A semigroup approach to the reconstruction theorem and the

multilevel Schauder estimate for singular modelled

distributions
arXiv:2408.04322v1 [math.PR] 8 Aug 2024

Masato Hoshino∗ and Ryoji Takano†

Abstract
We extend the semigroup approach used in [19, 17] to provide shorter proofs of
the reconstruction theorem and the multilevel Schauder estimate for singular modelled
distributions.

1 Introduction
The theory of regularity structures established by Hairer [12] provides a robust framework
adapted to a wide class of (subcritical) singular stochastic PDEs. One of the most impor-
tant concepts in this theory is the notion of modelled distributions, which are considered as
“generalized Taylor expansions” of the solutions to the underlying equations. The analytic
core of the theory is to prove two key theorems for modelled distributions: the reconstruc-
tion theorem [12, Theorem 3.10] and the multilevel Schauder estimate [12, Theorem 5.12].
The former theorem constructs a global distribution by gluing local distributions derived
from a given modelled distribution together. The latter translates an integral operator such
as the convolution operator with Green function into the operator on the space of mod-
elled distributions. Since Hairer first proved the reconstruction theorem, some alternative
proofs have been proposed using various approaches, such as Littlewood–Paley theory [11],
the heat semigroup approach [19, 2], the mollification approach [22], and the convolution
approach [9]. Inspired by [19], the first author of this paper proved both theorems by using
the operator semigroup in [17]. On the other hand, Caravenna and Zambotti [8] intro-
duced the notion of germs to describe the analytic core of the proof of the reconstruction
theorem, and later, they and Broux [5] proved the multilevel Schauder estimate at the level
∗
Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka,
560-8531, Japan. Email: hoshino@sigmath.es.osaka-u.ac.jp
†
Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka,
560-8531, Japan. Email: rtakano@sigmath.es.osaka-u.ac.jp

1
of germs. See also [13, 7, 16, 18, 20, 6, 23, 15] for extensions of the theorems into different
settings, such as Besov or Triebel–Lizorkin norms, or Riemannian manifolds.
In the aforementioned literatures, modelled distributions are often defined on the entire
space Rd to avoid technical difficulties related to boundary conditions. However, it is not
sufficient for applications. To apply the theory of regularity structures to parabolic equa-
tions, it is necessary to define modelled distributions on the time-space region (0, ∞) × Rd
allowing a singularity at the hyperplane {0} × Rd . This modified version of modelled dis-
tributions is called singular modelled distributions. In [12, Section 6], the reconstruction
theorem and the multilevel Schauder estimare were extended to the class of singular mod-
elled distributions. An extension to Besov norms is demonstrated in [14], and boundary
conditions on both time and space variables are considered in [10]. However, compared to
the case of modelled distributions without boundary conditions, there seems to be a less
number of studies on alternative proofs and extensions.
The aim of this paper is to extend the semigroup approach used in [17] and provide
alternative proofs of the reconstruction theorem (see Corollary 3.9) and the multilevel
Schauder estimate (see Corollary 4.6) for singular modelled distributions. The proofs use
arguments similar to [17], but require the following technical modifications.
(i) Following [17], we define Besov norms using the operator semigroup {Qt }t>0 . The as-
sociated integral kernel Qt (x, y) is inhomogeneous and has restricted regularities with
respect to x and y in general. Hence the equivalence between the norm associated
with {Qt }t>0 and the standard norm defined from Littlewood–Paley theory is uncer-
tain. For this reason, we need some nontrivial arguments to prove the uniqueness of
the reconstruction.
(ii) Since Qt is an integral operator defined over the entire spacetime, we always require
global bounds on models and modelled distributions, unlike the original definitions
in [12] that assume only local bounds. Consequently, in addition to the definition
of singular modelled distributions (see Definition 3.4) which is closer to the original
one, we use a different definition that assumes global bounds (see Proposition 3.5-
(iii)). For this reason, as for the existence of the reconstruction, we assume a stronger
condition “η − γ > −s1 ” for the parameters appearing in the definition of singular
modelled distributions than the condition “η > −s1 ” as in [12]. It is not actually
a serious problem in applications because we can switch to a small γ to apply the
reconstruction theorem.
Moreover, as an application, we discuss the parabolic Anderson model (PAM)



∂1 − a(x)∆ u(t, x) = b u(t, x) ξ(x) ((t, x) ∈ (0, ∞) × T2 )
with a spatial white noise ξ in the final section. Here b : R → R is in the class Cb3 and
a : T2 → R is an α-Hölder continuous function for some α ∈ (0, 1) and satisfies
C1 ≤ a(x) ≤ C2 (x ∈ T2 )

2
for some constants 0 < C1 < C2 . When a is a constant, the above equation is one of
the simplest examples of subcritical singular stochastic PDEs, as studied in [12, 7]. We
show that the equation with general coefficients as above can be renormalized, with the
spacetime dependent renormalization function (see Theorem 5.12). Such “non-translation
invariant” equations are more generally studied by [1, 21]. The aim of this paper is to
deepened the analytic core of [1], which uses the semigroup approach. On the other hand,
[21] is a direct extension of [12]. One of differences between this paper and [21] is in the
requirements of the smoothness of coefficients. [21] requires a bit smoothness of coefficients,
but in this paper the coefficients only need to have positive Hölder continuities.
This paper is organized as follows. In Section 2, we recall from [17] Besov norms asso-
ciated with the operator semigroup, and prove important inequalities used throughout this
paper. In Section 3, we recall the basics of regularity structures and prove the reconstruc-
tion theorem for singular modelled distributions. Section 4 is devoted to the proof of the
multilevel Schauder estimate for singular modelled distributions. In Section 5, we discuss
an application to the two-dimensional PAM.

Notations
The symbol N denotes the set of all nonnegative integers. Until Section 4, we fix anPinteger
d ≥ 1, the scaling s = (s1 , . . . , sd ) ∈ [1, ∞)d , and a number ℓ > 0. We define |s| = di=1 si .
For any multiindex k = (ki )di=1 ∈ Nd , any x = (xi )di=1 ∈ Rd , and any t > 0, we use the
following notations.
d
Y d
X d
X
k! := ki !, |k|s := si ki , kxks := |xi |1/si ,
i=1 i=1 i=1
Yd
xk := xki i , ts/ℓ x := (tsi /ℓ xi )di=1 , t−s/ℓ x := (t−si /ℓ xi )di=1 .
i=1

We define the set N[s] := {|k|s ; k ∈ Nd }, which will be used in Section 4. The parameter
t is not a physical time variable, but an auxiliary variable used to define regularities of
distributions. For multiindices k = (ki )di=1 and l = (li )di=1 , we write l ≤ k if li ≤ ki for any

Q

1 ≤ i ≤ d, and then define kl := di=1 klii .
We use the notation A . B for two functions A(x) and B(x) of a variable x, if there
exists a constant c > 0 independent of x such that A(x) ≤ cB(x) for any x.

2 Preliminaries
In this section, we introduce some function spaces and prove important inequalities used
throughout this paper. Until Section 4, we fix a nonnegative measurable function G : Rd →
R and define for any t > 0,

Gt (x) = t−|s|/ℓ G t−s/ℓ x .

3
2.1 Weighted Besov space
In this subsection, we recall from [17] some basics of Besov norms associated with the
operator semigroup. For simplicity, we consider only L∞ type norms.

Definition 2.1. A continuous function w : Rd → [0, 1] which is strictly positive outside a

set of Lebesgue measure 0 is called a weight. For any weight w, we define the weighted L∞
norm of a measurable function f : Rd → R by

kf kL∞ (w) := kf wkL∞ (Rd ) .

We denote by L∞ (w) the space of all measurable functions with finite L∞ (w) norms, and
define C(w) = C(Rd ) ∩ L∞ (w).

While we assumed that w(x) > 0 for every x ∈ Rd in [17], we impose a weaker condition
to consider a weight vanishing on the hyperplane {0} × Rd−1 in next subsection. Note that
k · kL∞ (w) is nondegenerate because w(x) > 0 for almost every x ∈ Rd . If w(x) > 0 for any
x ∈ Rd , then C(w) is a closed subspace of L∞ (w).

Definition 2.2. A weight w is said to be G-controlled if w(x) > 0 for any x ∈ Rd and
there exists a continuous function w∗ : Rd → [1, ∞) such that

w(x + y) ≤ w∗ (x)w(y) (2.1)

for any x, y ∈ Rd and

n
o
sup sup kxkns w∗ ts/ℓ x G(x) < ∞ (2.2)
0<t≤T x∈Rd

for any n ≥ 0 and T > 0.

From the properties (2.1) and (2.2), we have that

kGt ∗ f kL∞ (w) . kf kL∞ (w) (2.3)

uniformly over f ∈ L∞ (w) and t ∈ (0, T ] for any T > 0. This is a particular case of [17,
Lemma 2.4]. Next we introduce a semigroup of integral operators.

Definition 2.3. We call a family of continuous functions {Qt : Rd × Rd → R}t>0 a G-type

semigroup if it satisfies the following properties.

(i) (Semigroup property) For any 0 < s < t and x, y ∈ Rd ,

Z
Qt−s (x, z)Qs (z, y)dz = Qt (x, y).
Rd

4
 (ii) (Conservativity) For any x ∈ Rd ,
Z
lim Qt (x, y)dy = 1.
t↓0 Rd

(iii) (Upper G-type estimate) There exists a constant C1 > 0 such that, for any t > 0 and
x, y ∈ Rd ,
|Qt (x, y)| ≤ C1 Gt (x − y).

(iv) (Time derivative) For any x, y ∈ Rd , Qt (x, y) is differentiable with respect to t.

Moreover, there exists a constant C2 > 0 such that, for any t > 0 and x, y ∈ Rd ,
|∂t Qt (x, y)| ≤ C2 t−1 Gt (x − y).

We fix a G-type semigroup {Qt }t>0 until Section 4. If w is a G-controlled weight, the
linear operator on L∞ (w) defined by
Z
(Qt f )(x) :=: Qt (x, f ) := Qt (x, y)f (y)dy (f ∈ L∞ (w), x ∈ Rd )
Rd

is bounded in L∞ (w) uniformly over t ∈ (0, 1], by Definition 2.3-(iii) and the inequality
(2.3). As an important fact, Qt f is a continuous function for any f ∈ L∞ (w) and t > 0.
Moreover, if f ∈ C(w), we have
lim(Qt f )(x) = f (x) (2.4)
t↓0

for any x ∈ Rd . See [17, Proposition 2.8] for the proofs.

Definition 2.4. Let w be a G-controlled weight and let {Qt }t>0 be a G-type semigroup.
For every α ≤ 0, we define the Besov space C α,Q (w) as the completion of C(w) under the
norm
kf kC α,Q (w) := sup t−α/ℓ kQt f kL∞ (w) .
0<t≤1

By the property (2.4), the norm k · kC α,Q (w) is nondegenerate on C(w).

