Non-local quasilinear singular SPDEs

I. Bailleul & H. Eulry

Abstract. We study in this short note a counterpart to the quasilinear
generalized parabolic Anderson model (gPAM) on the $2$-dimensional torus where
the coefficients are nonlocal functionals of the solution. Under a positivity
assumption on the diffusion coefficient we give a local in time solution
theory within the framework of paracontrolled calculus.

## 1 – Introduction

We are interested in this work in giving a solution theory for some
quasilinear singular stochastic partial differential equations

$\partial_{t}u-A(f(u))\Delta u=B(g(u))\,\xi$ (1.1)

whose coefficients $A(f(u)),B(f(u))$ are nonlocal functionals of some
functions of the solution $u$. We do so here in the mildly singular setting of
the $2$-dimensional torus $\mathbb{T}^{2}$ for a space white noise $\xi$. We
need some notations to set the stage of our main result, Theorem 1 below.

Denote by $(e_{1},e_{2})$ the canonical basis of
$\mathbb{Z}^{2}\subset\mathbb{R}^{2}$, and for $i\in\\{1,2\\}$ and a function
$\sigma:\mathbb{T}^{2}\times\mathbb{Z}^{2}\rightarrow\mathbb{C}$ set

$(D_{i}\sigma)(x,k):=\sigma(x,k+e_{i})-\sigma(x,k),$

and ${\bf D}:=(D_{1},D_{2})$. Write $\mathcal{F}$ for the Fourier transform on
$\mathbb{T}^{2}$ and denote by
$\widehat{\sigma}(n,k)=(\mathcal{F}\sigma)(n,k)$ the Fourier transform of the
function $\sigma(\cdot,k)$ for each $k\in\mathbb{Z}^{2}$. Write $\langle
k\rangle$ for $1+|k|$. Given $s\in\mathbb{R}$ and $\alpha>0$ we define the
class $\Sigma_{\alpha}^{s}$ of functions
$\sigma:\mathbb{T}^{2}\times\mathbb{Z}^{2}\rightarrow\mathbb{C}$ such that

$\big{\|}\partial_{x}^{i}{\bf
D}^{j}\sigma(\cdot,k)\big{\|}_{\mathcal{C}^{\alpha}}\lesssim\langle
k\rangle^{s-|j|}$ (1.2)

for all multi-indices $i,j$ in $\mathbb{N}^{2}$, and for which there exists a
constant $0<\mu<1$ such that

$(\mathcal{F}\sigma)(n,k)=0\quad\textrm{whenever }\quad|n|>\mu\langle
k\rangle.$ (1.3)

For such a symbol, we can easily bound $\widehat{\sigma}$ uniformly in $(n,k)$
by

$\displaystyle{}\big{|}\widehat{\sigma}(n,k)\big{|}\lesssim\langle
n\rangle^{-4}\,\langle k\rangle^{s}\,\mathbf{1}_{|n|\leq\mu\langle k\rangle}.$
(1.4)

The convolution operator with a function $\mathcal{F}^{-1}(\sigma)$ on
$\mathbb{T}^{2}$ whose Fourier transform $\sigma(k)$ satisfies (1.2) provides
an elementary example of a function $\sigma\in\Sigma_{\alpha}^{0}$ – here it
is independent of $x$ and $\sigma$ is supported on the diagonal $\\{n=k\\}$.
Similarly, ad hoc conditions on a kernel $K$ ensure that the integral operator
$v\mapsto\int K(\cdot,y)v(y)dy$ is associated with a symbol
$\sigma\in\Sigma_{\alpha}^{s}$ for some $s$. A negative $s$ means an operator
that regularizes. For $a$ and $b$ in $\Sigma_{\alpha}^{s}$ we denote by $A$
and $B$ the pseudodifferential operators with symbols $a$ and $b$
respectively.

We say that $A$ preserves positivity if $A(v)>0$ whenever $v$ is bounded below
by a positive constant. Positivity preserving pseudodifferential operators
with a symbol in $\Sigma^{0}_{\alpha}$ for some $\alpha>0$ are given by some
kernels $\mu(x,\cdot)$ that are finite non-negative measures for all
$x\in\mathbb{T}^{2}$ such that testing $A(v)$ against a smooth test function
$\ell$ on $\mathbb{T}^{2}$ one has

$A(v)(\ell)=\iint v(y)\mu(x,dy)\ell(x)dx.$ (1.5)

This is the case of the integral operator with kernel $K$ when $K$ is bounded
below by a positive constant. In the renormalized equation (1.6) below we
regularize the space white noise $\xi$ into a smooth function

$\xi^{\varepsilon}:=\mathcal{F}^{-1}\big{(}\chi(\varepsilon|\cdot|)\,\mathcal{F}(\xi)(\cdot)\big{)}$

using a ‘smooth Fourier cut-off’ $\chi:[0,\infty)\rightarrow[0,1]$ of class
$C^{\infty}$, equal to $1$ on $[0,1]$ and $0$ on $[2,\infty)$.

###### Theorem 1 –

Pick two regularity exponents $2/3<\beta<\alpha<1$ and $s\leq 0$. Take two
symbols $a,b\in\Sigma_{\alpha}^{s}$ and assume that the operator $A$ preserves
positivity. Take further $f,g\in C^{3}_{b}(\mathbb{R},\mathbb{R})$.

  1. (a)

There exists two deterministic diverging functions
$c_{a},c_{b}:(0,1]\rightarrow C^{\infty}(\mathbb{T}^{2})$ and a positive
random time $T$ such that the solution $u^{\varepsilon}$ to the equation

$\partial_{t}u^{\varepsilon}-A\big{(}f(u^{\varepsilon})\big{)}\Delta
u^{\varepsilon}=B\big{(}g(u^{\varepsilon})\big{)}\,\xi^{\varepsilon}+c_{a}(\varepsilon)\bigg{(}\frac{B\big{(}g(u^{\varepsilon})\big{)}}{A\big{(}f(u^{\varepsilon})\big{)}}\bigg{)}^{2}f^{\prime}(u^{\varepsilon})-c_{b}(\varepsilon)\frac{B\big{(}g(u^{\varepsilon})\big{)}}{A\big{(}f(u^{\varepsilon})\big{)}}\,g^{\prime}(u^{\varepsilon})$
(1.6)

with initial condition $u_{0}\in\mathcal{C}^{\alpha}(\mathbb{T}^{2})$
converges in probability in the parabolic $\alpha$-Hölder space on
$[0,T]\times\mathbb{T}^{2}$, as $\varepsilon\in(0,1]$ goes to $0$.

  2. (b)

The singular PDE

$\partial_{t}u-A(f(u))\Delta u=B(g(u))\,\xi$

has a well-defined formulation in a space of paracontrolled functions, where
it has a unique solution $(u,u^{\prime})$. The limit of the $u^{\varepsilon}$
is given by $u$.

This statement gives back the result first proved in [15, 8, 5] by Otto &
Weber, Furlan & Gubinelli and Bailleul, Debussche & Hofmanová when $A$ and $B$
are the identity operators. The approach of Otto & Weber was later fully
developed in [14, 11, 12] to provide a variant of regularity structures well-
adapted to a certain class of quasilinear singular stochastic PDEs. It would
be interesting to see whether their approach can be twicked to handle nonlocal
equations as here. The approach of [5] was developped in a paracontrolled
setting in Bailleul & Mouzard’s work [6] and fully developped in a regularity
structure setting in Bailleul, Hoshino & Kusuoka’s work [7].

In the proof of Theorem 1 we obtain point (a) as a corollary of point (b). The
proof of point (b) involves two different tasks.

  * –

Build an analytical setting where to formulate the equation as a well-defined
fixed point problem.

  * –

Construct a random variable that plays the role of an ill-defined polynomial
functional of the noise. It is required as an independent ingredient in the
analytical setting.

We formulate the equation as a fixed point in a space of paracontrolled
functions in Section 2 . We prove therein that it is well-posed locally in
time. The construction of the polynomial functional of the noise is done in
Section 3.

Notations – Throughout we denote by $\mathcal{C}^{\gamma}(\mathbb{T}^{2})$ the
Besov space $B_{\infty,\infty}^{\gamma}(\mathbb{T}^{2})$, for any
$\gamma\in\mathbb{R}$. For $\alpha\in(0,1)$ and a finite positive time horizon
$T$ we set

$\mathscr{C}^{\alpha}_{T}:=C\big{(}[0,T],\mathcal{C}^{\alpha}(\mathbb{T}^{2})\big{)}\cap
C^{\alpha/2}\big{(}[0,T],L^{\infty}(\mathbb{T}^{2})\big{)}.$

This space coincides with the $\alpha$-Hölder parabolic space, with an
equivalent norm. We will denote by $A$ and $B$ the pseudodifferential
operators associated with the symbols $a$ and $b$ respectively. We will use
the symbol $\lesssim$ to denote an inequality that holds up to a
multiplicative positive constant that does not matter for our purpose. We
write $\lesssim_{c}$ to emphasize that this implicit constant depends only on
a parameter $c$.

## 2 – Paracontrolled formulation of the equation

The setting of paracontrolled calculus is now familiar enough that we can go
straight to the point. We refer the reader to Gubinelli & Perkowski’s lecture
notes [10] for an introduction. We refer to the original article [9] of
Gubinelli, Imkeller & Perkowski or the advanced work [3] of Bailleul &
Bernicot to see paracontrolled calculus in action in some situations that are
more involved than what we need here.

In its basic form a paracontrolled structure is a Banach space of functions of
the form

$v=\mathsf{P}_{v^{\prime}}X+v^{\sharp}$ (2.1)

where $v^{\prime}\in\mathcal{C}^{\alpha_{1}}(\mathbb{T}^{2})$ with
$\alpha_{1}>0$, and $X\in\mathcal{C}^{\alpha}(\mathbb{T}^{2})$, and
$v^{\sharp}\in\mathcal{C}^{\alpha_{2}}(\mathbb{T}^{2})$ with
$\alpha_{2}>\alpha$. Given that
$\mathsf{P}_{v^{\prime}}X\in\mathcal{C}^{\alpha}(\mathbb{T}^{2})$ from the
continuity estimate of Proposition 6 in Appendix A, one sees $v^{\sharp}$ as a
‘remainder’ term in the above decomposition of $v$. This paracontrolled
structure is stable by nonlinear functions, a consequence of Bony’s
paralinearization result, Proposition 9 in Appendix A. In the study of the
semilinear version of Equation (1.1), where $A=1$ say, we can formulate the
equation as a fixed point problem in some space of time-dependent
paracontrolled functions provided that we have an appropriate definition of
the product $X\xi$. An explicit choice of $X$ as a function of $\xi$ brings
back the question of that definition to a problem about random variables only.
Once this probability problem is understood, solving the equation itself is a
purely deterministic problem in a given $\omega$-dependent Banach space of
paracontrolled functions, where $\omega$ stands for the chance element. We
will follow here a similar strategy, following Bailleul, Debussche &
Hofmanová’s reformulation of the quasilinear equation (1.1) as a
‘perturbation’ of a semilinear equation [5]

$\partial_{t}u-A(f(u_{0}^{T}))\Delta u=B(g(u))\xi+D_{0}(u)\Delta u,$ (2.2)

where

$u_{0}^{T}:=e^{T\Delta}(u_{0})\in C^{\infty}(\mathbb{T}^{2})$

and

$D_{0}(u):=A(f(u))-A(f(u_{0}^{T})).$

In order to proceed like that we need first to make sure that both $A(f(v))$
and $B(g(v))$ have a paracontrolled structure if $v$ does and that this
structure is preserved by the equation.

