient for our result is the following:

**Lemma C.3.** For every $f ∈ D^{γ,w}$, there exist $f_± ∈ D^{σ,η}$ such that $f_+(x) = f(x)$ for $x ∈ R × D$ and $f_-(x) = f(x)$ for $x ∈ R × D^c$.

*Proof.* It follows from the definition of the spaces $D^{γ,w}$ that the restriction of $Q_{<σ}f$ to either $R × \bar{D}$ or $R × \bar{D}^c$ belongs to $D^{σ,η}$. In particular, components of $f$ of degree below $σ$ can be extended continuously to $(R \setminus \{0\}) × D$. (Note that $σ < 0$ though!) The claim then follows from an adaptation of Whitney's extension theorem to the setting of regularity structures, see for example [18, Thm 5.3.16]. □
**Remark C.4.** In general there is no reason for $f_+$ and $f_-$ to coincide on $\mathbf{R} \times \partial D$.

We define spaces $C^{\alpha, \eta}$ with $\eta \le \alpha < 0$ as consisting of those distributions $\zeta \in C^{\eta}(\mathbf{R}^{1+d}) \cap C^{\alpha}(\mathbf{R}^{1+d} \setminus P_0)$ such that

$$|\zeta(\psi_x^\lambda)| \lesssim \lambda^\alpha |x|^{\eta-\alpha},$$

uniformly over $x$ in compacts away from $P_0$, $2\lambda \in (0, 2 \wedge |x|_{P_0})$, and test functions $\psi \in \mathcal{B}$. Here and below we write $\mathcal{B}$ for the set of functions supported in the centred (parabolic) ball of radius 1 and with $r$ derivatives bounded by 1, where $r$ is some fixed sufficiently large value. Note that if $\eta > -2$, $\zeta \in C^{\alpha, \eta}$ is uniquely determined by its action on test functions supported outside $P_0$, see [11].

**Theorem C.5.** Under Assumptions C.1 and C.2, there then exists a unique continuous linear operator $\mathcal{R}: D_P^{\gamma, w} \to C^{\alpha, \eta}$ such that $\mathcal{R}f(\psi) = \hat{\mathcal{R}}f(\psi)$ for all $\psi$ supported in $\mathbf{R}^d \setminus P$, and such that one has the bound

$$(\mathcal{R}f - \Pi_x^+ f_+(x) - \Pi_x^- f_-(x))(\psi_x^\lambda) \lesssim \lambda^\sigma |x|_{P_0}^{\mu-\sigma} \quad (C.2)$$

uniformly over $x \in P_1 \setminus P_0$ (in compacts), over $\lambda \in (0, 1]$ such that $2\lambda \le |x|_{P_0}$, and over $\psi \in \mathcal{B}$.

**Remark C.6.** We did not specify the continuity of the map $\mathcal{R}$ with respect to different models. We will continue to omit continuity statements in the sequel, on one hand for the sake of easing the presentation, and on the other hand due to the fact that since all the operations discussed here are linear, the “linearisation trick” of [14, Prop 3.11] automatically implies all the required continuity properties.

*Proof.* The proof is very similar to that of [9, Thm 4.10], but due to the central role of the statement in our proof, we provide some detail. The uniqueness part is quite straightforward: take two $C^{\alpha, \eta}$ distributions $\xi_1, \xi_2$ that have the properties claimed for $\mathcal{R}f$ in the theorem. Their difference then vanishes away from $P$, and thanks to the bound (C.2), must belong to $C^{\alpha, \eta}$. Since $\eta > -2$ however, such a distribution necessarily vanishes.

To construct $\mathcal{R}$, we use essentially the same construction as in the proof of [11, Prop. 6.9]. Similarly to the construction of the functions $\phi_{x,n}$ performed there, we can find, for every $n \in \mathbf{N}$, a countable index set $\Xi_n$ and functions $\phi_{x,n}$ with $n \in \mathbf{N}$ and $x \in \Xi_n$ with the following properties. There exist constants $c_i > 0$ such that:

(i) For every $n \in \mathbf{N}$ and $x \in \Xi_n$ there exists $\psi \in \mathcal{B}$, $y \in \mathbf{R}^{d+1}$ with $|y|_{P_1} = 2^{-n}$ such that, setting $\lambda = 2^{-n-1}$, one has $\phi_{x,n} = c_1\lambda^{d+2}\psi_y^\lambda$.

(ii) For every ball $B$ of (parabolic) radius $\lambda$, there exist at most $c_2\lambda^{d+1}$ elements $x \in \Xi_n$ with $\text{supp}\phi_{x,n} \cap B \neq \emptyset$.

(iii) For every $y \in \mathbf{R}^{d+1}$ with $0 < |y|_{P_1} \le 1$, one has $\sum_{n \in \mathbf{N}} \sum_{x \in \Xi_n} \phi_{x,n}(y) = 1$.