Remark 2.5. As stated in [17, Proposition 2.14], for any α1 < α2 ≤ 0, the identity
ια1 : C(w) ֒→ C α1 ,Q (w) is uniquely extended to the continuous injection
ιαα21 : C α2 ,Q (w) ֒→ C α1 ,Q (w).
Moreover, for any α ≤ 0, the operator Qt : C(w) → C(w) is continuously extended to the
operator Qαt : C α,Q (w) → C(w) and they satisfy the relation
Qαt 1 ◦ ιαα21 = Qαt 2
for any α1 < α2 ≤ 0. For this compatibility, we can omit the letter α and use the notation
Qt to mean its extension Qαt regardless of its domain.

5
2.2 Temporal weights
In what follows, the first variable x1 in x = (x1 , x2 , . . . , xd ) ∈ Rd is regarded as the temporal
variable, and the others (x2 , . . . , xd ) are spatial variables, denoted by x′ = (x2 , . . . , xd ).
Accordingly, we denote s′ = (s2 , . . . , sd ). The aim of this paper is to extend the results in
[17] to norms allowing a singularity at the hyperplane {0} × Rd−1 . We define the weight
ω : Rd → [0, 1] by
ω(x) := |x1 |1/s1 ∧ 1
and set ω(x, y) := ω(x) ∧ ω(y). The following inequalities are used frequently throughout
this paper.

Lemma 2.6. Let w be a G-controlled weight. For any α ≥ 0 and β ∈ [0, s1 ), there exists
a constant C such that, for any t ∈ (0, 1] and x ∈ Rd we have
Z

ω(y)−β kx − ykαs w∗ (x − y)Gt (x − y)dy ≤ Ctα/ℓ ω(x)−β ∧ t−β/ℓ
Rd

and
Z
ω(x, y)−β kx − ykαs w∗ (x − y)Gt (x − y)dy ≤ Ctα/ℓ ω(x)−β .
Rd

Proof. The second inequality immediately follows from the first one because of the trivial
inequality ω(x, y)−β ≤ ω(x)−β + ω(y)−β . Hence we focus on the first inequality. To obtain
the bound Ct(α−β)/ℓ , we divide the integral into two parts. In the region {|y1 |1/s1 > t1/ℓ },
since ω(y)−β ≤ t−β/ℓ we have
Z
ω(y)−β kx − ykαs w∗ (x − y)Gt (x − y)dy
1/s
|y1 | 1 >t1/ℓ
Z
−β/ℓ
≤t kzkαs w∗ (z)Gt (z)dz
RdZ


≤ t(α−β)/ℓ kzkαs w∗ ts/ℓ z G(z)dz . t(α−β)/ℓ .
Rd

In the region {|y1 |1/s1 ≤ t1/ℓ }, by treating the temporal variable and spatial variables
separately, we have
Z
ω(y)−β kx − ykαs w∗ (x − y)Gt (x − y)dy
1/s
|y1 | 1 ≤t 1/ℓ
Z  Z 
≤ |y1 |−β/s1 dy1 sup k(z1 , z ′ )kαs w∗ (z1 , z ′ )Gt (z1 , z ′ )dz ′
|y1 |1/s1 ≤t1/ℓ Rd−1 z1 ∈R
 Z 
s1 /ℓ 1−β/s1 −s1 /ℓ s1 /ℓ s′ /ℓ ′ α ∗ s1 /ℓ s′ /ℓ ′ ′ ′
. (t ) t sup k(t z1 , t z )ks w (t z1 , t z )G(z1 , z )dz
Rd−1 z1 ∈R

6
  Z 
s1 /ℓ 1−β/s1 −s1 /ℓ+α/ℓ ′ s′ /ℓ ′
= (t ) t sup k(z1 , z )kαs ∗
w (t s1 /ℓ
z1 , t ′
z )G(z1 , z )dz ′
Rd−1 z1 ∈R

. t(α−β)/ℓ .

Therefore, we obtain the upper bound Ct(α−β)/ℓ . Moreover, by decomposing

ω(x)β . |x1 − y1 |β/s1 + ω(y)β . kx − ykβs + ω(y)β ,

we have
Z
β
ω(x) ω(y)−β kx − ykαs w∗ (x − y)Gt (x − y)dy
R d
Z
. {ω(y)−β kx − ykα+β
s + kx − ykαs } w∗ (x − y)Gt (x − y)dy
Rd
α/ℓ
.t .

This yields another bound Ctα/ℓ ω(x)−β .

From the above lemma, we obtain an inequality similar to (2.3).

Corollary 2.7. Let w be a G-controlled weight. For any β ∈ [0, s1 ), there exists a constant
C such that, for any f ∈ L∞ (ω β w) we have

sup kGt ∗ f kL∞ (ωβ w) + sup tβ/ℓ kGt ∗ f kL∞ (w) ≤ Ckf kL∞ (ωβ w) .
0<t≤1 0<t≤1

Proof. By Lemma 2.6, we have

Z
w(x)|(Gt ∗ f )(x)| ≤ ω(y)−β w∗ (x − y)Gt (x − y)ω(y)β w(y)|f (y)|dy
Rd

≤ C ω(x)−β ∧ t−β/ℓ kf kL∞ (ωβ w) .

We obtain the following assertions by arguments similar to [17].

Proposition 2.8. Let w be a G-controlled weight and let {Qt }t>0 be a G-type semigroup.
We consider the weight w̃ := ω β w for any fixed β ∈ [0, s1 ).
(i) For any f ∈ L∞ (w̃) and t > 0, the function Qt f belongs to C(w).
(ii) For any α ≤ 0, the Besov norm

kf kC α,Q (w̃) := sup t−α/ℓ kQt f kL∞ (w̃)

0<t≤1

is nondegenerate on C(w̃), so we can define C α,Q (w̃) as the completion of C(w̃) under
this norm.

7
(iii) For any α1 < α2 ≤ 0, the identity ι̃α1 : C(w̃) ֒→ C α1 ,Q (w̃) is uniquely extended to
the continuous injection ι̃αα21 : C α2 ,Q (w̃) ֒→ C α1 ,Q (w̃). For any α ≤ 0, the operator
Qt : C(w̃) → C(w̃) is continuously extended to the operator Q̃αt : C α,Q (w̃) → C(w̃),
where C(w̃) is the closure of C(w̃) under the norm k · kL∞ (w̃) . Moreover, they satisfy
Q̃αt 1 ◦ ι̃αα21 = Q̃αt 2 for any α1 < α2 ≤ 0.

(iv) For any α ≤ 0, the identity i : C(w) ֒→ C(w̃) is uniquely extended to the continuous
injection iα : C α,Q (w) ֒→ C α,Q (w̃). Moreover, the extensions Q̃αt : C α,Q (w̃) → C(w̃)
and Qαt : C α,Q (w) → C(w) defined in (iii) and Remark 2.5 satisfy the relation

i ◦ Qαt = Q̃αt ◦ iα .

Consequently, we can use the same notation Qt to denote both Qαt and Q̃αt .

(v) For any α ≤ 0, there exists a constant C > 0 such that, for any f ∈ C α,Q (w̃),
t ∈ (0, 1], and ε ∈ [0, ℓ], we have

k(Qt − id)f kC α−ε,Q (w̃) ≤ C tε/ℓ kf kC α,Q (w̃) .

The norm C α,Q (ω β w) is used in the proof of Theorem 3.7.

Proof. (i) We have Qt f ∈ L∞ (w) by Corollary 2.7. To show the continuity of (Qt f )(x)
with respect to x, it is sufficient to consider the case t = 1. By the property (2.2), for any
fixed R > 0 and n ≥ 0, the inequalities
ω(y)−β
w(x)|Q1 (x, y)f (y)| . w∗ (x − y)w(y)G(x − y)|f (y)| . kf kL∞ (w̃)
1 + kykns
R
hold uniformly over kxks ≤ R and y ∈ Rd . Since Rd ω(y)−β /(1 + kykns )dy < ∞ for n > |s|,
we have
Z
lim (Q1 f )(z)w(z) = lim Q1 (z, y)f (y)w(z)dy = (Q1 f )(x)w(x)
z→x Rd z→x

by Lebesgue’s convergence theorem. Since w is strictly positive and continuous, we have

limz→x (Q1 f )(z) = (Q1 f )(z).
(ii) It is sufficient to show that

lim(Qt f )(x) = f (x)

t↓0

for any f ∈ C(w̃) and x ∈ Rd . For any ε > 0, we can choose δ > 0 such that |f (y)−f (x)| < ε
if ky − xks < δ, and have
Z Z 


|w(x)(Qt f − f )(x)| = w(x) Qt (x, y) f (y) − f (x) dy + Qt (x, y)dy − 1 f (x)
Rd Rd

8
 Z Z
≤ w(x)ε Gt (x − y)dy + w(x) Gt (x − y)|f (y)|dy
ky−xks <δ ky−xks ≥δ
Z Z
+ w(x)|f (x)| Gt (x − y)dy + w(x)|f (x)| Qt (x, y)dy − 1 .
ky−xks ≥δ Rd

In the far right-hand side, the only nontrivial part is the second term. We bound it from
above by
Z
Gt (x − y)w∗ (x − y)|f (y)|w(y)dy
ky−xks ≥δ
Z
≤ kf kL∞ (w̃) ω(y)−β w∗ (x − y)Gt (x − y)dy
ky−xks ≥δ
Z
−s1
≤ kf kL∞ (w̃) δ ky − xkss1 ω(y)−β w∗ (x − y)Gt (x − y)dy
Rd
−s1 (s1 −β)/s1
. kf kL∞ (w̃) δ t .

Since β < s1 , we obtain the convergence as t ↓ 0.

The proofs of (iii) and (iv) are similar to [17, Proposition 2.14], and the proof of (v) is
similar to [17, Lemma 2.15].

3 Reconstruction of singular modelled distributions

In this section, we recall from [12] the definitions of regularity structures, models, and
singular modelled distributions, and prove the reconstruction theorem for singular modelled
distributions using the operator semigroup. For simplicity, we consider only regularity
structures, rather than general regularity-integrability structures as in [17]. Throughout
this and next sections, we fix a G-type semigroup {Qt }t>0 .

3.1 Regularity structures and models

Definition 3.1. A regularity structure T = (A, T, G) consists of the following objects.

(1) (Index set) A is a locally finite subset of R bounded below.

L
(2) (Model space) T = α∈A Tα is an algebraic sum of Banach spaces (Tα , k · kα ).

(3) (Structure group) G is a group of continuous linear operators on T such that, for
any Γ ∈ G and α ∈ A,
M
(Γ − id)Tα ⊂ T<α := Tβ .
β∈A, β<α

9
The smallest element α0 of A is called the regularity of T . For any α ∈ A, we denote by
Pα : T → Tα the canonical projection and write

kτ kα := kPα τ kα

for any τ ∈ T, by abuse of notation.

Following [17], we define the topology on the space of models by using {Qt }t>0 . For
two Banach spaces X and Y , we denote by L(X, Y ) the Banach space of all continuous
linear operators X → Y . When Y = R, we write X ∗ := L(X, R).

Definition 3.2. Let w be a G-controlled weight. A smooth model M = (Π, Γ) is a pair

of two families of continuous linear operators Π = {Πx : T → C(Rd )}x∈Rd and Γ =
{Γxy }x,y∈Rd ⊂ G with the following properties.

(1) (Algebraic conditions) Πx Γxy = Πy , Γxx = id, and Γxy Γyz = Γxz for any x, y, z ∈ Rd .

