# A stochastic reconstruction theorem

Hannes Kern
Technische Universität Berlin
Email<EMAIL_ADDRESS>
###### Abstract

In a recent landmark paper, Khoa Lê (2020) established a stochastic sewing
lemma which since has found many applications in stochastic analysis. He
further conjectured that a similar result may hold in the context of the
reconstruction theorem within Hairer’s regularity structures. The purpose of
this article is to provide such a stochastic reconstruction theorem. We also
discuss two variations of this theorem, motivated by different constructions
of stochastic integration against white noise. Our formulation makes use of
the distributional viewpoint of Caravenna–Zambotti (2021).

###### Contents

  1. 1 Introduction
    1. 1.1 Stochastic reconstruction in a nutshell
    2. 1.2 The structure of this paper
  2. 2 Preliminaries
    1. 2.1 Distributions and Hölder continuity
  3. 3 Main results
    1. 3.1 Stochastic Reconstruction
    2. 3.2 Two-sided Reconstruction
    3. 3.3 The covariance setting in $L_{2}$
  4. 4 Proofs of Theorem 22 and 30
    1. 4.1 Proof of Theorem 22
    2. 4.2 Proof of Theorem 30
  5. 5 Stochastic reconstruction is stochastic sewing in 1 dimension
  6. 6 Gaussian Martingale Measure
    1. 6.1 Gaussian martingale measures and Walsh-type Integration
    2. 6.2 Reconstruction of the Walsh-type Integral
  7. 7 Integration against white noise
  8. A Wavelet Bases
  9. B A BDG-type inequality
  10. C Kolmogorov’s continuity theorem for negative Hölder regularity

##  1 Introduction

###  1.1 Stochastic reconstruction in a nutshell

In recent years, the reconstruction theorem from Martin Hairer’s regularity
structure theory [Hai14] has gained popularity in the mathematical community,
not just within the said theory, but also as a standalone theorem (see [CZ20],
[ST18]).

It answers the following question: Given a family of distributions
$(F_{x})_{x\in\mathbb{R}^{d}}$, is it possible to find a single distribution
$f$, which around any point $x\in\mathbb{R}^{d}$ locally looks like $F_{x}$?
The most simple example for such a situation is given by the Young product:
Given an $0<\alpha$-Hölder continuous function $g$ and a distribution $h$,
which is $0>\beta$-Hölder continuous, with $\alpha+\beta>0$, it is possible to
define a canonical product $g\cdot h$ even for non-smooth $g$. E.g. [BCD11]
and [RS96] have used paraproducts to construct this product.

As it turns out, there is a clear canonical local approximation of $h\cdot g$:
We can approximate $g$ locally with its Taylor polynomial $T_{x}g$ around any
$x\in\mathbb{R}^{d}$, and set $F_{x}=h\cdot T_{x}g$, as $T_{x}g$ is a smooth
function. The only ingredient missing is a way to reconstruct the distribution
from its local approximations.

To return to the general setting, let us consider a family of distributions
$(F_{x})_{x\in\mathbb{R}^{d}}$ and let us formalize what it means to “locally
look like $F_{x}$” around $x$: For a smooth, compactly supported mollifier
$\psi$, the rescaled function $\psi_{x}^{\epsilon}$ approximates a Dirac
distribution $\delta_{x}$ as $\epsilon\to 0$. Since we formally want
“$F_{x}(x)=f(x)$” to hold, we will require that
$(f-F_{x})(\psi_{x}^{\epsilon})$ vanishes as $\epsilon\to 0$. To be precise,
the rigorous condition to “locally look like $f$” is given by

$\left|(f-F_{x})(\psi_{x}^{\epsilon})\right|\lesssim\epsilon^{\lambda}$ (1.1)

for some $\lambda>0$. For this to be possible, $(F_{x})_{x\in\mathbb{R}^{d}}$
will need to fulfill some kind of continuity assumption: For example, it
suffices that
$\left|(F_{x}-F_{y})(\psi_{x}^{\epsilon})\right|\lesssim\epsilon^{\alpha}\left|x-y\right|^{\gamma-\alpha}$
for some $\alpha<0$, which is the condition the germs in regularity structure
fulfill [Hai14]. Caravenna and Zambotti managed to find the optimal condition
called coherence in [CZ20], which ensures 1.1 and is given as follows:
$(F_{x})_{x\in\mathbb{R}^{d}}$ is called coherent, if

$\left|(F_{x}-F_{y})(\psi_{x}^{\epsilon})\right|\lesssim\epsilon^{\alpha}(\left|x-y\right|+\epsilon)^{\gamma-\alpha}$
(1.2)

for some $\alpha\in\mathbb{R}$ (typically negative) and a $\gamma>0$. Under
this assumption, they proved the following theorem:

######  Theorem 1 ([CZ20]).

Let $(F_{x})_{x\in\mathbb{R}^{d}}$ be a coherent germ for some $\gamma>0$.
Then there exists a unique distribution $f$ with

$\left|(f-F_{x})(\psi_{x}^{\epsilon})\right|\lesssim\epsilon^{\gamma}.$

######  Remark 2.

They also showed that it is possible to find such a distribution $f$ for
$\gamma<0$. However, this distribution is no longer unique. Also, note that
for $\gamma<0$, the bound $\epsilon^{\gamma}$ does explode, so the condition
is much weaker. In this article, we will concentrate on the $\gamma>0$ case.

At this point, the reconstruction theorem is a purely analytic tool. This is
surprising, as it is often applied in SPDEs, see for example [Bay+20],
[Bra19], [CFG17], [Hai14], and many more. In all those examples, the
reconstruction theorem only uses the analytic properties of the stochastic
processes, neglecting their stochastic properties.

This naturally leads to the question, of whether there exists a stochastic
version of the reconstruction theorem, especially if one considers its close
connection to the sewing lemma [FL06], [Gub04]: In rough paths theory, rough
integrals can be constructed with the sewing lemma (see [FH14],[Lyo98]),
whereas in regularity structure theory the reconstruction theorem is used to
construct the corresponding products. And in [Lê20], Khoa Lê showed the
existence of a stochastic version of the sewing lemma.

It would be interesting to generalize his techniques to the multidimensional
setting of the reconstruction theorem. Let us give two possible applications
such a stochastic reconstruction theorem could have:

  * •

In the paper [FHL21], Peter Fritz and Khoa Lê use the stochastic sewing lemma
to generate a mixed theory of rough paths and stochastic differential
equations to solve equations of the form

$dY_{t}=b(t,Y_{t})dt+\sigma(t,Y_{t})dB_{t}+f(t,Y_{t})d\mathbb{X}_{t}\,,$

where $B_{t}$ is a Brownian motion and $\mathbb{X}_{t}$ is a deterministic
rough path. This is outside of the scope of both Itô-calculus as well as rough
path theory since the construction of the integral $\int
f(t,Y_{t})d\mathbb{X}_{t}$ requires one to understand the integration of a
random process against a rough path. One can imagine similar problems for
SPDEs, for example, one can look at the multiplicative heat equation with
common noise

$du-\Delta udt=f(u)d\bar{W}+g(u)dW\,,$

for a two independent Brownian sheets $\bar{W}$ and $W$ with spatial dimension
$d=1$. One usually wants to treat the common noise $dW$ as a deterministic
distribution, so it would make sense to treat it with the help of regularity
structure theory. However, if one wants to keep $d\bar{W}$ as a stochastic
integral, one would need a mixed theory of SPDEs and regularity structures.
While we are not yet able to present such a theory, we hope that our
stochastic reconstruction theorem provides the first step toward it.

  * •

The stochastic sewing lemma has shown tremendously useful in analyzing
regularization by noise phenomena [HP21]. In recent years, there has been a
growing interest in studying such phenomenons in multi-parameter settings,
e.g. [BHR22]. This is currently done with a deterministic multi-parameter
sewing lemma, and the authors themselves point out that many more interesting
noises like fractional Brownian sheet [WX07] will most likely require a
stochastic multi-parameter technique. We conjecture that using the stochastic
reconstruction theorem instead of the deterministic multi-parameter sewing
lemma would allow one to extend the results of [BHR22] to more general noises.

######  Remark 3.

We need to point out that since the inception of this paper’s ideas, the
regularization by noise community has moved towards favoring sewing techniques
over reconstruction techniques. A stochastic multi-parameter sewing lemma is
being researched as we are writing this paper, and while we believe that there
is plenty enough reason to research both reconstruction and sewing techniques,
it is at the moment not clear if stochastic reconstruction will hold a general
advantage over stochastic multi-parameter sewing in the regularization by
noise field, aside from being first.

In this paper we present such a generalization of Lê’s results to the
multidimensional setting of the reconstruction theorem: Consider a family of
random distributions $(F_{x})_{x\in\mathbb{R}^{d}}$ with $2\leq p<\infty$ (see
Definition 11), which is adapted to a filtration
$(\mathcal{F}_{y})_{y\in\mathbb{R}}$ (see Definition 17). The main idea of
stochastic reconstruction is to enhance the coherence property 1.2 by an
additional condition on the conditional expectation of $F_{x}$ as follows:

$\displaystyle\left|\left|(F_{x}-F_{y})(\psi_{x}^{\epsilon})\right|\right|_{L_{p}}\lesssim\epsilon^{\alpha}(\left|x-y\right|+\epsilon)^{\gamma-\alpha}$
$\displaystyle\left|\left|E^{\mathcal{F}_{y_{1}}}(F_{x}-F_{y})(\psi_{x}^{\epsilon})\right|\right|_{L_{p}}\lesssim\epsilon^{\alpha}(\left|x-y\right|+\epsilon)^{\tilde{\gamma}-\alpha},$

where $y=(y_{1},\dots,y_{d}),x=(x_{1},\dots,x_{d})$ such that $y_{1}\leq
x_{1}$ and $\psi$ has a positive support (i.e.
$\mathrm{supp}(\psi)\in[0,\infty)^{d}$) and for some $\tilde{\gamma}>\gamma$,
so the conditional expectation is better behaved than the germ itself. We then
hope, that $(F_{x})_{x\in\mathbb{R}^{d}}$ can “borrow regularity” from its
conditional expectation.

In the case of stochastic sewing, this was achieved based on a Doob-Meyer
decomposition. Khoa Lê managed to borrow up to $\frac{1}{2}$ regularity from
the conditional expectation of the terms corresponding to
$(F_{x}-F_{y})(\psi_{x}^{\epsilon})$, which indicates that we should assume
$\tilde{\gamma}=\gamma+\frac{1}{2}$.

The interesting result of our analysis is that in the multidimensional
setting, this technique can be used repeatedly for several directions to gain
a higher improvement in regularity. We thus introduce the _stochastic
dimension_ $e\leq d$ of $(F_{x})$ as the number of directions, such that for
suitable filtrations $\mathcal{F}_{y_{i}}^{(i)}$ for $i=1,\dots,e\leq d$,

$\left|\left|E^{\mathcal{F}^{(i)}_{y_{i}}}(F_{x}-F_{y})(\psi_{x}^{\epsilon})\right|\right|_{L_{p}}\lesssim\epsilon^{\alpha}(\left|x-y\right|+\epsilon)^{\tilde{\gamma}-\alpha}\,.$

In this case, we can borrow $\frac{1}{2}$ regularity from all of those
directions, resulting in $\gamma=\tilde{\gamma}-\frac{e}{2}$, an improvement
of $\frac{e}{2}$ regularity. Thus, we say that $(F_{x})$ is stochastically
$\gamma$-coherent, if it fulfills certain adaptedness properties (see
Definition 17 for details), and for $i=1,\dots,e\leq d$

$\displaystyle\left|\left|(F_{x}-F_{y})(\psi_{x}^{\epsilon})\right|\right|_{L_{p}}\lesssim\epsilon^{\alpha}(\left|x-y\right|+\epsilon)^{\gamma-\frac{e}{2}-\alpha}$
(1.3)
$\displaystyle\left|\left|E^{\mathcal{F}^{(i)}_{y_{i}}}(F_{x}-F_{y})(\psi_{x}^{\epsilon})\right|\right|_{L_{p}}\lesssim\epsilon^{\alpha}(\left|x-y\right|+\epsilon)^{\gamma-\alpha}.$
(1.4)

Under this condition, we can formulate our main result:

######  Theorem 4 (Stochastic reconstruction).

Assume the adaptedness condition of Definition 17 and that
$(F_{x})_{x\in\mathbb{R}^{d}}$ is stochastically $\gamma$-coherent for some
$\gamma>0$ with stochastic dimension $e\leq d$. Then, there exists a unique
random distribution $f$, which fulfills

$\displaystyle\left|\left|(f-F_{x})(\psi_{x}^{\epsilon})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\epsilon^{\gamma-\frac{e}{2}}$
$\displaystyle\left|\left|E^{\mathcal{F}^{(i)}_{x_{i}}}(f-F_{x})(\psi_{x}^{\epsilon})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\epsilon^{\gamma}$

for $i=1,\dots,e$.

