# A first order description of a nonlinear SPDE in the spirit of rough paths

Florian Kunick  Max-Planck-Institut für Mathematik in den Naturwissenschaften
<EMAIL_ADDRESS>
###### Abstract.

We consider a nonlinear stochastic partial differential equation (SPDE) in
divergence form where the forcing term is a Gaussian noise, that is white in
time and colored in space such that the gradient of the solution is Hölder-
continuous, but not differentiable. Then, we prove a generalized Taylor
expansion of the difference between the solution to the SPDE and the solution
to its linearization around a fixed basepoint. The result is reminiscent of
the theory of (controlled) rough paths and agrees with the general
observation, that, in settings with a rough driver, subtracting the solution
to the linearized equation yields a more regular object.

###### Key words and phrases:

partial differential equations, stochastic partial differential equations,
rough paths theory, regularity theory

###### 2020 Mathematics Subject Classification:

60H15, 60L20, 35K10

##  1\. Introduction

Let $A:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be given. We consider the following
stochastic partial differential equation (SPDE)

(1.1) $\displaystyle\begin{cases}\partial_{t}u=\nabla\cdot A(\nabla u)+\xi\\\
u|_{t\leq 0}=0\end{cases}$

on ${\mathbb{R}}\times{\mathbb{R}}^{d}$. The forcing term $\xi$ is a space-
time Gaussian noise, which is white in time, and periodic, colored and
stationary in space. More precisely, we consider a Gaussian process $\xi$,
formally defined via its covariance

(1.2)
$\displaystyle\mathbb{E}\left[\xi(t,x)\xi(t^{\prime},x^{\prime})\right]=\delta(t-t^{\prime})K(x-x^{\prime}),$

where $K:{\mathbb{R}}^{d}\to{\mathbb{R}}$ is periodic (cf. Section 3), and
that for convenience is localized in the time interval $\left[0,1\right]$. The
spatial covariance function $K$ is chosen in such a way that the solution $v$
to the linearized equation, i.e. the stochastic heat equation (SHE)

(1.3) $\displaystyle\begin{cases}\partial_{t}v=\Delta v+\xi\\\ v|_{t\leq
0}=0,\end{cases}$

is not only a (continuous) function, but differentiable, and we have

(1.4) $\displaystyle\left[\nabla v\right]_{\alpha}<\infty\ \mathrm{a.s.}$

for some $\alpha\in\left(\frac{1}{2},1\right)$111This is in contrast to the
SHE with space-time white noise forcing, where already in dimension 2 the
solution is not a function anymore. For the precise notation we refer to
Section 2. Since $v$ is a linear functional of the Gaussian field $\xi$ – and
thus is itself Gaussian – more is true. Indeed, in [6, Lemma 3] Gaussian
moments for $v$ were established, meaning that there exists a $C_{0}>0$ such
that

(1.5)
$\displaystyle\mathbb{E}\left[\mathrm{exp}\left(\frac{1}{C_{0}}\left[\nabla
v\right]^{2}_{\alpha}\right)\right]<\infty.$

Moreover, we assume that the non-linearity $A$ is elliptic (cf. (2.1)). Under
these assumptions, it is expected that we should also have

(1.6) $\displaystyle\left[\nabla u\right]_{\alpha}<\infty\ \mathrm{a.s.}$

and in fact [6, Corollary 1] yields the (a priori) estimate

(1.7) $\displaystyle\left[\nabla u\right]_{\alpha}\lesssim\left[\nabla
v\right]_{\alpha}^{\frac{\alpha}{\alpha_{0}}}+\left[\nabla v\right]_{\alpha}$

for an $\alpha_{0}\in\left(0,1\right)$ as in [6, Lemma 1]222coming from the
DeGiorgi–Nash theorem. Here, we note that the implicit constant in $\lesssim$
only depends and will depend on the dimension, $\alpha$ and the ellipticity of
$A$ (cf. Section 2).

In particular, we want to stress that for spatially colored noise, (1.1) is
_not_ a singular SPDE. In fact, the SPDE (1.1) with space-time white noise is,
in general, not subcritical in any space-dimension, in the sense that by
zooming in on small scales, the nonlinear terms blow up. For a reference
concerning singular (semi-linear) SPDEs and the notion of subcriticality we
refer to [3].

We should mention, that there are no probabilistic arguments in this paper and
if $\xi$ is a distribution of suitable regularity such that (1.4) holds, the
same result can be obtained. Nevertheless, we have chosen to consider the case
where $\xi$ is a (Gaussian) noise, on the one hand because we are heavily
building on the work [6]333where the analysis is also purely deterministic
except for (1.5) and on the other hand because a typical example of a
distribution with a negative Hölder-regularity is a realization of some
Gaussian field with a short correlation length. Thus, everything in this
article can be seen as a pathwise analysis, which is in the spirit of the
theory of regularity structures (cf. [3]), where there is a clear distinction
between the deterministic and probabilistic steps. Usually, the probabilistic
arguments involve the construction of singular products along with the
corresponding renormalization. In our setting no renormalization is necessary
and hence it appears to be natural that the arguments are purely
deterministic.

In [6, p.70, Theorem 1], it is shown that, under suitable assumptions on $A$ –
that for convenience we recall in Section 2 – a (unique) solution to (1.1)
exists such that (1.7) and thus (1.6) hold. Due to the rough setting, the
authors introduce a spatial increment operator $\delta_{y}$ (cf. (2.7)) with
which they linearize equation (1.1)444as opposed to taking the derivative of
the equation due to the low regularity of the noise. Then it is convenient
(and a guiding principle) to subtract the increment $\delta_{y}v$ (where $v$
solves (3.1)) in order to get rid of the noise $\xi$. Hence
$\delta_{y}\left(u-v\right)$ satisfies a variable, but linear coefficient
equation and the celebrated DeGiorgi–Nash theorem555a localized version
thereof, to be precise yields an a priori estimate for its Hölder-norm for
some $\alpha_{0}\in\left(0,1\right)$. Postprocessing this estimate in turn
establishes (1.6) for $\alpha=\alpha_{0}$. In the next step, this estimate is
upgraded using standard $C^{1+\alpha}$-Schauder theory as well as the
stochastic estimate (1.5), thus yielding (1.6) for any $\alpha\in(0,1)$. In
general, (1.6) is sharp; in case of Brownian motion it is well-known that its
paths have Hölder–regularity of at most $\frac{1}{2}$ but not better. We want
to address the following question: Is there a way to give a finer description
of the regularity of $u$?

In this work we intend to give such a regularity statement in the following
way. It is often the case that the difference of the solution to a non-linear
equation with a rough driving signal and the solution to the linearized
equation is more regular. A prominent example that has been treated in the
recent years is the $\phi^{4}_{2}$-model (cf. [1]). Indeed, its solution
$\varphi$ and the solution $f$ to the stochastic heat equation in dimension
$2$ are distributions but their difference $\varphi-f$ is smoother, in
particular a function. This has been exploited in [1] in order to construct a
solution which is local in time. In fact, this is also a guiding principle for
the theory of (controlled) rough paths (cf. [2]) and ultimately in the theory
of regularity structures (cf. [3]), where this concept has been vastly
generalized.

Our main result is that $\nabla u$ is _modelled_ after $\nabla v$ which
essentially means that a certain _a priori_ -estimate holds (cf. (1.16)). In
the context of singular SPDEs such a modelling assumption is used in the
following way. First, one constructs a singular product on the level of the
linear, but irregular model, i.e. $\nabla v$ in this case. In a second step,
one constructs the singular product on the level of the solution $\nabla u$
using the singular product on the level of the model and the assumption that
$\nabla u$ is modelled after $\nabla v$. Due to the nonlinear nature of the
problem it is not sufficient to just consider the solution $v$ to the SHE, but
for all space-time points $z=(t^{\prime},x^{\prime})$ we consider the solution
to an anisotropic SHE.