(2) (Analytic conditions) For any γ ∈ R,



kΠkγ,w := max sup sup t−α/ℓ w(x) Qt x, Πx (·) T∗α
α∈A, α<γ 0<t≤1 x∈Rd
!
−α/ℓ |Qt (x, Πx τ )|
= max sup sup sup t w(x) <∞
α∈A, α<γ 0<t≤1 x∈Rd τ ∈Tα \{0} kτ kα

and
w(x)kΓyx kL(Tα ,Tβ )
kΓkγ,w := max sup
α,β∈A x,y∈Rd , x6=y w∗ (y − x)ky − xkα−β
s
β<α<γ
w(x)kΓyx τ kβ
= max sup sup < ∞.
α,β∈A x,y∈Rd , x6=y τ ∈Tα \{0} w∗ (y − x)ky − xkα−β
s kτ kα
β<α<γ

We write |||M |||γ,w := kΠkγ,w + kΓkγ,w . In addition, for any two smooth models M (i) =
(Π(i) , Γ(i) ) with i ∈ {1, 2}, we define the pseudo-metrics

|||M (1) ; M (2) |||γ,w := kΠ(1) − Π(2) kγ,w + kΓ(1) − Γ(2) kγ,w

by replacing Π and Γ above with Π(1) − Π(2) and Γ(1) − Γ(2) respectively. Finally, we define
the space Mw (T ) as the completion of the set of all smooth models, under the pseudo-
metrics |||·; ·|||γ,w for all γ ∈ R. We call each element of Mw (T ) a model for T . We still
use the notation M = (Π, Γ) to denote a generic model.

10
Remark 3.3. As stated in [17, Proposition 3.3], if there exist two G-controlled weights w1
and w2 that satisfy
 
sup kxkns w∗ (x)w1 (x) + sup kxkns w1∗ (x)w2 (x) < ∞
x∈Rd x∈Rd

for any n ≥ 0, and such that ww1 and ww2 are also G-controlled, then we can regard Πx
as a continuous linear operator from T to C α0 ∧0,Q (ww1 ), where α0 is the regularity of T .
More precisely, for any α < γ and τ ∈ Tα we have

sup (ww2 )(x)kΠx τ kC α0 ∧0,Q (ww1 ) . kΠkγ,w (1 + kΓkγ,w )kτ kα .

x∈R2

In what follows, we assume the existence of w1 and w2 as above, and regard Πx τ as an

element of C α0 ∧0,Q (ww1 ) for any τ ∈ T.

3.2 Singular modelled distributions

Throughout the rest of this section, we fix a regularity structure T of regularity α0 , and also
fix G-controlled weights w and v such that wv is also G-controlled. Recall the definitions
of functions ω(x) and ω(x, y) from Section 2.2.

Definition 3.4. Let M = (Π, Γ) ∈ Mw (T ). For any γ ∈ R and η ≤ γ, we define Dvγ,η (Γ)
as the space of all functions f : (R \ {0}) × Rd−1 → T<γ such that

v(x)kf (x)kα
L f Mγ,η,v := max sup < ∞,
α<γ x∈(R\{0})×Rd−1 ω(x)(η−α)∧0
v(x)k∆Γyx f kα
kf kγ,η,v := max sup < ∞,
α<γ
x,y∈(R\{0})×Rd−1 , x6=y ω(x, y)η−γ v ∗ (x − y)ky − xkγ−α
s
ky−xks ≤ω(x,y)

where ∆Γyx f := f (y) − Γyx f (x). We write |||f |||γ,η,v := L f Mγ,η,v + kf kγ,η,v . We call each
element of Dvγ,η (Γ) a singular modelled distribution.
In addition, for any two models M (i) = (Π(i) , Γ(i) ) ∈ Mw (T ) and singular modelled dis-
tributions f (i) ∈ Dvγ,η (Γ(i) ) with i ∈ {1, 2}, we define |||f (1) ; f (2) |||γ,η,v := L f (1) − f (2) Mγ,η,v +
kf (1) ; f (2) kγ,η,v by

v(x)kf (1) (x) − f (2) (x)kα

L f (1) − f (2) Mγ,η,v := max sup ,
α<γ x∈(R\{0})×Rd−1 ω(x)(η−α)∧0
v(x)k∆Γyx f (1) − ∆Γyx f (2) kα
kf (1) ; f (2) kγ,η,v := max sup .
α<γ
x,y∈(R\{0})×Rd−1 , x6=y ω(x, y)η−γ v ∗ (x − y)ky − xkγ−α
s
ky−xks ≤ω(x,y)

11
 In [12], the topologies of the space of models and the space of modelled distributions
are defined by the family of pseudo-metrics parametrized by compact subsets K of Rd ,
where x and y in the above definitions are restricted within K. In this paper, we employ
weight functions w and v instead of such local bounds.
We consider the relations between Dvγ,η under varying parameters γ, η, as well as the
relation between Dvγ,η and a variant. We say that the function u : Rd → R is symmetric if
u(−x) = u(x) for any x ∈ Rd .

Proposition 3.5. Let M = (Π, Γ) ∈ Mw (T ) and η ≤ γ.

(i) For any θ ≤ η, we have the continuous embedding Dvγ,η (Γ) ֒→ Dvγ,θ (Γ).

(ii) Assume that w∗ is symmetric. For each α ∈ R, we denote by P<α : T → T<α the
canonical projection. For any η ≤ δ ≤ γ, the map P<δ extends to a continuous linear
map Dvγ,η (Γ) → Dwvδ,η
(Γ). In precise, we have the inequality

kP<δ f kδ,η,wv . kΓkγ,w L f Mγ,η,v + kf kγ,η,v .

(iii) Instead of the norm kf kγ,η,v , we define

v(x)k∆Γyx f kα
kf k#
γ,η,v := max sup .
α<γ x,y∈(R\{0})×Rd−1 , x6=y ω(x, y)η−γ v ∗ (x − y)ky − xkγ−α
s

Then the inequality kf kγ,η,v ≤ kf k# ∗

γ,η,v obviously holds. Conversely, if w is symmet-
ric, then we also have

kf k#
γ,η∧α0 ,wv . (1 + kΓkγ,w )L f Mγ,η,v + kf kγ,η,v .

Proof. (i) The assertion immediately follows from the inequalities ω(x)(η−α)∧0 ≤ ω(x)(θ−α)∧0
and ω(x, y)η−γ ≤ ω(x, y)θ−γ .
(ii) For any x, y ∈ (R \ {0}) × Rd−1 such that ky − xks ≤ ω(x, y) and any α < δ, we
decompose
X
(wv)(x)k∆Γyx P<δ f kα ≤ v(x)k∆Γyx f kα + (wv)(x) kΓyx Pβ f (x)kα =: A1 + A2 .
β∈[δ,γ)

For A1 , by definition of the norm kf kγ,η,v we have

A1 ≤ kf kγ,η,v v ∗ (x − y)ω(x, y)η−γ ky − xkγ−α

s
≤ kf kγ,η,v v ∗ (x − y)ω(x, y)η−δ ky − xksδ−α .

12
For A2 , by definitions of the model and the norm L f Mγ,η,v we have
X
A2 ≤ w(x)kΓyx kL(Tβ ,Tα ) v(x)kf (x)kβ
β∈[δ,γ)
X
≤ kΓkγ,w L f Mγ,η,v w∗ (y − x) ky − xksβ−α ω(x)η−β
β∈[δ,γ)

≤ kΓkγ,w L f Mγ,η,v w (x − y)ky − xksδ−α ω(x, y)η−δ .

∗

Thus we obtain the desired inequality for kP<δ f kδ,η,wv .

(iii) It is sufficient to show the estimate of ∆Γyx f on the region ky − xks > ω(x, y). For
any α < γ we decompose
X
(wv)(x)k∆Γyx f kα ≤ v(x)kf (y)kα + (wv)(x) kΓyx Pβ f (x)kα =: B1 + B2 .
β∈[α,γ)

For B1 , by definition of the norm L f Mγ,η,v we have

B1 ≤ v ∗ (x − y)v(y)kf (y)kα
≤ L f Mγ,η,v v ∗ (x − y)ω(y)(η−α)∧0
≤ L f Mγ,η,v v ∗ (x − y)ω(x, y)(η−α)∧0
≤ L f Mγ,η,v v ∗ (x − y)ω(x, y)η∧α−γ ky − xkγ−α
s .
For B2 , by an argument similar to A2 in the proof of (ii), we have
X
B2 ≤ kΓkγ,w L f Mγ,η,v w∗ (y − x) ky − xksβ−α ω(x)(η−β)∧0
β∈[α,γ)

. kΓkγ,w L f Mγ,η,v w (x − y)ky − xkγ−α

∗
s ω(x, y)η∧α−γ .
Thus we obtain the desired inequality.
We also recall the definition of reconstruction.
Definition 3.6. Let M = (Π, Γ) ∈ Mw (T ). For any η ≤ γ and f ∈ Dvγ,η (Γ), we say that
Λ ∈ C ζ,Q (wv) with some ζ ≤ 0 is a reconstruction of f for M , if it satisfies
 
JΛKγ,η,wv := sup sup t−γ/ℓ ω(x)γ−η (wv)(x)|Qt (x, Λx )| < ∞,
0<t≤1 x∈(R\{0})×Rd−1

where Λx := Λ − Πx f (x). Furthermore, for any M (i) = (Π(i) , Γ(i) ) ∈ Mw (T ), f (i) ∈

Dvγ,η (Γ(i) ), and any reconstructions Λ(i) ∈ C ζ,Q(wv) of f (i) for M (i) with i ∈ {1, 2}, we
define


JΛ(1) ; Λ(2) Kγ,η,wv := sup sup t−γ/ℓ ω(x)γ−η (wv)(x) Qt x, Λ(1)x − Λx
(2)
,
0<t≤1 x∈(R\{0})×Rd−1

(i) (i)
where Λx := Λ(i) − Πx f (i) (x) for each i ∈ {1, 2}.

13
3.3 Reconstruction theorem
In this subsection, we provide a short proof of the reconstruction theorem. First, we
prove the theorem for the subclass Dvγ,η (Γ)# of Dvγ,η (Γ) consisting of all functions f :
(R \ {0}) × Rd−1 → T<γ such that

|||f |||# #
γ,η,v := L f Mγ,η,v + kf kγ,η,v < ∞.

In addition, for any M (i) = (Π(i) , Γ(i) ) ∈ Mw (T ) and f (i) ∈ Dvγ,η (Γ(i) )# with i ∈ {1, 2},
we define |||f (1) ; f (2) |||#
γ,η,v := L f
(1) − f (2) M
γ,η,v + kf
(1) ; f (2) k#
γ,η,v similarly to Definition 3.4.