We further discuss two more variations of Theorem 4. In the above version of
the reconstruction theorem, we assume $\psi$ to have positive support and
$x_{i}\leq y_{i}$ for $i=1,\dots,e$ in the stochastic coherence property. This
is motivated by Itô calculus, where one needs to carefully separate between
information “in the future” and “in the past”. In the deterministic theory
however, this is not necessary and the classical reconstruction theorem
[CZ20], [Hai14] do not have similar assumptions. We show that as long as one
carefully chooses the point $y_{i}$ in the conditional expectation
$E^{\mathcal{F}^{(i)}_{y_{i}}}$, one can get rid of the assumptions $x_{i}\leq
y_{i}$ and that $\psi$ has a positive support in Theorem 26.

The second variation we are discussing is reconstruction in the covariance
setting: If $p=2$, we can replace the additional condition in the coherence
property 1.4 with the following property:

$\left|E\left(\prod_{i=1}^{2}(F_{x_{i}}-F_{y_{i}})((\psi_{i})_{y_{i}}^{\epsilon_{i}})\right)\right|\lesssim\prod_{i=1}^{2}\epsilon_{i}^{\alpha}(\left|x_{i}-y_{i}\right|+\epsilon_{i})^{\gamma-\alpha}\,,$
(1.5)

for points $x_{1},y_{1},x_{2},y_{2}\in\mathbb{R}^{d}$ such that
$\left|x_{1}-x_{2}\right|,\left|y_{1}-y_{2}\right|$ are big enough (see
Section 3.3 for details), two test functions $\psi_{1},\psi_{2}$ and two
parameters $\epsilon_{1},\epsilon_{2}$. While this condition on the covariance
of $(F_{x}-F_{y})(\psi_{y}^{\epsilon})$ looks rather technical, it allows one
to entirely get rid of the filtrations $\mathcal{F}^{(i)}_{y_{i}}$! Indeed,
1.3 and 1.5 together suffice to construct a reconstruction of the germ $F_{x}$
without any adaptability assumptions. To our knowledge, this is a new idea,
even in the one-dimensional setting of stochastic sewing. Let us stress that
this is no technical detail: The filtrations we use in this paper need to have
an orthogonality condition (also often called commuting condition in the
literature, see [Kho06], chapter 11 or [Wal86a]). While this is a standard
assumption in the literature of multi-parameter martingales, in practice, it
can be hard to verify: For example for anisotropic Gaussian fields $X(s)$
[BE03], the covariance $E(X(s)X(t))$ is easy to calculate, but the filtrations
generated by $(X(t))_{t\in\mathbb{R}^{d}}$ are not well understood and hard to
work with. We conjecture that for noises in this class, the covariance setting
will prove to be a useful tool.

Perhaps an even easier example can be found in Section 7: In this section, we
use stochastic reconstruction to construct the Walsh integral $\int
X(s)\xi(ds)$ against white noise. To use Theorem 4, we would need to make an
unusually strong assumption on the adaptedness on $X$: We would need $X(s)$ to
be $\mathcal{F}_{s_{i}}^{(i)}$-measurable for all $i=1\dots,d$, while it is
well known that adaptedness in time, i.e. $X(s)$ is
$\mathcal{F}^{(1)}_{s_{1}}$-measurable suffices to construct the integral. The
stochastic reconstruction in the covariance setting is to our knowledge the
easiest way to construct this integral with reconstruction techniques.

######  Remark 5.

It should be noted that the stochastic reconstruction theorem in the
covariance setting is not strictly stronger than the classical one: In the
covariance setting, we make no distinction whether
$(x_{1})_{i}\leq(x_{2})_{i}$ or $(x_{1})_{i}\geq(x_{2})_{i}$, i.e. we do not
differentiate between the “past” and “future”. For white noise, this does not
matter since $\xi(1_{[s,t]})$ is both independent of the past before $s$ and
the future after $t$. However, if one only expects an improved regularity for
conditioning on the past and no improvement for the future, it is highly
likely that Theorem 4 is better suited for that situation.

Our last main result reads as follows:

######  Theorem 6.

Let $p=2$. Assume that $(F_{x})_{x\in\mathbb{R}^{d}}$ is stochastically
$\gamma$-coherent for some $\gamma>0$ in the covariance setting, in the sense
that 1.3 and 1.5 hold. Then there exists a unique distribution, which fulfills

$\displaystyle\left|\left|(f-F_{x_{1}})(\psi_{x_{1}}^{\lambda})\right|\right|_{L_{2}}$
$\displaystyle\lesssim\lambda^{\gamma-\frac{E}{2}}$
$\displaystyle\left|E\left(\prod_{i=1}^{2}(f-F_{x_{i}})(\psi_{x_{i}}^{\lambda})\right)\right|$
$\displaystyle\lesssim\lambda^{2\gamma}\,.$

###  1.2 The structure of this paper

This paper is structured as follows: In Section 2, we will formally introduce
distribution and scalings. The proofs of our main results make heavy use of
wavelet techniques as well as a BDG-type inequality, but since they are not
necessary for the presentation of the results themselves, we moved them to the
appendix. Both of these parts mainly follow [Hai14] and [Mey93].

Section 3 will introduce the stochastic reconstruction theorem in Theorem 22
as well as the two advertised modifications in Theorem 26 and Theorem 30. The
section focuses on presenting the results themselves, while the proofs can be
found in Section 4.

In Section 5, we show that in one dimension the stochastic reconstruction
theorem becomes the stochastic sewing lemma under weak additional assumptions
needed for the distributional framework.

The last two sections of this paper are dedicated to showing simple examples,
of how the stochastic reconstruction theorem could be applied. In Section 6,
we consider a Gaussian martingale measure $W_{t}(A)$ and a stochastic process
$X(t,x)$, and show how the Walsh type integral [Wal86]

$\int_{0}^{\infty}\int_{\mathbb{R}^{d}}X(s,x)W(ds,dx)$

can be reconstructed as a product between the process $X$ and the distribution
$W(ds,dx)$. This closely resembles the construction at the very start of this
paper and shows how stochastic integration can be seen as a stochastic version
of the Young product.

Since the above integral only assumes martingale properties in one dimension,
this case only yields stochastic dimension $e=1$. To give an example with
$e>1$, we show a similar reconstruction for integration against white noise in
Section 7 and reconstruct the integral

$\int_{\mathbb{R}^{d}}X(z)\xi(dz),$

which can be found in [Qua12].

##  2 Preliminaries

###  2.1 Distributions and Hölder continuity

A property that is typical for the theory of SPDEs is that most solutions are
too irregular to be viewed as functions and need to be seen as distributions.
Let us quickly introduce those: Let $C_{c}^{\infty}$ be the space of smooth,
compactly supported test functions, and let $C_{c}^{r}$ be the space of
compactly supported, $r$ times continuously differentiable functions. We equip
those spaces with the following $C_{c}^{r}$ norms: For any multiindex
$k=(k_{1},\dots,k_{n})$ and $k_{1}+\dots+k_{n}$ times continuously
differentiable function $\psi$, let
$\psi^{(k)}:=\frac{\partial^{k_{1}+\dots+k_{n}}}{\partial
x_{1}^{k_{1}}\dots\partial x_{d}^{k_{d}}}\psi$ be a partial derivative of
$\psi$. Then, for some $r\in\mathbb{N}$, we set

$\left|\left|\psi\right|\right|_{C_{c}^{r}}=\sum_{k_{1}+\dots+k_{d}\leq
r}\left|\left|\psi^{(k)}\right|\right|_{\infty}.$

A distribution on this space is defined as follows:

######  Definition 7.

For any $r\in\mathbb{N}$, a _distribution_ (of order at most $r$) is a linear
map $f:C_{c}^{r}\to\mathbb{R}$, such that for any compact set $K$ there is a
constant $C(K)$, such that for any $\psi\in C_{c}^{r}$ with support in $K$, it
holds that

$\left|f(\psi)\right|\leq C(K)\left|\left|\psi\right|\right|_{C_{c}^{r}}.$

Further, for any $p<\infty$, we call a linear map $f:C_{c}^{r}\to
L_{p}(\Omega)$ a _random distribution_ , if it fulfills

$\left|\left|f(\psi)\right|\right|_{L_{p}}\leq
C(K)\left|\left|\psi\right|\right|_{C_{c}^{r}}.$

We use the notation
$\left|f(\psi)\right|\lesssim\left|\left|\psi\right|\right|_{C_{c}^{r}}$,
where we allow constants hidden in $\lesssim$ to depend on underlying compact
sets.

We need to extend the notion of Hölder continuity to distributions. This
requires us to measure the local behavior of distributions, which can be
achieved with the help of localizations of test functions. To this end, we
call the family of maps $(S^{\lambda})_{\lambda\geq 0}$ a _scaling_ associated
to the vector $s\in\mathbb{R}^{d}$ with $s_{i}>0$ for all $i=1,\dots,d$, if it
is given by

$\displaystyle S^{\lambda}:\mathbb{R}^{d}$ $\displaystyle\to\mathbb{R}^{d}$
$\displaystyle(x_{1},\dots,x_{d})$
$\displaystyle\mapsto(\lambda^{s_{1}}x_{1},\dots,\lambda^{s_{d}}x_{d}).$

The rank of $S$ is given by
$\left|S\right|=\left|s\right|:=s_{1}+\dots+s_{d}$. If $\mathbb{R}^{d}$ has a
scaling $S$ assigned to it, we also equip the space with the seminorm

$\left|x\right|=\left|x_{1}\right|^{\frac{1}{s_{1}}}+\dots+\left|x_{d}\right|^{\frac{1}{s_{d}}},$

which is often referred to as a homogeneous norm, although it is not a norm in
the strict sense. Note that it has the property, that
$\left|S^{\lambda}x\right|=\lambda\left|x\right|$.

To give two short examples, the canonical scaling $x\mapsto\lambda x$ is
simply given by the map $S$ associated to $s=(1,\dots,1)$, while the parabolic
scaling is associated to $s=(2,1,\dots,1)$.

With this in mind, we can now take a look at the localization of a test
function: Let $\psi\in C_{c}^{r}$ be a test function, $x\in\mathbb{R}^{d}$ and
$\lambda\in(0,1)$. Then, the _localized_ test function $\psi_{x}^{\lambda}$
(under the scaling $S$) is given by

$\psi_{x}^{\lambda}(y):=\frac{1}{\lambda^{\left|S\right|}}\psi\left(S^{\frac{1}{\lambda}}(y-x)\right).$

In the case where $S$ is the canonical scaling associated to $(1,\dots,1)$,
this is simply given by the classical notion of localized test functions:

$\psi_{x}^{\lambda}(y):=\frac{1}{\lambda^{d}}\psi\left(\frac{y-x}{\lambda}\right).$

The localization has the following properties:

  * •

Assume $\mathrm{supp}(\psi)\subset[-C,C]^{d}$. Then the support of
$\psi^{\lambda}_{x}$ lies in
$[x-\lambda^{s_{1}}C,x+\lambda^{s_{1}}C]\times\dots\times[x-\lambda^{s_{d}}C,x+\lambda^{s_{d}}C]$,
and

  * •

it holds that
$\int_{\mathbb{R}^{d}}\left|\psi(y)\right|dy=\int_{\mathbb{R}^{d}}\left|\psi_{x}^{\lambda}(y)\right|dy$.

These properties justify thinking of $\psi_{x}^{\lambda}$ as the density of a
measure on $\mathbb{R}^{d}$, which is highly concentrated around $x$ for small
$\lambda$. Indeed, as $\lambda\to 0$, this converges to the Dirac measure
$\delta_{x}$: For all continuous functions $f:\mathbb{R}^{d}\to\mathbb{R}$

$\langle f,\psi_{x}^{\lambda}\rangle\xrightarrow{\lambda\to 0}f(x),$ (2.1)

where $\langle f,g\rangle=\int_{\mathbb{R}^{d}}f(x)g(x)dx$ is simply the
$L_{2}(\mathbb{R}^{d})$ scalar product.