Let $a(t^{\prime},x^{\prime}):=DA(\nabla
u(t^{\prime},x^{\prime}))\in{\mathbb{R}}^{d\times d}$. Then we write
$v_{a(t^{\prime},x^{\prime})}$ for the solution to the anisotropic stochastic
heat equation, i.e. $v_{a(t^{\prime},x^{\prime})}$ solves

(1.8)
$\displaystyle\begin{cases}\partial_{t}v_{a(t^{\prime},x^{\prime})}=\nabla\cdot
a(t^{\prime},x^{\prime})\nabla v_{a(t^{\prime},x^{\prime})}+\xi\\\
v_{a(t^{\prime},x^{\prime})}|_{t\leq 0}=0\end{cases}$

with $\xi$ as in (1.3). By the $C^{1+\alpha}$-Schauder theory developed in [6]
we get the following uniform666in the basepoint estimate.

######  Lemma 1.1.

There exists a solution $v_{a(t^{\prime},x^{\prime})}$ to (1.8) where
$a(t^{\prime},x^{\prime})=DA(\nabla u(t^{\prime},x^{\prime}))$ and $A$
satisfies (2.1), (2.2) as well as (2.3), and we have

(1.9)
$\displaystyle\sup_{\left(t^{\prime},x^{\prime}\right)\in{\mathbb{R}}\times{\mathbb{R}}^{d}}\left[\nabla
v_{a(t^{\prime},x^{\prime})}\right]_{\alpha}\lesssim\left[\nabla
v\right]_{\alpha}^{\frac{\alpha}{\alpha_{0}}}+\left[\nabla v\right]_{\alpha}.$

Then we can state our main result.

######  Theorem 1.2.

Let $u$ be a solution to the equation

(1.10) $\displaystyle\begin{cases}\partial_{t}u=\nabla\cdot A(\nabla u)+\xi\\\
u|_{t\leq 0}=0\end{cases}$

on ${\mathbb{R}}\times{\mathbb{R}}^{d}$ where $A$ satisfies (2.1), (2.2) and
(2.3), and where $\xi$ is a Gaussian noise that is white in time, periodic,
stationary and colored in space such that the corresponding solution $v$ to
the stochastic heat equation (1.3) satisfies $[\nabla v]_{\alpha}<\infty$ for
some $\alpha\in(\frac{1}{2},1)$. Moreover, let
$a(t^{\prime},x^{\prime})=DA(\nabla
u(t^{\prime},x^{\prime}))\in{\mathbb{R}}^{d\times d}$ and let
$v_{a(t^{\prime},x^{\prime})}$ be a solution to (1.8). Then there exists a
family of symmetric matrices
$(B(t^{\prime},x^{\prime}))_{(t^{\prime},x^{\prime})}$ such that for all
$x,x^{\prime}\in\mathbb{R}^{d}$ and $t,t^{\prime}\in{\mathbb{R}}$ it holds
a.s. that

(1.11) $\displaystyle\left\lvert\nabla u(t,x)-\nabla
u(t^{\prime},x^{\prime})-\left(\nabla v_{a(t^{\prime},x^{\prime})}(t,x)-\nabla
v_{a(t^{\prime},x^{\prime})}(t^{\prime},x^{\prime})\right)-B(t^{\prime},x^{\prime})(x-x^{\prime})\right\rvert$
(1.12) $\displaystyle\lesssim d^{2\alpha}((t,x),(t^{\prime},x^{\prime}))$

and $\lesssim$ denotes $\leq C$ where $C$ depends only on $\left[\nabla
v\right]_{\alpha}$, $\lambda$ as well as $\Lambda$, $\alpha$ and $d$.

One way to think of (1.11) is as a generalized Taylor expansion. Indeed,
rewriting (1.11) slightly as

(1.13) $\displaystyle\left\lvert(\nabla u(t,x)-\nabla
v_{a(t^{\prime},x^{\prime})}(t,x))-(\nabla u(t^{\prime},x^{\prime})-\nabla
v_{a(t^{\prime},x^{\prime})}(t^{\prime},x^{\prime}))-B(t^{\prime},x^{\prime})(x-x^{\prime})\right\rvert$
(1.14) $\displaystyle\lesssim d^{2\alpha}((t,x),(t^{\prime},x^{\prime})).$

the symmetric matrix $B(t^{\prime},x^{\prime})$ plays the role of the Hessian
of the function
$u(t^{\prime},\cdot)-v_{a(t^{\prime},x^{\prime})}(t^{\prime},\cdot)$ at the
basepoint $(t^{\prime},x^{\prime})$. Since $2\alpha>1$ it is natural, that an
affine correction appears. Moreover, since $2\alpha>1$, the matrix $B$ can be
seen to be unique and does not depend on $(t,x)$, or more precisely it does
not depend on $d\left((t,x),(t^{\prime},x^{\prime})\right)$. Note that (1.13)
is essentially the modelledness condition777on the level of the solution and
not the gradient for $\sigma\equiv 1$ of [7, Definition 3.1, p.880], which is
an extension of controlled rough paths (cf. [2]). In [7], they construct
singular products on the level of the stochastic heat equation using
probabilistic arguments. They then use the modelledness condition [7,
Definition 3.1, p.880] to lift these singular products to the non-linear
setting. Since (1.1) is not singular, the regularity theory in [6] does not
rely on such a modelledness. Nevertheless 1.2 states that such a modelledness
holds true. Of course, (1.11) also implies that $\nabla
u(t^{\prime},\cdot)-\nabla v_{a(t^{\prime},x^{\prime})}(t^{\prime},\cdot)$ is
differentiable in $x^{\prime}$ as well as the improved Hölder-regularity in
time $\left\lvert\nabla u(t,x^{\prime})-\nabla
v_{a(t^{\prime},x^{\prime})}(t,x^{\prime})-\left(\nabla
u(t^{\prime},x^{\prime})-\nabla
v_{a(t^{\prime},x^{\prime})}(t^{\prime},x^{\prime})\right)\right\rvert\lesssim\left\lvert
t-t^{\prime}\right\rvert^{\alpha}$ at $z=\left(t^{\prime},x^{\prime}\right)$.

From now on, we set
$w_{a(t^{\prime},x^{\prime})}:=u-v_{a(t^{\prime},x^{\prime})}$. Then we define
the _modelling constant_

(1.15) $\displaystyle
M:=\sup_{z=(t^{\prime},x^{\prime})}\inf_{B}\sup_{r>0}r^{-2\alpha}\left\lVert\nabla
w_{a(t^{\prime},x^{\prime})}-B_{x^{\prime}}\right\rVert_{P_{r}(z)},$

where the infimum ranges over all affine functions
$B_{x^{\prime}}(x):=B(x-x^{\prime})+b$ with $B\in{\mathbb{R}}^{d\times d}$
being symmetric and $b\in{\mathbb{R}}^{d}$, and $P_{r}(z)$ is the parabolic
cylinder of radius $r$ defined in Section 2. In order to prove 1.2 we show
that we have

(1.16) $\displaystyle M\leq C(d,\lambda,\Lambda,\alpha,\left[\nabla
v\right]_{\alpha}).$

Then it is straightforward to see that the optimal choice is $b=\nabla
w_{a(t^{\prime},x^{\prime})}(t^{\prime},x^{\prime})$ and hence (1.16) yields
1.2. Note that the definition of $M$ is essentially the same as the definition
of the modelling constant in [7, Definition 3.1]. Moreover, we want to make
the following connection. Apart from the dependence of
$w_{a(t^{\prime},x^{\prime})}$ on the basepoint $z=(t^{\prime},x^{\prime})$,
the semi-norm $M$ bears close resemblance to the Hölder-norm defined in [4,
Section 3.3] where Hölder-regularity is measured in terms of how well a
function is approximated by polynomials.