(Note that the sum appearing in the last claim always converges since, by the first two properties, it is guaranteed to only contain finitely many terms.)

We now write $\mathcal{R}_\pm$ for the reconstruction operators for $D^\gamma$ spaces with $\gamma < 0$ associated to the models $\Pi^\pm$ as in the second part of [11, Thm 3.10], we fix $y \in \mathbf{R}^{d+1} \setminus P_0$, $\lambda \le 1 \wedge |y|_{P_0}/C$ for some large enough (but fixed) constant $C$, and $\psi \in \mathcal{B}$, and we define
$$ (\mathcal{R}f)(\psi_y^\lambda) = (\mathcal{R}_+f_+ + \mathcal{R}_-f_-)(\psi_y^\lambda) \\ + \sum_{n \ge 0} \sum_{x \in \Xi_n} (\tilde{\mathcal{R}}f - \mathcal{R}_+f_+ - \mathcal{R}_-f_-)(\phi_{x,n}\psi_y^\lambda). $$

If one also has $\lambda \le |y|/P_1$, then the second sum only contains finitely many terms and one has $(\mathcal{R}f)(\psi) = (\tilde{\mathcal{R}}f)(\psi)$. Since in this case it follows from [9, Def. 3.1] combined with [11, Lem. 6.7] that one has the bound

$$ |(\tilde{\mathcal{R}}f)(\psi_y^\lambda)| \lesssim |y|^{\eta-\alpha} \lambda^\alpha, $$

it remains to consider the convergence of the second term. Since we can restrict ourselves to the case $|y|/P_1 \le 2\lambda$, we can assume without loss of generality that $y \in P_1 \setminus P_0$. In particular, the terms in the sum vanish unless $2^{-n} \lesssim \lambda$. We also note that for these terms, one can write

$$ \phi_{x,n} \psi_y^\lambda = \bar{\lambda}^{d+2} \lambda^{-d-2} \tilde{\psi}_z^{\bar{\lambda}}, $$

for some $z$, some $\tilde{\psi} \in \mathcal{B}$, and $\bar{\lambda} = 2^{-n-1}$. Provided that $C$ is sufficiently large, properties (i) and (ii) guarantee furthermore that $z$ is such that $\bar{\lambda} \simeq |z|/P_1 \le |z|/P_0$. It follows from this that, when restricted to the support of $\tilde{\psi}_z^{\bar{\lambda}}$, one has $\|Q_{<\delta}f\|_{D^\delta} \lesssim |z|^{\eta-\sigma} \bar{\lambda}^{\sigma-\delta}$, provided that $\delta \in [\sigma, \gamma]$, so that [11, Lem. 6.7] yields again

$$ \begin{aligned} |(\tilde{\mathcal{R}}f - \Pi_z f(z))(\tilde{\psi}_z^{\bar{\lambda}})| & \lesssim |z|^{\eta-\sigma} \bar{\lambda}^{\sigma}, \\ |(\mathcal{R}_{\pm}f_{\pm} - \Pi_z^{\pm} f_{\pm})(\tilde{\psi}_z^{\bar{\lambda}})| & \lesssim |z|^{\eta-\sigma} \bar{\lambda}^{\sigma}. \end{aligned} $$

Finally, since the support of $\tilde{\psi}$ is either fully contained in $\mathbf{R} \times D$ or fully contained in $\mathbf{R} \times D^c$, it follows that

$$ \begin{aligned} |(\Pi_z f(z) - \Pi_z^+ f_+(z) - \Pi_z^- f_-(z))(\tilde{\psi}_z^{\bar{\lambda}})| &= \sum_{\sigma \le \alpha < \gamma} |(\Pi_z Q_\alpha f(z))(\tilde{\psi}_z^{\bar{\lambda}})| \\ &\lesssim |z|^{\eta-\sigma} \sum_{\sigma \le \alpha < \gamma} |z|^{\sigma-\alpha} \bar{\lambda}^{\alpha} \lesssim |z|^{\eta-\sigma} \bar{\lambda}^{\sigma}, \end{aligned} $$

and we conclude that

$$ |(\tilde{\mathcal{R}}f - \mathcal{R}_+f_+ - \mathcal{R}_-f_-)(\phi_{x,n}\psi_y^\lambda)| \lesssim |z|^{\eta-\sigma} \lambda^{-d-2} 2^{-(\sigma+d+2)n}. $$

Since there are at most $2^{n(d+1)}$ such terms and since $\sigma > -1$ by assumption, both the convergence of the sum and the required bound (C.2) follow. □

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M. HAIRER

Imperial College London,
London,
UK.

e-mail: m.hairer@imperial.ac.uk

and

É. PARDOUX

Aix Marseille Univ, CNRS,
Centrale Marseille,
12M, Marseille,
France.

e-mail: etienne.pardoux@univ-amu.fr

(Received November 7, 2019 / Accepted September 14, 2020)

Published online October 6, 2020

© The Author(s) (2020)