### 2.1 – Stability of the paracontrolled structure.

Recall the notation of the modified paraproduct $\overline{\mathsf{P}}$ from
Appendix A. We now fix $X$ for the remainder of this chapter

$X=-\Delta^{-1}(\xi),$

with null mean on $\mathbb{T}^{2}$. Pick

$\frac{2}{3}<\beta<\alpha<1$

and define the space $\mathbf{C}^{\beta}_{\alpha,T}(X)$ as the set of all
functions
$(u,u^{\prime})\in\mathscr{C}^{\alpha}_{T}\times\mathscr{C}^{\beta}_{T}$ such
that

$u^{\sharp}:=\big{(}u-\overline{\mathsf{P}}_{u^{\prime}}X\big{)}\in\mathscr{C}^{\alpha}_{T}\qquad\text{and}\qquad\sup_{0<t\leq
T}t^{\frac{2\beta-\alpha}{2}}\|u^{\sharp}(t)\|_{\mathcal{C}^{2\beta}}<+\infty$
(2.3)

which we equip with the norm

$\|(u,u^{\prime})\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}:=\|u^{\prime}\|_{\mathscr{C}^{\beta}_{T}}+\|u^{\sharp}\|_{\mathscr{C}^{\alpha}_{T}}+\sup_{0<t\leq
T}t^{\frac{2\beta-\alpha}{2}}\|u^{\sharp}(t)\|_{\mathcal{C}^{2\beta}}.$

We will look for a solution to Equation (1.1), equivalently Equation (2.2), as
an element of $\mathbf{C}^{\beta}_{\alpha,T}(X)$. The reformulation (2.2) of
Equation (1.1) involves a number of operations. We need to check how the a
priori paracontrolled structure of a solution to (2.2) interacts with these
operations.

a) Pseudodifferential operators and paraproducts. The class of symbols
$\Sigma^{s}_{\alpha}$ is defined from the constraints (1.2) and (1.3). For
$\alpha_{1}>0$ and $h_{1}\in\mathcal{C}^{\alpha_{1}}(\mathbb{T}^{2})$ we check
that the operator $\mathsf{P}_{h_{1}}(\cdot)$ has a symbol in the class
$\Sigma^{0}_{\alpha_{1}}$. We read on the condition (1.2) that the class of
operators associated to $\Sigma^{s}_{\alpha}$ is a subclass of the set of
pseudodifferential operators $\Psi^{s}_{11}$. The operators in this class send
continuously any $\mathcal{C}^{\gamma}(\mathbb{T}^{2})$ space into
$\mathcal{C}^{\gamma-s}(\mathbb{T}^{2})$ if $\gamma>0$ and $(\gamma-s)>0$ are
non-integers – this is the classical Schauder estimate.

One gets as follows the structure (1.5) of the operators with a symbol in
$\Sigma^{0}_{\alpha}$ that are positivity preserving. For
$v\in\mathcal{C}^{\gamma}(\mathbb{T}^{2})$ with $\gamma$ one has from the
classical Schauder estimate $A(v)\in\mathcal{C}^{\gamma}(\mathbb{T}^{2})$. Now
for $v\in\mathcal{C}^{\gamma}(\mathbb{T}^{2})$, since $2\|v\|_{\infty}\pm
v\geq\|v\|_{\infty}>0$ the positivity preserving property yields
$A(2\|v\|_{\infty}\pm v)\geq 0$, that is $\pm A(v)\leq 2\|v\|_{\infty}A({\bf
1})$, i.e. $|A(v)|\leq 2\|v\|_{\infty}A({\bf 1})$. For any
$x\in\mathbb{T}^{2}$ the linear form $A(\cdot)(x)$ on
$\mathcal{C}^{\gamma}(\mathbb{T}^{2})$ thus has a continuous extension into a
linear form on $C^{0}(\mathbb{T}^{2})$. Riesz representation theorem then
gives the existence of a non-negative measure $\mu(x,\cdot)$ such that
$A(\cdot)(x)$ is the integration map against the measure $\mu(x,\cdot)$.

To check that the paracontrolled structure (2.1) is stable by the nonlocal
maps $A$ associated with some symbols $\sigma\in\Sigma^{s}_{\alpha_{1}}$ we
can use the following commutator estimate from Proposition 3.8 in Bailleul,
Dang, Ferdinand & Tô’s work [4], together with the classical Schauder
estimate.

######  Proposition 2 –

Pick $s\leq 0,\alpha_{1}\in(0,1)$ and $\alpha_{2}>0$. For any
$\sigma\in\Sigma^{s}_{\alpha_{1}}$ and
$h_{1}\in\mathcal{C}^{\alpha_{1}}(\mathbb{T}^{2})$, for any
$h_{2}\in\mathcal{C}^{\alpha_{2}}(\mathbb{T}^{2})$ one has

$\big{\|}Op(\sigma)(\mathsf{P}_{h_{1}}h_{2})-\mathsf{P}_{h_{1}}\big{(}Op(\sigma)(h_{2})\big{)}\big{\|}_{\mathcal{C}^{\alpha_{1}+\alpha_{2}-s}}\lesssim_{a}\|h_{1}\|_{\mathcal{C}^{\alpha_{1}}}\|h_{2}\|_{\mathcal{C}^{\alpha_{2}}}.$

b) Structural elements in the right hand side of (2.2). From Proposition 9 on
paralinearization, Proposition 8 on iterated paraproducts and the continuity
result for the corrector in Proposition 7, one can expand the first ill-
defined product

$\displaystyle{}B(f(u))\,\xi=\mathsf{P}_{B(f(u))}\xi+\mathsf{P}_{\xi}B(f(u))+u^{\prime}g^{\prime}(u)\mathsf{\Pi}(B(X),\xi)+g^{\prime}(u)\mathsf{\Pi}(B(u^{\sharp}),\xi)+\sharp_{B}$
(2.4)

where the ill-defined product $\mathsf{\Pi}(B(X),\xi)$ is to be understood as
its renormalized counterpart defined in Section 3 – note that
$u^{\prime}g^{\prime}(u)\mathsf{\Pi}(B(X),\xi)$ is the only term featuring
this ill-defined product. The term

$\displaystyle\sharp_{B}$
$\displaystyle=\mathsf{C}(u^{\prime}g^{\prime}(u),B(X),\xi)+\mathsf{C}(g^{\prime}(u),u^{\sharp},\xi)+\mathsf{\Pi}\big{(}[B,\mathsf{P}_{u^{\prime}g^{\prime}(u)}](X)\,,\,\xi\big{)}+\mathsf{\Pi}\big{(}[B,\mathsf{P}_{g^{\prime}(u)}](u^{\sharp})\,,\,\xi\big{)}$
$\displaystyle\qquad+\mathsf{\Pi}\big{(}B\big{(}\mathsf{P}_{g^{\prime}(u)}(\overline{\mathsf{P}}_{u^{\prime}}X)-\mathsf{P}_{u^{\prime}g^{\prime}(u)}X\big{)}\,,\,\xi\big{)}+\mathsf{\Pi}\big{(}B\big{(}g(u)-\mathsf{P}_{g^{\prime}(u)}u\big{)},\xi\big{)}$

denotes an element of the space
$C(\big{[}0,T],\mathcal{C}^{2\alpha+\beta-2}(\mathbb{T}^{2})\big{)}$ that is
polynomial in $(u,u^{\prime})$ and depends continuously on $\xi$. Given
$\gamma\in\mathbb{R}$ write $=_{\gamma}$ to mean equality up to an element of
$C\big{(}[0,T],\mathcal{C}^{\gamma}(\mathbb{T}^{2})\big{)}$. One also has for
the Laplace term

$\begin{split}D_{0}(u)\Delta
u&=D_{0}(u)\big{(}\Delta\overline{\mathsf{P}}_{u^{\prime}}X+\Delta
u^{\sharp}\big{)}\\\
&=D_{0}(u)\big{(}-\overline{\mathsf{P}}_{u^{\prime}}\xi+[\Delta,\overline{\mathsf{P}}_{u^{\prime}}]X+\Delta
u^{\sharp}\big{)}\\\
&=_{\alpha+\beta-2}D_{0}(u)\big{(}-\overline{\mathsf{P}}_{u^{\prime}}\xi+\Delta
u^{\sharp}\big{)}\\\
&=_{\alpha+\beta-2}-\mathsf{P}_{D_{0}(u)}\big{(}\overline{\mathsf{P}}_{u^{\prime}}\xi\big{)}-\mathsf{\Pi}\big{(}D_{0}(u),\overline{\mathsf{P}}_{u^{\prime}}\xi\big{)}+D_{0}(u)\Delta
u^{\sharp}\\\
&=_{\alpha+\beta-2}-\mathsf{P}_{D_{0}(u)u^{\prime}}\xi-\mathsf{\Pi}\big{(}A(f(u)),\overline{\mathsf{P}}_{u^{\prime}}\xi\big{)}+D_{0}(u)\Delta
u^{\sharp}\\\
&=_{\alpha+\beta-2}-\mathsf{P}_{D_{0}(u)u^{\prime}}\xi-u^{\prime}\mathsf{\Pi}\big{(}A(f(u)),\xi\big{)}+D_{0}(u)\Delta
u^{\sharp},\end{split}$

where we used successively the paracontrolled expansion of $u$, the
commutation lemma 11 to get rid of the
$[\Delta,\overline{\mathsf{P}}_{u^{\prime}}]$ term and switch back and forth
between $\overline{\mathsf{P}}$ and $\mathsf{P}$, the commutation property 8
to merge the paraproducts, the fact that, as
$D_{0}(u)=A(f(u))-A(f(u_{0}^{T}))$ with $u_{0}^{T}$ smooth, so is
$A(f(u_{0}^{T}))$, and the
$\mathsf{\Pi}\big{(}A(f(u_{0}^{T})),\overline{\mathsf{P}}_{u^{\prime}}\xi\big{)}$
term has $C_{T}\mathcal{C}^{\alpha+\beta-2}$ regularity, and finally the
corrector estimate 7 for the remaining resonant term. The resonant product in
the last step can be expanded as follows using the paracontrolled expansion of
$u$ and paralinearization lemma 10

$\displaystyle{}\mathsf{\Pi}\big{(}A(f(u)),\xi\big{)}=_{2\alpha+\beta-2}u^{\prime}f^{\prime}(u)\mathsf{\Pi}\big{(}A(X),\xi\big{)}+f^{\prime}(u)\mathsf{\Pi}\big{(}A(u^{\sharp}),\xi\big{)}.$
(2.5)