For a true distribution $g$ (i.e. there is no function $\tilde{g}$, such that
$g(\psi)=\langle\tilde{g},\psi\rangle$ for all $\psi\in C_{c}^{r}$),
$g(\psi_{x}^{\lambda})$ is bound to diverge as $\lambda\to 0$. (If it were to
converge, we could set $\tilde{g}(x)=g(\delta_{x})=\lim_{\lambda\to
0}g(\psi_{x}^{\lambda})$) The speed of divergence is going to give us a way to
define the regularity of $g$, which will help us to define Hölder spaces of
negative regularity.

Let us first introduce Hölder continuity for positive regularities: For an
$\alpha\in(0,1]$, and a compact set $K\subset\mathbb{R}^{d}$, let

$C^{\alpha}(K):=\\{f\in C^{0}(\mathbb{R}^{d})\leavevmode\nobreak\
|\leavevmode\nobreak\ \exists C>0\forall x,y\in K:\left|f(x)-f(y)\right|\leq
C\left|x-y\right|^{\alpha}\\},$

and we call $C^{\alpha}:=\bigcap_{K\subset\mathbb{R}^{d}}C^{\alpha}(K)$ the
set of (locally) $\alpha$-Hölder continuous functions, where the intersection
is taken over all compact sets $K$ in $\mathbb{R}^{d}$. We also refer to these
spaces simply as the space of Hölder-continuous functions (and drop the
“locally”), as every function and distribution in this paper will be only
locally Hölder continuous. In consistence with the $\lesssim$ notation, we
write $\left|f(x)-f(y)\right|\lesssim\left|x-y\right|^{\alpha}$ for $f\in
C^{\alpha}$.

For random processes, we define the space of $\alpha$-Hölder continuous
processes by

$C^{\alpha}(L_{p}(\Omega),K):=\\{X:K\to L_{P}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\
\left|\left|X(x)-X(y)\right|\right|_{L_{p}}\lesssim\left|x-y\right|^{\alpha}\\}\,,$

where $C^{\alpha}(L_{p})$ is defined similarly to before.

As mentioned above, for $\alpha<0$, we measure the Hölder regularity of a
distribution $f$ by measuring the speed of divergence of
$f(\psi_{x}^{\lambda})$ for $\lambda\to 0$. More specifically, we define the
space $C^{\alpha}$ for negative $\alpha$ as follows:

######  Definition 8.

Let $\alpha<0$, and $r\in\mathbb{N}$ with $r-1\leq\left|\alpha\right|<r$. A
distribution $f:C_{c}^{r}\rightarrow\mathbb{R}$ is in the space $C^{\alpha}$,
if for any compact set $K\subset\mathbb{R}^{d}$, there is some constant
$C(K)$, such that for every distribution $\psi$ with
$\left|\left|\psi\right|\right|_{C_{c}^{r}}=1$ and compact support in
$[-1,1]^{d}$ and for all $x\in K$ the following holds:

$\left|f(\psi_{x}^{\lambda})\right|\leq C(K)\lambda^{\alpha}.$ (2.2)

For a random distribution, we say that $f\in C^{\alpha}(L_{p}(\Omega))$, if it
holds that

$\left|\left|f(\psi_{x}^{\lambda})\right|\right|_{L_{p}}\leq
C(K)\lambda^{\alpha}.$ (2.3)

######  Remark 9.

Note that any random $C_{c}^{r}$-distribution is in
$C^{-\left|S\right|-r}(L_{p})$, since
$\left|\left|\psi_{x}^{\lambda}\right|\right|_{C_{c}^{r}}\lesssim\lambda^{-\left|S\right|-r}\left|\left|\psi\right|\right|_{C_{c}^{r}}$.

######  Remark 10.

In general, for a random distribution $f$ and any $\omega\in\Omega$,
$\psi\mapsto f(\psi)(\omega)$ is not a distribution. However, it is possible
to show a version of Kolmogorov’s continuity theorem for negative Hölder
spaces: Given an $f\in C^{\alpha}(L_{p})$, it holds that for almost all
$\omega\in\Omega$, $f(\cdot,\omega)\in C^{\hat{\alpha}}(L_{p})$ for a small
enough $\hat{\alpha}<\alpha$. By the previous remark, it follows that any
random distribution $f$ is a.s. a deterministic distribution on some space
$C_{c}^{\hat{r}}$ for a large enough $\hat{r}$. A detailed statement of this
can be found in the appendix, Lemma 64.

##  3 Main results

In this section, we are going to present our main result, the stochastic
reconstruction theorem, as well as two variations of it. Since the proofs are
rather technical, we postpone them to a separate section.

###  3.1 Stochastic Reconstruction

Let us start by rigorously defining the notion of random germ:

######  Definition 11.

We call $(F_{z})_{z\in\mathbb{R}^{d}}$ a _random germ_ , if for all
$z\in\mathbb{R}^{d}$, $F_{z}:C_{c}^{r}\to L_{p}(\Omega)$ is a continuous,
linear map for some fixed number $1\leq p<\infty$.

Recall that Martin Hairer showed in [Hai14] that the reconstructing sequence

$f_{n}(\psi):=\sum_{y\in\Delta_{n}}F_{y}(\phi_{y}^{n})\langle\phi_{y}^{n},\psi\rangle$
(3.1)

for a wavelet function $\phi$ (see the appendix for details) converges to a
limiting distribution, whenever the germ $(F_{z})_{z\in\mathbb{R}^{d}}$
fulfills a property called $\gamma$-coherence for a $\gamma>0$. (Note that
Hairer did not namely mention the coherence property. It was introduced by
Caravenna and Zambotti in [CZ20]. However, Hairer’s proof works for 3.1 under
the coherence assumption of Caravenna-Zambotti.)

If the germ $(F_{x})_{x\in\mathbb{R}^{d}}$ is a random germ, it is very
reasonable to assume random cancellations in the above sum and thus
convergence for certain $\gamma<0$, if the germ has some form of martingale
properties. As a motivation, we want to start with a short example: Consider
space-time white noise $\xi(t,x)$ (see Section 7 for a formal introduction) in
$1+1$ dimension and an $\alpha$-Hölder continuous deterministic function $X\in
C^{\alpha}(\mathbb{R}^{2})$ for some $\alpha>0$. We show in Section 7 that,
for the germ $(F_{(t,x)})_{(t,x)\in\mathbb{R}^{2}}$ defined by

$F_{(t,x)}(\psi):=X(t,x)\xi(\psi)=X(t,x)\int\int\psi(s,y)\xi(s,y)dsdy,$ (3.2)

the reconstructing sequence $f_{n}(\psi)$ converges in $L_{p}(\Omega)$ for
$2\leq p<\infty$ to

$f_{n}(\psi)\xrightarrow{n\to\infty}\int\int X(s,y)\psi(s,y)\xi(s,y)dsdy,$

where the right-hand side is a Walsh-type integral against white noise, see
[Wal86] for reference. It is also not hard to see, that $F_{(t,x)}$ is only
$(-1+\alpha)^{-}$-coherent (which reflects $\xi$ having regularity $(-1)^{-}$,
if we use the canonical scaling $s=(1,1)$, and $X$ having regularity
$\alpha$). So this example shows a reconstruction with a $\gamma>-1$, an
improvement of $+1$ over classic reconstruction. If one compares this to the
stochastic sewing lemma in which one usually gets an improvement of
$+\frac{1}{2}$, it seems as if one can apply the sewing lemma twice in the
above construction. This makes a lot of sense, as the white noise has
martingale properties in both space and time directions.

However, since we also wish to cover more general noises, which might not have
these properties in all directions (e.g. white noise in time, colored noise in
space), we will split our dimensions $1,\dots,d$ into two types of directions:
Those, in which we have certain stochastic properties, and those, in which we
lack such properties. We call the number of directions, which have such
properties the _stochastic dimension_ of the germ.

Before we formally introduce this concept, let us make two quick remarks:

######  Remark 12.

The above construction also works for a stochastic process $X(t,x)$, which is
adapted to the sigma-algebra $\mathcal{F}_{t}$ defined in 3.4, i.e. $X(t,x)$
is $\mathcal{F}_{t}$ measurable for all $(x,t)\in\mathbb{R}^{2}$, which is
$\alpha$-Hölder continuous in the following sense:

$\left|\left|X(t,x)-X(s,y)\right|\right|_{L_{p}}\lesssim\left|(t,x)-(s,y)\right|^{\alpha}.$

######  Remark 13.

It should be noted that for the above germ, Caravenna-Zambotti’s
reconstructing sequence given by

$f_{n}(\psi)=\int_{\mathbb{R}^{d}}F_{z}(\rho_{z}^{\epsilon_{n}})\psi(z)dz$
(3.3)

converges to the same limit for deterministic functions $X$. To see this, note
that dominated convergence and the Itô-isometry ([Qua12], equation (121))
imply
$f_{n}(\psi)=\int_{\mathbb{R}^{d}}(\rho^{\epsilon_{n}}(-\cdot)*(X\psi))(y)\xi(y)dy$.
One then shows
$\rho^{\epsilon_{n}}(-\cdot)*(X\psi)\xrightarrow{\epsilon_{n}\to 0}X\psi$ in
$L_{2}(\mathbb{R}^{d})$. Using again the Itô-isometry, we get that
$f_{n}(\psi)$ converges to $\int_{\mathbb{R}^{d}}X(y)\psi(y)\xi(y)dy$ in
$L_{2}$.

Because of this, we suspect that the main result of this paper can also be
achieved using their mollification techniques.

Let us fix the stochastic setting for reconstruction: Using our example above
as a motivation, we see that $(F_{(t,x)})_{(t,x)\in\mathbb{R}^{2}}$ is a
random germ over the probability space $(\Omega,\mathcal{F},P)$, where
$\mathcal{F}=\sigma(\xi(\psi)|\psi\in L_{2}(\mathbb{R}^{2}))$ is the sigma-
algebra generated by white noise. This sigma-algebra comes with a natural
filtration, which one can write informally as
“$\mathcal{F}_{t}=\\{\xi(s,x)|s\leq t\\}$”, but is more formally given by

$\mathcal{F}_{t}=\sigma(\xi(\psi)|\mathrm{supp}(\psi)\subset(-\infty,t]\times\mathbb{R}).$
(3.4)

Let now $t\in\mathbb{R}$ and $\psi$ with a support in
$(-\infty,s]\times\mathbb{R}$ be fixed. It then holds that $\xi(\psi)$ is
$\mathcal{F}_{s}$-measurable and we assume that $X(\cdot,x)$ is adapted to
$(\mathcal{F}_{t})_{t\in\mathbb{R}}$, i.e. $X(t,x)$ is
$\mathcal{F}_{t}$-measurable for all $x\in\mathbb{R}$. It immediately follows,
that $F_{t,x}(\psi)=X(t,x)\xi(\psi)$ is $\mathcal{F}_{\max(s,t)}$-measurable.

Of course, a similar construction can be made in the spatial direction $x$
with respect to the filtration
$\mathcal{F}_{x}^{\text{spatial}}:=\sigma(\xi(\psi)|\mathrm{supp}(\psi)\subset\mathbb{R}\times(-\infty,x])$.
This gives us two different filtrations, and our germ is adapted to both in
the sense that for $t,x\in\mathbb{R}$ and $\psi$ with support in
$(-\infty,s]\times(-\infty,y]$, $F_{t,x}(\psi)$ is $\mathcal{F}_{\max(s,t)}$
and $\mathcal{F}_{\max(x,y)}^{\text{spatial}}$-measurable, as long as $X$ is
adapted to both filtrations.

We want to generalize this adaptedness property: Let $(\Omega,\mathcal{F},P)$
be a complete probability space. As in the example above, consider $e$ many
filtrations $(\mathcal{F}_{t}^{(i)})_{t\in\mathbb{R}}$, $i=1,\dots,e$. We make
the following assumptions about these filtrations:

  * •

Completeness: We assume, that for each $i=1,\dots,e$, $t\in\mathbb{R}$,
$\mathcal{F}^{(i)}_{t}$ is complete.

  * •

Right-continuity: We assume, that for $i=1,\dots,e$,
$(\mathcal{F}_{t}^{(i)})_{t\in\mathbb{R}}$ is a right continuous filtration.

  * •

Orthogonality: We assume that conditioning in one direction does not change
measurability along the other directions. More precisely, let $X$ be a
$\mathcal{F}_{z_{i}}^{(i)}$-measurable random variable for some $1\leq i\leq
e$ and $z_{i}\in\mathbb{R}$. We assume that $E^{\mathcal{F}_{z_{j}}^{(j)}}X$
is still $\mathcal{F}_{z_{i}}^{(i)}$-measurable for all $1\leq j\leq e,j\neq
i$ and all $z_{j}\in\mathbb{R}$.