##  2\. Notation and assumptions

Our notation and assumptions are the same as in [6, Section 2]. For the
convenience of the reader we will recall them here. The non-linearity
$A:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ is assumed to be continuously
differentiable and uniformly elliptic in the sense that there exists
$\lambda>0$ such that

(2.1) $\displaystyle\eta\cdot DA(x)\eta\geq\lambda|\eta|^{2}\ \text{for all}\
x,\eta\in\mathbb{R}^{d}$

and we have

(2.2) $\displaystyle|DA(x)\eta|\leq|\eta|\ \text{for all}\
x,\eta\in\mathbb{R}^{d}$

where by $DA$ we denote the Jacobian of $A$. Moreover, we assume that there
exists $\Lambda>0$ such that

(2.3) $\displaystyle|DA(x)-DA(y)|\leq\Lambda|x-y|\ \text{for all}\
x,y\in{\mathbb{R}}^{d}.$

For $\alpha\in(0,1)$ the semi-norm $\left[\cdot\right]_{\alpha}$ is defined as

(2.4) $\displaystyle[f]_{\alpha}:=\sup_{z\neq
z^{\prime}\in{\mathbb{R}}\times{\mathbb{R}}^{d}}\frac{|f(z)-f(z^{\prime})|}{d^{\alpha}(z,z^{\prime})}<\infty$

for all space-time functions
$f:{\mathbb{R}}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ or vector fields
$f:{\mathbb{R}}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ where

(2.5) $\displaystyle
d((t,x),(t^{\prime},x^{\prime})):=|t-t^{\prime}|^{\frac{1}{2}}+|x-x^{\prime}|$

denotes the _Carnot–Caratheodory_ metric and by abuse of notation
$\lvert\cdot\rvert$ refers either to the absolute value or the Euclidean norm
on ${\mathbb{R}}^{d}$. Naturally, the space $C^{\alpha}$ denotes all functions
$f$ such that $\left[f\right]_{\alpha}<\infty$. For $r>0$ and
$z=(t^{\prime},x^{\prime})$, by

(2.6) $\displaystyle P_{r}(z):=(t^{\prime}-r^{2},t^{\prime})\times
B_{r}(x^{\prime})$

we denote the parabolic cylinder centered around $z$ with radius $r$. Then
$\left\lVert\cdot\right\rVert_{P_{r}(z)}$ is the supremum norm on $P_{r}(z)$.
We will also frequently write $[f]_{\alpha,P_{r}(z)}$ which is the same semi-
norm as in (2.4) restricted to $P_{r}(z)$.

For $y\in\mathbb{R}^{d}$ we define the spatial increment operator $\delta_{y}$
as

(2.7) $\displaystyle\delta_{y}f(t,x):=f(t,x+y)-f(t,x)$

where $f$ is either a scalar or a vector field. Then, by the mean value
theorem, we can linearize our non-linearity according to

(2.8) $\displaystyle\delta_{y}A(\nabla u)=a_{y}\nabla\delta_{y}u$

where

(2.9) $\displaystyle a_{y}(t,x)=\int_{0}^{1}DA(\theta\nabla
u(t,x+y)+(1-\theta)\nabla
u(t,x))\mathop{}\\!\mathup{d}\theta\in{\mathbb{R}}^{d\times d}.$

The assumptions on the Jacobian $DA$ translate to estimates on $a_{y}$ as
follows. We have

(2.10) $\displaystyle\eta\cdot a_{y}(t,x)\eta\geq\lambda|\eta|^{2}\ \text{for
all}\ x,\eta\in\mathbb{R}^{d},t\in{\mathbb{R}}$

as well as

(2.11) $\displaystyle|a_{y}(t,x)\eta|\leq|\eta|\ \text{for all}\
x,\eta\in\mathbb{R}^{d},t\in{\mathbb{R}}$

and

(2.12) $\displaystyle[a_{y}]_{\alpha}\leq[\nabla u]_{\alpha}.$

Let $\psi$ be a smooth, positive and radially symmetric mollifier that
satisfies $\operatorname{supp}\psi\subset B_{1}(0)$ as well as
$\int_{{\mathbb{R}}^{d}}\psi\mathop{}\\!\mathup{d}x=1$. The radial symmetry
assumption also ensures that first moments vanish, i.e. for all $i=1,\dots,d$
it holds $\int_{{\mathbb{R}}^{d}}\psi(x)x_{i}\mathop{}\\!\mathup{d}x=0$. Then
we write $\psi_{r}(x):=\frac{1}{r^{d}}\psi\left(\frac{x}{r}\right)$ and we
define for any function $f$

(2.13) $\displaystyle f_{r}:=f_{r,0}:=f*\psi_{r}$

as well as for $i=1,\dots,d$

(2.14) $\displaystyle f_{r,i}:=f*\left(\partial_{i}\psi\right)_{r}.$

For vector fields888such as a gradient this notation is to be understood
entrywise. Moreover, we will always write $B_{x^{\prime}}$ for the affine
function

(2.15) $\displaystyle B_{x^{\prime}}(x):=B(x-x^{\prime})+b$

where $B\in{\mathbb{R}}^{d\times d}$ and $b\in{\mathbb{R}}^{d}$.

##  3\. The stochastic heat equation

In the work [6], they consider a stationary, spatially periodic Gaussian noise
$\xi$ on ${\mathbb{R}}\times\mathbb{R}^{d}$, localized in the time interval
$\left[0,1\right]$, that is white in time and colored in space, and such that
the corresponding stochastic heat equation

(3.1) $\displaystyle\begin{cases}\partial_{t}v=\Delta v+\xi\\\ v|_{t\leq
0}=0\end{cases}$

satisfies

(3.2) $\displaystyle\left[\nabla v\right]_{\alpha}<\infty\ \mathrm{a.s.}$

for some $\alpha\in(\frac{1}{2},1)$. More specifically, they consider a
Gaussian field $\xi=\left(\xi(f)\right)_{f}$ with $1-$periodic, positive
definite covariance function $K:{\mathbb{R}}^{d}\to{\mathbb{R}}$ such that for
smooth test-functions $f,g$

(3.3)
$\displaystyle\mathbb{E}[\xi(f)\xi(g)]=\int_{0}^{1}\int_{\left[0,1\right]^{d}}\int_{\left[0,1\right]^{d}}f(t,x)K(x-y)g(t,y)\mathop{}\\!\mathup{d}x\mathop{}\\!\mathup{d}y\mathop{}\\!\mathup{d}t$

and $\xi(f)$ is normally distributed with mean zero. Such a Gaussian field is
easily seen to exist by Kolmogorov’s consistency theorem. The fact that $K$
only depends on one variable yields stationarity of $\xi$. Periodicity of $K$
translates to periodicity of $\xi$. Positive definiteness of $K$ implies that
its Fourier transform $\hat{K}$ is real-valued and non-negative and requiring
that $K$ is symmetric, i.e. $K(-x)=K(x)$, yields $\hat{K}(-k)=\hat{K}(k)$.
Most importantly, requiring that the Fourier transform of $K$ satisfies

(3.4) $\displaystyle\hat{K}(k)\lesssim(1+|k|^{2})^{-\frac{s}{2}}$

for all $k\in(2\pi\mathbb{Z})^{d}$, where $s=2\alpha+d\in(d,d+2)$ finally
implies (3.2) (cf. [6, Lemma 3]).

Now we give the proof for Lemma 1.1.