Note that $-(u^{\prime})^{2}f^{\prime}(u)\mathsf{\Pi}\big{(}A(X),\xi\big{)}$
is the only term in this decomposition of $D_{0}(u)\Delta u$ featuring the
renormalized product $\mathsf{\Pi}\big{(}A(X),\xi\big{)}$. This gives

$D_{0}(u)\Delta u=-\mathsf{P}_{D_{0}(u)u^{\prime}}\xi-u^{\prime
2}f^{\prime}(u)\mathsf{\Pi}\big{(}A(X),\xi\big{)}+u^{\prime}f^{\prime}(u)\mathsf{\Pi}\big{(}A(u^{\sharp}),\xi\big{)}+D_{0}(u)\Delta
u^{\sharp}+\sharp_{A}$

and

$\displaystyle\sharp_{A}$
$\displaystyle=D_{0}(u)\,[\Delta,\overline{\mathsf{P}}_{u^{\prime}}](X)+\big{(}\mathsf{P}_{D_{0}(u)u^{\prime}}\xi-
D_{0}(u)\,\overline{\mathsf{P}}_{u^{\prime}}X\big{)}-\mathsf{P}_{\overline{\mathsf{P}}_{u^{\prime}}\xi}D_{0}(u)+\mathsf{\Pi}\big{(}\overline{\mathsf{P}}_{u^{\prime}}\xi,A(f(u_{0}^{T}))\big{)}$
$\displaystyle\quad-\big{\\{}\mathsf{\Pi}\big{(}\overline{\mathsf{P}}_{u^{\prime}}\xi,A(f(u))\big{)}-u^{\prime}\mathsf{\Pi}\big{(}\xi,A(f(u))\big{)}\big{\\}}-u^{\prime}\,\mathsf{C}\big{(}u^{\prime}f^{\prime}(u),A(X),\xi\big{)}+u^{\prime}\,\mathsf{C}\big{(}f^{\prime}(u),u^{\sharp},\xi\big{)}$
$\displaystyle\quad+u^{\prime}\,\mathsf{\Pi}\big{(}[A,\mathsf{P}_{u^{\prime}f^{\prime}(u)}](X),\xi\big{)}+u^{\prime}\,\mathsf{\Pi}\big{(}[A,\mathsf{P}_{f^{\prime}(u)}](u^{\sharp}),\xi\big{)}$
$\displaystyle\quad+u^{\prime}\,\mathsf{\Pi}\big{(}A\big{(}\mathsf{P}_{f^{\prime}(u)}\overline{\mathsf{P}}_{u^{\prime}}X-\mathsf{P}_{u^{\prime}f^{\prime}(u)}X\big{)},\xi\big{)}+u^{\prime}\,\mathsf{\Pi}\big{(}A\big{(}f(u)-\mathsf{P}_{f^{\prime}(u)}u\big{)},\xi\big{)}$

is an element of
$C\big{(}[0,T],\mathcal{C}^{2\alpha+\beta-2}(\mathbb{T}^{2})\big{)}$ that is
polynomial and continuous in $(u,u^{\prime})$ and continuous in $\xi$. Note
that although $(\Delta u^{\sharp})(t)$ is an element of
$\mathcal{C}^{2\beta-2}(\mathbb{T}^{2})$ at every positive time $t$ it cannot
be considered as a remainder term as its $\mathcal{C}^{2\beta-2}$-norm blows
up when $t>0$ goes to $0$. Using Lemma 14 we can then rewrite Equation (2.2)
as a system of coupled equation

$\begin{split}\partial_{t}u-A(f(u_{0}^{T}))\Delta
u&=\mathsf{P}_{B(g(u))-A(f(u))u^{\prime}}\xi+\big{\\{}A(f(u))-A(f(u_{0}^{T}))\big{\\}}\Delta
u^{\sharp}\\\
&\quad+g^{\prime}(u)\mathsf{\Pi}\big{(}B(u^{\sharp}),\xi\big{)}-u^{\prime}f^{\prime}(u)\,\mathsf{\Pi}\big{(}A(u^{\sharp}),\xi\big{)}+\sharp_{1}(u,u^{\prime})\end{split}$

and

$\begin{split}\partial_{t}u^{\sharp}-A(f(u_{0}^{T}))\Delta
u^{\sharp}&=\mathsf{P}_{B(g(u))-A(f(u))u^{\prime}}\xi+\big{\\{}A(f(u))-A(f(u_{0}^{T}))\big{\\}}\Delta
u^{\sharp}\\\
&\quad+g^{\prime}(u)\mathsf{\Pi}\big{(}B(u^{\sharp}),\xi\big{)}-u^{\prime}f^{\prime}(u)\,\mathsf{\Pi}\big{(}A(u^{\sharp}),\xi\big{)}+\sharp_{2}(u,u^{\prime})\end{split}$

where $\sharp_{1}(u,u^{\prime})$ and $\sharp_{2}(u,u^{\prime})$ are some
elements of
$C\big{(}[0,T],\mathcal{C}^{\alpha+\beta-2}(\mathbb{T}^{2})\big{)}$ that are
non-linear functions of $u$ that are multilinear in $(u^{\prime},u^{\sharp})$
and depend continuously on the enhanced noise data

$\big{(}\xi,\mathsf{\Pi}(A(X),\xi),\mathsf{\Pi}(B(X),\xi)\big{)}.$

We emphasized in Equations (2.4) and (2.5) the dependency on the renormalized
products $\mathsf{\Pi}(B(X),\xi)$ and $\mathsf{\Pi}(A(X),\xi)$ respectively.
This will be useful when we investigate the consistency of the renormalization
procedure from a dynamical point of view.

c) Fixed point formulation. We are now able to formulate equation (2.2) as a
fixed point problem. We define on $\mathbf{C}^{\beta}_{\alpha,T}(X)$ a map
$\Phi$ setting

$\Phi\big{(}(u,u^{\prime})\big{)}=(v,v^{\prime})$

where

$v^{\prime}:=\frac{B(g(u))-\big{\\{}A(f(u))-A(f(u_{0}^{T}))\big{\\}}u^{\prime}}{A(f(u_{0}^{T}))}$

and $v$ solves the equation

$\begin{split}\partial_{t}v-A(f(u_{0}^{T}))\Delta
v&=\mathsf{P}_{A(f(u_{0}^{T}))v^{\prime}}\xi+\big{\\{}A(f(u))-A(f(u_{0}^{T}))\big{\\}}\Delta
u^{\sharp}+g^{\prime}(u)\,\mathsf{\Pi}\big{(}B(u^{\sharp}),\xi\big{)}\\\
&\quad-u^{\prime}f^{\prime}(u)\,\mathsf{\Pi}\big{(}A(u^{\sharp}),\xi\big{)}+\sharp_{1}(u,u^{\prime})\end{split}$
(2.6)

with the initial value

$v(0)=u_{0}.$

Note that in view of Lemma 4 below, $(v,v^{\prime})$ is an element of
$\mathbf{C}^{\beta}_{\alpha,T}(X)$. Furthermore the equation on the remainder
$v^{\sharp}$ writes

$\begin{split}\partial_{t}v^{\sharp}-A(f(u_{0}^{T}))\Delta
v^{\sharp}&=\big{\\{}A(f(u))-A(f(u_{0}^{T}))\big{\\}}\Delta
u^{\sharp}+g^{\prime}(u)\,\mathsf{\Pi}\big{(}B(u^{\sharp}),\xi\big{)}\\\
&\quad-u^{\prime}f^{\prime}(u)\,\mathsf{\Pi}\big{(}A(u^{\sharp}),\xi\big{)}+\sharp_{2}(u,u^{\prime})\end{split}$
(2.7)

with initial value

$v^{\sharp}(0)=u_{0}-{\sf
P}_{v^{\prime}(0)}X=u_{0}-\overline{\mathsf{P}}_{v^{\prime}(0)}X.$

Denote by $\mathbf{\mathcal{B}}_{T}(r)$ the following ball of
$\mathbf{C}^{\beta}_{\alpha,T}(X)$

$\mathbf{\mathcal{B}}_{T}(r):=\left\\{(u,u^{\prime})\in\mathbf{C}^{\beta}_{\alpha,T}(X)\,;u(0)=u_{0}\,,u^{\prime}(0)=\frac{B(g(u_{0}))}{A(f(u_{0}))}\,,\|u\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}\leq
r\right\\}.$

We prove below that for a sufficiently large radius $r$ and sufficiently small
time horizon $T$ the map $\Phi$ is a contraction of
$\mathbf{\mathcal{B}}_{T}(r)$.

### 2.2 – Short time estimate.

The technical part of the argument is handled by the following Schauder-like
estimate found in Lemma 5 of [5].

######  Lemma 3 –

Pick $u_{0}\in\mathcal{C}^{\alpha}(\mathbb{T}^{2})$ and $w\in
C^{2}(\mathbb{T}^{2})$ bounded below by some positive constant. Let also
$\phi_{1},\phi_{2}$ be such that

$\phi_{1}\in
C\left((0,T],\mathcal{C}^{2\beta-2}(\mathbb{T}^{2})\right)\quad\text{and}\quad\ell_{1}(T):=\sup_{0<t\leq
T}t^{\frac{2\beta-\alpha}{2}}\|\phi_{1}(t)\|_{\mathcal{C}^{2\beta-2}}<+\infty,$

and

$\phi_{2}\in
C\left((0,T],\mathcal{C}^{\alpha+\beta-2}(\mathbb{T}^{2})\right)\quad\text{and}\quad\ell_{2}(T):=\sup_{0<t\leq
T}t^{\frac{2\beta-\alpha}{2}}\|\phi_{2}(t)\|_{\mathcal{C}^{\alpha+\beta-2}}<+\infty.$

Then for $T$ small enough the solution of the equation

$\partial_{t}u-w\Delta u=\phi_{1}+\phi_{2}$

with initial data $u_{0}$ satisfies the estimate

$\displaystyle\|u\|_{\mathscr{C}^{\alpha}}+\sup_{0<t\leq
T}t^{\frac{2\beta-\alpha}{2}}\|u(t)\|_{\mathcal{C}^{2\beta}}\lesssim\|u_{0}\|_{\mathcal{C}^{\alpha}}+\ell_{1}(T)+T^{\frac{\alpha-\beta}{2}}\ell_{2}(T),$

where the implicit constant depends only on $\|w\|_{\mathcal{C}^{\alpha}}$.

We obtain as a consequence of Lemma 3 the stability of the mapping $\Phi$ as a
function form the space of paracontrolled functions
$\mathbf{C}^{\beta}_{\alpha,T}(X)$ into itself.

######  Lemma 4 –

Let $u_{0},w,\phi_{1},\phi_{2}$ be as in Lemma 3 and let
$u^{\prime}\in\mathcal{C}^{\beta}$. Denote by $u$ the solution to

$\partial_{t}u-w\Delta u=\mathsf{P}_{wu^{\prime}}\xi+\phi_{1}+\phi_{2}$

starting from $u(0)=u_{0}$. Then
$(u,u^{\prime})\in\mathbf{C}^{\beta}_{\alpha,T}(X)$.