Let us give a short intuition, of what we need each of these properties for.
The reconstructing sequence 3.1 will be a limit in $L_{p}$, so completeness is
necessary to ensure that the limit, as well as modifications of the limit, are
still $\mathcal{F}_{t}^{(i)}$ measurable for all $i$ and large enough $t$. A
closer look at the reconstructing sequence reveals that $f_{n}(\psi)$ is only
$\mathcal{F}^{(i)}_{y_{i}+\epsilon_{n}}$-measurable for $\psi$ with support in
$(-\infty,y_{1}]\times\dots\times(-\infty,y_{d}]$ and a certain sequence
$\epsilon_{n}\xrightarrow{n\to\infty}0$. Right continuity then assures that
the limit is $\mathcal{F}^{(i)}_{y_{i}}$-measurable.

Orthogonality is used more subtly: The proof requires us to use the BDG-type
inequality B.1 inductively over all stochastic directions. However, without
any assumption on how the filtrations $(\mathcal{F}_{x_{i}}^{(i)})_{x_{i}}$,
$i=1,\dots,e$ interact with each other,
$E^{\mathcal{F}_{x_{i}}^{(i)}}F_{x}(\psi)$ will not have any adaptedness
property with regards to the filtrations $(\mathcal{F}_{x_{j}}^{(j)})_{x_{j}}$
for $i\neq j$. So orthogonality ensures that we can use B.1 several times with
respect to different filtrations.

It should also be noted that if $e=1$, orthogonality is automatically
fulfilled as one only has one filtration.

######  Remark 14.

Another way of formulating orthogonality is requiring that conditioning on
$\mathcal{F}^{(i)}$ and $\mathcal{F}^{(j)}$ commutes, i.e. for all $1\leq
i,j\leq e$ with $i\neq j$, $z_{i},z_{j}\in\mathbb{R}$ and for a random
variable $X$, it holds that
$E^{\mathcal{F}_{s_{j}}^{(j)}}E^{\mathcal{F}_{s_{i}}^{(i)}}X=E^{\mathcal{F}_{s_{i}}^{(i)}}E^{\mathcal{F}_{s_{j}}^{(j)}}X$.

######  Remark 15.

This setting can be seen as a generalization of the setting used in [Imk85],
[Wal86a] to construct multiparameter martingales: The authors start from a
multiparameter filtration $(\mathcal{F}_{x})_{x\in\mathbb{R}^{d}}$ and
constructs

$\mathcal{F}_{y}^{(i)}:=\sigma\left(\bigcup_{x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{d}\in\mathbb{R}}\mathcal{F}_{(x_{1},\dots,x_{i-1},y,x_{i+1},\dots,x_{d})}\right)\,.$

In this case, orthogonality is equivalent to conditional independence of
$\mathcal{F}_{y}^{(i)}$, $i=1\dots,d$, see hypothesis (F4) of [Wal86a] on page
348. We prefer to directly work with the filtrations
$(\mathcal{F}_{x}^{(i)})$, as our proofs do not use any additional structure
of the multiparameter setting.

######  Remark 16.

We want to mention that the condition of orthogonality does not come for free:
While it is a standard assumption in multiparameter martingale theory, e.g.
the noises constructed in [BE03] do not generate such filtrations. For a rule
of thumb, Brownian sheet and deterministic kernel integrated over Brownian
sheet $B^{H}_{t}:=\int_{(-\infty,t]^{d}}K(t,s)dB_{s}$ like fractional Brownian
sheet (see [WX07] for the definition of $B^{H}$ in dimension $d>1$ and [Pic11]
for the kernel $K$) will be adapted to orthogonal filtrations, while more
general Gaussian noises like the spectral density construction of [BE03] will
not generate such filtrations. Theorem 30 is better suited to deal with those
cases.

We denote by $\pi_{i}$ the projection onto the $i$-th coordinate. With this
notation, we can now define the term stochastic dimension:

######  Definition 17.

Let $(F_{x})_{x\in\mathbb{R}^{d}}$ be a random germ in $d$ dimensions, and let
$e\leq d$. Let $(\mathcal{F}^{(i)}_{z})_{z\in\mathbb{R}},i=1,\dots,e$ be
filtrations of some $\sigma$-Algebra $\mathcal{F}$, which fulfill the above
three properties of completeness, right-continuity, and orthogonality.

We say that $(F_{x})$ is of _stochastic dimension_ $e$, if for $i=1,\dots,e$:

For any test function $\psi$ with support
$\mathrm{supp}(\psi)\subset\mathbb{R}^{i-1}\times(-\infty,y]\times\mathbb{R}^{d-i}$,
it holds that

$F_{x}(\psi)\text{ is
}\mathcal{F}^{(i)}_{\max(\pi_{i}(x),y)}\text{-measurable}.$ (3.5)

We further say that a random distribution $f$ has stochastic dimension $e$, if
for any test function $\psi$ with support
$\mathrm{supp}(\psi)\subset\mathbb{R}^{i-1}\times(-\infty,y]\times\mathbb{R}^{d-i}$

$f(\psi)\text{ is }\mathcal{F}^{(i)}_{y}\text{-measurable}.$

######  Remark 18.

As most readers have undoubtedly noticed, we use a two-directional setting in
the sense that we always use $t\in\mathbb{R}$ instead of $t\in[0,\infty)$. In
one dimension, our example 3.2 becomes a two-sided Brownian motion, not a
classical one.

Since this whole theory highly depends on recentering and scaling operations,
the origin point does not have any special role, which makes this two-
directional setting more natural. However, one can use the locality of this
theory to achieve a one-sided setting, by restricting oneself to compact sets
$K\subset[0,\infty)\times\mathbb{R}^{d-1}$ in Definition 19 and Theorem 22.

Since we work with a scaling $S$ associated to some $s\in\mathbb{R}^{d}_{+}$,
we do not consider $e$ to be the number of stochastic dimensions, but rather
consider

$E:=\sum_{i=1}^{e}s_{i}\leq\left|S\right|.$

With the definition of the stochastic dimension in mind, we can now formulate
a stochastic version of the coherence property. Since we aim to get
$L_{p}$-convergence, let us fix now some $2\leq p<\infty$ and define
stochastic coherence as follows:

######  Definition 19.

Let $(F_{x})_{x\in\mathbb{R}^{d}}$ be a random germ with stochastic dimension
$e\leq d$ and let $E$ be as above. We call $(F_{x})$ _stochastically
$\gamma$-coherent_, if there is an $\alpha\leq\gamma$, such that the following
holds for all test functions $\psi\in C_{c}^{r}(\mathbb{R}^{d})$ with
$\left|\left|\psi\right|\right|_{C_{c}^{r}}=1$ and
$\mathrm{supp}(\psi)\subset[0,\tilde{R}]^{e}\times[-\tilde{R},\tilde{R}]^{d-e}$
for some $\tilde{R}>0$, for the stochastic directions $i=1,\dots,e$, for any
compact set $K\subset\mathbb{R}^{d}$ and for any $x,y\in K$ with
$\pi_{i}(x)\leq\pi_{i}(y)$:

$\displaystyle\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(x)}}(F_{x}-F_{y})(\psi_{y}^{\epsilon})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\epsilon^{\alpha}(\left|x-y\right|+\epsilon)^{\gamma-\alpha}$
(3.6)
$\displaystyle\left|\left|(F_{x}-F_{y})(\psi_{y}^{\epsilon})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\epsilon^{\alpha}(\left|x-y\right|+\epsilon)^{\gamma-\frac{E}{2}-\alpha},$
(3.7)

where the constant appearing in $\lesssim$ is allowed to depend on the compact
set $K$ and the radius $\tilde{R}$.

We sometimes write $(\gamma,\alpha)-coherent$, if we need the explicit value
of $\alpha$. Let us stress that the constant appearing in $\lesssim$ (both in
the above definition and the results throughout this paper) should not depend
on $\psi$, as long as $\tilde{R}$ and
$\left|\left|\psi\right|\right|_{C_{c}^{r}}$ are fixed.

######  Remark 20.

In the usual construction of stochastic integrals, it is far more natural to
use left-side approximations than two-sided ones. Thus, 3.2 is only a useful
approximation, if the support of $\psi$ is in $[t,\infty)\times[x,\infty)$. We
therefore only consider test functions with positive support in the stochastic
directions in the definition above and Theorem 22. It is however possible to
get a two-sided stochastic reconstruction, which will be discussed in Section
3.2.

######  Remark 21.

It is possible to allow $\alpha=\alpha(K)$ to depend on the compact set $K$.
In this case, one simply has to replace $\alpha$ with $\alpha(\tilde{K})$ in
the proofs, for some fattening $\tilde{K}$ of $K$ which depends on the support
of the wavelet.

Note that each stochastic dimension reduces the coherence required by
$(F_{x})$ in 3.7 by $\frac{1}{2}$, resulting in a total reduction of
$\frac{E}{2}$, under the corresponding scaling. In particular, if the scaling
is just the canonical scaling $s=(1,\dots,1)$, we get a reduction by
$\frac{e}{2}$. Further note that the support of $\psi_{y}^{\epsilon}$ lies
completely on the right-hand side of the time we condition on, $\pi_{i}(x)$,
with respect to the direction $i$.

Recall that a random distribution $\tilde{f}$ is a modification of $f$, if for
all test functions $\psi\in C_{c}^{r}$, $f(\psi)=\tilde{f}(\psi)$ almost
surely. With this in mind, we can formulate our main result:

######  Theorem 22 (Stochastic Reconstruction).

Let $(F_{x})_{x\in\mathbb{R}^{d}}$ be a random germ for some $2\leq p<\infty$
with stochastic dimension $e\leq d$, which is stochastically
$(\gamma,\alpha)$-coherent for some $\gamma>0$ and $\alpha<0$. Set
$\tilde{r}:=r\cdot\min_{i=1,\dots,d}s_{i}$. Assume that our parameters fulfill
$\alpha+\tilde{r}>0$ and $\gamma-\frac{E}{2}+\tilde{r}>0$.

Then, there is a unique random distribution $f$ (up to modifications) with
stochastic dimension $e$ with regard to the same filtrations
$\mathcal{F}^{(i)}$ as our germ, such that the following holds for
$i=1,\dots,e$, any test function $\psi\in C_{c}^{r}$ with
$\left|\left|\psi\right|\right|_{C_{c}^{r}}=1$ and whose support is in
$\mathrm{supp}(\psi)\subset[\tilde{C},\tilde{R}]^{e}\times[-\tilde{R},\tilde{R}]^{d-e}$
for some $0\leq\tilde{C}\leq\tilde{R}$ and $\lambda\in(0,1]$:

$\displaystyle\left|\left|(f-F_{x})(\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\lambda^{\gamma-\frac{E}{2}}$ (3.8)
$\displaystyle\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(x)}}(f-F_{x})(\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\lambda^{\gamma},$ (3.9)

where the constant in $\lesssim$ again depends on the compact set
$K\subset\mathbb{R}^{d}$ with $x\in K$ and the constants
$\tilde{C},\tilde{R}$.

Note that the support of $\psi$ does have to be strictly positive. This is
necessary to “fit in” a wavelet between the zero and the support of $\psi$. If
we set $\tilde{C}=2R$ (recall that the support of our wavelet is in
$[C,R]^{d}$), the constant in $\lesssim$ generated by $\tilde{C}$ disappears.

######  Remark 23.

The restrictions to $\psi$ might seem strange to the reader. Indeed, it seems
that this theorem only makes a statement about test functions with (strictly)
positive supports in the stochastic directions. This is not true since the
localizations $\psi_{y}^{\epsilon}$ can have negative supports in all
directions depending on $y$. The support does only serve to separate the point
of localization of $\psi$ and the index of the germ $F_{y}$. Another way to
formulate the result is to use a $\psi$ with any compact support, and take a
look at $F_{(y-S^{\lambda}C)}(\psi_{y}^{\lambda})$ for a certain vector $C$
depending on the support of $\psi$ and which directions are supposed to be
stochastic.

######  Remark 24.

If one is given a germ $(F_{x})_{x\in\mathbb{R}^{d}}$ such that
$\alpha+\tilde{r}$ or $\gamma-\frac{E}{2}+\tilde{r}$ is smaller than $0$, one
can always restrict $F_{x}$ to a space $C_{c}^{\hat{r}}$ for a large enough
$\hat{r}$ to fulfill the conditions on the parameters.