Theorem 3.7. Let γ > 0 and η ∈ (γ − s1 , γ]. Then for any M = (Π, Γ) ∈ Mw (T )
and f ∈ Dvγ,η (Γ)# , there exists a unique reconstruction Rf ∈ C ζ,Q(wv) of f for M with
ζ := η ∧ α0 ∧ 0 and it holds that

kRf kC ζ,Q (wv) . kΠkγ,w |||f |||#

γ,η,v , (3.1)
JRf Kγ,η,wv . kΠkγ,w kf k#
γ,η,v . (3.2)

Moreover, there is an affine function CR > 0 of R > 0 such that

kRf (1) − Rf (2) kC ζ,Q (wv) ≤ CR kΠ(1) − Π(2) kγ,w + |||f (1) ; f (2) |||#
γ,η,v ,


JRf (1) ; Rf (2) Kγ,η,wv ≤ CR kΠ(1) − Π(2) kγ,w + kf (1) ; f (2) k# γ,η,v

for any M (i) = (Π(i) , Γ(i) ) ∈ Mw (T ) and f (i) ∈ Dvγ,η (Γ(i) ) with i ∈ {1, 2} such that
|||M (i) |||γ,w ≤ R and |||f (i) |||#
γ,η,v ≤ R.

Proof. The proof is carried out by a method similar to that of [17, Theorem 4.1], but we
have to treat the temporal weight more carefully. For t > 0 and 0 < s ≤ t ∧ 1, we define
the functions
Z

 Qt−s (x, y)Qs y, Πy f (y) dy, s < t,
t
Rs f (x) := Rd


Qt x, Πx f (x) , s = t.

Note that

X

(wv)(x) Qt x, Πx f (x) ≤ w(x) Qt x, Πx (·) T∗α
v(x)kf (x)kα
α<γ
X
≤ kΠkγ,w L f Mγ,η,v tα/ℓ ω(x)(η−α)∧0 .
α<γ

Thus, by Proposition 2.8-(i), for any s ∈ (0, t) we have Rts f ∈ C(wv) and
X
kRts f kL∞ (wv) . kΠkγ,w L f Mγ,η,v sα/ℓ (t − s)(η−α)∧0 . (3.3)
α<γ

14
We separate the proof into four steps.
(1) Cauchy property. Set Fx := Πx f (x). By the definition of norms, we have

(wv)(y)|Qt (x, Fy − Fx )|


= (wv)(y) Qt x, Πx Γxy f (y) − f (x)
X
≤ w∗ (y − x) w(x)kQt (x, Πx (·))kT∗α v(y)kΓxy f (y) − f (x)kα (3.4)
α<γ
X
≤ kΠkγ,w kf k# ∗ ∗
γ,η,v (w v )(y − x) tα/ℓ ω(x, y)η−γ ky − xkγ−α
s .
α<γ

By the semigroup property, for any 0 < u < s < t ∧ 1 we have

(wv)(x)|Rts f (x) − Rtu f (x)|

Z
≤ (w∗ v ∗ )(x − y)(wv)(y)|Qt−s (x, y)Qs−u (y, z)Qu (z, Fy − Fz )|dydz
(Rd )2
X Z
# α/ℓ
. kΠkγ,w kf kγ,η,v u (w∗ v ∗ )(x − y)(w∗ v ∗ )(y − z)
α<γ (Rd )2

× Gt−s (x − y)Gs−u (y − z)ω(y, z)η−γ ky − zkγ−α

s dydz.

By applying the second inequality of Lemma 2.6 to the integral with respect to z and then
applying the first inequality of Lemma 2.6 to the integral with respect to y, we obtain

(wv)(x)|Rts f (x) − Rtu f (x)|

X Z
# α/ℓ (γ−α)/ℓ
. kΠkγ,w kf kγ,η,v u (s − u) (w∗ v ∗ )(x − y)Gt−s (x − y)ω(y)η−γ dy
α<γ (Rd )2
X
. kΠkγ,w kf k#
γ,η,v uα/ℓ (s − u)(γ−α)/ℓ ω(x)η−γ .
α<γ

Consequently, when u ∈ [s/2, s) we have the inequality

(wv)(x)|Rts f (x) − Rtu f (x)| . kΠkγ,w kf k#

γ,η,v ω(x)
η−γ γ/ℓ
s . (3.5)

Similarly to the proof of [17, Theorem 4.1], we can also extend it into u ∈ (0, s/2) by
decomposing
∞
X
|Rts f (x) − Rtu f (x)| ≤ |Rt(s/2n )∧u f (x) − Rt(s/2n+1 )∧u f (x)|.
n=0

The same inequality for the case s = t ≤ 1 can be obtained by a similar argument. In the
end, the inequality (3.5) holds for any 0 < u < s ≤ t ∧ 1.

15
(2) Convergence as s ↓ 0. Note that Qs Rtu f = Rt+s u f follows from the semigroup
property. By the inequality (3.5), for any 0 < u < s ≤ t/2 we have

(wv)(x)|Rts f (x) − Rtu f (x)|

Z
≤ (w∗ v ∗ )(x − y)(wv)(y)|Qt/2 (x, y)(Rt/2 t/2
s f − Ru f )(y)|dy
R d
Z (3.6)
. kΠkγ,w kf k# γ,η,v s γ/ℓ
(w∗ v ∗ )(x − y)Gt/2 (x − y)ω(y)η−γ dy
Rd
# γ/ℓ (η−γ)/ℓ
. kΠkγ,w kf kγ,η,v s t .

Since γ > 0, this implies that {Rts f }0<s≤t/2 is Cauchy in C(wv) as s ↓ 0. We denote its
limit by
Rt0 f := lim Rts f.
s↓0

We also have Qs Rt0 f = Rt+s t t+s

0 f by taking the limit u ↓ 0 in Qs Ru f = Ru f .

(3) Convergence as t ↓ 0. Combining the Cauchy property (3.6) and the bound (3.3)
with s = t/2, we have

kRt0 f kL∞ (wv) ≤ kRtt/2 f kL∞ (wv) + kRtt/2 f − Rt0 f kL∞ (wv)
. kΠkγ,w |||f |||#
γ,η,v t
(η∧α0 )/ℓ
.

Since Qs Rt0 f = Rt+s

0 f , this implies

sup kRt0 f kC η∧α0∧0,Q (wv) . kΠkγ,w |||f |||#

γ,η,v .
0<t≤1

From here onward, in exactly the same way as the part (4) of the proof of [17, Theorem
4.1], we can show the existence of Rf ∈ C ζ,Q(wv) with ζ = η ∧ α0 ∧ 0 which satisfies the
bound (3.1) and
lim kRf − Rt0 f kC ζ−ε,Q (wv) = 0
t↓0

for any ε ∈ (0, ℓ]. Moreover, we have Qt Rf = Rt0 f by taking the limit s ↓ 0 in Qt Rs0 f =
Rt+s
0 f . We have another bound (3.2) by letting u ↓ 0 and s = t in the inequality (3.5).

(4) Uniqueness. Let Λ, Λ′ ∈ C ζ,Q(wv) be reconstructions of f for M . By the property of

reconstruction, g := Λ − Λ′ satisfies

sup ω(x)γ−η (wv)(x)|Qt g(x)| . tγ/ℓ .

x∈Rd

Set w̃ := ω γ−η wv. By Proposition 2.8-(iv) and (v), for any ε ∈ (0, ℓ] we have

kgkC ζ−ε,Q (w̃) ≤ k(Qt − id)gkC ζ−ε,Q (w̃) + kQt gkC ζ−ε,Q (w̃)

16
 . tε/ℓ kgkC ζ,Q (w̃) + kQt gkL∞ (w̃)
. tε/ℓ kgkC ζ,Q (wv) + tγ/ℓ .

By taking the limit t ↓ 0, we have g = 0 in C ζ−ε,Q(w̃). By Proposition 2.8-(iii) and (iv),

we also have g = 0 in C ζ,Q (wv).

The following result is used in Section 5.

Proposition 3.8. In addition to the setting of Theorem 3.7, we assume that the model M
is smooth in the sense of Definition 3.2 and
|(Πx τ )(x)|
sup sup w(x) <∞
x∈Rd τ ∈Tα \{0} kτ kα

for any α ∈ A. Then the reconstruction Rf of f ∈ Dvγ,η (Γ)# is realized as a continuous

function on (R \ {0}) × Rd−1 such that


(Rf )(x) = Πx f (x) (x)

for any x ∈ (R \ {0}) × Rd−1 .

Proof. Set Λ(x) = Πx f (x) (x). Since (Πx τ )(x) = limt↓0 Qt (x, Πx τ ) = 0 if τ ∈ Tα with
α > 0, we have
X

(wv)(x)|Λ(x)| ≤ w(x) Πx (·) (x) T∗ v(x)kf (x)kα
α
α≤0
X
. ω(x)(η−α)∧0 . ω(x)η∧0 .
α≤0

Since η > −s1 , we have Λ ∈ C η∧0,Q (wv) ⊂ C ζ,Q (wv) by Corollary 2.7. Moreover, since
Z


(wv)(x)|Qt (x, Λx )| = (wv)(x) Qt (x, y)Πy f (y) − Γyx f (x) (y)dy
Rd
XZ

. w∗ (x − y)Gt (x − y)w(y) Πy (·) (y) T∗ v(x)kf (y) − Γyx f (x)kα dy
d α
α≤0 R
XZ
. (w∗ v ∗ )(x − y)Gt (x − y)ky − xkγ−α
s ω(x, y)η−γ dy
d
α≤0 R
X
. t(γ−α)/ℓ ω(x)η−γ . tγ/ℓ ω(x)η−γ ,
α≤0

we have JΛKγ,η,wv < ∞. Hence Rf = Λ by the uniqueness of the reconstruction.

Combining Theorem 3.7 with Proposition 3.5-(iii), we have the desired result.

17
Corollary 3.9. Assume that w2 v is also G-controlled. If γ > 0 and η ∧ α0 ∈ (γ − s1 , γ],
then for any M = (Π, Γ) ∈ Mw (T ) and f ∈ Dvγ,η (Γ), there exists a unique reconstruction
Rf ∈ C η∧α0 ∧0,Q (w2 v) of f for M and it holds that
kRf kC η∧α0 ∧0,Q (w2 v) . kΠkγ,w (1 + kΓkγ,w )|||f |||γ,η,v ,
JRf Kγ,η∧α0 ,w2v . kΠkγ,w (1 + kΓkγ,w )kf kγ,η,v .
The local Lipschitz estimates similar to the latter part of Theorem 3.7 also hold.

4 Multilevel Schauder estimate

This section is devoted to the proof of the multilevel Scahuder estimate for singular mod-
elled distributions. After recalling from [17] the basics of regularizing kernels in the first
subsection, we prove the multilevel Scahuder estimate in the second subsection.

4.1 Regularizing kernels

We recall from [17, Section 5.1] the definition of regularizing kernels.
Definition 4.1. Let β̄ > 0. A β̄-regularizing (integral) kernel admissible for {Qt }t>0 is
a family of continuous functions {Kt : Rd × Rd → R}t>0 which satisfies the following
properties for some constants δ > 0 and CK > 0.
(i) (Convolution with Q) For any 0 < s < t and x, y ∈ Rd ,
Z
Kt−s (x, z)Qs (z, y)dz = Kt (x, y).
Rd

(ii) (Upper estimate) For any k ∈ Nd with |k|s < δ, the k-th partial derivative of Kt (x, y)
with respect to x exists, and we have for any t > 0 and x, y ∈ Rd ,
|∂xk Kt (x, y)| ≤ CK t(β̄−|k|s )/ℓ−1 Gt (x − y).