###### Proof of Lemma 1.1.

For any $(t^{\prime},x^{\prime})\in{\mathbb{R}}\times{\mathbb{R}}^{d}$ we
apply Theorem 1 of [6] to $A$ given by

(3.5) $\displaystyle A(\nabla u)=a(t^{\prime},x^{\prime})\nabla u$

where $u=v_{a(t^{\prime},x^{\prime})}$999In this case $A$ is non-
deterministic, but since the analysis is pathwise and all the constants depend
on $A$ only through the ellipticity constants $\lambda,\Lambda$, which are
deterministic, the proofs are unaffected.. Thus we get a unique solution
$v_{a(t^{\prime},x^{\prime})}$ that satisfies $\left[\nabla
v_{a(t^{\prime},x^{\prime})}\right]_{\alpha}<\infty$ a.s.. Then we can apply
Corollary 1 of [6] which yields uniformly in $(t^{\prime},x^{\prime})$ the
estimate

(3.6) $\displaystyle\left[\nabla
v_{a(t^{\prime},x^{\prime})}\right]_{\alpha}\leq
C(d,\lambda,\Lambda,\alpha)(\left[\nabla v\right]_{\alpha}+\left[\nabla
v\right]_{\alpha}^{\frac{\alpha}{\alpha_{0}}})$

and hence the conclusion. ∎

##  4\. Deterministic estimates

From now on we fix a space-time point $z=(t^{\prime},x^{\prime})$. We set
$w_{a(t^{\prime},x^{\prime})}:=u-v_{a(t^{\prime},x^{\prime})}$, where we
recall that $a(t^{\prime},x^{\prime})=DA(\nabla u(t^{\prime},x^{\prime}))$.
For notational convenience we will drop the subscripts referring to
$a(t^{\prime},x^{\prime})$.

In the first step, for simplicity, we focus on the spatial part of the
modelling constant. The proof is elementary and essentially an extension of
the proof of Lemma 1 in [6].

######  Proposition 4.1.

Let $x^{\prime}\in{\mathbb{R}}^{d}$. For $f:\mathbb{R}^{d}\to{\mathbb{R}}$
such that $\nabla f\in C^{\alpha}$101010only in space in this case, we have

(4.1) $\displaystyle\sup_{r>0}\frac{1}{r^{2\alpha}}\inf_{B}\left\lVert\nabla
f-B_{x^{\prime}}\right\rVert_{B_{r}(x^{\prime})}$
$\displaystyle\lesssim\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq
l}\inf_{k\in{\mathbb{R}}^{d}}\left\lVert\nabla\delta_{y}f-k\right\rVert_{B_{l}(x^{\prime})}=:N$

where $B_{x^{\prime}}(x)=B(x-x^{\prime})+b$, $B\in{\mathbb{R}}^{d\times d}$
symmetric, $b\in{\mathbb{R}}^{d}$.

###### Proof.

First of all, we assume that $f$ is smooth. We denote by
$\left\\{e_{i}\right\\}_{i=1,\dots,d}$ the standard orthonormal basis of
${\mathbb{R}}^{d}$. Let $k=k(y,l)$ be the (near) optimal constant for $N$. Fix
$i,j=1,\dots,d$ and $l>0$. Note that for all $y_{1},y_{2}$ we have the
identity
$\delta_{y_{1}+y_{2}}\partial_{i}f=\delta_{y_{2}}\partial_{i}f(\cdot+y_{1})+\delta_{y_{1}}\partial_{i}f$.

Then we estimate

(4.2) $\displaystyle\left\lvert
k_{i}(2le_{j},2l)-2k_{i}(le_{j},2l)\right\rvert$ $\displaystyle\leq\left\lVert
k_{i}(2le_{j},2l)-\delta_{2le_{j}}\partial_{i}f\right\rVert_{B_{2l}(x^{\prime})}+\left\lVert
k_{i}(le_{j},2l)-\delta_{le_{j}}\partial_{i}f\right\rVert_{B_{2l}(x^{\prime})}$
(4.3) $\displaystyle+\left\lVert
k_{i}(le_{j},2l)-\delta_{le_{j}}\partial_{i}f\right\rVert_{B_{l}(x^{\prime})}$
(4.4) $\displaystyle\leq 3(2l)^{2\alpha}N$

and similarly

(4.5) $\displaystyle\left\lvert k_{i}(le_{j},2l)-k_{i}(le_{j},l)\right\rvert$
$\displaystyle\leq\left\lVert
k_{i}(le_{j},2l)-\delta_{le_{j}}\partial_{i}f\right\rVert_{B_{2l}(x^{\prime})}+\left\lVert
k_{i}(le_{j},l)-\delta_{le_{j}}\partial_{i}f\right\rVert_{B_{l}(x^{\prime})}$
(4.6) $\displaystyle\leq 2(2l)^{2\alpha}N.$

Combining these two estimates yields via the triangle inequality

(4.7) $\displaystyle\left\lvert
k_{i}(2le_{j},2l)-2k_{i}(le_{j},l)\right\rvert\lesssim Nl^{2\alpha}.$

Hence for any $l>0$ and $n\in{\mathbb{N}}$ we get

(4.8)
$\displaystyle\left\lvert\frac{k_{i}(\frac{l}{2^{n}}e_{j},\frac{l}{2^{n}})}{\frac{l}{2^{n}}}-\frac{k_{i}(\frac{l}{2^{n+1}}e_{j},\frac{l}{2^{n+1}})}{\frac{l}{2^{n+1}}}\right\rvert\lesssim
Nl^{2\alpha-1}\left(2^{-n}\right)^{2\alpha-1}.$

Using our assumption that $2\alpha-1>0$ we see that the corresponding sequence
is Cauchy and thus there exists $a_{ij}(l)\in{\mathbb{R}}$ such that

(4.9)
$\displaystyle\frac{k_{i}(\frac{l}{2^{n}}e_{j},\frac{l}{2^{n}})}{\frac{l}{2^{n}}}\to
a_{ij}(l)\ \mathrm{for}\ n\to\infty$

and, by dyadic summation111111again using the fact that $2\alpha-1>0$, this
yields

(4.10)
$\displaystyle\left\lvert\frac{k_{i}(le_{j},l)}{l}-a_{ij}(l)\right\rvert\lesssim
Nl^{2\alpha-1}.$

Note that $a_{ij}(l)$ is constant on dyadics, i.e.

(4.11) $\displaystyle a_{ij}(l)=a_{ij}(2^{-m}l)$

for all $m\in{\mathbb{N}}$. Feeding the estimate (4.10) into $N$ we
consequently have

(4.12)
$\displaystyle\left\lVert\frac{1}{l}\delta_{le_{j}}\partial_{i}f-a_{ij}(l)\right\rVert_{B_{l}(x^{\prime})}\lesssim
Nl^{2\alpha-1}$

and, since $f$ is smooth, we infer that

(4.13) $\displaystyle a_{ij}(l)\to\partial_{ij}f(x^{\prime})\ \mathrm{for}\
l\to 0.$

Hence we conclude by (4.11) that $a_{ij}$ is constant and we have
$a_{ij}=\partial_{ij}f(x^{\prime})$. Moreover, the estimate

(4.14)
$\displaystyle\left\lVert\frac{1}{l}\delta_{le_{j}}\partial_{i}f-\partial_{ij}f(x^{\prime})\right\rVert_{B_{l}(x^{\prime})}\lesssim
Nl^{2\alpha-1}$

holds. Now let $x\in B_{r}(x^{\prime})$ and we set $y:=x-x^{\prime}$. Then we
estimate componentwise using (4.14)

(4.15)
$\displaystyle\left\lvert\partial_{i}f(x)-\partial_{i}f(x^{\prime})-\nabla\partial_{i}f(x^{\prime})\cdot\left(x-x^{\prime}\right)\right\rvert$
(4.16)
$\displaystyle\leq\sum_{k=1}^{d}\left\lvert\delta_{y_{k}e_{k}}\partial_{i}f(x^{\prime}_{1},\dots,x^{\prime}_{k+1}+y_{k+1},\dots,x^{\prime}_{d}+y_{d})-\partial_{ki}f(x^{\prime})y_{k}\right\rvert$
(4.17) $\displaystyle\lesssim N\sum_{k=1}^{d}\left\lvert
y_{k}\right\rvert^{2\alpha}\lesssim Nr^{2\alpha}.$