Before we prove this theorem, let us discuss two modifications of it:

###  3.2 Two-sided Reconstruction

The theorem we presented above is modeled after left-sided approximations of
stochastic integrals. However, it is also possible (and closer to the original
reconstruction Theorem) to have a two-sided reconstruction, i.e. to use test
functions $\psi$ with support in $[-\tilde{R},\tilde{R}]^{d}$ for some
$\tilde{R}>0$. In this setting, it no longer makes sense to condition
$F_{x}(\psi_{x}^{\lambda})$ onto $\mathcal{F}^{(i)}_{\pi_{i}(x)}$, since the
conditioning would not have any effect on test functions with purely negative
support. Instead, we are going to condition it on the left boundary of the
support of $\psi_{x}^{\lambda}$, i.e. onto
$\mathcal{F}^{(i)}_{\pi_{i}(x)-\lambda^{s_{i}}\tilde{R}}$.

######  Definition 25 (two-sided stochastic coherence).

Let $(F_{x})_{x\in\mathbb{R}^{d}}$ be a stochastic germ with stochastic
dimension $e\leq d$. We call $(F_{x})$ _two-sided stochastically
$\gamma$-coherent_, if there is an $\alpha<\gamma$, such that the following
holds for all $\psi\in C_{c}^{r}$ with
$\left|\left|\psi\right|\right|_{C_{c}^{r}}=1$ and support in
$[-\tilde{R},\tilde{R}]^{d}$ for some $\tilde{R}>0$, for the stochastic
directions $i=1,\dots,e$, for any compact set $K\subset\mathbb{R}^{d}$ and for
any $x,y\in K$:

$\displaystyle\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(y)-\lambda^{s_{i}}\tilde{R}}}(F_{x}-F_{y})(\psi_{y}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\lambda^{\alpha}(\left|x-y\right|+\lambda)^{\gamma-\alpha}$
$\displaystyle\left|\left|(F_{x}-F_{y})(\psi_{y}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\lambda^{\alpha}(\left|x-y\right|+\lambda)^{\gamma-\frac{E}{2}-\alpha},$

where the constant in $\lesssim$ is allowed to depend on the compact set $K$
and $\tilde{R}$.

Note that this definition also does not have the condition, that
$\pi_{i}(x)\leq\pi_{i}(y)$, but rather uses arbitrary $x,y$. We want to add
that there is no cheap way to get from the one-sided coherence property to the
two-sided, or vice versa: If one replaces $\psi$ with a shifted function
$\psi_{\vec{r}}$ to get a test function with a purely positive support, and
substitute $y$ with $\tilde{y}=y-S^{\lambda}\vec{r}$, such that
$\psi_{y}^{\lambda}=(\psi_{\vec{r}})_{\tilde{y}}^{\lambda}$, one gets
$F_{\tilde{y}+S^{\lambda}\vec{r}}$ instead of $F_{\tilde{y}}$. This highlights
the main difference between the two versions: In the one-sided reconstruction,
$F_{y}$ only gets tested against the future, while in the two-sided version,
$F_{y}$ sits “in the middle” of the support of the test function. This makes
the two-sided version closely connected to Stratonovich integration. Since we
are interested in the martingale properties, the one-sided version (which is
closely connected to Itô integration) is more natural.

Using this property, one can show the following theorem:

######  Theorem 26 (two sided stochastic reconstruction).

Let $(F_{x})_{x\in\mathbb{R}^{d}}$ be a random germ for some $2\leq p<\infty$
with stochastic dimension $e\leq d$, which is two-sided stochastically
$(\gamma,\alpha)$-coherent for some $\gamma>0$, $\alpha<0$. Let
$\tilde{r}:=r\cdot\min_{i=1,\dots,d}(s_{i})$ and assume $\alpha+\tilde{r}>0$
and $\gamma-\frac{E}{2}+\tilde{r}>0$.

Then, where is a unique (up to modifications) random distribution $f$ with
stochastic dimension $e$ with respect to the same filtrations as the germ
$(F_{x})$, such that the following holds for $i=1,\dots,e$ and any test
function $\psi$ with support in $[-\tilde{R},\tilde{R}]^{d}$ for some
$\tilde{R}>0$ and $\left|\left|\psi\right|\right|_{C_{c}^{r}}=1$ and
$\lambda\in(0,1]$:

$\displaystyle\left|\left|(f-F_{x})(\psi_{x}^{\lambda})\right|\right|_{L_{2}}$
$\displaystyle\lesssim\lambda^{\gamma-\frac{E}{2}}$
$\displaystyle\left|\left|E^{\mathcal{F}^{(i)}_{\check{S}_{i}(x)}}(f-F_{x})(\psi_{x}^{\lambda})\right|\right|_{L_{2}}$
$\displaystyle\lesssim\lambda^{\gamma},$

where $\check{S}_{i}(x):=\pi_{i}(x)-\lambda^{s_{i}}(\tilde{R}+\tilde{C})$ for
some $\tilde{C}>0$, and the constant in $\lesssim$ is allowed to depend on the
compact set $K\subset\mathbb{R}^{d}$ with $x\in K$ and on
$\tilde{C},\tilde{R}$.

###### Proof.

The proof is essentially the same as for the one-sided version. However, one
can use the more usual wavelets with supports fulfilling
$\mathrm{supp}(\phi),\mathrm{supp}(\hat{\phi})\subset[-R,R]^{d}$. ∎

######  Remark 27.

Note that this theorem has a similar fattening happening in the uniqueness-
property: We do not condition on the left boundary
$\pi_{i}(x)-\lambda^{s_{i}}\tilde{R}$, but we leave a “gap”, in which enough
wavelets fit. If one sets $\tilde{C}=4R$, then $\tilde{C}$ does not contribute
a constant in the above inequalities.

###  3.3 The covariance setting in $L_{2}$

Theorem 22 contains an assumption, which might be stronger than one would
expect at first: Namely that $F_{x}(\psi)$ is
$\mathcal{F}_{\max(\pi_{i}(x),y)}^{(i)}$ measurable for $i=1,\dots,d$ for test
functions $\psi$ with the respective support, for a set of filtrations which
have the orthogonality property. As mentioned in Remark 16, this adaptedness
condition does not come for free. Furthermore, one can find several
interesting examples, in which stochastic integrals have been constructed,
even if one does violate this property:

  * •

In the example 3.2, we have to assume that $X$ is adapted to all
$\mathcal{F}^{(i)}_{x}$ in the above mentioned way. However, it is well known
that the construction of the Walsh integral should work for all processes $X$,
which are adapted to the time-filtration $\mathcal{F}^{(1)}_{t}$ [Wal86]. A
more throughout discussion of this can be found in Section 7.

  * •

More generally, Malliavin calculus [Nua06] successfully constructs the
Skorohod integral over white noise, even for non-adapted processes $X$.

  * •

The technique used in [HP21] to analyze regularization by noise only uses
local non-determinism properties of the underlying noise for their
construction, which only need the conditional covariances
$\mathrm{cov}(X(t)\leavevmode\nobreak\ |\leavevmode\nobreak\
X(s_{1}),\dots,X(s_{n}))$ to be known for a Gaussian noise $X$. These
techniques should be possible for certain noises defined through their
spectral densities [BE03], even if they do not generate orthogonal
filtrations.

All of the above examples can circumvent the use of conditional expectation by
using the covariances of the underlying Gaussian processes in some way. This
motivated us to consider the case, in which the germ $F_{x}(\psi)$ is not
adapted to orthogonal filtrations, but we instead know its covariance
$E(F_{x}(\psi_{1})F_{y}(\psi_{2}))$.

Let us take a look at how the conditional expectation is used in the proof of
Theorem 22: We use it to establish an inequality of the form
$\left|\left|\sum_{x\in\Delta_{n}}g_{x}\right|\right|_{L_{p}}\lesssim\left(\sum_{x\in\Delta_{n}}\left|\left|g_{x}\right|\right|^{2}_{L_{p}}\right)^{\frac{1}{2}}$
plus some remainder (see estimate 4.11) for a technical random term $g_{x}$
defined in 4.10. The remainder gets controlled by the additional property 3.6
of the stochastic coherence property, which allows us to get a better bound
for $\left|\left|\sum_{x\in\Delta_{n}}g_{x}\right|\right|_{L_{p}}$ than simply
pulling the norm into the sum.

If $p=2$, there is no need for such a bound, as one can explicitly calculate
$\left|\left|\sum_{x\in\Delta_{n}}g_{x}\right|\right|_{L_{2}}^{2}=\sum_{x\in\Delta_{n}}\left|\left|g_{x}\right|\right|_{L_{2}}^{2}+\sum_{x\neq
y}E(g_{x}g_{y})$. So what we need is an upper bound on $\sum_{x\neq
y}E(g_{x}g_{y})$ to replace the stochastic machinery used in Theorem 22. So we
need an additional property that allows us to bound $E(g_{x}g_{y})$ for $x\neq
y$.

In practice, this should be a bound on the term
$\Pi_{i=1}^{2}(F_{x_{i}}-F_{y_{i}})((\psi_{i})_{y_{i}}^{\epsilon_{i}})$ for
$(x_{1},y_{1})$ and $(x_{2},y_{2})$ “far enough” from each other, and two test
functions $\psi_{1},\psi_{2}$ in $C_{c}^{r}$. To quantify how far apart
$(x_{1},y_{1})$ and $(x_{2},y_{2})$ need to be, we introduce the following
concept:

######  Definition 28.

We call the following intervals effective supports of the corresponding
expressions:

  * •

For $(F_{x}-F_{y})(\psi)$, the effective support is the smallest possible
intervall
$[a,b]:=[\pi_{1}(a),\pi_{1}(b)]\times\dots\times[\pi_{d}(a),\pi_{d}(b)]\subset\mathbb{R}^{d}$,
such that $x,y\in[a,b]$ and $\mathrm{supp}(\psi)\subset[a,b]$.

  * •

Given the reconstruction $f$ of Theorem 30, and a test function $\psi$ with
support in $[0,\tilde{R}]^{d}$, we say that the effective support of
$(f-F_{x})(\psi_{x}^{\lambda})$ is given by
$[\pi_{1}(x),\pi_{1}(x)+2\lambda^{s_{1}}\tilde{R}]\times\dots\times[\pi_{d}(x),\pi_{d}(x)+2\lambda^{s_{d}}\tilde{R}]$.

Furthermore, given an $e\leq d$, we call
$[a,b]_{e}:=[\pi_{1}(a),\pi_{1}(b)]\times\dots\times[\pi_{e}(a),\pi_{e}(b)]$
as well as
$[\pi_{1}(x),\pi_{1}(x)+2\lambda^{s_{1}}\tilde{R}]\times\dots\times[\pi_{e}(x),\pi_{e}(x)+2\lambda^{s_{e}}\tilde{R}]$
the stochastic effective support of the respective expressions.

Note that the effective support of $(F_{x}-F_{y})(\psi)$ is precisely the
smallest interval that the expression sees. For
$(f-F_{x})(\psi_{x}^{\lambda})$ however, $x$ and
$\mathrm{supp}(\psi_{x}^{\lambda})$ are contained in the smaller interval
$[\pi_{1}(x),\pi_{1}(x)+\lambda^{s_{1}}\tilde{R}]\times\dots\times[\pi_{d}(x),\pi_{d}(x)+\lambda^{s_{d}}\tilde{R}]$
without the factor $2$. The reason we use this factor is that the
reconstruction theorem causes a “fattening” effect, where we need some extra
distance between two test functions to put in another wavelet. The easiest way
to do this is by extending the effective support to be slightly bigger.
Instead of the factor $2$, one can use any factor $\eta\geq 1$, at the cost
that the constants appearing in $\lesssim$ will depend on $\eta$.

With these definitions, we can now give a precise meaning to points
$(x_{1},y_{1})$ $(x_{2},y_{2})$ being “far enough” from each other. We want
them to be so far apart, that the (stochastic) effective supports of
$(F_{x_{i}}-F_{y_{i}})(\psi_{y_{i}}^{\epsilon_{i}})$ are disjoint. This is
motivated by two observations:

  1. 1.