  * Proof –

Set $u^{\\#}:=u-\overline{\mathsf{P}}_{u^{\prime}}X$, it only remains to check
that $u^{\sharp}$ has the regularity matching the definition (2.3) to ensure
that $(u,u^{\prime})$ has indeed a paracontrolled structure. Note that in view
of Lemma 14 the term

$(\partial_{t}-w\Delta)\overline{\mathsf{P}}_{u^{\prime}}X-\mathsf{P}_{wv^{\prime}}(-\Delta
X)$

belongs to $C_{T}\mathcal{C}^{\alpha+\beta-2}$. As such $u^{\sharp}$ satisfies
the equation

$\displaystyle\partial_{t}u^{\sharp}-w\Delta u^{\sharp}$
$\displaystyle=\partial_{t}u-w\Delta
u-(\partial_{t}-w\Delta)\big{(}\overline{\mathsf{P}}_{u^{\prime}}X\big{)}$
$\displaystyle=\mathsf{P}_{wu^{\prime}}\xi+\phi_{1}+\phi_{2}-\big{(}(\partial_{t}-w\Delta)\overline{\mathsf{P}}_{u^{\prime}}X-\mathsf{P}_{wv^{\prime}}(\xi)\big{)}-\mathsf{P}_{wv^{\prime}}\xi$
$\displaystyle=_{\alpha+\beta-2}\phi_{1}+\phi_{2}$

starting from $u^{\sharp}(0)$. Lemma 3 then finishes the proof. $\rhd$

$\bullet$ First to control $v^{\prime}$ let
$(u_{1},u_{1}^{\prime}),(u_{2},u_{2}^{\prime})\in\mathbf{\mathcal{B}}_{T}(r)$,
we control the difference $v_{1}^{\prime}-v_{2}^{\prime}$ writing

$A(f(u_{0}^{T}))\big{(}v_{1}^{\prime}-v_{2}^{\prime}\big{)}=B(g(u_{1}))-B(g(u_{2}))-\big{\\{}A(f(u_{1}))-A(f(u_{0}^{T}))\big{\\}}u_{1}^{\prime}+\big{\\{}A(f(u_{2}))-A(f(u_{0}^{T}))\big{\\}}u_{2}^{\prime}.$

First use the fact that $B$ gains $-s$ derivatives and Lemma 9 to get

$\displaystyle\big{\|}B(g(u_{1}))-B(g(u_{2}))\big{\|}_{\mathscr{C}^{\beta}}$
$\displaystyle\lesssim$
$\displaystyle\big{\|}B(g(u_{1}))-B(g(u_{2}))\big{\|}_{\mathscr{C}^{\beta-s}}$
$\displaystyle\lesssim$
$\displaystyle\big{(}1+\|u_{1}\|_{\mathscr{C}^{\beta}}\big{)}\|u_{1}-u_{2}\|_{\mathscr{C}^{\beta}}$
$\displaystyle\lesssim$ $\displaystyle
T^{\frac{\alpha-\beta}{2}}\big{(}1+\|u_{1}\|_{\mathscr{C}^{\alpha}}\big{)}\|u_{1}-u_{2}\|_{\mathscr{C}^{\alpha}},$

where we used the assumption that $\beta-s<\alpha$. Then writing

$A_{i}:=A(f(u_{i}))-A(f(u_{0}^{T}))$

we have

$\displaystyle\big{\|}A_{1}u_{1}^{\prime}-A_{2}u_{2}^{\prime}\big{\|}_{\mathscr{C}^{\beta}}$
$\displaystyle\lesssim$
$\displaystyle\|A_{1}(u_{1}^{\prime}-u_{2}^{\prime})\|_{\mathscr{C}^{\beta}}+\|(A_{1}-A_{2})u_{2}^{\prime}\|_{\mathscr{C}^{\beta}}$
$\displaystyle\lesssim$
$\displaystyle\|A_{1}\|_{C_{T}L^{\infty}}\|u_{1}^{\prime}-u_{2}^{\prime}\|_{\mathscr{C}^{\beta}}+\|A_{1}\|_{\mathscr{C}^{\beta}}\|u_{1}^{\prime}-u_{2}^{\prime}\|_{C_{T}L^{\infty}}$
$\displaystyle\quad+\|A_{1}-A_{2}\|_{C_{T}L^{\infty}}\|u_{2}^{\prime}\|_{\mathscr{C}^{\beta}}+\|A_{1}-A_{2}\|_{\mathscr{C}^{\beta}}\|u_{2}^{\prime}\|_{C_{T}L^{\infty}}.$

The difference $A_{1}-A_{2}=A(f(u_{1}))-A(f(u_{2}))$ is dealt with in the same
way as $B$ above, with $A_{i}$ estimated per se as we cannot extract a
contracting factor from the difference
$\|u^{\prime}_{1}-u^{\prime}_{2}\|_{\mathscr{C}^{\beta}}$. We have

$A_{i}=\big{\\{}A(f(u_{i}))-A\big{(}f(u(0))\big{)}\big{\\}}+\big{\\{}A\big{(}f(u(0))\big{)}-A(f(u_{0}^{T}))\big{\\}},$

thus reasoning as for the $B$ term we can use the continuity properties of $A$
and the paralinearization Lemma 9 to obtain

$\big{\|}A(f(u_{i}))-A\big{(}f(u(0))\big{)}\big{\|}_{\mathscr{C}^{\beta}}\lesssim
T^{\frac{\alpha-\beta}{2}}\big{(}1+\|u_{i}\|_{\mathscr{C}^{\alpha}}\big{)}\|u_{i}-u(0)\|_{\mathscr{C}^{\alpha}}.$

In a similar way, using the classical estimate (A.2) on
$\text{Id}-e^{T\Delta}$ one gets

$\big{\|}A\big{(}f(u(0))\big{)}-A(f(u_{0}^{T}))\big{\|}_{\mathcal{C}^{\beta}}\lesssim
T^{\frac{\alpha-\beta}{2}}\big{(}1+\|u_{0}\|_{\mathcal{C}^{\alpha}}\big{)}\|u_{0}\|_{\mathcal{C}^{\alpha}}.$

In the end one has

$\big{\|}A_{1}u_{1}^{\prime}-A_{2}u_{2}^{\prime}\big{\|}_{\mathscr{C}^{\beta}}\leq
T^{\frac{\alpha-\beta}{2}}P\big{(}\|u_{i}\|_{\mathscr{C}^{\alpha}},\|u_{i}^{\prime}\|_{\mathscr{C}^{\beta}},\|u_{0}\|_{\mathcal{C}^{\alpha}}\big{)}\Big{(}\|u_{1}-u_{2}\|_{\mathscr{C}^{\alpha}}+\|u^{\prime}_{1}-u^{\prime}_{2}\|_{\mathscr{C}^{\beta}}\Big{)},$

for some constant
$P(\|u_{i}\|_{\mathscr{C}^{\alpha}},\|u_{i}^{\prime}\|_{\mathscr{C}^{\beta}},\|u_{0}\|_{\alpha})$
that grows at most polynomially as a function of its arguments.

$\bullet$ We now deal with the remainder term $v^{\sharp}$. Recall from (2.6)
and (2.7) the notations for the functions $\sharp_{1}$ and $\sharp_{2}$. Note
that $v_{1}^{\sharp}-v_{2}^{\sharp}$ is a solution of the equation

$\begin{split}\big{(}\partial_{t}-A(f(u_{0}^{T}))\Delta\big{)}\big{(}v_{1}^{\sharp}-v_{2}^{\sharp}\big{)}&=\big{\\{}A(f(u_{1}))-A(f(u_{0}^{T}))\big{\\}}\Delta
u_{1}^{\sharp}-\big{\\{}A(f(u_{2}))-A(f(u_{0}^{T}))\big{\\}}\Delta
u_{2}^{\sharp}\\\
&\quad+g^{\prime}(u_{1})\,\mathsf{\Pi}\big{(}B(u_{1}^{\sharp}),\xi\big{)}-g^{\prime}(u_{2})\,\mathsf{\Pi}\big{(}B(u_{2}^{\sharp}),\xi\big{)}\\\
&\quad+u_{2}^{\prime}f^{\prime}(u_{2})\,\mathsf{\Pi}\big{(}A(u_{2}^{\sharp}),\xi\big{)}-u_{1}^{\prime}f^{\prime}(u_{1})\,\mathsf{\Pi}\big{(}A(u_{1}^{\sharp}),\xi\big{)}\\\
&\quad+\sharp_{2}(u_{1},u_{1}^{\prime})-\sharp_{2}(u_{2},u_{2}^{\prime})\end{split}$

with null initial condition. Set

$\phi_{1}:=\big{\\{}A(f(u_{1}))-A(f(u_{0}^{T}))\big{\\}}\Delta
u_{1}^{\sharp}-\big{\\{}A(f(u_{2}))-A(f(u_{0}^{T}))\big{\\}}\Delta
u_{2}^{\sharp}$

and

$\begin{split}\phi_{2}&:=g^{\prime}(u_{1})\,\mathsf{\Pi}\big{(}B(u_{1}^{\sharp}),\xi\big{)}-g^{\prime}(u_{2})\,\mathsf{\Pi}\big{(}B(u_{2}^{\sharp}),\xi\big{)}\\\
&\quad+u_{2}^{\prime}f^{\prime}(u_{2})\,\mathsf{\Pi}\big{(}A(u_{2}^{\sharp}),\xi\big{)}-u_{1}^{\prime}f^{\prime}(u_{1})\,\mathsf{\Pi}\big{(}A(u_{1}^{\sharp}),\xi\big{)}+\sharp_{2}(u_{1},u_{1}^{\prime})-\sharp_{2}(u_{2},u_{2}^{\prime}).\end{split}$

We now check that $\phi_{1}$ and $\phi_{2}$ satisfy the regularity assumptions
of Lemma 3.

– Concerning $\phi_{1}(t)$, the estimate in
$\mathcal{C}^{2\beta-2}(\mathbb{T}^{2})$ is similar to what we did for
$v_{1}^{\prime}-v_{2}^{\prime}$ so one has

$\|\phi_{1}(t)\|_{\mathcal{C}^{2\beta-2}}\lesssim
T^{\frac{\alpha-\beta}{2}}Q(\|u_{i}\|_{\mathscr{C}^{\alpha}},\|u_{0}\|_{\mathcal{C}^{\alpha}})\left(\|u_{1}-u_{2}\|_{\mathscr{C}^{\alpha}}\|u_{1}^{\sharp}(t)\|_{\mathcal{C}^{2\beta-2}}+\|u_{1}^{\sharp}(t)-u_{2}^{\sharp}(t)\|_{\mathcal{C}^{2\beta-2}}\right)$

and therefore

$\sup_{0<t\leq
T}t^{\frac{2\beta-\alpha}{2}}\|\phi_{1}(t)\|_{\mathcal{C}^{2\beta-2}}\lesssim
T^{\frac{\alpha-\beta}{2}}Q\big{(}\|u_{i}\|_{\mathscr{C}^{\alpha}},\|u_{0}\|_{\mathcal{C}^{\alpha}}\big{)}\|u_{1}-u_{2}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}$

for yet another positive constant $Q(\cdots)$ with polynomial growth with
respect to its parameters.