Now we drop the assumption that $f$ is smooth. To this end, let
$\varepsilon>0$. Let $k$ be the (near) optimal constant for $f$ in $N$. Then
we have for all $x\in B_{l}(x^{\prime})$ and $\left\lvert y\right\rvert\leq l$
by the triangle inequality

(4.18)
$\displaystyle\left\lvert\delta_{y}f_{\varepsilon}(x)-k\right\rvert\leq\left\lVert\delta_{y}f-k\right\rVert_{B_{2l}(x^{\prime})}$

and hence

(4.19) $\displaystyle\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq
l}\inf_{k^{\varepsilon}}\left\lVert\delta_{y}f_{\varepsilon}-k^{\varepsilon}\right\rVert_{B_{l}(x^{\prime})}$
$\displaystyle\leq\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq
l}\left\lVert\delta_{y}f_{\varepsilon}-k\right\rVert_{B_{l}(x^{\prime})}$
(4.20) $\displaystyle\leq\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq l}\left\lVert\delta_{y}f-k\right\rVert_{B_{l}(x^{\prime})}.$

Note that, since $\partial_{i}f$ is Hölder-continuous,
$\partial_{i}f_{\varepsilon}$ converges uniformly to $\partial_{i}f$ and thus,
for fixed $r$, it holds that $\inf_{B^{\varepsilon}}\left\lVert\nabla
f_{\varepsilon}-B^{\varepsilon}_{x^{\prime}}\right\rVert_{B_{r}(x^{\prime})}$
converges to $\inf_{B}\left\lVert\nabla
f-B_{x^{\prime}}\right\rVert_{B_{r}(x^{\prime})}$. Dividing by $r^{2\alpha}$
and taking the supremum over $r$ yields lower semicontinuity of the seminorm

(4.21) $\displaystyle\sup_{r>0}\frac{1}{r^{2\alpha}}\inf_{B}\left\lVert\nabla
f-B_{x^{\prime}}\right\rVert_{B_{r}(x^{\prime})}$
$\displaystyle\leq\liminf_{\varepsilon\to
0}\sup_{r>0}\frac{1}{r^{2\alpha}}\inf_{B^{\varepsilon}}\left\lVert\nabla
f_{\varepsilon}-B^{\varepsilon}_{x^{\prime}}\right\rVert_{B_{r}(x^{\prime})}.$

We conclude

(4.22) $\displaystyle\sup_{r>0}\frac{1}{r^{2\alpha}}\inf_{B}\left\lVert\nabla
f-B_{x^{\prime}}\right\rVert_{B_{r}(x^{\prime})}$
$\displaystyle\stackrel{{\scriptstyle\eqref{lsc}}}{{\leq}}\liminf_{\varepsilon\to
0}\sup_{r>0}\frac{1}{r^{2\alpha}}\inf_{B^{\varepsilon}}\left\lVert\nabla
f_{\varepsilon}-B^{\varepsilon}_{x^{\prime}}\right\rVert_{B_{r}(x^{\prime})}$
(4.23)
$\displaystyle\stackrel{{\scriptstyle\eqref{smoothest}}}{{\lesssim}}\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq
l}\inf_{k^{\varepsilon}}\left\lVert\delta_{y}f_{\varepsilon}-k^{\varepsilon}\right\rVert_{B_{l}(x^{\prime})}$
(4.24)
$\displaystyle\stackrel{{\scriptstyle\eqref{bound}}}{{\lesssim}}\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq
l}\inf_{k}\left\lVert\delta_{y}f-k\right\rVert_{B_{l}(x^{\prime})}.$

∎

In order to include time we extend 4.1 to space-time functions via the
following interpolation inequality.

######  Corollary 4.2.

For $f:{\mathbb{R}}\times{\mathbb{R}}^{d}\to{\mathbb{R}}$ such that $\nabla
f\in C^{\alpha}$ we have

(4.25) $\displaystyle\sup_{r>0}\frac{1}{r^{2\alpha}}\inf_{B}\left\lVert\nabla
f-B_{x^{\prime}}\right\rVert_{P_{r}(z)}\lesssim$
$\displaystyle\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq
l}\inf_{k\in{\mathbb{R}}^{d}}\left\lVert\nabla\delta_{y}f-k\right\rVert_{P_{l}(z)}$
(4.26) $\displaystyle+\sum_{i=0}^{d}\sup_{r>0}r^{1-2\alpha}\sup_{\left\lvert
y\right\rvert\leq
r}\left\lVert\partial_{t}\left(\delta_{y}f\right)_{r,i}\right\rVert_{P_{r}(z)}$

###### Proof.

Let $\left(t,x\right)\in P_{r}(z)$. By 4.1 there exist a symmetric matrix
$B(t^{\prime},x^{\prime})$ and a vector $b(t^{\prime},x^{\prime})$ such that

(4.27) $\displaystyle\left\lvert\nabla
f(t^{\prime},x)-\left(B(t^{\prime},x^{\prime})\left(x-x^{\prime}\right)+b(t^{\prime},x^{\prime})\right)\right\rvert$
$\displaystyle\lesssim
r^{2\alpha}\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert y\right\rvert\leq
l}\inf_{k=k(t)}\left\lVert\nabla\delta_{y}f(t^{\prime},\cdot)-k\right\rVert_{B_{l}(x^{\prime})}$
(4.28) $\displaystyle\lesssim
r^{2\alpha}\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert y\right\rvert\leq
l}\inf_{k\in{\mathbb{R}}^{d}}\left\lVert\nabla\delta_{y}f-k\right\rVert_{P_{l}(z)}.$

Via the triangle inequality we split the remainder into a spatial increment
and a temporal increment

(4.29) $\displaystyle\left\lvert\nabla f(t,x)-\nabla
f(t^{\prime},x)\right\rvert\leq$ $\displaystyle\left\lvert\nabla
f(t,x)-\left(\nabla f\right)_{r}(t,x)\right\rvert+\left\lvert\left(\nabla
f\right)_{r}(t,x)-\left(\nabla f\right)_{r}(t^{\prime},x)\right\rvert$ (4.30)
$\displaystyle+\left\lvert\nabla f(t^{\prime},x)-\left(\nabla
f\right)_{r}(t^{\prime},x)\right\rvert.$

Again by 4.1 there exist a symmetric matrix $B(t,x^{\prime})$ and a vector
$b(t,x^{\prime})$ such that

(4.31) $\displaystyle\left\lvert\nabla
f(t,x)-\left(B(t,x^{\prime})\left(x-x^{\prime}\right)+b(t,x^{\prime})\right)\right\rvert$
$\displaystyle\lesssim
r^{2\alpha}\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert y\right\rvert\leq
l}\inf_{k\in{\mathbb{R}}^{d}}\left\lVert\nabla\delta_{y}f-k\right\rVert_{P_{l}(z)}.$

and, in order to estimate the spatial increment, we appeal to radial symmetry
of our mollifier, which implies

(4.32) $\displaystyle
B(t,x^{\prime})(x-x^{\prime})=\int_{{\mathbb{R}}^{d}}\psi_{r}(x-\zeta)B(t,x^{\prime})(\zeta-x^{\prime})\mathop{}\\!\mathup{d}\zeta$

and, hence again by 4.1 and the triangle inequality

(4.33) $\displaystyle\left\lvert\nabla f(t,x)-\left(\nabla
f\right)_{r}(t,x)\right\rvert$ (4.34) $\displaystyle=\left\lvert\nabla
f(t,x)-b(t,x^{\prime})-B(t,x^{\prime})\left(x-x^{\prime}\right)-\int_{{\mathbb{R}}^{d}}\psi_{r}(x-\zeta)\left(\nabla
f(t,\zeta)-b(t,x^{\prime})-B(t,x^{\prime})\left(\zeta-x^{\prime}\right)\right)\mathop{}\\!\mathup{d}\zeta\right\rvert$
(4.35) $\displaystyle\lesssim
r^{2\alpha}\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert y\right\rvert\leq
l}\inf_{k\in{\mathbb{R}}^{d}}\left\lVert\nabla\delta_{y}f-k\right\rVert_{P_{l}(z)}.$