If we think of the example of reconstructing the Walsh integral against white
noise 3.2, we have that white noise $\xi(\psi)$ is independent to
$\mathcal{F}^{(i)}_{x_{i}}$ if
$\mathrm{supp}(\psi)\subset\mathbb{R}^{i-1}\times[x_{i},\infty)\times\mathbb{R}^{d-i+1}$,
and has expected value $E(\xi(\psi))=0$. This implies that
$\left|E\left(\prod_{i=1}^{2}(F_{x_{i}}-F_{y_{i}})(\psi_{i})\right)\right|=0$,
where $F_{x}(\psi)=X(x)\xi(\psi)$ and whenever the effective stochastic
supports of this terms are disjoint. So in this example, disjoint effective
stochastic supports give us a true improvement in the regularity over simply
applying Hölder’s inequality to the product.

  2. 2.

Let $x_{i},y_{i},\psi_{i},i=1,2$ be such that the effective stochastic
supports of $(F_{x_{i}}-F_{y_{i}})(\psi_{i})$ are disjoint and assume w.l.o.g.
$\pi_{1}(x_{1})\leq\pi_{1}(x_{2})$. Then the disjoint effective stochastic
supports imply that $\pi_{1}(y_{1})\leq\pi_{1}(x_{2})$ and
$\mathrm{supp}(\psi_{1})\subset(-\infty,\pi_{1}(x_{2})]$ which shows that
$(F_{x_{1}}-F_{y_{1}})(\psi_{1})$ is
$\mathcal{F}^{(1)}_{\pi_{1}(x_{2})}$-measurable. It follows that

$\displaystyle\left|E\left(\prod_{i=1}^{2}(F_{x_{i}}-F_{y_{i}})(\psi_{i})\right)\right|$
$\displaystyle=\left|E\left((F_{x_{1}}-F_{y_{1}})(\psi_{1})E^{\mathcal{F}_{\pi_{1}(x_{2})}^{(1)}}(F_{x_{2}}-F_{y_{2}})(\psi_{2})\right)\right|$
$\displaystyle\leq\left|\left|(F_{x_{1}}-F_{y_{1}})(\psi_{1})\right|\right|_{L_{2}}\left|\left|E^{\mathcal{F}_{\pi_{1}(x_{2})}^{(1)}}(F_{x_{2}}-F_{y_{2}})(\psi_{2})\right|\right|_{L_{2}}\,.$

So if we assume that inequality 3.6 from the classical setting holds, disjoint
effective stochastic supports also give us better bounds on
$E\left(\left|\prod_{i=1}^{2}(F_{x_{i}}-F_{y_{i}})(\psi_{i})\right|\right)$
than simply applying Hölder’s inequality.

Let us formally write down the coherence property in this setting:

######  Definition 29 (stochastic coherence in the covariance setting).

Let $(F_{x})_{x\in\mathbb{R}^{d}}$ be a stochastic germ with $p=2$ and let
$e\leq d$. We call $(F_{x})$ stochastically $\gamma$-coherent in the
covariance setting, if there is an $\alpha<\gamma$, such that the following
holds for all test functions $\psi_{1},\psi_{2}\in C_{c}^{r}(\mathbb{R}^{d})$
with $\left|\left|\psi_{i}\right|\right|_{C_{c}^{r}}=1$ and
$\mathrm{supp}(\psi_{i})\subset[0,\tilde{R}]^{e}\times[-\tilde{R},\tilde{R}]^{d-e}$
for some $\tilde{R}>0$ for $i=1,2$, for any compact set
$K\subset\mathbb{R}^{d}$ and $x_{1},x_{2},y_{1},y_{2}\in K$ with
$\pi_{i}(x_{j})\leq\pi_{i}(y_{j})$ for $i=1,\dots,e,j=1,2$, such that the
stochastic effective supports of
$(F_{x_{i}}-F_{y_{i}})((\psi_{i})_{y_{i}}^{\epsilon_{i}})$, $i=1,2$ are
disjoint and $\epsilon_{1},\epsilon_{2}>0$:

$\displaystyle\left|\left|(F_{x_{1}}-F_{y_{1}})((\psi_{1})_{y_{1}}^{\epsilon_{1}})\right|\right|_{L_{2}}$
$\displaystyle\lesssim\epsilon_{1}^{\alpha}(\left|x-y\right|+\epsilon)^{\gamma-\frac{E}{2}-\alpha}$
(3.10)
$\displaystyle\left|E\left(\prod_{i=1}^{2}(F_{x_{i}}-F_{y_{i}})((\psi_{i})_{y_{i}}^{\epsilon_{i}})\right)\right|$
$\displaystyle\lesssim\prod_{i=1}^{2}\epsilon_{i}^{\alpha}(\left|x_{i}-y_{i}\right|+\epsilon_{i})^{\gamma-\alpha}\,,$
(3.11)

where the constant appearing in $\lesssim$ is allowed to depend on $K$ and
$\tilde{R}$.

Note that the stochastic dimension $e$ of our problem is no longer an
intrinsic property of the germ $(F_{x})_{x\in\mathbb{R}^{d}}$, but only
appears in the coherence property. If there is any room for confusion, we will
write that $(F_{x})_{x\in\mathbb{R}^{d}}$ is stochastically
$(\gamma,\alpha,e)$ coherent in the covariance setting.

We want to stress that this condition is not a generalization of 3.6, not even
in $p=2$: While one can show a bound on $\sum_{x\neq y}E(g_{x}g_{y})$ under
the conditions of Definition 19, it is in general not true, that
$\left|E(g_{x}g_{y})\right|\lesssim\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(x)}}g_{x}\right|\right|_{L_{2}}\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(y)}}g_{y}\right|\right|_{L_{2}}$
for $\pi_{i}(x)<\pi_{i}(y)$. So 3.6 does not imply 3.11.

Let us now formulate the main result of this section:

######  Theorem 30 (stochastic reconstruction in the covariance setting).

Let $(F_{x})_{x\in\mathbb{R}^{d}}$ be a random germ for some $p\geq 2$ which
is stochastically $(\gamma,\alpha,e)$-coherent in the covariance setting for
some $\gamma>0,\alpha<0$ and $e\leq d$. Let
$\tilde{r}:=r\cdot\min_{i=1,\dots,d}(s_{i})$ and assume $\alpha+\tilde{r}>0$
and $\gamma-\frac{E}{2}+\tilde{r}>0$.

Then, where is a unique (up to modifications) random distribution $f$, such
that the following holds for any test function $\psi\in C_{c}^{r}$ with
$\left|\left|\psi\right|\right|_{C_{c}^{r}}=1$ and with support
$\mathrm{supp}(\psi)\subset[\tilde{C},\tilde{R}]^{e}\times[-\tilde{R},\tilde{R}]^{d-e}$
for some $0<\tilde{C}<\tilde{R}$, and for any compact set
$K\subset\mathbb{R}^{d}$, $x_{1},x_{2}\in K$ and $\lambda\in(0,1]$ such that
the stochastic effective supports of $(f-F_{x_{i}})(\psi)$ are disjoint for
$i=1,2$:

$\displaystyle\left|\left|(f-F_{x_{1}})(\psi_{x_{1}}^{\lambda})\right|\right|_{L_{2}}$
$\displaystyle\lesssim\lambda^{\gamma-\frac{E}{2}}$ (3.12)
$\displaystyle\left|E\left(\prod_{i=1}^{2}(f-F_{x_{i}})(\psi_{x_{i}}^{\lambda})\right)\right|$
$\displaystyle\lesssim\lambda^{2\gamma}$ (3.13)

where the constant in $\lesssim$ again depends on $K$ and the constants
$\tilde{C},\tilde{R}$.

Condition 3.13 is the weakest additional condition we could find, which still
uniquely characterizes the reconstruction. However, it would be closer to the
coherence condition to analyze the expression
$\prod_{i=1}^{2}(f-F_{x})((\psi_{i})_{x_{i}}^{\lambda_{i}})$ for different
test functions $\psi_{1},\psi_{2}$ and different
$\lambda_{1},\lambda_{2}\in(0,1]$. A simple corollary of the proof of Theorem
30 shows that 3.13 still holds in this setting:

######  Corollary 31.

Let $(F_{x})$ be as in Theorem 30 and let $f$ be the reconstructed random
distribution. Let $\psi_{1},\psi_{2}\in C_{c}^{r}$ be two test functions with
$\left|\left|\psi_{i}\right|\right|_{C_{c}^{r}}=1$ and with support
$\mathrm{supp}(\psi_{i})\subset[\tilde{C},\tilde{R}]^{e}\times[-\tilde{R},\tilde{R}]^{d-e}$
for some $0<\tilde{C}<\tilde{R}$, $i=1,2$. For any compact set
$K\subset\mathbb{R}^{d}$, $x_{1},x_{2}\in K$ and
$\lambda_{1},\lambda_{2}\in(0,1]$ such that the stochastic effective supports
of $(f-F_{x_{i}})((\psi_{i})_{x_{i}}^{\lambda_{i}})$ are disjoint, it holds
that:

$\left|E\left(\prod_{i=1}^{2}(f-F_{x_{i}})((\psi_{i})_{x_{i}}^{\lambda_{i}})\right)\right|\lesssim\lambda_{1}^{\gamma}\lambda_{2}^{\gamma}\,,$

where the constant in $\lesssim$ depends on $K,\tilde{C}$ and $\tilde{R}$.

Since the proof of this result relies on the proof of Theorem 30, we will
present it directly after said proof in Section 4.

##  4 Proofs of Theorem 22 and 30

The proofs in this section make use of technical tools, which we did not yet
introduce, namely the BDG-type inequality B.1 and wavelet techniques. As we
will make heavy use of wavelet techniques especially, let us fix a wavelet
bases $(\phi,\Phi)$ here, as well as two constants $0<C<R$ such that
$\mathrm{supp}(\phi),\mathrm{supp}(\hat{\phi})\subset[C,R]^{d}$ holds for all
$\hat{\phi}\in\Phi$. A proper introduction to wavelets can be found in the
appendix.

###  4.1 Proof of Theorem 22

The main idea of the proof is to take the reconstructing sequence

$f^{(n)}(\psi):=\sum_{x\in\Delta_{n}}F_{x}(\phi_{x}^{n})\langle\phi_{x}^{n},\psi\rangle\,,$
(4.1)

and show the following three properties under the assumptions of Theorem 22

  1. 1.

$f^{(n)}$ converges to some distribution $f$ for $n\to\infty$.

  2. 2.

$f$ fulfills inequalities 3.8 and 3.9.

  3. 3.

There is at most one distribution $f$ fulfilling inequalities 3.8 and 3.9.

Our strategy for the proof is to decompose
$f^{(l)}=f^{(n)}+\sum_{k=n}^{l-1}(f^{(k+1)}-f^{(k)})$ and split a test
function $\psi$ into $\psi=P_{n}\psi+\sum_{m=n}^{\infty}\hat{P}_{m}\psi$, so
that it suffices to test $f^{(n+1)}-f^{(n)}$ against $P_{n}\psi$ and
$\hat{P}_{n}\psi$. We first show bounds for these terms in Lemma 32 and 36,
before proving properties 1 and 2 in Lemma 37 and Lemma 38. Property 3 will be
shown in Lemma 39.

Before we start with the proofs, let us recall the assumptions of Theorem 22:

  1. i)

$(F_{x})$ is a random germ with stochastic dimension $e\leq d$.

  2. ii)

$(F_{x})$ is stochastically $(\gamma,\alpha)$-coherent for some
$\gamma>0,\alpha<0$.

  3. iii)

$r$ is big enough, such that
$\alpha+\tilde{r}>0,\gamma-\frac{E}{2}+\tilde{r}>0$, for $\tilde{r}$ as in
Theorem 22.