– For $\phi_{2}$, as the function $\sharp_{2}$ is locally lipschitz in
$(u,u^{\prime})$ it will not cause any issue in the estimate and only the
first two terms need a special treatment as they involve a $u^{\sharp}$
factor. We only consider the $B$-term as they are both dealt with in the same
way. We need to estimate

$\left\|\big{(}g^{\prime}(u_{1})\,\mathsf{\Pi}\big{(}B(u_{1}^{\sharp}),\xi\big{)}-g^{\prime}(u_{2})\,\mathsf{\Pi}\big{(}B(u_{2}^{\sharp}),\xi\big{)}\big{)}(t)\right\|_{\mathcal{C}^{\alpha+\beta-2}}$

for a positive time $t$. An elementary splitting gives for this quantity the
upper bound

$\begin{split}\left\|\big{(}g^{\prime}(u_{1})-g^{\prime}(u_{2})\big{)}(t)\,\mathsf{\Pi}\big{(}B(u_{1}^{\sharp}),\xi\big{)}(t)\right\|_{\mathcal{C}^{\alpha+\beta-2}}&+\big{\|}\big{(}g^{\prime}(u_{1})-g^{\prime}(u_{2})\big{)}(t)\big{\|}_{\mathcal{C}^{\beta}}\|u_{1}^{\sharp}(t)\|_{\mathcal{C}^{2\beta}}\\\
&+\|g^{\prime}(u_{2})(t)\|_{\mathcal{C}^{\beta}}\big{\|}(u_{1}^{\sharp}-u_{2}^{\sharp})(t)\big{\|}_{\mathcal{C}^{2\beta}}.\end{split}$

Note that as $g$ is $C^{3}_{b}$ one has

$\|g^{\prime}(u_{2})(t)\|_{\mathcal{C}^{\beta}}\leq
C\big{(}\|u_{2}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}\big{)}$

and

$\big{\|}\big{(}g^{\prime}(u_{1})-g^{\prime}(u_{2})\big{)}(t)\big{\|}_{\mathcal{C}^{\beta}}\leq
C\big{(}\|u_{i}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}\big{)}\|u_{1}-u_{2}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}$

for yet again polynomial constants. One therefore has

$\sup_{0<t\leq
T}t^{\frac{2\beta-\alpha}{2}}\|\phi_{2}(t)\|_{\mathcal{C}^{\alpha+\beta-2}}\leq
C\big{(}\|u_{i}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}\big{)}\|u_{1}-u_{2}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}.$

Finally one can apply the modified Schauder estimate from Lemma 3 together
with the classical Schauder estimates to recover an estimate on $v_{1}-v_{2}$

$\begin{split}\|v_{1}-v_{2}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}&=\|v_{1}^{\prime}-v_{2}^{\prime}\|_{\mathscr{C}^{\beta}}+\big{\|}v_{1}^{\sharp}-v_{2}^{\sharp}\big{\|}_{\mathscr{C}^{\alpha}}+\sup_{0<t\leq
T}t^{\frac{2\beta-\alpha}{2}}\big{\|}u_{1}^{\sharp}(t)-u_{2}^{\sharp}(t)\big{\|}_{\mathcal{C}^{2\beta}}\\\
&\leq
C\Big{(}\|u_{1}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)},\|u_{2}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}\Big{)}\,T^{\frac{\alpha-\beta}{2}}\|u_{1}-u_{2}\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}\end{split}$
(2.8)

with the constant $C(\cdot)$ depending continuously on the enhanced noise and
polynomially on $u_{i}\in\mathbf{C}^{\beta}_{\alpha,T}(X)$. Here it is crucial
that in Lemma 3 the constant depends only on the $\mathcal{C}^{\alpha}$ norm
of $w$ as we want an estimate on $w=u_{0}^{T}$ that does not explode as $T>0$
goes to $0$. To conclude note that

$u:=\overline{\mathsf{P}}_{\frac{B(g(u_{0}))}{A(f(u_{0}))}}X+u_{0}^{T}$

defines an element of $\mathbf{C}^{\beta}_{\alpha,T}(X)$ so if
$\widetilde{u}\in\mathbf{\mathcal{B}}_{T}(r)$ and

$\max\Big{(}\|u\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)},\|\Phi(u)\|_{\mathbf{C}^{\beta}_{\alpha,T}(X)}\Big{)}<\frac{r}{2}$

then

$\big{\|}\Phi(\widetilde{u})\big{\|}_{\mathbf{C}^{\beta}_{\alpha,T}(X)}\leq\frac{r}{2}+2rC(r)T^{\frac{\alpha-\beta}{2}}\leq
r$

if $T$ is chosen small enough, depending on $r$. This means that for $r$ large
enough and $T$ small enough, both depending only on the size of the initial
value $u_{0}\in\mathcal{C}^{\alpha}(\textbf{{T}}^{2})$ and the problem data

$\Big{(}\xi,\mathsf{\Pi}\big{(}A(X),\xi\big{)},\mathsf{\Pi}\big{(}B(X),\xi\big{)},f,g\Big{)},$

the map $\Phi$ sends the ball $\mathbf{\mathcal{B}}_{T}(r)$ into itself and is
a contraction from (2.8). As such it has a unique fixed point in that ball and
the equation

$\partial_{t}u-A(f(u))\Delta u=B(g(u))\,\xi$

with initial condition $u(0)=u_{0}\in\mathcal{C}^{\alpha}(\mathbb{T}^{2})$ has
a unique paracontrolled solution on a small time interval $[0,T]$.

## 3 – Renormalization matters

We construct in Subsection 3.1 the enhanced noise; it was used as a data in
the analytic setting of Section 2. The renormalization process used for that
purpose comes with a dynamical interpretation of the solution to the fixed
point problem for $\Phi$. The renormalized equation is described in Subsection
3.2.

### 3.1 – Enhanced noise.

Following the expansions and given an arbitrary symbol $a$ of order $s\leq 0$
we need to make sense of the resonant product
$\mathsf{\Pi}\big{(}A(X),\xi\big{)}$ as an element of
$\mathcal{C}^{2\alpha-s-2}(\mathbb{T}^{2})$. This is done via a
renormalization procedure described here. Let
$\chi:[0,\infty)\rightarrow[0,1]$ be of class $C^{\infty}$, equal to $1$ on
$[0,1]$ and $0$ on $[2,\infty)$. Set for any distribution $u$ on
$\mathbb{T}^{2}$

$R^{\varepsilon}(u):=\mathcal{F}^{-1}\big{(}\chi(\varepsilon|\cdot|)\,\widehat{u}\,\big{)}$

and recall that we regularize the space white noise $\xi$ into a smooth
function using that smooth Fourier cut-off function

$\xi^{\varepsilon}:=R^{\varepsilon}(\xi).$

###### Theorem 5 –

For any $a\in\Sigma^{s}_{0}$ with associated operator $A$ there exists some
deterministic diverging function $c:\varepsilon\in(0,1]\to
C^{\infty}(\mathbb{T}^{2})$ such that

$\mathsf{\Pi}\big{(}A(X^{\varepsilon}),\xi^{\varepsilon}\big{)}-c^{\varepsilon}$

converges in
$L^{r}\big{(}\Omega;\mathcal{C}^{2\alpha-2-s}(\mathbb{T}^{2})\big{)}$ for any
$1\leq r<\infty$ to a limit $:\mathsf{\Pi}(AX,\xi):$ in
$\mathcal{C}^{2\alpha-2-s}(\mathbb{T}^{2})$.

  * Proof –

First note that

$\xi^{\varepsilon}\overset{\mathcal{C}^{\alpha-2}}{\longrightarrow}\xi\quad\text{and}\quad
X^{\varepsilon}\overset{\mathcal{C}^{\alpha}}{\longrightarrow}X$

almost surely and in $L^{r}(\Omega)$ for any $1\leq r<\infty$. This is due to
continuity of $(-\Delta)^{-1}$ for $X^{\varepsilon}$, and the
hypercontractivity and Besov embedding arguments used below can also be used
to prove the convergence of $\xi^{\varepsilon}$. Note that due to the
commutator estimate from Proposition 3.7 in [4] one has
$[A,R^{\varepsilon}](X)\in\mathcal{C}^{2\alpha-s}(\mathbb{T}^{2})$. This
brings us back to looking at

$\mathsf{\Pi}\big{(}(A(X))^{\varepsilon},\xi^{\varepsilon}\big{)}-c^{\varepsilon}$

which is what we look at below. Note that if $\widehat{a}=\widehat{a}^{x}$ is
the Fourier transform of $a$ with respect to $x$, then

$(AX)^{\varepsilon}=\sum_{n\in\mathbb{Z}^{2}}\chi(\varepsilon|n|)\widehat{AX}(n)e_{n}=\sum_{n,p\in\mathbb{Z}^{2}}\chi(\varepsilon|n|)\widehat{X}(n-p)\widehat{a}(p,n-p)e_{n}=:\sum_{p\in\mathbb{Z}^{2}}(AX)^{\varepsilon}_{p},$

where $e_{n}$ is the function $e_{n}(x)=e^{in\cdot x}$. We shall carry out the
renormalization procedure for each $(AX)^{\varepsilon}_{p}$ and then pass to
the limit by keeping careful track of the $p$-dependency.
Fix some $p\in\mathbb{N}$ and $\varepsilon>0$, the very definition of
$\mathsf{\Pi}$ yields

$\mathsf{\Pi}\big{(}(AX)^{\varepsilon}_{p},\xi^{\varepsilon}\big{)}=\sum_{|i-j|\leq
1}\sum_{k,l\in\mathbb{Z}^{2},l\neq
p}\rho_{i}(k)\,\rho_{j}(l)\,\chi(\varepsilon|k|)\,\chi(\varepsilon|l|)\,\frac{\widehat{a}(p,l-p)}{|l-p|^{2}}\,\widehat{\xi}(k)\,\widehat{\xi}(l-p)e_{k+l}.$

Set

$\displaystyle{}c_{p}(\varepsilon):=\mathbb{E}\left[\mathsf{\Pi}\big{(}(AX)^{\varepsilon}_{p},\xi^{\varepsilon}\big{)}\right]=\sum_{|i-j|\leq
1}\sum_{k\in\mathbb{Z}^{2}\setminus\\{0\\}}\rho_{i}(k)\,\rho_{j}(p-k)\,\chi(\varepsilon|k|)\,\chi(\varepsilon|p-k|)\,\frac{\widehat{a}(p,k)}{|k|^{2}}\,e_{p}$
(3.1)

and

$Y_{p}^{\varepsilon}:=(AX)_{p}^{\varepsilon}-c_{p}(\varepsilon).$

We prove that $Y_{p}^{\varepsilon}$ converges towards some $Y_{p}$ in
$\mathcal{C}^{2\alpha-2}(\mathbb{T}^{2})$, in $L^{r}(\Omega)$ for any $r\geq
1$. Let $q\geq-1$ and $\varepsilon,\eta>0$. Then