In order to treat the time difference, we show

(4.36) $\displaystyle\left\lVert\partial_{t}\left(\nabla
f\right)_{r}\right\rVert_{P_{r}(z)}\lesssim\sum_{i=0}^{d}\frac{1}{r}\sup_{\left\lvert
y\right\rvert\leq
r}\left\lVert\partial_{t}\left(\delta_{y}f\right)_{r,i}\right\rVert_{P_{r}(z)}.$

Let $i=1,\dots,d$. For any $\hat{t}\in{\mathbb{R}}$ we compute, appealing to
the mean value theorem in space and then Fubini’s theorem

(4.37)
$\displaystyle\left(\partial_{i}f\right)_{r}(\hat{t},x)-\left(\frac{1}{r}\delta_{re_{i}}f\right)_{r}(\hat{t},x)$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\psi_{r}(x-\zeta)\left(\partial_{i}f(\hat{t},\zeta)-\int_{0}^{1}\partial_{i}f(\hat{t},\zeta+\theta
re_{i})\mathop{}\\!\mathup{d}\theta\right)\mathop{}\\!\mathup{d}\zeta$ (4.38)
$\displaystyle=-\int_{0}^{1}\int_{{\mathbb{R}}^{d}}\psi_{r}(x-\zeta)\delta_{\theta
re_{i}}\partial_{i}f(\hat{t},\zeta)\mathop{}\\!\mathup{d}\zeta\mathop{}\\!\mathup{d}\theta.$

Integration by parts then yields

(4.39)
$\displaystyle\left(\partial_{i}f\right)_{r}(\hat{t},x)-\left(\frac{1}{r}\delta_{re_{i}}f\right)_{r}(\hat{t},x)=-\int_{0}^{1}\left(\frac{1}{r}\delta_{\theta
re_{i}}f\right)_{r,i}(\hat{t},x)\mathop{}\\!\mathup{d}\theta.$

Taking the time derivative proves (4.36).

By the mean value theorem in time, for any $x\in{\mathbb{R}}^{d}$ we have

(4.40)
$\displaystyle\left(\partial_{i}f\right)_{r}(t,x)-\left(\partial_{i}f\right)_{r}(t^{\prime},x)=(t-t^{\prime})\int_{0}^{1}\partial_{t}\left(\partial_{i}f\right)_{r}(\delta
t+(1-\delta)t^{\prime},x)\mathop{}\\!\mathup{d}\delta.$

and combining (4.40) with (4.36) finally yields

(4.41) $\displaystyle\left\lvert\left(\nabla f\right)_{r}(t,x)-\left(\nabla
f\right)_{r}(t^{\prime},x)\right\rvert\lesssim
r^{2}\left\lVert\partial_{t}\left(\nabla
f\right)_{r}\right\rVert_{P_{r}(z)}\lesssim\sum_{i=0}^{d}r\sup_{\left\lvert
y\right\rvert\leq
r}\left\lVert\partial_{t}\left(\delta_{y}f\right)_{r,i}\right\rVert_{P_{r}(z)}.$

Hence we have proven that for $\left(t,x\right)\in P_{r}(z)$

(4.42) $\displaystyle\left\lvert\nabla
f(t,x)-\left(B(t^{\prime},x^{\prime})\cdot\left(x-x^{\prime}\right)+b(t^{\prime},x^{\prime})\right)\right\rvert\lesssim$
$\displaystyle r^{2\alpha}\sup_{l>0}\frac{1}{l^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq
l}\inf_{k\in{\mathbb{R}}^{d}}\left\lVert\nabla\delta_{y}f-k\right\rVert_{P_{l}(z)}$
(4.43)
$\displaystyle+r^{2\alpha}\sum_{i=0}^{d}\sup_{l>0}l^{1-2\alpha}\sup_{\left\lvert
y\right\rvert\leq
l}\left\lVert\partial_{t}\left(\delta_{y}f\right)_{l,i}\right\rVert_{P_{l}(z)}$

∎

Let $y\in{\mathbb{R}}^{d}$. Since $\delta_{y}u$ satisfies

(4.44) $\displaystyle\partial_{t}\delta_{y}u-\nabla\cdot
a_{y}\nabla\delta_{y}u=\partial_{t}\delta_{y}v-\nabla\cdot
a(t^{\prime},x^{\prime})\nabla\delta_{y}v$

we see that $\delta_{y}w$ satisfies the equation

(4.45) $\displaystyle\partial_{t}\delta_{y}w-\nabla\cdot
a_{y}\nabla\delta_{y}w=\nabla\cdot(a_{y}-a(t^{\prime},x^{\prime}))\nabla\delta_{y}v.$

Then, we can write this as a constant coefficient equation

(4.46) $\displaystyle\partial_{t}\delta_{y}w-\nabla\cdot
a(t^{\prime},x^{\prime})\nabla\delta_{y}w=\nabla\cdot g$

by introducing

(4.47) $\displaystyle g:=(a_{y}-a(t^{\prime},x^{\prime}))\nabla\delta_{y}u.$

Note that (4.46) is now invariant under affine spatial translations, i.e. of
functions of the form $\mathrm{aff}(x):=b\cdot x+a$ for some
$b\in{\mathbb{R}}^{d},a\in{\mathbb{R}}$. Hence, we can apply the
$C^{1+\alpha}$ interior Schauder estimate [5, Theorem 4.8], using this
invariance, as well as parabolic rescaling, to obtain that, for all $l>0$ and
all space-time points $z=(t^{\prime},x^{\prime})$, it holds

(4.48)
$\displaystyle\quad\quad\,l^{\alpha}\inf_{\mathrm{aff}}[\nabla(\delta_{y}w-\mathrm{aff})]_{\alpha,P_{l}(z)}+\inf_{\mathrm{aff}}\left\lVert\nabla(\delta_{y}w-\mathrm{aff})\right\rVert_{P_{l}(z)}$
$\displaystyle\lesssim l^{\alpha}\left[g\right]_{\alpha,P_{2l}(z)}$ (4.49)
$\displaystyle+l^{-1}\inf_{\mathrm{aff}}\left\lVert\delta_{y}w-\mathrm{aff}\right\rVert_{P_{2l}(z)}$

which is equivalent to

(4.50)
$\displaystyle\quad\quad\,l^{\alpha}[\nabla\delta_{y}w]_{\alpha,P_{l}(z)}+\inf_{k\in{\mathbb{R}}^{d}}\left\lVert\nabla\delta_{y}w-k\right\rVert_{P_{l}(z)}$
$\displaystyle\lesssim l^{\alpha}\left[g\right]_{\alpha,P_{2l}(z)}$ (4.51)
$\displaystyle+l^{-1}\inf_{\mathrm{aff}}\left\lVert\delta_{y}w-\mathrm{aff}\right\rVert_{P_{2l}(z)}.$

This suggests that we need to further estimate the right hand side which is
captured in the next Proposition and which is an extension of [6, p.73, (34)].
The proof is similar.

######  Proposition 4.3.

Let $z=(t^{\prime},x^{\prime})$. For
$f:{\mathbb{R}}\times\mathbb{R}^{d}\to{\mathbb{R}}$ continuously
differentiable and $y\in{\mathbb{R}}^{d}$ we have the estimate

(4.52)
$\displaystyle\inf_{\mathrm{aff}}\left\lVert\delta_{y}f-\mathrm{aff}\right\rVert_{P_{l}(z)}\lesssim|y|\inf_{B}\left\lVert\nabla
f-B_{x^{\prime}}\right\rVert_{P_{2l}(z)}$

where $B_{x^{\prime}}(x)=B(x-x^{\prime})+b$, $B\in{\mathbb{R}}^{d\times d}$ is
symmetric and $b\in{\mathbb{R}}^{d}$.