With this in mind, we can state our first technical lemma. Since
$\left|\left|(f^{(n+1)}-f^{(n)})(P_{n}\psi)\right|\right|_{L_{p}}$ is rather
technical to bound, we do so in a separate lemma and start with the other
necessary bounds. As it turns out, instead of estimating
$\left|\left|(f^{(n+1)}-f^{(n)})(\hat{P}_{n}\psi)\right|\right|_{L_{p}}$,
estimating
$\left|\left|(f^{(n+1)}-f^{(n)}-F_{x})(\hat{P}_{n}\psi)\right|\right|_{L_{p}}$
allows us to combine the proofs of properties 1 and 2, so this is the term we
want to bound from above:

######  Lemma 32.

Let $(F_{x})$ be as in Theorem 22, choose some $\tilde{C}>0$ and let
$B:=\min_{i=1,\dots,e}\left(\frac{\tilde{C}}{2R}\right)^{-\frac{1}{s_{i}}}$.
Then it holds that for any $x$ in some compact $K\subset\mathbb{R}^{d}$,
$B2^{-n}\leq\lambda\leq 1$ and any test function $\psi\in C_{c}^{r}$ with
support in $[\tilde{C},\infty)^{e}\times\mathbb{R}^{d-e}$, we have:

$\displaystyle\left|\left|(f^{(n+1)}-f^{(n)}-F_{x})(\hat{P}_{n}\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\lesssim
2^{-n(\alpha+\tilde{r})}\lambda^{\gamma-\frac{E}{2}-\alpha-\tilde{r}}\left|\left|\psi\right|\right|_{C_{c}^{r}}$
(4.2)
$\displaystyle\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(x)}}(f^{(n+1)}-f^{(n)}-F_{x})(\hat{P}_{n}\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\lesssim
2^{-n(\alpha+\tilde{r})}\lambda^{\gamma-\alpha-\tilde{r}}\left|\left|\psi\right|\right|_{C_{c}^{r}}$
(4.3)
$\displaystyle\left|\left|E^{\mathcal{F}_{\pi_{i}(x)}^{(i)}}(f^{(n+1)}-f^{(n)})(P_{n}\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\lesssim
2^{-n\gamma}\left|\left|\psi\right|\right|_{C_{c}^{r}}\,,$ (4.4)

for $i=1,\dots,e$.

###### Proof.

Let us start with inequality 4.2. Since $\hat{P}_{n}\psi\in(V^{n})^{\perp
V_{n+1}}$, it follows that $f^{(n)}(\hat{P}_{n}\psi)=0$. Using the definition
of $f^{(n)}$ as well as
$F_{x}(\hat{\phi}_{y}^{n})=\sum_{z\in\Delta_{n+1}}F_{x}(\phi_{z}^{n+1})\langle\phi_{z}^{n+1},\hat{\phi}_{y}^{m}\rangle$,
we get

$\displaystyle(f^{(n+1)}-f^{(n)}-F_{x})(\hat{P}_{n}\psi_{x}^{\lambda})$
$\displaystyle=\sum_{y\in\Delta_{n}}\sum_{\hat{\phi}\in\Phi}(f^{(n+1)}-F_{x})(\hat{\phi}_{y}^{n})\langle\hat{\phi}_{y}^{n},\psi_{x}^{\lambda}\rangle$
$\displaystyle=\sum_{y\in\Delta_{n}}\sum_{z\in\Delta_{n+1}}\sum_{\hat{\phi}\in\Phi}(F_{z}-F_{x})(\phi_{z}^{n+1})\langle\phi_{z}^{n+1},\hat{\phi}_{y}^{n}\rangle\langle\hat{\phi}_{y}^{n},\psi_{x}^{\lambda}\rangle\,.$

For all non-zero terms in this sum, the supports of $\hat{\phi}_{y}^{n}$ and
the support of $\psi_{x}^{\lambda}$ need to overlap, as well as the supports
of $\phi_{z}^{n+1}$ and $\hat{\phi}_{z}^{n}$. Together with
$\mathrm{supp}(\psi)\in[\tilde{C},\infty)^{d}$ and
$\mathrm{supp}(\phi),\mathrm{supp}(\hat{\phi})\in[C,R]$, this implies that all
non-zero terms fulfill
$\pi_{i}(z)\geq\pi_{i}(y)-2^{-(n+1)s_{i}}R\geq\pi_{i}(x)+\lambda^{s_{i}}\cdot\tilde{C}-2^{-ns_{i}}\cdot
2R\geq\pi_{i}(x)$ by our choice of $\lambda$ for $i=1\dots,e$, so we can use
stochastic coherence. Note that the number of $z$, such that
$\langle\phi_{z}^{n+1},\hat{\phi}_{y}^{n}\rangle\neq 0$ for any fixed $y$ is
of order $1$, and the number of $y$, such that
$\langle\hat{\phi}_{y}^{n},\psi_{x}^{\lambda}\rangle\neq 0$ is of order
$2^{n\left|S\right|}\lambda^{\left|S\right|}$. Furthermore, all non-zero terms
will have $\left|z-x\right|\lesssim\lambda+2^{-n}\lesssim\lambda$. Applying
A.5 to $\left|\langle\phi_{z}^{n+1},\hat{\phi}_{y}^{n}\rangle\right|$ as well
as A.6 to $\left|\langle\hat{\phi}_{y}^{n},\psi_{x}^{\lambda}\rangle\right|$
and using stochastic coherence, we get:

$\displaystyle\left|\left|\hat{P}_{n}(f^{(n+1)}-f^{(n)}-F_{x})(\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\leq\sum_{y\in\Delta_{n}}\sum_{z\in\Delta_{n+1}}\sum_{\hat{\phi}\in\Phi}\underbrace{\left|\left|(F_{z}-F_{x})(\phi_{z}^{n+1})\right|\right|_{L_{p}}}_{\lesssim
2^{-n\frac{\left|S\right|}{2}-n\alpha}\lambda^{\gamma-\frac{E}{2}-\alpha}}$
$\displaystyle\qquad\qquad\qquad\qquad\times\underbrace{\left|\langle\phi_{z}^{n+1},\hat{\phi}_{y}^{n}\rangle\right|}_{\lesssim
1}\underbrace{\left|\langle\hat{\phi}_{y}^{n},\psi_{x}^{\lambda}\rangle\right|}_{\lesssim
2^{-n\frac{\left|S\right|}{2}-n\tilde{r}}\lambda^{-{\left|S\right|}-\tilde{r}}\left|\left|\psi\right|\right|_{C_{c}^{r}}}$
$\displaystyle\lesssim
2^{-n\alpha-n\tilde{r}}\lambda^{\gamma-\frac{E}{2}-\alpha-\tilde{r}}\left|\left|\psi\right|\right|_{C_{c}^{r}}\,,$

which shows 4.2. 4.3 follows analogously. To show 4.4, we decompose

$(f^{(n+1)}-f^{(n)})(P_{n}\psi_{x}^{\lambda})=\sum_{y\in\Delta_{n}}(f^{(n+1)}-f^{(n)})(\phi_{y}^{n})\langle\phi_{y}^{n},\psi_{x}^{\lambda}\rangle$
(4.5)

and calculate

$\displaystyle(f^{(n+1)}-f^{(n)})(\phi_{x}^{n})$
$\displaystyle=\sum_{z\in\Delta_{n+1}}F_{z}(\phi_{z}^{n+1})\langle\phi_{z}^{n+1},\phi_{x}^{n}\rangle-\sum_{y\in\Delta_{n}}F_{y}(\phi_{y}^{n})\langle\phi_{y}^{n},\phi_{x}^{n}\rangle$
$\displaystyle=\sum_{z\in\Delta_{n+1}}\sum_{k\in\Delta_{n+1}}F_{z}(\phi_{z}^{n+1})\langle\phi_{z}^{n+1},a_{k}^{(n+1)}\phi_{x+k}^{n+1}\rangle-
F_{x}(\phi_{x}^{n})$
$\displaystyle=\sum_{k\in\Delta_{n+1}}a_{k}^{(n+1)}F_{x+k}(\phi_{x+k}^{(n+1)})-\sum_{k\in\Delta_{n+1}}a_{k}^{(n+1)}F_{x}(\phi_{x+k}^{(n+1)})$
$\displaystyle=\sum_{k\in\Delta_{n+1}}a_{k}^{(n+1)}(F_{x+k}-F_{x})(\phi_{k+x}^{(n+1)}).$
(4.6)

Note that $a_{k}^{(n+1)}\neq 0$ only for $\left|k\right|\leq dR2^{-n}$, so the
sum is finite. Furthermore, $\pi_{i}(k)\geq 0$ for $i=1,\dots,n$ and all non-
zero terms and the support of $\phi$ is positive, so we can use stochastic
reconstruction to show:

$\displaystyle\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(x)}}(f^{(n+1)}-f^{(n)})(\phi_{x}^{n})\right|\right|_{L_{p}}$
$\displaystyle\leq\sum_{k\in\Delta_{n+1}}\left|a_{k}^{(n+1)}\right|\left|\left|E^{\mathcal{F}_{\pi_{i}(x)}^{(i)}}(F_{x+k}-F_{x})(\phi_{k+x}^{(n+1)})\right|\right|_{L_{p}}$
$\displaystyle\lesssim
2^{-\frac{(n+1)\left|S\right|}{2}-\alpha(n+1)}(\left|k\right|+2^{-n-1})^{\gamma-\alpha}$
$\displaystyle\lesssim 2^{-n\gamma-n\frac{\left|S\right|}{2}}\,.$ (4.7)

Putting this into 4.5 gives us

$\displaystyle\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(x)}}(f^{(n+1)}-f^{(n)})(P_{n}\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\leq\sum_{y\in\Delta_{n}}\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(x)}}(f^{(n+1)}-f^{(n)})(\phi_{x}^{n})\right|\right|_{L_{p}}\underbrace{\left|\langle\phi_{y}^{n},\psi_{x}^{\lambda}\rangle\right|}_{\leq
2^{-n\frac{\left|S\right|}{2}}\lambda^{-\left|S\right|}\left|\left|\psi\right|\right|_{C_{c}^{r}}}$
$\displaystyle\lesssim
2^{-n\gamma}\left|\left|\psi\right|\right|_{C_{c}^{r}}\,,$

where we used A.5 as well as the fact that the order of non-zero terms is
given by $2^{n\left|S\right|}\lambda^{\left|S\right|}$. ∎

The last inequality we need for the main result is a bound on
$\left|\left|(f^{(n+1)}-f^{(n)})(P_{n}\psi)\right|\right|$. If one uses the
techniques used in Lemma 32, this ends up being $\lesssim
2^{-n(\gamma-\frac{E}{2})}$, which is not good enough for our result. One can
improve this bound with the use of the BDG type inequality B.1.

Since we will this technique both in the proof of existence and uniqueness of
the reconstruction, let us formulate the technique itself in its own lemma:

######  Lemma 33.

Let $h$ be a random distribution, and assume there exists a constant
$\tilde{R}>0$ such that for $i=1,\dots,e,n\in\mathbb{N}$ and $x\in\Delta_{n}$,
$h(\phi_{x}^{n})$ is $\mathcal{F}_{\hat{S}_{i}(x)}^{(i)}$-measurable for
$\hat{S}_{i}(x)=\pi_{i}(x)+2^{-ns_{i}}\tilde{R}$. Further, assume that $h$
fulfills the inequalities

$\displaystyle\left|\left|h(\phi_{x}^{n})\right|\right|_{L_{p}}$
$\displaystyle\lesssim 2^{-n\gamma-n\frac{\left|S\right|-E}{2}}$ (4.8)
$\displaystyle\left|\left|E^{\mathcal{F}_{\pi_{i}(x)}^{(i)}}h(\phi_{x}^{n})\right|\right|_{L_{p}}$
$\displaystyle\lesssim 2^{-n\gamma-n\frac{\left|S\right|}{2}}\,,$ (4.9)

for $i=1,\dots,e$. Then it holds that for each $x_{0}$ in any constant
$K\subset\mathbb{R}^{d}$, $1\leq\lambda\leq 2^{-n}$ and $\psi\in C_{c}^{r}$
with support in $[-\hat{R},\hat{R}]^{d}$ for some $\hat{R}>0$, we have

$\left|\left|h(P_{n}\psi_{x_{0}}^{\lambda})\right|\right|_{L_{p}}\lesssim
2^{-n\gamma}\lambda^{-\frac{E}{2}}\left|\left|\psi\right|\right|_{C_{c}^{r}}\,,$

where the constant in $\lesssim$ is allowed to depend on $K,\tilde{R}$ and
$\hat{R}$.

######  Remark 34.

Note that any $h$ with stochastic dimension $e$ will automatically fulfill the
measurability condition given above. The reason we want to work with this
weaker condition is that we want to apply the above lemma to
$h=f^{(n+1)}-f^{(n)}$, which is in general not adapted but has a “fattening of
the support” happening. More rigorously, given a test function $\psi$ with
support in $(-\infty,y]\times\mathbb{R}^{d-1}$, $(f^{(n+1)}-f^{(n)})(\psi)$ is
not $\mathcal{F}_{\pi_{1}(y)}^{(1)}$ measurable, but
$\mathcal{F}_{\pi_{1}(y)+R2^{-ns_{1}}}^{(1)}$.

######  Remark 35.

We want to stress that this lemma is the only place, where we need B.1. Thus,
this can be seen as the technique that gives us the improvement in regularity
over classical reconstruction.