$\displaystyle\Delta_{q}(Y^{\varepsilon}_{p}-Y^{\eta}_{p})$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}|i-j|\leq 1\\\
k,l\in\mathbb{Z}^{2},l\neq
p\end{subarray}}\rho_{q}(k+l)\,\rho_{i}(k)\,\rho_{j}(l)\,\frac{\widehat{a}(p,l-p)}{|l-p|^{2}}\Big{(}\chi(\varepsilon|k|)\,\chi(\varepsilon|l|)-\chi(\eta|k|)\,\chi(\eta|l|)\Big{)}$
$\displaystyle\qquad\qquad\times\left(\widehat{\xi}(k)\,\widehat{\xi}(l)-\delta_{k,p-l}\right)e_{k+l}.$

When computing the expectation
$\mathbb{E}\big{[}|\Delta_{q}(Y^{\varepsilon}_{p}-Y^{\eta}_{p})|^{2}\big{]}$
we will come across a term in the form of

$\mathbb{E}\left[\left(\widehat{\xi}(k)\,\widehat{\xi}(l-p)-\delta_{k,p-l}\right)\left(\overline{\widehat{\xi}(k^{\prime})}\,\overline{\widehat{\xi}(l^{\prime}-p)}-\delta_{k^{\prime},p-l^{\prime}}\right)\right].$

This can be handled using the properties of the kernel of
$\widehat{\xi}(\cdot)$ and Isserlis’ formula as $\widehat{\xi}(k)$ are
gaussian

$\mathbb{E}\left[\left(\widehat{\xi}(k)\,\widehat{\xi}(l-p)-\delta_{k,p-l}\right)\left(\overline{\widehat{\xi}(k^{\prime})}\,\overline{\widehat{\xi}(l^{\prime}-p)}-\delta_{k^{\prime},p-l^{\prime}}\right)\right]=\delta_{k,k^{\prime}}\,\delta_{l,l^{\prime}}+\delta_{k,l^{\prime}-p}\,\delta_{l,k^{\prime}+p}$

hence we can keep only the terms where
$(k,l)\in\left\\{(k^{\prime},l^{\prime}),(l^{\prime}-p,k^{\prime}+p)\right\\}$.
This allows to get rid of the $e_{k+l-k^{\prime}-l^{\prime}}$ terms and get

$\displaystyle\mathbb{E}\left[\big{|}\Delta_{q}(Y^{\varepsilon}_{p}-Y^{\eta}_{p})\big{|}^{2}\right]=\sum_{\begin{subarray}{c}|i_{1}-j_{1}|\leq
1\\\ |i_{2}-j_{2}|\leq
1\end{subarray}}\sum_{\begin{subarray}{c}k,l\in\mathbb{Z}^{2},l\neq p\\\
k^{\prime},l^{\prime}\in\mathbb{Z}^{2},l^{\prime}\neq
p\end{subarray}}\rho_{q}(k+l)^{2}\,\rho_{i_{1}}(k)\,\rho_{i_{2}}(k^{\prime})\,\rho_{j_{1}}(l)\,\rho_{j_{2}}(l^{\prime})$
$\displaystyle\hskip
56.9055pt\times\Big{\\{}\chi(\varepsilon|k|)\,\chi(\varepsilon|l|)-\chi(\eta|k|)\,\chi(\eta|l|)\Big{\\}}\Big{\\{}\chi(\varepsilon|k^{\prime}|)\chi(\varepsilon|l^{\prime}|)-\chi(\eta|k^{\prime}|)\chi(\eta|l^{\prime}|)\Big{\\}}$
$\displaystyle\hskip
56.9055pt\times\frac{1}{|l-p|^{2}}\,\frac{1}{|l^{\prime}-p|^{2}}\,\widehat{a}(p,l-p)\,\overline{\widehat{a}(p,l^{\prime}-p)}\big{(}\delta_{k,k^{\prime}}\,\delta_{l,l^{\prime}}+\delta_{k,l^{\prime}-p}\,\delta_{l,k^{\prime}+p}\big{)}.$

Since in both sums $|i-j|\leq 1$ and $\text{supp}(\rho)$ is a fixed annulus,
we can actually keep only the terms where $|k|\sim|l|$ (resp.
$|k^{\prime}|\sim|l^{\prime}|$), that is $\theta|l|\leq|k|\leq\theta^{-1}|l|$
for some $\theta>0$ that does not depend on $i_{1},i_{2},j_{1},j_{2}$ and
bound the sum on $i_{1},i_{2},j_{1},j_{2}$ by a constant independent on
$k,l,k^{\prime},l^{\prime}$, leaving us with an upper bound for
$\mathbb{E}\left[\big{|}\Delta_{q}(Y_{p}^{\varepsilon}-Y_{p}^{\eta})(x)\big{|}^{2}\right]$
of the form

$\displaystyle\sum_{\begin{subarray}{c}k,l\in\mathbb{Z}^{2},l\neq p\\\
\theta|l|\leq|k|\leq\theta^{-1}|l|\end{subarray}}\rho_{q}(k+l)^{2}\,\frac{\Psi_{1}^{\varepsilon,\eta}(k,l)}{|l-p|^{4}}\,\big{|}\widehat{a}(p,l-p)\big{|}^{2}$
$\displaystyle\hskip
85.35826pt+\sum_{\begin{subarray}{c}k,l\in\mathbb{Z}^{2},k\neq 0,l\neq p\\\
\theta|l|\leq|k|\leq\theta^{-1}|l|\end{subarray}}\rho_{q}(k+l)^{2}\,\frac{\Psi_{2}^{\varepsilon,\eta}(k,l,p)}{|k|^{2}|l-p|^{2}}\,\big{|}\widehat{a}(p,l-p)\widehat{a}(p,k)\big{|}$

where

$\Psi_{1}^{\varepsilon,\eta}(k,l)=\big{|}\chi(\varepsilon|k|)\,\chi(\varepsilon|l|)-\chi(\eta|k|)\,\chi(\eta|l|)\big{|}^{2}$

and

$\Psi_{2}^{\varepsilon,\eta}(k,l,p)=\Big{|}\chi(\varepsilon|k|)\,\chi(\varepsilon|l|)-\chi(\eta|k|)\,\chi(\eta|l|)\Big{|}\,\Big{|}\chi(\varepsilon|l+p|)\,\chi(\varepsilon|k-p|)-\chi(\eta|l+p|)\,\chi(\eta|k-p|)\Big{|}.$

Apply Estimate (1.4) for $k=l-p$, we have

$\left|\frac{\widehat{a}(p,l-p)}{|l-p|^{2}}\right|\lesssim\langle
p\rangle^{-4}\,\langle l\rangle^{s-2}\,\mathbf{1}_{|p|\leq\mu(1+|l-p|)}$

so that, up to the $\langle p\rangle^{-4}$ factor, $p$ only shows up in
$\Psi_{2}\mathbf{1}_{|p|\leq\mu(1+|l-p|)}$. Note that thanks to the size
constraint on $|p|$ and $\chi$ being smooth and bounded, we can easily derive
a bound for $\Psi_{1}$ and $\Psi_{2}$ (uniformly in $p$) as

$|\Psi_{1}^{\varepsilon,\eta}(k,l)|\vee|\Psi_{2}^{\varepsilon,\eta}(k,l,p)|\lesssim_{\gamma}|\varepsilon-\eta|^{\gamma}\big{(}|k|^{\gamma}+|l|^{\gamma}\big{)}$

for any $\gamma\in[0,2]$. Fix $\delta>0$ to be a small parameter and write
$n=k+l$. As $|k|\leq\theta^{-1}|l|$, $|n|\leq(1+\theta^{-1})|l|$, this yields

$\displaystyle\mathbb{E}\left[\big{|}\Delta_{q}(Y_{p}^{\varepsilon}-Y_{p}^{\eta})(x)\big{|}^{2}\right]$
$\displaystyle\lesssim$
$\displaystyle\frac{|\varepsilon-\eta|^{\gamma}}{\langle
p\rangle^{8}}\sum_{n\in\mathbb{Z}^{2}}\rho_{q}(n)^{2}\sum_{\begin{subarray}{c}k+l=n\\\
\theta|l|\leq|k|\leq\theta^{-1}|l|\end{subarray}}\frac{|k|^{\gamma}+|l|^{\gamma}}{\langle
l\rangle^{4-2s}}$ $\displaystyle\lesssim$
$\displaystyle\frac{|\varepsilon-\eta|^{\gamma}}{\langle
p\rangle^{8}}\sum_{n\in\mathbb{Z}^{2}}\frac{\rho_{q}(n)^{2}}{\langle
n\rangle^{2-\delta-2s}}\sum_{\begin{subarray}{c}k+l=n\\\
\theta|l|\leq|k|\leq\theta^{-1}|l|\end{subarray}}\frac{1}{\langle
l\rangle^{2+\delta-\gamma}}$ $\displaystyle\lesssim$
$\displaystyle\frac{|\varepsilon-\eta|^{\gamma}}{\langle
p\rangle^{8}}\sum_{n\in\mathbb{Z}^{2}}\frac{\rho_{q}(n)^{2}}{\langle
n\rangle^{2-\delta-2s}}$

where the last inequality is valid as soon as $\delta>\gamma$, say
$\gamma=\delta/2$. Then it is only a matter of counting how many
$\mathbb{Z}^{2}$ points belong to a $2^{q}$-sized annulus to finally obtain
the bound

$\mathbb{E}\left[\big{|}\Delta_{q}(Y_{p}^{\varepsilon}-Y_{p}^{\eta})(x)\big{|}^{2}\right]\lesssim
2^{q(\delta+2s)}|\varepsilon-\eta|^{\delta/2}\,\langle p\rangle^{-8}$

for any $\delta>0$. Using Gaussian hypercontractivity, this yields

$\mathbb{E}\left[\big{|}\Delta_{q}(Y_{p}^{\varepsilon}-Y_{p}^{\eta})(x)\big{|}^{r}\right]\lesssim
2^{q\frac{r}{2}(\delta+2s)}|\varepsilon-\eta|^{r\delta/4}\,\langle
p\rangle^{-4r}$

so that multiplying both sides by $2^{qr(2\alpha-2-s+2/r)}$ and summing over
$q\geq-1$, we recover the $B^{2\alpha-2+2/r}_{r,r}(\mathbb{T}^{2})$ norm