###### Proof.

For $B\in{\mathbb{R}}^{d\times d}$ symmetric we define

(4.53)
$\displaystyle\widetilde{f}(t,x):=f(t,x)-\left(\frac{1}{2}(x-x^{\prime})\cdot
B(x-x^{\prime})+b\cdot(x-x^{\prime})\right).$

Then we compute

(4.54) $\displaystyle\nabla\widetilde{f}(t,x)=\nabla
f(t,x)-(B(x-x^{\prime})+b)$

as well as

(4.55)
$\displaystyle\delta_{y}\widetilde{f}(t,x)=\delta_{y}f(t,x)-\left(y\cdot
B(x-x^{\prime})+\frac{1}{2}y\cdot By+b\cdot y\right).$

Notice that for any $y$ the map $x\mapsto
y^{T}B(x-x^{\prime})+\frac{1}{2}y^{T}By+b\cdot y$ is again affine. Thus, by
the mean value theorem, we have
$\left\lVert\delta_{y}\widetilde{f}\right\rVert_{P_{l}(z)}\leq|y|\left\lVert\nabla\widetilde{f}\right\rVert_{P_{2l}(z)}$
which yields the statement. ∎

###  4.1. Proof of the main theorem

We are now able to prove our main theorem.

###### Proof of Theorem 1.2.

In the following $B$ always denotes a symmetric matrix. We write
$M=\sup_{z=(t,x)}M_{z}$ where
$M_{z}=\inf_{B}\sup_{r>0}r^{-2\alpha}\left\lVert\nabla
w_{a(t^{\prime},x^{\prime})}-B_{x^{\prime}}\right\rVert_{P_{r}(z)}$. For now
we assume that $M_{z}$ is finite for any $z$. By 4.2, we have

(4.56) $\displaystyle\sup_{r>0}\frac{1}{r^{2\alpha}}\inf_{B}\left\lVert\nabla
w-B_{x^{\prime}}\right\rVert_{P_{r}(z)}$
$\displaystyle\lesssim\sup_{r>0}\frac{1}{r^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq
r}\inf_{k}\left\lVert\nabla\delta_{y}w-k\right\rVert_{P_{r}(z)}$ (4.57)
$\displaystyle+\sum_{i=0}^{d}\sup_{r>0}r^{1-2\alpha}\sup_{\left\lvert
y\right\rvert\leq
r}\left\lVert\partial_{t}\left(\delta_{y}w\right)_{r,i}\right\rVert_{P_{r}(z)}.$

Let $\left\lvert y\right\rvert\leq r\leq l$. First, we focus on the second
term on the right hand side of (4.56). To this end, we recall (cf. (4.46))
that $\delta_{y}w$ satisfies the constant coefficient equation

(4.58) $\displaystyle\partial_{t}\delta_{y}w=\nabla\cdot
a(t^{\prime},x^{\prime})\nabla\delta_{y}w+\nabla\cdot g$

where

(4.59) $\displaystyle g=(a_{y}-a(t^{\prime},x^{\prime}))\nabla\delta_{y}u$

and we estimate for any $i=0,\dots,d$

(4.60)
$\displaystyle\left\lVert\partial_{t}(\delta_{y}w)_{r,i}\right\rVert_{P_{r}(z)}$
$\displaystyle\lesssim
r^{\alpha-1}\left[a(t^{\prime},x^{\prime})\nabla\delta_{y}w+g\right]_{\alpha,P_{2r}(z)}$
(4.61) $\displaystyle\lesssim
r^{\alpha-1}\left(\left[\nabla\delta_{y}w\right]_{\alpha,P_{2r}(z)}+\left[g\right]_{P_{2r}(z)}\right).$

Hence we have

(4.62) $\displaystyle\sup_{r>0}\frac{1}{r^{2\alpha}}\inf_{B}\left\lVert\nabla
w-B_{x^{\prime}}\right\rVert_{P_{r}(z)}\lesssim$
$\displaystyle\sup_{r>0}\frac{1}{r^{2\alpha}}\sup_{\left\lvert
y\right\rvert\leq
r}\inf_{k}\left\lVert\nabla\delta_{y}w-k\right\rVert_{P_{r}(z)}$ (4.63)
$\displaystyle+\sup_{r>0}r^{-\alpha}\sup_{\left\lvert y\right\rvert\leq
r}\left(\left[\nabla\delta_{y}w\right]_{\alpha,P_{2r}(z)}+\left[g\right]_{P_{2r}(z)}\right).$

For $\left\lvert y\right\rvert\leq r\leq l$ we estimate by (4.50)

(4.64)
$\displaystyle\inf_{k}\left\lVert\nabla\delta_{y}w-k\right\rVert_{P_{r}(z)}\leq
r^{\alpha}\left[\nabla\delta_{y}w\right]_{\alpha,P_{l}(z)}\lesssim
r^{\alpha}\left[g\right]_{\alpha,P_{2l}(z)}+\frac{r^{\alpha}}{l^{1+\alpha}}\inf_{\mathrm{aff}}\left\lVert\delta_{y}w-\mathrm{aff}\right\rVert_{P_{2l}(z)}$

as well as

(4.65)
$\displaystyle\left[\nabla\delta_{y}w\right]_{\alpha,P_{l}(z)}\lesssim\left[g\right]_{\alpha,P_{2l}(z)}+\frac{1}{l^{1+\alpha}}\inf_{\mathrm{aff}}\left\lVert\delta_{y}w-\mathrm{aff}\right\rVert_{P_{2l}(z)}.$

Now we turn to the estimate of $\left[g\right]_{\alpha,P_{2l}(z)}$. We appeal
to the Lipschitz continuity of $DA$ (cf. (2.3)) and recall the definition of
$a(t^{\prime},x^{\prime})$ as well as $a_{y}$ in order to estimate

(4.66) $\displaystyle|a_{y}(t,x)-a(t^{\prime},x^{\prime})|$
$\displaystyle\lesssim\int_{0}^{1}|\theta\nabla u(t,x+y)+(1-\theta)\nabla
u(t,x)-\nabla u(t^{\prime},x^{\prime})|\mathop{}\\!\mathup{d}\theta$ (4.67)
$\displaystyle\lesssim\int_{0}^{1}\theta|\nabla u(t,x+y)-\nabla
u(t,x)|+|\nabla u(t,x)-\nabla
u(t^{\prime},x^{\prime})|\mathop{}\\!\mathup{d}\theta$ (4.68)
$\displaystyle\lesssim|y|^{\alpha}+d((t,x),(t^{\prime},x^{\prime}))^{\alpha}$

and thus, for $|y|\leq r\leq l$ we get

(4.69) $\displaystyle\left\lVert
a_{y}-a(t^{\prime},x^{\prime})\right\rVert_{P_{2l}(z)}\lesssim l^{\alpha}.$

By (2.12), we have

(4.70)
$\displaystyle[a_{y}-a(t^{\prime},x^{\prime})]_{\alpha,P_{2l}(z)}=[a_{y}]_{\alpha,P_{2l}(z)}\lesssim[\nabla
u]_{\alpha,P_{3l}(z)}.$

Moreover, it holds that

(4.71)
$\displaystyle\left\lVert\nabla\delta_{y}u\right\rVert_{P_{2l}(z)}\lesssim
l^{\alpha}[\nabla u]_{\alpha,P_{3l}(z)}$

as well as

(4.72) $\displaystyle[\nabla\delta_{y}u]_{\alpha,P_{2l}(z)}\lesssim[\nabla
u]_{\alpha,P_{3l}(z)}.$