(iii) (Hölder continuity) For any k ∈ Nd with |k|s < δ, any t > 0 and x, y, h ∈ Rd with
khks ≤ t1/ℓ ,
X hl k+l δ−|k|s (β̄−δ)/ℓ−1
∂xk Kt (x + h, y) − ∂x Kt (x, y) ≤ CK khks t Gt (x − y).
l!
|l|s <δ−|k|s

We fix a β̄-regularizing kernel {Kt }t>0 throughout this section. For any f ∈ L∞ (w)
with a G-controlled weight w and any |k|s < δ, we define
Z
k k
(∂ Kt f )(x) :=: ∂ Kt (x, f ) := ∂xk Kt (x, y)f (y)dy.
Rd
R1
Moreover, we write ∂ k Kf := 0 ∂ k Kt f dt if the integral makes sense.

18
Lemma 4.2. Let w and v be G-controlled weights such that w2 and wv are also G-
controlled. Let T = (A, T, G) be a regularity-integrability structure and let M = (Π, Γ) ∈
Mw (T ).

(i) [17, Lemma 5.4] For any α ≤ 0, |k|s < δ, and f ∈ L∞ (w), we have

k∂ k Kt f kL∞ (w) . CK t(α+β̄−|k|s )/ℓ−1 kf kC α,Q (w) ,

where the implicit proportional constant depends

R 1 k only on G and w. Consequently, if
k
|k|s < (α + β̄) ∧ δ, the integral ∂ Kf := 0 ∂ Kt f dt converges in C(w).

(ii) [17, Lemma 5.6] For any α < γ, τ ∈ Tα , |k|s < δ, and t ∈ (0, 1], we have
(α+β̄−|k|s )/ℓ−1
k∂ k Kt (x, Πx τ )kL∞ 2 . CK t
x (w )
kΠkγ,w (1 + kΓkγ,w )kτ kα ,

where the implicit proportional constant depends onlyR on G, w, and A. Consequently,

1
if |k|s < (α + β̄) ∧ δ, the integral ∂ k K(x, Πx τ ) := 0 ∂ k Kt (x, Πx τ )dt converges for
any x ∈ R .d

(iii) Let γ ∈ R, η ∈ (γ − s1 , γ], and ζ ≤ 0. For any f ∈ Dvγ,η (Γ)# and its reconstruction
Λ ∈ C ζ,Q(wv), |k|s < δ, and t ∈ (0, 1], we have


(wv)(x)|∂ k Kt (x, Λx )| . CK t(γ+β̄−|k|s )/ℓ−1 ω(x)η−γ JΛKγ,η,wv + kΠkγ,w kf k#
γ,η,v ,

where the implicit proportional constant depends only on R 1G, w, v, and A. Conse-
quently, if |k|s < (γ + β̄) ∧ δ, the integral ∂ k K(x, Λx ) := 0 ∂ k Kt (x, Λx )dt converges
for any x ∈ (R \ {0}) × Rd−1 .
Proof. We prove only (iii). By Definition 4.1-(i), we can decompose
Z
k
|∂ Kt (x, Λx )| ≤ ∂ k Kt/2 (x, y)Qt/2 (y, Λy )dy
Rd
Z


+ ∂ k Kt/2 (x, y)Qt/2 y, Πy f (y) − Πx f (x) dy .
Rd

For the first term, by Definition 4.1-(ii) and by the property of reconstruction, we have
Z
(wv)(x) ∂ k Kt/2 (x, y)Qt/2 (y, Λy )dy
R d
Z
(β̄−|k|s )/ℓ−1
. CK t (w∗ v ∗ )(x − y)Gt/2 (x − y)(wv)(y)|Qt/2 (y, Λy )|dy
R d
Z
(γ+β̄−|k|s )/ℓ−1
. CK t JΛKγ,η,wv ω(y)η−γ (w∗ v ∗ )(x − y)Gt/2 (x − y)dy
Rd
(γ+β̄−|k|s )/ℓ−1 η−γ
. CK t ω(x) JΛKγ,η,wv .

19
For the second term, by using the inequality (3.4) obtained in the proof of Theorem 3.7
with x and y swapped, we have
Z


(wv)(x) ∂ k Kt/2 (x, y)Qt/2 y, Πy f (y) − Πx f (x) dy
Rd
Z
(β̄−|k|s )/ℓ−1


. CK t Gt/2 (x − y)(wv)(x) Qt/2 y, Πy f (y) − Πx f (x) dy
Rd Z
X
. CK kΠkγ,w kf k# γ,η,v t (α+β̄−|k|s )/ℓ−1
ω(x, y)η−γ ky − xkγ−α
s (w∗ v ∗ )(x − y)Gt/2 (x − y)dy
α<γ Rd

. CK t(γ+β̄−|k|s )/ℓ−1 ω(x)η−γ kΠkγ,w kf k#

γ,η,v .

4.2 Compatible models and multilevel Schauder estimate

We recall from [17, Section 5.2] the notions of abstract integrations and compatible models.
Hereafter, we use the polynomial structure generated by dummy variables X1 , . . . , Xd as in
[12, Section 2].

Definition 4.3. Let T¯ = (Ā, T̄, Ḡ) be a regularity structure satisfying the following prop-
erties.

(1) N[s] ⊂ Ā.

Qd ki
(2) For each α ∈ N[s], the space T̄α contains all X k := i=1 Xi with |k|s = α.

(3) The subspace span{X k }k∈Nd of T̄ is closed under Ḡ-actions.

Let T = (A, T, G) be another regularity structure. A continuous linear operator I : T → T̄

is called an abstract integration of order β ∈ (0, β̄] if

I : Tα → T̄α+β

for any α ∈ A. For a fixed G-controlled weight w, we say that the pair (M, M̄ ) of two
models M = (Π, Γ) ∈ Mw (T ) and M̄ = (Π̄, Γ̄) ∈ Mw (T¯ ) is compatible for I if it satisfies
the following properties.

(i) For any k ∈ Nd ,

X k
(Π̄x X k )(·) = (· − x)k , Γ̄yx X k = (y − x)l X k−l .
l
l≤k

20
 (ii) For each x ∈ Rd , we define the linear map J (x) : T<δ−β → span{X k }|k|s <δ ⊂ T̄ by
setting
X Xk k
J (x)τ = ∂ K(x, Πx τ ) (4.1)
k!
|k|s <α+β

for any α ∈ A such that α + β < δ and τ ∈ Tα . Then for any τ ∈ T<δ−β ,



Γ̄yx I + J (x) τ = I + J (y) Γyx τ.

In addition, if the regularity α0 of T is greater than −β̄ and

X (· − x)k k
(Π̄x Iτ )(·) = K(·, Πx τ ) − ∂ K(x, Πx τ ) (4.2)
k!
|k|s <α+β

for any τ ∈ Tα with α + β < δ, then we say that the pair (M, M̄ ) is K-admissible for I.

In (4.1) and (4.2), the function K(·, Πx τ ) and the coefficients ∂ k K(x, Πx τ ) are well-
defined by Lemma 4.2. The following theorem is the second main result of this paper.

Theorem 4.4. Let T and T¯ be regularity structures satisfying the setting of Definition 4.3
and let I : T → T̄ be an abstract integration of order β ∈ (0, β̄]. Let w and v be G-controlled
weights such that w2 v is also G-controlled. Given (Π, Γ) ∈ Mw (T ), f ∈ Dvγ,η (Γ)# with
γ + β̄ < δ and η ∈ (γ − s1 , γ], and its reconstruction Λ ∈ C ζ,Q(wv), we define the functions
X Xk k
N (x; f, Λ) = ∂ K(x, Λx )
k!
|k|s <γ+β

and
Kf (x) := If (x) + J (x)f (x) + N (x; f, Λ)
for x ∈ (R \ {0}) × Rd−1 . We assume ζ ≤ η ∧ α0 and either of the following conditions.

(1) β < β̄.


(2) β = β̄ and α + β̄ ; α ∈ A ∪ {γ, ζ} ∩ N[s] = ∅.


Then for any compatible pair of models M = (Π, Γ), M̄ = (Π̄, Γ̄) ∈ Mw (T )×Mw (T¯ ) and
any singular modelled distribution f ∈ Dvγ,η (Γ)# , the function Kf belongs to Dw
γ+β,ζ+β
2v (Γ̄)# ,
and we have

L Kf Mγ+β,ζ+β,w2v . kIkL f Mγ,η,v + CK kΠkγ,w (1 + kΓkγ,w )|||f |||#
γ,η,v
(4.3)
+ kΛkC ζ,Q (wv) + JΛKγ,η,wv ,

21
 
kKf k#
γ+β,ζ+β,w 2 v
. kIkkf k# #
γ,η,v + CK kΠkγ,w (1 + kΓkγ,w )kf kγ,η,v + JΛKγ,η,wv , (4.4)

where kIk is the operator norm from T<γ to T̄<γ+β , and the implicit proportional constant
depends only on G, w, v, γ, η, and A. Moreover, there is a quadratic function CR > 0 of
R > 0 such that
 
|||Kf (1) ; Kf (2) |||# 2
γ+β,ζ+β,w v
≤ C R |||M (1)
; M (2)
|||γ,w + |||f (1) (2) #
; f |||γ,η,v ,

for any M (i) = (Π(i) , Γ(i) ) ∈ Mw (T ) and M̄ (i) = (Π̄(i) , Γ̄(i) ) ∈ Mw (T¯ ) such that (M (i) , M̄ (i) )
is compatible and any f (i) ∈ Dvγ,η (Γ(i) ) with i ∈ {1, 2} such that |||M (i) |||γ,w ≤ R and
|||f (i) |||#
γ,η,v ≤ R.

Proof. The proof is carried out by a method similar to that of [17, Theorem 5.12], but we
have to prove (4.3) more carefully than [17]. For the I term, by the continuity of I we
immediately have

v(x)kIf (x)kα ≤ v(x)kIkkf (x)kα−β ≤ kIkL f Mγ,η,v ω(x)(η+β−α)∧0

for any α < γ + β. For the J and N terms, we decompose

X Xk k
J (x)f (x) + N (x, f ; Λ) = A (x),
k!
|k|s <γ+β

where X

Ak (x) = ∂ k K x, Πx Pα f (x) + ∂ k K(x, Λx ).
α∈[α0 ,γ), |k|s <α+β
R1
We further define the decomposition Ak (x) = 0 Akt (x)dt according to the integral form
R1
K = 0 Kt dt, where Akt is defined in the k

same way as kA with K replaced by Kt . By using
k
Lemma 4.2-(ii) for ∂ Kt x, Πx Pα f (x) and (iii) for ∂ Kt (x, Λx ), we have
X
(w2 v)(x)|Akt (x)| . L1 ω(x)(η−α)∧0 t(α+β̄−|k|s )/ℓ−1
α∈[α0 ,γ], |k|s <α+β


where L1 := CK kΠkγ,w (1 + kΓkγ,w )|||f |||#
γ,η,v + JΛKγ,η,wv . Since all powers of t above are
greater than −1, we have
Z ω(x)ℓ X
2
(w v)(x) |Akt (x)|dt . L1 ω(x)η∧α+β̄−|k|s
0 α∈[α0 ,γ], |k|s <α+β

. L1 ω(x)(η∧α0 +β−|k|s )∧0 .