$\mathbb{E}\Big{[}\big{\|}Y_{p}^{\varepsilon}-Y_{p}^{\eta}\big{\|}_{B^{2\alpha-2-s+2/r}_{r,r}}^{r}\Big{]}\lesssim\langle
p\rangle^{-4r}\left(\sum_{q\geq-1}2^{qr(2\alpha-2+\delta/2+2/r)}\right)|\varepsilon-\eta|^{r\delta/4}$

where the series converges if $r>\frac{2}{1-\alpha}$ and $\delta<2-2\alpha$
for instance. Combined with the continuous embedding of
$B^{2\alpha-2-s+2/r}_{r,r}(\mathbb{T}^{2})$ into
$\mathcal{C}^{2\alpha-2-s}(\mathbb{T}^{2})$ this yields the estimate

$\displaystyle{}\mathbb{E}\left[\left\|Y_{p}^{\varepsilon}-Y_{p}^{\eta}\right\|_{\mathcal{C}^{2\alpha-2-s}}^{r}\right]\lesssim\langle
p\rangle^{-4r}\,|\varepsilon-\eta|^{r\frac{\delta}{4}}$ (3.2)

and proves that $(Y_{p}^{\varepsilon})$ is a Cauchy, therefore convergent,
sequence in $L^{r}(\Omega;\mathcal{C}^{2\alpha-2-s}(\mathbb{T}^{2}))$ for any
$r$ large enough – hence for any $r\geq 1$ from hypercontractivity. This
proves the existence of the renormalized product
$:\mathsf{\Pi}((AX)_{p},\xi):$ for each $p\in\mathbb{Z}^{2}$. Note that we
kept explicit track of $p$. The dominated convergence theorem ensures that the
renormalization tranfers to the whole series $\sum_{p}Y^{\varepsilon}_{p}$.
This gives the $L^{r}(\Omega)$ convergence in
$\mathcal{C}^{2\alpha-2-s}(\mathbb{T}^{2})$ of the renormalized product for
all $1\leq r<\infty$. $\rhd$

Remark – Note that applying the Kolmogorov continuity criterion on the
estimate (3.2) for $r\delta>2$ yields the existence of a modification of the
process $\varepsilon\mapsto Y_{p}^{\varepsilon}$ that is a continuous
function. The convergence of the renormalized product then also happens almost
surely if we allow for a change of the probability space.

### 3.2 – Renormalized equation.

As for the convergence part of the claim, fix $u_{0}\in\mathcal{C}^{\alpha}$,
note that the fixed point argument we run holds locally uniformly in the
enhanced noise data

$\Xi:=\Big{(}\xi,\mathsf{\Pi}\big{(}A(X),\xi\big{)},\mathsf{\Pi}\big{(}B(X),\xi\big{)}\Big{)},$

hence so does the fixed point solution we obtain. We shall denote by

$\Xi\mapsto\mathscr{S}(\Xi)$

the solution mapping corresponding to the paracontrolled equation starting
from $u_{0}$; it is a continuous function of the enhanced noise data $\Xi$. If
$\zeta$ is a smooth noise, $\zeta$ has a natural enhancement $Z$. Write
$\widetilde{\mathscr{S}}:\zeta\mapsto v$ for the solution map corresponding to
the locally in time well-posed quasilinear equation

$\partial_{t}v-A(f(v))\Delta u=B(g(v))\zeta$

with initial condition $u_{0}$. The map $\mathscr{S}$ extends
$\widetilde{\mathscr{S}}$ in so far as

$\mathscr{S}(Z)=\widetilde{\mathscr{S}}(\zeta).$

Denote by

$\zeta\in
C^{\infty}([0,T]\times\mathbb{T}^{2})\mapsto\mathscr{S}^{\varepsilon}(\zeta)$

the solution map corresponding to the locally in time well-posed equation

$\partial_{t}u^{\varepsilon}-A\big{(}f(u)\big{)}\Delta
u=B\big{(}g(u)\big{)}\,\zeta+c_{a}(\varepsilon)\bigg{(}\frac{B\big{(}g(u)\big{)}}{A\big{(}f(u)\big{)}}\bigg{)}^{2}f^{\prime}(u)-c_{b}(\varepsilon)\frac{B\big{(}g(u)\big{)}}{A\big{(}f(u)\big{)}}\,g^{\prime}(u)$

with initial condition $u_{0}$. Then we have the identity

$\mathscr{S}^{\varepsilon}(\xi^{\varepsilon})=\mathscr{S}\Big{(}\xi^{\varepsilon},\mathsf{\Pi}\big{(}A(X^{\varepsilon}),\xi^{\varepsilon}\big{)}-c_{a}(\varepsilon),\mathsf{\Pi}\big{(}B(X^{\varepsilon}),\xi^{\varepsilon}\big{)}-c_{b}(\varepsilon)\Big{)}.$

Indeed, in view of Equations (2.4) and (2.5), replacing the formal resonant
products $\mathsf{\Pi}(B(X),\xi)$ and $\mathsf{\Pi}(A(X),\xi)$ therein by
their approximations
$\mathsf{\Pi}\big{(}B(X^{\varepsilon}),\xi^{\varepsilon}\big{)}-c_{b}(\varepsilon)$
and
$\mathsf{\Pi}\big{(}A(X^{\varepsilon}),\xi^{\varepsilon}\big{)}-c_{a}(\varepsilon)$,
respectively, and then regrouping the (well-defined) products together we
obtain that Equation (2.2) driven by the noise data

$\Big{(}\xi^{\varepsilon},\mathsf{\Pi}\big{(}A(X^{\varepsilon}),\xi^{\varepsilon}\big{)}-c_{a}(\varepsilon),\mathsf{\Pi}\big{(}B(X^{\varepsilon}),\xi^{\varepsilon}\big{)}-c_{b}(\varepsilon)\Big{)}$

writes

$\partial_{t}u-A\big{(}f(u)\big{)}\Delta
u=B\big{(}g(u)\big{)}\,\xi^{\varepsilon}+c_{a}(\varepsilon)\bigg{(}\frac{B\big{(}g(u)\big{)}}{A\big{(}f(u)\big{)}}\bigg{)}^{2}f^{\prime}(u)-c_{b}(\varepsilon)\frac{B\big{(}g(u)\big{)}}{A\big{(}f(u)\big{)}}\,g^{\prime}(u).$

Note that since $\mathscr{S}$ depends locally uniformly on the noise data,
there exists a random time $T$ that does not depend on $\varepsilon$ for which
the equation above starting from $u_{0}$ is well-posed in
$\mathscr{C}^{\alpha}_{T}$. Together with the continuity of $\mathscr{S}$ and
Theorem 5, this completes the proof of the convergence.

Remark – If we ask further that the action of $A$ and $B$ is diagonal – if for
instance $A$ and $B$ are convolution operators, then a strong consequence of
Theorem 5 is that the renormalization function is a constant. It would be
interesting to see whether the renormalization function can be constant in
some other cases.

##  Appendix A Estimates from paracontrolled calculus

We gather in this section the estimates used in the exposition above. Let
$\chi$ and $\rho$ be smooth compactly supported radial functions on
$\mathbb{R}^{d}$. Assume $\chi$ is supported in a ball and $\rho$ is supported
in an annulus. Set $\rho_{j}(\cdot):=\rho(2^{-j}\cdot)$ for $j\geq 0$ and
$\rho_{-1}:=\chi$. One can find $\chi$ and $\rho$ such that

$\sum_{j\geq-1}\rho_{j}\equiv 1\quad\text{and}\quad\rho_{j}\rho_{i}\equiv 0\
\text{for}\ |i-j|\geq 0$

The $i$-th Littlewood-Paley projector is defined as
$\Delta_{j}f:=\mathcal{F}^{-1}(\rho_{j}\widehat{f}\,)$. In those terms one has
the formal decomposition

$\displaystyle uv$ $\displaystyle=$
$\displaystyle\sum_{i<j-1}(\Delta_{i}u)(\Delta_{j}v)+\sum_{|i-j|\leq
1}(\Delta_{i}u)(\Delta_{j}v)+\sum_{i>j+1}(\Delta_{i}u)(\Delta_{j}v)$
$\displaystyle=:$
$\displaystyle\mathsf{P}_{u}v+\mathsf{\Pi}(u,v)+\mathsf{P}_{v}u$

The operator $\mathsf{P}$ is called paraproduct operator and the operator
$\mathsf{\Pi}$ the resonant operator. One can think of $\mathsf{P}_{u}v$ as a
function or distribution that globally looks like $v$, but with lower
frequencies modulated by $u$. This decomposition allows for a precise track of
regularity with the following continuity result.

######  Proposition 6 –

One has

$\|\mathsf{P}_{u}v\|_{B^{\alpha\wedge
0+\beta}_{p,q}}\lesssim\|u\|_{B^{\alpha\wedge
0}_{p_{1},q}}\|v\|_{B^{\beta}_{p_{2},q}}$

for any $\alpha,\beta\in\mathbb{R}$ and $1\leq p_{1},p_{2},p,q\leq\infty$ such
that $\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}$, and

$\|\mathsf{\Pi}(u,v)\|_{B^{\alpha+\beta}_{p,q}}\lesssim\|u\|_{B^{\alpha}_{p_{1},q}}\|v\|_{B^{\beta}_{p_{2},q}}$

for any $\alpha,\beta\in\mathbb{R}$ such that $\alpha+\beta>0$.

In view of the regularity condition $\alpha+\beta>0$, it is clear that the
ill-defined part of a product comes from its resonant term. This can however
be circumvented for functions having themselves a paracontrolled expansion.
This is the purpose of the following operator, first introduced in [9], see
Lemma 2.4 therein. Define the operator

$\mathsf{C}:(u,v,w)\mapsto\mathsf{\Pi}(\mathsf{P}_{u}v,w)-u\,\mathsf{\Pi}(v,w)$

on the set of smooth functions on $\mathbb{T}^{2}$.

######  Proposition 7 –

Pick some regularity exponents $\alpha\in(0,1)$, $\beta,\gamma\in\mathbb{R}$
such that $\beta+\gamma<0$ and $\alpha+\beta+\gamma>0$. The operator
$\mathsf{C}$ has a unique extension into a bounded trilinear operator from
$\mathcal{C}^{\alpha}(\mathbb{T}^{2})\times\mathcal{C}^{\beta}(\mathbb{T}^{2})\times\mathcal{C}^{\gamma}(\mathbb{T}^{2})$
to $\mathcal{C}^{\alpha+\beta+\gamma}(\mathbb{T}^{2})$.

With these results at hand, we could already almost tackle linear singular
SPDEs such as the Parabolic Anderson Model (PAM)

$\partial_{t}u-\Delta u=u\xi$

on the 2-dimensional torus $\mathbb{T}^{2}$. The idea is then to work in a
random subspace of functions having a paracontrolled expansion

${}u=\mathsf{P}_{u^{\prime}}X+u^{\sharp}$ (A.1)

where $X$ depends only on the noise $\xi$ and the remainder $u^{\sharp}$ has
better regularity than $u$ itself. We then run a fixed point argument on the
pair $(u^{\prime},u^{\sharp})$ rather than $u$. Note that the corrector
$\mathsf{C}$ is defined to get around singular products, but other commutators
and correctors might be needed to investigate well-defined products that may
not be written in a good form for the analysis. A thorough study of these
commutators in a very general framework can be found in the works of Bailleul
& Bernicot [2] and [3]. We only retain the following result on the merging
operator that can be found in Theorem 3.6 of [2].