Summing up, we arrive at

(4.73)
$\displaystyle\left[g\right]_{\alpha,P_{2l}(z)}=[(a_{y}-a(t^{\prime},x^{\prime}))\nabla\delta_{y}u]_{\alpha,P_{2l}(z)}$
$\displaystyle\leq[a_{y}-a(t^{\prime},x^{\prime})]_{\alpha,P_{3l}(z)}\left\lVert\nabla\delta_{y}u\right\rVert_{P_{3l}(z)}$
(4.74) $\displaystyle+\left\lVert
a_{y}-a(t^{\prime},x^{\prime})\right\rVert_{P_{3l}(z)}[\nabla\delta_{y}u]_{\alpha,P_{3l}(z)}$
(4.75) $\displaystyle\lesssim l^{\alpha}[\nabla u]_{\alpha,P_{3l}(z)}$

for $|y|\leq r\leq l$. Combining (4.73) with (4.64) yields for $|y|\leq r\leq
l$

(4.76) $\displaystyle
r^{\alpha}\left[\nabla\delta_{y}w\right]_{\alpha,P_{l}(z)}$
$\displaystyle\lesssim l^{2\alpha}\left[\nabla
u\right]_{\alpha,P_{3l}(z)}+\frac{r^{\alpha}}{l^{1+\alpha}}\inf_{\mathrm{aff}}\left\lVert\delta_{y}w-\mathrm{aff}\right\rVert_{P_{2l}(z)}.$

Then using 4.3 we further estimate (4.76) to the effect that

(4.77) $\displaystyle
r^{\alpha}\left[\nabla\delta_{y}w\right]_{\alpha,P_{l}(z)}$
$\displaystyle\lesssim l^{2\alpha}\left[\nabla
u\right]_{\alpha,P_{3l}(z)}+\left(\frac{r}{l}\right)^{1+\alpha}\inf_{B}\left\lVert\nabla
w-B_{x^{\prime}}\right\rVert_{P_{4l}(z)}.$

Summing up, we get by (4.60) and (4.77)

(4.78) $\displaystyle\frac{1}{r^{2\alpha}}\sup_{\left\lvert y\right\rvert\leq
r}\inf_{k}\left\lVert\nabla\delta_{y}w-k\right\rVert_{P_{r}(z)}+\sum_{i=0}^{d}r^{1-2\alpha}\sup_{\left\lvert
y\right\rvert\leq
r}\left\lVert\partial_{t}\left(\delta_{y}w\right)_{r}\right\rVert_{P_{r}(z)}$
(4.79) $\displaystyle\lesssim\left(\frac{l}{r}\right)^{2\alpha}\left[\nabla
u\right]_{\alpha,P_{3l}(z)}+\left(\frac{r}{l}\right)^{1-\alpha}\frac{1}{l^{2\alpha}}\inf_{B}\left\lVert\nabla
w-B_{x^{\prime}}\right\rVert_{P_{4l}(z)}.$

Let $K\geq 1$ and set $l=Kr$. By introducing
$M^{\prime}_{z}:=\sup_{r>0}\frac{1}{r^{2\alpha}}\inf_{B}\left\lVert\nabla
w-B_{x^{\prime}}\right\rVert_{P_{r}(z)}$ we estimate combining (4.56) and
(4.78)

(4.80) $\displaystyle M^{\prime}_{z}\lesssim K^{2\alpha}\left[\nabla
u\right]_{\alpha}+K^{\alpha-1}M_{z}$

Thus, after using (1.7) as well as Lemma 1.1 we arrive at

(4.81) $\displaystyle M^{\prime}_{z}\lesssim K^{2\alpha}C\left(\left[\nabla
v\right]_{\alpha}\right)+K^{\alpha-1}M_{z}$

where the implicit constant in $\lesssim$ depends only on $d,\lambda,\Lambda$
and $\alpha$ and $C\left(\left[\nabla v\right]_{\alpha}\right)$ depends
polynomially on $\left[\nabla v\right]_{\alpha}$. Since $\alpha>\frac{1}{2}$,
using the reasoning in Step 3 in the proof of Lemma 3.6 of [7] the matrix $B$
is independent of $r$, i.e.

(4.82) $\displaystyle M_{z}\lesssim M^{\prime}_{z}$

and hence

(4.83) $\displaystyle M_{z}\lesssim K^{2\alpha}C\left(\left[\nabla
v\right]_{\alpha}\right)+K^{\alpha-1}M_{z}.$

Choosing $K$ sufficiently large, and since $\alpha<1$, as well as the
assumption that $M_{z}$ is finite, we can absorb $K^{\alpha-1}M_{z}$ into the
left hand side and end up with

(4.84) $\displaystyle M_{z}\lesssim C(\left[\nabla v\right]_{\alpha})$

and taking the supremum in $z$ we get

(4.85) $\displaystyle M\lesssim C(\left[\nabla v\right]_{\alpha}).$

In order to get rid of the assumption that $M_{z}$ is finite we appeal to the
following approximation argument. Up to now we made the abbreviation
$v=v_{a(t^{\prime},x^{\prime})}$ but from now on by $v$ we mean the solution
to the SHE (3.1). As in the proof of [6, Theorem 1], we can consider smooth
approximations $u_{\varepsilon}$, $v_{a(t^{\prime},x^{\prime}),\varepsilon}$
and $v_{\varepsilon}$ to $u$, $v_{a(t^{\prime},x^{\prime})}$ and $v$ such that

(4.86)
$\displaystyle\nabla\left(u_{\varepsilon}-v_{a(t^{\prime},x^{\prime}),\varepsilon}\right)\to\nabla\left(u-v_{a(t^{\prime},x^{\prime})}\right)=\nabla
w$

uniformly on compact sets and such that $\left[\nabla
v_{\varepsilon}\right]_{\alpha}\lesssim\left[\nabla v\right]_{\alpha}$. Then
the corresponding constant $M_{z,\varepsilon}$ is finite for all $z$ and hence
(4.84) applies and we have

(4.87) $\displaystyle M_{\varepsilon}\lesssim C(\left[\nabla
v_{\varepsilon}\right]_{\alpha})\lesssim C(\left[\nabla v\right]_{\alpha}).$

Hence, by the same reasoning as in 4.1, we conclude by lower-semicontinuity

(4.88) $\displaystyle M\lesssim\liminf_{\varepsilon\to
0}M_{\varepsilon}\lesssim C(\left[\nabla v\right]_{\alpha}).$

In other words, (4.88) means that for all $r>0$ and all space-time points
$z=(t^{\prime},x^{\prime})$ there exists a family of symmetric matrices
$\left(B(t^{\prime},x^{\prime})\right)_{(t^{\prime},x^{\prime})}$ and a family
of vectors $\left(b(t^{\prime},x^{\prime})\right)_{(t^{\prime},x^{\prime})}$,
such that we have

(4.89) $\displaystyle|\nabla u(t,x)-\nabla
v_{a(t^{\prime},x^{\prime})}(t,x)-(B(t^{\prime},x^{\prime})(x-x^{\prime})+b(t^{\prime},x^{\prime}))|\lesssim
r^{2\alpha}\ \text{for}\ (t,x)\in P_{r}(z).$

Letting $r\to 0$ implies $(t,x)\to(t^{\prime},x^{\prime})$ and hence the
optimal $b$ is of the form

(4.90) $\displaystyle b(t^{\prime},x^{\prime})=\nabla
u(t^{\prime},x^{\prime})-\nabla
v_{a(t^{\prime},x^{\prime})}(t^{\prime},x^{\prime}).$

Then we can choose $r=d((t,x),(t^{\prime},x^{\prime}))$ and thus

(4.91) $\displaystyle|\nabla u(t,x)-\nabla
v_{a(t^{\prime},x^{\prime})}(t,x)-(\nabla u(t^{\prime},x^{\prime})-\nabla
v_{a(t^{\prime},x^{\prime})}(t^{\prime},x^{\prime}))-B(t^{\prime},x^{\prime})(x-x^{\prime})|$
(4.92) $\displaystyle\lesssim d^{2\alpha}((t,x),(t^{\prime},x^{\prime})),$

which we wanted to prove. ∎

## Acknowledgments

The author would like to thank Felix Otto for helpful discussions and for
bringing this problem to his attention.

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