22
For the integral over ω(x)ℓ < t ≤ 1, we use another decomposition
X

Akt (x) = ∂ k Kt x, Λ) − ∂ k Kt x, Πx Pα f (x)
α∈[α0 ,γ), |k|s ≥α+β

and consider the two terms in the right hand side separately. For the first term, by the
assumption that Λ ∈ C ζ,Q(wv) and by Lemma 4.2-(i), we have
(wv)(x)|∂ k Kt x, Λ)| . CK kΛkC ζ,Q (wv) t(ζ+β̄−|k|s )/ℓ−1 .
If ζ + β̄ − |k|s 6= 0, we have
Z 1
t(ζ+β̄−|k|s )/ℓ−1 dt . ω(x)(ζ+β̄−|k|s )∧0 . ω(x)(ζ+β−|k|s )∧0 .
ω(x)ℓ

Otherwise, since ζ + β − |k|s < ζ + β̄ − |k|s = 0 by assumption we have

Z 1 Z 1
t(ζ+β̄−|k|s )/ℓ−1 dt . t(ζ+β−|k|s )/ℓ−1 dt . ω(x)ζ+β−|k|s .
ω(x)ℓ ω(x)ℓ

In either case, we obtain the desired estimate. For the remaining term, by Lemma 4.2-(ii)
we have
X

(w2 v)(x) ∂ k Kt x, Πx Pα f (x)
α∈[α0 ,γ), |k|s ≥α+β
X
. L2 ω(x)(η−α)∧0 t(α+β̄−|k|s )/ℓ−1 ,
α∈[α0 ,γ), |k|s ≥α+β

where L2 := CK kΠkγ,w (1 + kΓkγ,w )L f Mγ,η,v . For α such that |k|s > α + β, we easily have
Z 1 Z 1
(η−α)∧0 (α+β̄−|k|s )/ℓ−1 (η−α)∧0
ω(x) t dt . ω(x) t(α+β−|k|s )/ℓ−1 dt
ω(x)ℓ ω(x)ℓ
η∧α+β−|k|s
. ω(x) .
If there exists α such that |k|s = α + β, then since 0 = α + β − |k|s < α + β̄ − |k|s by
assumption, we have
Z 1
(η−α)∧0
ω(x) t(α+β̄−|k|s )/ℓ−1 dt . ω(x)(η−α)∧0 = ω(x)η∧α+β−|k|s .
ω(x)ℓ

Consequently, we obtain
Z 1
2
(w v)(x) |Akt (x)|dt . {CK kΛkC ζ,Q (wv) + L2 }ω(x)(ζ+β−|k|s )∧0 .
ω(x)ℓ

The proof of (4.4) is completely the same as that of [17, Theorem 5.12] except the
existence of the factor ω(x, y)η−γ .

23
 The following theorem is obtained similarly to [17, Theorem 5.13], so we omit the proof.

Theorem 4.5. In addition to the setting of Theorem 4.4, we assume that ζ + β̄ > 0
and that (M, M̄ ) is K-admissible for I. Then KΛ ∈ C(wv) is a reconstruction of Kf ∈
γ+β,ζ+β
Dw 2v (Γ̄)# and


JKΛKγ+β,ζ+β,w2v . CK JΛKγ,η,wv + kΠkγ,w kf k#
γ,η,v .

A similar local Lipschitz estimate to the latter part of Theorem 4.4 also holds.

Combining Theorem 4.4 with Proposition 3.5-(iii), we have the desired result.

Corollary 4.6. In addition to the setting of Theorem 4.4, assume that w3 v is G-controlled


and that α0 > γ − s1 . Then for any compatible pair of models M = (Π, Γ), M̄ = (Π̄, Γ̄) ∈
Mw (T ) × Mw (T¯ ) and any singular modelled distribution f ∈ Dvγ,η (Γ), the function Kf
γ+β,ζ+β
belongs to Dw 3v (Γ̄), and we have

L Kf Mγ+β,ζ+β,w3v . kIkL f Mγ,η,v + CK kΠkγ,w (1 + kΓkγ,w )2 |||f |||γ,η,v
+ kΛkC ζ,Q (wv) + JΛKγ,η,wv ,

kKf kγ+β,ζ+β,w3 v . kIk kΓkγ,w L f Mγ,η,v + kf kγ,η,v

+ CK kΠkγ,w (1 + kΓkγ,w )2 kf kγ,η,v + JΛKγ,η,wv .

A similar local Lipschitz estimate to the latter part of Theorem 4.4 also holds.

5 Parabolic Anderson model

In this section, we study the parabolic Anderson model (PAM)



∂1 − a(x′ )∆ + c u(x) = b u(x) ξ(x′ ) (x ∈ (0, ∞) × T2 ) (5.1)

with a spatial white noise ξ defined on a probability space (Ω, F, P). Recall that x1 in x =
(x1 , x2 , x3 ) denotes the temporal variable and x′ = (x2 , x3 ) denotes the spatial variables.
Throughout this section, we fix the function b : R → R in the class Cb3 , and the function
a : T2 → R which is α-Hölder continuous for some α ∈ (0, 1) and satisfies

C1 ≤ a(x′ ) ≤ C2 (x′ ∈ T2 )

for some constants 0 < C1 < C2 . The constant c > 0 in the left hand side of (5.1) is fixed
later (see Propositions 5.1 and 5.2). We prove the renormalizability of (5.1) in Section 5.6.
Note that we fix d = 3, s = (2, 1, 1), and ℓ = 4 throughout this section.

24
5.1 Preliminaries
We denote by e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1) the canonical basis vectors of
R3 . We define Cb (R × T2 ) as the set of all bounded continuous functions f : R3 → R such
that
f (x + ei ) = f (x)
for any x ∈ R3 and i ∈ {2, 3}. For any β > 0, we define Csβ (R×T2 ) as the set of all elements
f ∈ Cb (R × T2 ) such that ∂xk f ∈ Cb (R × T2 ) for any |k|s < β, and if |k|s < β ≤ |k|s + si ,
we have
|∂ k f (x + hei ) − ∂ k f (x)| . |h|(β−|k|s )/si
for any x ∈ R3 and h ∈ R.
We denote by Px1 (x′ , y ′ ) the fundamental solution of the parabolic operator ∂1 −a∆+c.
Moreover, we introduce an additional variable t > 0 and the anisotropic parabolic operator


∂t − L := ∂t − ∂1 − a(x′ )∆ (∂1 + ∆) (t > 0, x ∈ R3 ).

We also denote by Qt (x, y) the fundamental solution of ∂t − L + c. First, we recall from

[4, Appendix A] some properties of Px1 (x′ , y ′ ) and Qt (x, y).
Proposition 5.1 ([4, Theorem 57]). For any C > 0, we define the function G(C) on R3
by 

G(C) (x) = exp − C |x1 |2 + |x2 |4/3 + |x3 |4/3 .
For sufficiently large c > 0, {Qt }t>0 is a G(C) -type semigroup for some constant C > 0, in
the sense of Definition 2.3.
In what follows, we fix C > 0 and write G = G(C) . For any G-controlled weight w and
any ζ ≤ 0, we can define the Besov space C ζ,Q(w) in the sense of Definition 2.4. We denote
by C ζ,Q(R × T2 ) the closure of Cb (R × T2 ) in the space C ζ,Q(1) with the flat weight w = 1.
Proposition 5.2. For sufficiently large c > 0, we have the following.
(i) [4, Theorems 61 and 62] Let β ∈ (0, α). For any g ∈ Csβ (R × T2 ), we can define the
function on R × T2 by
Z


(∂1 − a∆ + c)−1 g (x) := Px1 −y1 (x′ , y ′ )g(y)dy.
(−∞,x1 ]×R2

Then h = (∂1 − a∆ + c)−1 g is the unique solution of (∂1 − a∆ + c)h = g such that
h ∈ Csβ+2 (R × T2 ) and limx1 →−∞ h(x) = 0.

(ii) [4, Theorem 64] The operator c − L has the inverses of the form
Z ∞ Z 1
−1
(c − L) f = Qt f dt = Qt f dt + Q1 (c − L)−1 f.
0 0

25
 For any ζ ∈ (−4, 0) \ Z, the map (c − L)−1 is extended to the continuous operator
from C ζ,Q(R × T2 ) to Csζ+4 (R × T2 ).
(iii) [4, Theorem 6] We can decompose (∂1 − a∆ + c)−1 = K + S, where
Z 1 Z 1
K :=: Kt dt := − (∂1 + ∆)Qt dt
0 0

and
S := K1 (c − L)−1 + c(∂1 − a∆ + c)−1 (1 + ∂1 + ∆)(c − L)−1 .
Then {Kt }t>0 is a 2-regularizing kernel admissible for {Qt }t>0 in the sense of Def-
inition 4.1, where δ ∈ (2, 2 + α) in the condition (iii). Moreover, for any ζ ∈
α∧(ζ+2)+2−ε
(−2, 0)\{−1} and ε > 0, S is continuous from C ζ,Q (R×T2 ) to Cs (R×T2 ).

5.2 Regularity structure associated with PAM

Following [12], we prepare the regularity structure associated with PAM (5.1).
Definition 5.3. For any fixed ε ∈ (0, 1), we define the regularity structure T = (A, T, G)
of regularity α0 := −1 − ε as follows.
(1) (Index set) A = {−1 − ε, −2ε, −ε, 0, 1 − ε, 1}.
(2) (Model space) T is an eight dimensional linear space spanned by the symbols

Ξ, I(Ξ)Ξ, X2 Ξ, X3 Ξ, 1, I(Ξ), X2 , X3 .
L
The direct sum decomposition T = α∈A Tα is given by

T−1−ε = span{Ξ}, T−2ε = span{I(Ξ)Ξ}, T−ε = span{X2 Ξ, X3 Ξ},

T0 = span{1}, T1−ε = span{I(Ξ)}, T1 = span{X2 , X3 }.

(3) (Structure group) G is an Abel group consisting of all linear functionals on the sub-
space span{I(Ξ), X2 , X3 }. The action of g ∈ G to T is defined by

Γg 1 = 1, Γg Xi = Xi + g(Xi )1, Γg Xi Ξ = Xi Ξ + g(Xi )Ξ,

Γg Ξ = Ξ, Γg I(Ξ) = I(Ξ) + g(I(Ξ))1, Γg I(Ξ)Ξ = I(Ξ)Ξ + g(I(Ξ))Ξ.

By definition, the subspace S := span{1, I(Ξ), X2 , X3 } is invariant under the action

of G. Hence (A, S, G) also forms a regularity structure. In what follows, let T be the
regularity structure given in Definition 5.3 with fixed ε.
We consider the models and modelled distributions as in Section 3 with slight modifi-
cations. For any r ≥ 0, we define the weight function

vr (x) = e−r|x1 | .