######  Proposition 8 –

For $0<\beta\leq\alpha<1$ and
$h_{1}\in\mathcal{C}^{\alpha}(\mathbb{T}^{2}),h_{2}\in
C^{\beta}(\mathbb{T}^{2})$ and $h_{3}\in\mathcal{C}^{\alpha}(\mathbb{T}^{2})$
one has

$\big{\|}\mathsf{P}_{h_{1}}\big{(}\mathsf{P}_{h_{2}}h_{3}\big{)}-\mathsf{P}_{h_{1}h_{2}}h_{3}\big{\|}_{\mathcal{C}^{\alpha+\beta}}\lesssim\|h_{1}\|_{\mathcal{C}^{\alpha}}\|h_{2}\|_{\mathcal{C}^{\beta}}\|h_{3}\|_{\mathcal{C}^{\alpha}}.$

As for the non-linear case, we need to further investigate the action of non-
linear functions on paraproducts, this is done via Bony’s paralinearization
lemma. For $f\in C^{2}_{b}(\mathbb{R})$ and
$u\in\mathcal{C}^{\alpha}(\mathbb{T}^{2})$ with $\alpha>0$ set

$R_{f}(u):=f\circ u-\mathsf{P}_{f^{\prime}\circ u}u.$

then we have the paralinearization estimate – see for instance Theorem 5.2.5
in [13] or Theorem 2.92 in [1].

######  Proposition 9 –

For $f\in C^{2}_{b}(\mathbb{R})$ and
$u\in\mathcal{C}^{\alpha}(\mathbb{T}^{2})$ with $\alpha>0$ one has

$\left\|R_{f}(u)\right\|_{\mathcal{C}^{2\alpha}}\lesssim\|f\|_{C^{2}_{b}}(1+\|u\|_{\mathcal{C}^{\alpha}}^{2})$

Moreover we have a local-Lipschitz estimate

$\big{\|}f\circ u-f\circ
v\big{\|}_{\mathcal{C}^{\alpha}}\leq\|f\|_{C^{2}_{b}}\big{(}1+\|u\|_{\mathcal{C}^{\alpha}}\big{)}\|u-v\|_{\mathcal{C}^{\alpha}}$

This can be further improved as in Lemma C.1. of [9] to the special case of
functions having an expansion with a smoother remainder term as in (A.1).

######  Proposition 10 –

Pick $0<\beta\leq\alpha<1$ such that $\alpha+\beta>1$. For
$u\in\mathcal{C}^{\alpha}$ and $v\in\mathcal{C}^{\alpha+\beta}$, for $f\in
C^{3}_{b}(\mathbb{R})$, one has

$\Big{\|}f(u+v)-\mathsf{P}_{f^{\prime}(u+v)}(u+v)\Big{\|}_{\mathcal{C}^{\alpha+\beta}}\lesssim\|f\|_{C^{3}_{b}}\big{(}1+\|u\|^{1+\beta/\alpha}_{\mathcal{C}^{\alpha}}+\|v\|^{2}_{L^{\infty}}\big{)}(1+\|v\|_{\mathcal{C}^{\alpha+\beta}})$

To take into account the time dependency we now define Gubinelli, Imkeller &
Perkowski’s modified paraproduct $\overline{\mathsf{P}}$ on parabolic
functions as in [9]. Fix some smooth function
$\varphi:\mathbb{R}\to\mathbb{R}_{+}$ with compact support and integral $1$.
For $f\in C_{T}C^{\infty}$ and $i\geq-1$ define for $0\leq t\leq T$

$Q_{i}(f)(t):=\int_{\mathbb{R}}2^{2i}\varphi\left(2^{2i}(t-s)\right)f\big{(}(s\wedge
T)\vee 0\big{)}ds$

and

$\overline{\mathsf{P}}_{f}g:=\sum_{i<j-1}\Delta_{i}\big{(}Q_{j}(f)\big{)}(\Delta_{j}g).$

The operator $\overline{\mathsf{P}}$ satisfies the following uniform in time
continuity estimates

$\left\|\left(\overline{\mathsf{P}}_{f}g\right)(t)\right\|_{\mathcal{C}^{\alpha}}\lesssim\|f\|_{C_{T}L^{\infty}}\|g(t)\|_{\mathcal{C}^{\alpha}}.$

Denote by $\mathcal{L}$ the heat operator $\partial_{t}-\Delta$ and write
$\|\cdot\|_{C_{T}\mathcal{C}^{\beta}}$ for the natural norm on
$C\big{(}[0,T],\mathcal{C}^{\beta}(\mathbb{T}^{2})\big{)}$. Similarly we write
$\|\cdot\|_{C^{\alpha/2}_{T}L^{\infty}}$ for the natural norm on
$C^{\alpha/2}\big{(}[0,T],L^{\infty}(\mathbb{T}^{2})\big{)}$, then the
modified paraproduct $\overline{\mathsf{P}}$ satisfies the following
commutation estimates – see Lemma 5.1 in [9].

######  Proposition 11 –

Pick two regularity exponents $\alpha\in(0,1),\beta\in\mathbb{R}$ and a finite
positive time horizon $T$. For $u\in\mathscr{C}^{\alpha}_{T}$ and $v\in
C\big{(}[0,T],\mathcal{C}^{\beta}(\mathbb{T}^{2})\big{)}$ one has

  1. (a)

$\;\left\|\mathcal{L}\left(\overline{\mathsf{P}}_{u}v\right)-\overline{\mathsf{P}}_{u}(\mathcal{L}(v))\right\|_{C_{T}\mathcal{C}^{\alpha+\beta-2}}\lesssim\|u\|_{\mathscr{C}^{\alpha}_{T}}\|v\|_{C_{T}\mathcal{C}^{\beta}}$,

  2. (b)

$\;\left\|\overline{\mathsf{P}}_{u}v-\mathsf{P}_{u}v\right\|_{C_{T}\mathcal{C}^{\alpha+\beta}}\lesssim\|u\|_{C_{T}^{\alpha/2}L^{\infty}}\|v\|_{C_{T}\mathcal{C}^{\beta}}.$

The Schauder estimates for the inverse heat operator $\mathcal{L}^{-1}$ take
the following form – see e.g. Lemma 2.5 in [16] and references therin.

######  Proposition 12 –

The following statements holds true.

  1. (a)

For $t>0$, $\alpha\in\mathbb{R}$ and $\delta\geq 0$ we have

$\left\|e^{t\Delta}(u)\right\|_{\mathcal{C}^{\alpha+\delta}}\lesssim
t^{-\delta/2}\|u\|_{\mathcal{C}^{\alpha}}\quad\text{and}\quad\left\|e^{t\Delta}(u)\right\|_{\mathcal{C}^{\delta}}\lesssim
t^{-\delta/2}\|u\|_{L^{\infty}}.$

  2. (b)

For $\alpha\in(0,1)$, $u\in\mathcal{C}^{\alpha}$, then for any $t\geq 0$ one
has

$\left\|e^{t\Delta}(u)-u\right\|_{L^{\infty}}\lesssim
t^{\alpha/2}\|u\|_{\mathcal{C}^{\alpha}}.$ (A.2)

  3. (c)

Let $u\in C\big{(}[0,T],\mathcal{C}^{\alpha}(\mathbb{T}^{2})\big{)}$ for some
$\alpha\in\mathbb{R}$ and $T>0$, then for any $\gamma\in[0,1)$ and $t\in(0,T]$
we have

$t^{\gamma}\left\|\left(\mathcal{L}^{-1}(u)\right)(t)\right\|_{\mathcal{C}^{\alpha+2}}\lesssim\sup_{s\in[0,t]}s^{\gamma}\|u(s)\|_{\mathcal{C}^{\alpha}}.$

Moreover if $\alpha\in(-2,0)$ then one has

$\left\|\mathcal{L}^{-1}(u)\right\|_{C_{T}^{(\alpha+2)/2}L^{\infty}}\lesssim\|u\|_{C_{T}\mathcal{C}^{\alpha}}.$

  4. (d)

For $\alpha\in(0,2)$ we have the $T$-uniform estimate

$\left\|\mathcal{L}^{-1}(u)\right\|_{\mathscr{C}^{\alpha}_{T}}\lesssim(1+T)\|u\|_{C_{T}\mathcal{C}^{\alpha-2}}.$

  5. (e)

For $\alpha\in(0,2)$, uniformly in $\varepsilon\in(0,1)$ and $T$, we have

$\left\|\mathcal{L}^{-1}(u)\right\|_{\mathscr{C}^{\alpha}_{T}}\lesssim
T^{\varepsilon}\|u\|_{C_{T}\mathcal{C}^{\alpha-2+2\varepsilon}}.$

  6. (f)

For $\delta\in[0,1)$, $\alpha\in\mathbb{R}$ and $\beta\in[-\alpha,2-\alpha)$
one has

$\sup_{0\leq s\leq
t}s^{\delta}\big{\|}\mathcal{L}^{-1}(w)(s)\big{\|}_{\mathcal{C}^{\alpha+\beta}}\lesssim
T^{\frac{\alpha-\beta}{2}}\sup_{0\leq s\leq
t}s^{\delta}\|w(s)\|_{\mathcal{C}^{2\alpha-2}}.$

The following elementary statement is useful to get a small factor and produce
contractive maps.

######  Proposition 13 –

For $0<\beta<\alpha<2$ and $u\in\mathscr{C}^{\alpha}_{T}$ that is null at time
$0$ one has

$\|u\|_{\mathscr{C}^{\beta}_{T}}\lesssim
T^{\frac{\alpha-\beta}{2}}\|u\|_{\mathscr{C}^{\alpha}_{T}}.$

Finally the following commutation lemma is useful whenever we deal with a
weighted heat operator. See Lemma 2 in [5] for a proof.

######  Lemma 14 –

For $v\in\mathscr{C}^{\beta}_{T}$ and
$w\in\mathcal{C}^{2\beta}(\mathbb{T}^{2})$ one has

$\left\|(\partial_{t}-w\Delta)\left(\overline{\mathsf{P}}_{v}X\right)-\mathsf{P}_{wv}\big{(}-\Delta
X)\right\|_{C_{T}\mathcal{C}^{\alpha+\beta-2}}\lesssim\big{(}1+\|w\|_{\mathcal{C}^{2\beta}}\big{)}\|v\|_{\mathscr{C}^{\beta}_{T}}\|X\|_{\mathcal{C}^{\alpha}}.$

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$\bullet$ I. Bailleul – Univ Brest, CNRS, LMBA - UMR 6205, F- 29238 Brest,
France.
E-mail<EMAIL_ADDRESS>
$\bullet$ H. Eulry – Univ Rennes, CNRS, IRMAR - UMR 6625, F- 35000 Rennes,
France.
E-mail<EMAIL_ADDRESS>