26
It is easy see that vr satisfies the inequality (2.1) with vr∗ (x) := er|x1 | and vr is G-controlled.
Moreover, vr satisfies the assumption of Remark 3.3 with w1 = v2r and w2 = v3r .
Definition 5.4. We say that the smooth model M ∈ Mvr (T ) (defined on R3 ) is admissible
if it satisfies the following properties.
(i) For any x, y ∈ R3 and i ∈ {2, 3}, we have



Πx+ei (·) (y + ei ) = Πx (·) (y), Γ(y+ei )(x+ei ) = Γyx .

(ii) For any x, y ∈ R3 and i ∈ {2, 3}, we have

(Πx 1)(y) = 1, (Πx Xi )(y) = yi − xi ,
Γyx 1 = 1, Γyx Xi = Xi + (yi − xi )1.

(iii) We write ΠΞ = Πx Ξ since it is independent of x. For any x ∈ Rd , we have

Πx I(Ξ) = K(·, ΠΞ) − K(x, ΠΞ).

(iv) For any τ ∈ {Ξ, I(Ξ)Ξ, X2 Ξ, X3 Ξ, 1, I(Ξ), X2 , X3 }, we have

sup vr (x)|(Πx τ )(x)| < ∞.
x∈Rd

We define the closed subspace Mrad (T ) of Mvr (T ) as the completion of the set of smooth
admissible models.
In the sense of Definition 4.3, the linear operator I : T → T defined by
(
I(Ξ) (τ = Ξ)
Iτ =
0 (τ ∈ {I(Ξ)Ξ, X2 Ξ, X3 Ξ, 1, I(Ξ), X2 , X3 })

is an abstract integration of order 2, and for any M ∈ Mrad (T ), the pair (M, M ) is
K-admissible for I. Therefore, we can define the operator K by Corollary 4.6.
The weight function vr is used only to ensure the global bound of the model M defined
from the white noise. Since we study the local-in-time solution theory of (5.1), the flat
weight v0 = 1 is sufficient for the definition of singular modelled distributions.
Definition 5.5. For any interval I ⊂ R and any η ≤ γ, we define D γ,η (I; Γ) as the space
of all functions f : (I \ {0}) × T2 → T<γ such that
kf (x)kα
L f Mγ,η;I := max sup < ∞,
α<γ x∈(I\{0})×T2 ω(x)(η−α)∧0
k∆Γyx f kα
kf kγ,η;I := max sup < ∞.
α<γ
x,y∈(I\{0})×T2 , x6=y ω(x, y)η−γ ky − xkγ−α
s
ky−xks ≤ω(x,y)

We denote by D γ,η (I, S; Γ) the subspace of S-valued functions in the class D γ,η (I; Γ).

27
5.3 Convolution operators
We can rewrite the equation (5.1) in the form
Z

u(x) = Px1 (x′ , y ′ )u0 (y ′ )dy ′ + (∂1 − a∆ + c)−1 1(0,∞)×R2 b(u)ξ (x). (5.2)
R2

In this subsection, we prepare some operators to reformulate the equation (5.2) at the level
of singular modelled distributions.R
First, the function P u0 (x) := R2 Px1 (x′ , y ′ )u0 (y ′ )dy ′ can be lifted to the singular mod-
elled distribution taking values in the polynomial structure. For any sufficiently regular
function f on (R \ {0}) × R2 , we define the T-valued function
Lf (x) := f (x)1 + (∂2 f )(x)X2 + (∂3 f )(x)X3
of x ∈ (R \ {0}) × R2 .
Lemma 5.6 ([4, Lemma 29]). Let θ ∈ (0, 1) and u0 ∈ C θ (T2 ). Then the lift L(P u0 ) of the
function 1x1 >0 P u0 (x) is in the class D γ,θ for any γ ∈ (0, 2) and we have
kL(P u0 )kγ,θ;(0,t) . ku0 kC θ (T2 )
for any t > 0.
Next, to lift the second term on the right hand side of (5.2), we prepare two lemmas.
The first one is used to “extend” the domain of singular modelled distributions from (0, t)×
T2 to R × T2 .
Lemma 5.7. We fix a smooth non-increasing function χ : (0, ∞) → [0, 1] such that
(
1 (0 < t ≤ 1),
χ(t) =
0 (t ≥ 2).

For each t > 0, we define the function χt : R3 → R by setting χt (x) = 1x1 >0 χ(x1 /t). Let
M = (Π, Γ) ∈ Mrad (T ) with some r > 0 and let γ ∈ (0, 1 − ε) and η ≤ γ. For any t ∈ (0, 1]
and any f ∈ D γ,η ((0, 2t); Γ), we define the function


(Et f )(x) = P<γ (Lχt )(x) · f (x) ,
where a (partial) product (·) on T is defined by
1·τ =τ (τ ∈ {Ξ, I(Ξ)Ξ, X2 Ξ, X3 Ξ, 1}), Xi · Ξ = Xi Ξ (i ∈ {2, 3}).
(Other products do not appear due to the assumption on γ.) Then the function Et f belongs
to D γ,η∧α0 (R; Γ) and satisfies
|||Et f |||γ,η∧α0 ;R ≤ C(1 + kΓkγ,vr )|||f |||γ,η;(0,2t)
for some constant C > 0 independent of t. Moreover, (Et f )|(0,t]×T2 = f |(0,t]×T2 .

28
Proof. We can check that |||Lχt |||γ ′ ,0;R . 1 for any γ ′ ∈ (1, 2) by definition, so by applying
the continuity of the multiplication of modelled distributions [12, Proposition 6.12], we
have
|||Et f |||γ,η∧α0 ;(0,2t) . |||f |||γ,η;(0,2t) .
We can extend it into |||Et f |||γ,η∧α0 ;(0,2t] . |||f |||γ,η;(0,2t) by the uniform continuity. To show
that Et f ∈ D γ,η∧α0 ((0, ∞); R), we pick x ∈ [2t, ∞) × T2 and y ∈ (0, 2t) × T2 . By setting
z = (2t, y ′ ) we have

k(Et f )(y) − Γyx (Et f )(x)kα

≤ k(Et f )(y) − Γyz (Et f )(z)kα + kΓyz (Et f )(z) − Γyx (Et f )(x)kα
≤ kEt f kγ,η∧α0 ;(0,2t] ω(y)η∧α0 −γ ky − zkγ−α
s
. |||f |||γ,η;(0,2a) ω(x, y)η∧α0 −γ ky − xkγ−α
s .

For the case that x ∈ (0, 2t) × T2 and y ∈ [2t, ∞) × T2 , by the properties of models we have

vr (x)k(Et f )(y) − Γyx (Et f )(x)kα = vr (x)kΓyx {Γxy (Et f )(y) − (Et f )(x)}kα
X
≤ kΓkγ,vr vr∗ (y − x) ky − xksβ−α kΓxy (Et f )(y) − (Et f )(x)kβ
α≤β<γ

. kΓkγ,vr |||f |||γ,η;(0,2t) vr∗ (y − x)ω(x, y)η∧α0 −γ ky − xkγ−α

s .

Note that the supremum in the definition of the norm k · kγ,η;I is taken over ky − xks ≤
ω(x, y). Since |y1 | ≤ 1 + |x1 | ≤ 3 in this region, the factors vr (x) and vr∗ (y − x) are
bounded both above and below. Thus we can ignore these weights and have Et f ∈
D γ,η∧α0 ((0, ∞); Γ). On the other hand, Et f ∈ D γ,η∧α0 ((−∞, 0); Γ) is obvious from the
definition. Since ky − xks ≤ ω(x, y) implies that x1 and y1 have the same sign, we obtain
the assertion.

Next, we recall from [12] a variant of the norm of singular modelled distributions.
The following result holds for any singular modelled distributions on Rd taking values in
arbitrary regularity structures and any models.

Lemma 5.8 ([12, Lemma 6.5]). Let η ≤ γ and r ≥ 0, and let I ⊂ R be an interval. For
any functions f : (I \ {0}) × T2 → T<γ , we define

kf (x)kα
L f M◦γ,η;I := max sup .
α<γ x∈(I\{0})×T2 ω(x)η−α

Then the inequality L f Mγ,η;I ≤ L f M◦γ,η;I obviously holds. Conversely, if

lim Pα f (x) = 0
x1 →0

29
holds for any α < η, then there exists a polynomial p(·) such that, for any M ∈ Mrad (T )
and f ∈ D γ,η (I; Γ), we have

L f M◦γ,η;I . p(kΓkγ,vr )|||f |||γ,η;I .

In the end, we can lift the operator (∂1 − a∆ + c)−1 to the level of singular modelled
distributions. Recall the decomposition (∂1 − a∆ + c)−1 = K + S from Proposition 5.2-(iii)

Theorem 5.9. Let γ ∈ (0, α ∧ (1 − ε)), η ∈ (γ − 2, γ], r ≥ 0, and t ∈ (0, 1]. For any
M = (Π, Γ) ∈ Mrad (T ), f ∈ D γ,η ((0, 2t); Γ), and δ ∈ (0, γ + 2], we define the function

Ptδ f := P<δ {K(Et f ) + L(S(REt f ))}.

Then Ptδ f ∈ Dvδ,η∧α

3r
0 +2
(R; Γ). If M is smooth and admissible in the sense of Definition
5.4, then we have
R(Ptδ f )(x) = (∂1 − a∆ + c)−1 (REt f )(x). (5.3)
Moreover, there exists a polynomial p(·) such that, for any κ ≥ 0 we have

|||Ptδ f |||δ,η∧α0 +2−κ;(0,2t) ≤ p(|||M |||γ,vr ) tκ/2 |||f |||γ,η;(0,2t) . (5.4)

Finally, there exists a polynomial q(·) such that

|||Ptδ f (1) ; Ptδ f (2) |||δ,η∧α0 +2−κ;(0,2t) ≤ q(R) tκ/2 |||M (1) ; M (2) |||γ,vr + |||f (1) ; f (2) |||γ,η;(0,2t)

for any M (i) ∈ Mrad (T ) and f (i) ∈ D γ,η ((0, 2t); Γ(i) ) with i ∈ {1, 2} such that |||M (i) |||γ,vr ≤
R and |||f (i) |||γ,η;(0,2t) ≤ R.

Proof. In the proof of inequalities, we only use definitions of norms of models and singular
modelled distributions. Therefore, due to the density argument, we can assume that M is
smooth and admissible.
We know KEt f ∈ Dvγ+2,η∧α 3r
0 +2
(R; Γ) from Corollary 4.6, and REt f ∈ C η∧α0 ,Q (v2r ) from
Corollary 3.9. Moreover, since Et f (x) vanishes outside [0, 2] × T2 , we can modify the proof
of Theorem 3.7 and obtain REt f ∈ C η∧α0 ,Q (R × T2 ). Then since S(REt f ) ∈ Csγ+2 (R × T2 )
by Proposition 5.2-(iii), we have L(S(REt f )) ∈ D γ+2,γ+2 (Γ). Hence Ptδ f ∈ Dvδ,η∧α 3r
0 +2
(Γ)
by Proposition 3.5-(ii). The identity (5.3) follows from Theorem 4.5 and the definition of
L(S(REt f )).
Note that |||Ptδ f |||δ,η∧α0 +2;(0,2t) ≤ Cr |||Ptδ f |||δ,η∧α0 +2,v3r for some r-dependent constant
Cr . We show (5.4) for κ > 0 by applying Lemma 5.8. By definition, the only α ∈ A
smaller than η ∧ α0 + 2 (≤ 1 − ε) is α = 0. Since M is smooth, by Proposition 3.8, the
T0 -component of Ptδ f (x) is equal to


Πx (Ptδ f )(x) (x) = (RPtδ f )(x) = (∂1 − a∆ + c)−1 (REt f )(x).

30


Since (REt f )(y) = Πy (Et f )(y) (y) = 0 vanishes on y ∈ (−∞, 0) × T2 , we also have
Z
−1
(∂1 − a∆ + c) (REt f )(x) = Px1 −y1 (x′ , y ′ )(REt f )(y)dy.
[0,x1 ]×R2

Note that, in the proof of Proposition 3.8, we obtained

|REt f (y)| . ω(y)η∧α0 .
Since η ∧ α0 > −2, we can show that
Z x1
|(∂1 − a∆ + c)−1 (REt f )(x)| . |y1 |(η∧α0 )/2 dy1 → 0
0
as x1 ↓ 0. Therefore, by Lemma 5.8 we have
|||Ptδ f |||γ,η∧α0 +2−κ;(0,2t) . |||Ptδ f |||◦γ,η∧α0 +2−κ;(0,2t)
. tκ/2 |||Ptδ f |||◦γ,η∧α0 +2;(0,2t) . tκ/2 |||Ptδ f |||γ,η∧α0 +2;(0,2t) ,
where ||| · |||◦γ,η;I := L · M◦γ,η;I + k · kγ,η;I . The proof of the local Lipschitz estimate is a slight
modification.

5.4 Solution theory for PAM

We show the local-in-time well-posedness of the equation


U = L(P u0 ) + Ptγ b(U )Ξ (5.5)
in the class D γ,η ((0, 2t), S; Γ) with some appropriate choices of γ and η. The term L(P u0 )
and the operator Ptγ was defined in the previous subsection. The only missing factor f (U )
is the lift of the composition map u 7→ b(u) defined in [12, Proposition 6.13]. In the present
case, for sufficiently small ε and any U ∈ D γ,η ((0, 2t), S; Γ) with γ ∈ (1, 2) and η ∈ [0, γ] of
the form
U (x) = u(x)1 + v(x)I(Ξ) + u2 (x)X2 + u3 (x)X3 ,
we can define b(U ) ∈ D γ,η ((0, 2t), S; Γ) by the concrete form
b(U )(x) = b(u(x))1 + b′ (u(x)){v(x)I(Ξ) + u2 (x)X2 + u3 (x)X3 }.
Then the map U 7→ b(U ) is locally Lipschitz continuous.
Theorem 5.10. Assume ε ∈ (0, α ∧ (1/4)) and let θ ∈ (0, 1 − ε). Then there exists
a function t0 : (0, ∞)2 → (0, 1] such that, for any R1 , R2 > 0 the following assertion
holds: For any u0 ∈ C θ (T2 ) such that ku0 kC θ (T2 ) ≤ R1 , and any M ∈ Mrad (T ) such that
kM kγ,vr ≤ R2 , the equation (5.5) with t = t0 (R1 , R2 ) and γ = 1 + 2ε has a unique solution
U in the class D 1+2ε,θ ((0, 2t), S; Γ). Moreover, the mapping
St : (u0 , M ) 7→ U
is Lipschitz continuous on the space {u0 ; ku0 kC θ ≤ R1 } × {M ; kM kγ,wr ≤ R2 }.

31
Proof. The proof is a standard fixed point argument. Note that, the following operators
are well-defined and locally Lipschitz continuous.

• ([12, Proposition 6.13]) U ∈ D 1+2ε,θ ((0, 2t), S; Γ) 7→ b(U ) ∈ D 1+2ε,θ ((0, 2t), S; Γ).

• ([12, Proposition 6.12]) V ∈ D 1+2ε,θ ((0, 2t), S; Γ) 7→ V Ξ ∈ D ε,θ−1−ε((0, 2t); Γ).

• (Theorem 5.9) W ∈ D ε,θ−1−ε ((0, 2t); Γ) 7→ Pt1+2ε W ∈ D 1+2ε,1−ε ∈ ((0, 2t), S; Γ).


Therefore, by setting F (U ) = L(P u0 ) + Pt1+2ε b(U )Ξ , we have

|||F (U )|||1+2ε,θ;(0,2t) . ku0 kC θ + t(1−ε−θ)/2 |||f (U )Ξ|||ε,θ−1−ε

. ku0 kC θ + t(1−ε−θ)/2 |||f (U )|||1+2ε,θ
. ku0 kC θ + t(1−ε−θ)/2 p(|||U |||1+2ε,θ )

for some polynomial p(·). From this inequality, we can find a large R > 0 depending on u0
and M and show that F maps a ball of radius R in D 1+2ε,θ ((0, 2t), S; Γ) into itself. From
here onward, we can show the assertion by an argument similar to [12, Theorem 7.8].

5.5 Convergence of models

In this subsection, we define the sequence of smooth admissible models associated with
regularized noises and show its probabilistic convergence.
R We fix an even function ̺ :
R2 → [0, 1] in the Schwartz class and such that R2 ̺(x)dx = 1, and set ̺n (x) = 22n ̺(2n x)
for each n ∈ N. We define the smooth approximation of the spatial white noise ξ by
Z
ξn (x) = ρfn (x − y)ξ(y)dy, (x ∈ T2 )
T2
P
where ρ fn (x) := k∈Z2 ρn (x + k). For
fn denotes the spatial periodization of ρn defined by ρ
such ξn , we can define the smooth admissible model M n = (Πn , Γn ) ∈ Mrad (T ) by

(Πnx Ξ)(y) = ξn (y ′ ), (Πnx Xi Ξ)(y) = (yi − xi )ξn (y ′ ),

Πnx I(Ξ)Ξ (y) = Kξn (y) − Kξn (x) ξn (y ′ ) − Cn (y),

where the function Cn is defined by

Z
 
Cn (x) = E (Kξn )(x)ξn (x′ ) = K(x, y)cn (x′ − y ′ )dy
R3

with cn (x′ − y ′ ) := E[ξn (x′ )ξn (y ′ )] = ̺f

∗2 (x′ − y ′ ). Then Γn = Γ n (see Definition 5.3) is
n yx gyx
given by
n n
gyx (I(Ξ)) = Kξn (y) − Kξn (x), gyx (Xi ) = yi − xi .

32
Theorem 5.11. For any r > 0, the sequence {M n }n∈N of models defined above converges
in Lp (Ω, Mrad (T )) for any p ∈ [1, ∞) and almost surely in Mrad (T ).

Proof. In view of the inductive proof as in [3], it is sufficient to show the uniform bounds
 
E Qt (x, Πnx τ ) . tβ/4 (5.6)

for any β ∈ {−1−ε, −2ε, −ε} and τ ∈ Tβ . The integral operator used in [3] is homogeneous
in the sense that Qt (x, y) depends only on x − y, but this assumption is used only to
prove the above estimate. Since ξ is a centered Gaussian, we have only to show (5.6) for
τ = I(Ξ)Ξ. By definition,
Z
 n

E Qt (x, Πx τ ) = − Qt (x, y)E[(Kξn )(x)ξn (y ′ )]dy
R 3
Z
=− Qt (x, y)K(x, z)cn (z ′ − y ′ )dydz.
(R3 )2

R1
To estimate this integral, we decompose K = 0 Ks ds and set
Z
n
It,s (x) = − Qt (x, y)Ks (x, z)cn (z ′ − y ′ )dydz.
(R3 )2

By the Gaussian estimates of Qt and Ks , their time integral is estimated as

Z Z
(C)
|Qt (x, y)|dy1 . ht (x′ − y ′ ), |Ks (x, z)|dz1 . s−1/2 h(C) ′ ′
s (x − z ),
R R

(C) 4 /t)1/3 +(|x 4 /t)1/3 }

for some constant C > 0, where ht (x′ ) := t−1/2 e−C{(|x2 | 3| . Thus we have

n (C)
|It,s (x)| . s−1/2 (ht ∗ h(C)
s ∗ |cn |)(0).

(C) (C) (c)

Since |ht ∗ hs (x)| . ht+s (x) for some constant 0 < c < C (see [4, Lemma 55] for
instance), we have
n
|It,s (x)| . s−1/2 (t + s)−1/2 .
Since, for any ε > 0, we have
Z 1 Z t Z 1
n −1/2 −1/2
|It,s (x)|ds . s t ds + s−1 ds . − log t . t−ε/2 ,
0 0 t

we obtain the desired estimate (5.6) for τ = I(Ξ)Ξ.

33
5.6 Renormalization of PAM
For a fixed initial condition u0 ∈ C θ (T2 ) and the sequence of random models {M n } con-
structed in the previous subsection, we denote by
Un = St (u0 , M n )
the solution of the equation (5.5) with γ = 1 + 2ε and with the random time
 
n
t = t0 ku0 kC θ (T2 ) , sup kM kγ,vr .
n∈N

Combining Theorem 5.11 with Theorem 5.10, we have the following theorem.
Theorem 5.12. For each n ∈ N, we denote by Rn the reconstruction operator associated
with M n . Then the function un = Rn (Et Un ) converges in L∞ ((0, t) × T2 ) as n → ∞ and
solves the equation




∂1 − a(x′ )∆ un (x) = b un (x) ξn (x′ ) − Cn (x)(bb′ ) un (x) (5.7)
on x ∈ (0, t) × T2 .


Proof. On the region x ∈ (0, t) × T2 , since un (x) = Πnx Un (x) (x), we can assume that Un
is of the form
Un (x) = un (x)1 + vn (x)I(Ξ) + u2,n (x)X2 + u3,n (x)X3 . (5.8)
The convergence of {un } in L∞ ((0, t) × T2 ) follows from the convergence of {Un } and the
definition of the norm L · Mγ,η;(0,t) .
For any x ∈ (0, t) × T2 , the function b(Un )(x) is of the form
b(Un )(x) = b(un (x))1 + b′ (un (x)){vn (x)I(Ξ) + u2,n (x)X2 + u3,n (x)X3 },


and then Pt1+2ε b(U )Ξ is of the form


Pt1+2ε b(U )Ξ (x) = wn (x)1 + b(un (x))I(Ξ) + w2,n (x)X2 + w3,n (x)X3
for some functions wn , w2,n , and w3,n . For Un to solve the equation (5.5), the coefficient
vn (x) in (5.8) must be equal to b(un (x)) for any x ∈ (0, t) × T2 . By Theorem 5.9, the
function un satisfies
Z
un (x) = P u0 (x) + Px1 −y1 (x′ , y ′ )(Rn Et f (Un )Ξ)(y)dy.
[0,x1 ]×R2

On the region y ∈ (0, t) × T2 , we obtain from the definition of Πnx I(Ξ)Ξ that



(REt b(Un )Ξ)(y) = Πy Et b(Un )(y)Ξ (y) = Πy b(Un )(y)Ξ (y)
= b(un (y))ξn (y ′ ) − Cn (y)(bb′ )(un (y)).
Thus un satisfies the equation (5.7) on the region (0, t) × T2 .

34
Acknowledgements. The first author is supported by JSPS KAKENHI Grant Number
23K12987.

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