###### Proof.

We decompose

$h(P_{n}\psi_{x_{0}}^{\lambda})=\sum_{x\in\Delta_{n}}\underbrace{h(\phi_{x}^{n})\langle\phi_{x}^{n},\psi_{x_{0}}^{\lambda}\rangle}_{=:g_{x}}.$
(4.10)

Our strategy will be to use B.1 on the outermost sum and then iterate this
step over all stochastic directions. To this end, we specify the definition

$\Delta_{n}^{a,b}:=\left\\{\sum_{i=a}^{b}2^{-ns_{i}}l_{i}\vec{e}_{i}\leavevmode\nobreak\
\bigg{|}\leavevmode\nobreak\ l_{i}\in\mathbb{Z}\right\\}$

to be the rescaled mesh in the variables $a$ to $b$. Unfortunately, naively
applying B.1 does not work for our sum: If we set
$Z_{y}:=\sum_{x\in\Delta_{n}^{2,d}}g_{(y,x)}$, so that
$h(P_{n}\psi_{x_{0}}^{\lambda})=\sum_{y\in\Delta_{n}^{1,1}}Z_{y}$, we recall
that $Z_{y}$ is $\mathcal{F}^{(1)}_{\hat{S}_{1}(y)}$-measurable. B.1 would now
allow us to condition $Z_{y}$ onto
$\mathcal{F}_{\hat{S}_{1}(y)-2^{-ns_{1}}}^{(1)}$, which results in a random
variable we do not control.

Instead, we are going to split the sum into $\lceil\tilde{R}\rceil^{e}$ many
sums in which the nets have a bigger mesh, to get the right conditioning. To
this end, let $0\leq r_{i}\leq\lceil\tilde{R}\rceil-1,i=1,\dots,e$ be natural
numbers and set

$r_{n}^{(k)}:=S^{2^{-n}}(0,\dots,0,r_{k+1},\dots,r_{e},0,\dots,0)\in\Delta_{n}^{k+1,d}$

for $k=0,\dots,e$. We define the rescaled and shifted net for each such
$r_{n}^{(k)}$ by

$\Delta_{n}^{a,b}(r^{(k)}_{n})=\left\\{\sum_{i=a}^{b}\lceil C_{i}\rceil\cdot
2^{-ns_{i}}l_{i}\vec{e}_{i}+r_{n}^{(k)}\leavevmode\nobreak\
\bigg{|}\leavevmode\nobreak\ l_{i}\in\mathbb{Z}\right\\}\,.$

$\Delta_{n}^{1,2}$$\Delta_{n}^{1,1}$$\Delta_{n}^{2,2}$
$\Delta_{n}^{1,2}(r_{n}^{(0)})$

As these definitions are rather technical, we have illustrated
$\Delta_{n}^{1,2},\Delta_{n}^{1,1},\Delta_{n}^{2,2}$ as well as
$\Delta_{n}^{1,2}(r_{n}^{(0)})$ in the case $e=d=2$, $\lceil\tilde{R}\rceil=2$
and $r=(1,1)$. We split the expression

$h(P_{n}\psi_{x_{0}}^{\lambda})=\sum_{r_{n}^{(0)}}\sum_{x\in\Delta_{n}^{1,d}(r_{n}^{(0)})}g_{x}$

into $\lceil\tilde{R}\rceil^{e}$ many sums, where the first sum runs over all
possible $r_{n}^{(0)}$. It suffices to find a bound for the sum over $x$ as
the sum over $r_{n}^{(0)}$ will merely add the constant factor
$\lceil\tilde{R}\rceil^{e}$. Using B.1, we now do indeed get

$\displaystyle\left|\left|\sum_{x\in\Delta_{n}^{1,d}(r_{n}^{(0)})}g_{x}\right|\right|_{L_{p}}\lesssim$
$\displaystyle\sum_{y\in\Delta_{n}^{1,1}(r_{1})}\left|\left|\sum_{x\in\Delta_{n}^{2,d}(r_{n}^{(1)})}E^{\mathcal{F}^{(1)}_{y}}g_{(y,x)}\right|\right|_{L_{p}}$
$\displaystyle+\left(\sum_{y\in\Delta_{n}^{1,1}(r_{1})}\left|\left|\sum_{x\in\Delta_{n}^{2,d}(r_{n}^{(1)})}g_{(y,x)}-E^{\mathcal{F}^{(1)}_{y}}g_{(y,x)}\right|\right|_{L_{p}}^{2}\right)^{\frac{1}{2}},$
(4.11)

We want to iterate this step over all stochastic directions. To this end, we
use the simplified notation $(y,x):=y+x$ for
$y\in\Delta_{n}^{a,b},x\in\Delta_{n}^{c,d}$ with $[a,b]\cup[c,d]=\emptyset$
and define

$\displaystyle g_{x}$
$\displaystyle:=h(\phi_{x}^{n})\langle\phi_{x}^{n},\psi_{x_{0}}^{\lambda}\rangle$
$\displaystyle x\in\Delta_{n}^{1,d}$ $\displaystyle g_{y,x}$
$\displaystyle:=g_{(y,x)}-E^{\mathcal{F}^{(1)}_{y}}g_{(y,x)}$ $\displaystyle
x\in\Delta_{n}^{2,d},y\in\Delta_{n}^{1,1}$ $\displaystyle g_{y_{1},y_{2},x}$
$\displaystyle=g_{y_{1},(y_{2},x)}-E^{\mathcal{F}^{(2)}_{y_{2}}}g_{y_{1},(y_{2},x)}$
$\displaystyle
x\in\Delta_{n}^{3,d},y_{1}\in\Delta^{1,1}_{n},y_{2}\in\Delta_{n}^{2,2}$
$\displaystyle\vdots$ $\displaystyle g_{y_{1},\dots,y_{e},x}$
$\displaystyle:=g_{y_{1},\dots,y_{e-1}(y_{e},x)}-E^{\mathcal{F}^{(e)}_{y_{e}}}g_{y_{1},\dots,y_{e-1}(y_{e},x)}$
$\displaystyle
x\in\Delta_{n}^{e+1,d},y_{1}\in\Delta_{n}^{1,1},\dots,y_{e}\in\Delta_{n}^{e,e},$

with the goal of finding bounds for
$\left|\left|\sum_{x\in\Delta_{n}^{i+1,d}(r_{n}^{(i)})}g_{y_{1},\dots,y_{i},x}\right|\right|_{L_{p}}$
using a backward induction over $i=0,\dots,e$. Note that since
$(\mathcal{F}_{t}^{(i)})_{t}$, $i=1,\dots,e$ are orthogonal,
$g_{y_{1},\dots,y_{i},x}$ has the same adaptedness properties as $g_{x}$ for
$i=1,\dots,e$, so we can apply B.1 to
$\left|\left|\sum_{x\in\Delta_{n}^{i+1,d}(r_{n}^{(i)})}g_{y_{1},\dots,y_{i},x}\right|\right|_{L_{p}}$.

Recall, that A.5 gives us

$\left|\langle\phi_{x}^{n},\psi_{x_{0}}^{\lambda}\rangle\right|\lesssim
2^{-\frac{n\left|S\right|}{2}}\lambda^{-\left|S\right|}\left|\left|\psi\right|\right|_{C_{c}^{r}}.$
(4.12)

Together with 4.8, this implies

$\displaystyle\left|\left|g_{x}\right|\right|_{L_{p}}$
$\displaystyle\leq\left|\left|h(\phi_{x}^{n})\right|\right|_{L_{p}}\left|\langle\phi_{x}^{n},\psi_{x_{0}}^{\lambda}\rangle\right|$
$\displaystyle\lesssim
2^{-n\gamma-n\frac{\left|S\right|-E}{2}}2^{-n\frac{\left|S\right|}{2}}\lambda^{-\left|S\right|}\left|\left|\psi\right|\right|_{C_{c}^{r}}$
$\displaystyle=2^{-n\gamma-n\left|S\right|+n\frac{E}{2}}\lambda^{-\left|S\right|}\left|\left|\psi\right|\right|_{C_{c}^{r}},$
(4.13)

and analogously, using 4.9

$\left|\left|E^{\mathcal{F}^{(1)}_{y}}g_{(y,x)}\right|\right|_{L_{p}}\lesssim
2^{-n\gamma-n\left|S\right|}\lambda^{-\left|S\right|}\left|\left|\psi\right|\right|_{C_{c}^{r}}.$
(4.14)

Observe that for any $i=1,\dots,e$, $g_{y_{1},\dots,y_{i},x}$ is just a finite
linear combination of $g_{(y_{1},\dots,y_{i},x)}$ conditioned on different
sigma-algebras. By the contraction properties of the conditional expectation,
4.13 and 4.14 thus hold for any $g_{y_{1},\dots,y_{i},x},i=0,\dots,e$.

We claim, that the following holds for $i=0,\dots,e$:

$\left|\left|\sum_{x\in\Delta_{n}^{i+1,d}(r_{n}^{(i)})}g_{y_{1},\dots,y_{i},x}\right|\right|_{L_{p}}\lesssim
2^{-n\gamma-n\frac{E_{i}}{2}}\lambda^{-\frac{E+E_{i}}{2}}\left|\left|\psi\right|\right|_{C_{c}^{r}},$

where $E_{i}:=\sum_{j=1}^{i}s_{j}$. As said above, we show this claim by
backwards induction. Note that the $g_{x}\neq 0$ implies
$\langle\phi_{x}^{n},\psi_{x_{0}}^{\lambda}\rangle\neq 0$, which is only true
for $\cong 2^{ns_{i}}\lambda^{s_{i}}$ many $x$ for every coordinate
$i=1,\dots,d$. Thus, using 4.13, we get the induction start

$\sum_{x\in\Delta_{n}^{e+1,d}(r_{n}^{(e)})}\left|\left|g_{y_{1},\dots,y_{e},x}\right|\right|_{L_{p}}\lesssim
2^{-n\gamma-n\left|S\right|+n\frac{E}{2}}\lambda^{-\left|S\right|}\cdot
2^{n(\left|S\right|-E)}\lambda^{\left|S\right|-E}\left|\left|\psi\right|\right|_{C_{c}^{r}}=2^{-n\gamma-n\frac{E}{2}}\lambda^{-E}\left|\left|\psi\right|\right|_{C_{c}^{r}}.$
(4.15)

For the induction step, observe that

$\displaystyle\sum_{x\in\Delta_{n}^{i+1,d}(r_{n}^{(i)})}\left|\left|E^{\mathcal{F}^{(i)}_{y_{i}}}g_{y_{1},\dots,y_{i-1},(y_{i},x)}\right|\right|_{L_{p}}$
$\displaystyle\lesssim
2^{-n\gamma-n\left|S\right|}\lambda^{-\left|S\right|}\cdot
2^{n(\left|S\right|-E_{i})}\lambda^{\left|S\right|-E_{i}}\left|\left|\psi\right|\right|_{C_{c}^{r}}$
$\displaystyle=2^{-n\gamma-
nE_{i}}\lambda^{-E_{i}}\left|\left|\psi\right|\right|_{C_{c}^{r}}$

holds, and that summing over the squares and taking the square root has the
effect of multiplying a factor of
$2^{\frac{s_{i}n}{2}}\lambda^{\frac{s_{i}}{2}}$:

$\left(\sum_{k=1}^{\lfloor
2^{s_{i}n}\lambda^{s_{i}}\rfloor}a^{2}\right)^{\frac{1}{2}}\leq
2^{\frac{s_{i}n}{2}}\lambda^{\frac{s_{i}}{2}}a.$

Now, using 4.11, we get

$\displaystyle\left|\left|\sum_{x_{\in}\Delta_{n}^{i+1,d}(r_{n}^{(i)})}g_{y_{1},\dots,y_{i},x}\right|\right|_{L_{p}}$
$\displaystyle\lesssim\underbrace{\sum_{y_{i+1}\in\Delta_{n}^{i+1,i+1}(\pi_{i+1}(r_{n}))}\sum_{x\in\Delta_{n}^{i+2,d}(r_{n}^{(i+1)})}\left|\left|E^{\mathcal{F}^{(i+1)}_{y_{i+1}}}g_{y_{1},\dots,y_{i},(y_{i+1},x)}\right|\right|_{L_{p}}}_{\lesssim
2^{-n\gamma-
nE_{i}}\lambda^{-E_{i}}\left|\left|\psi\right|\right|_{C_{c}^{r}}}$
$\displaystyle\qquad+\Bigg{(}\sum_{y_{i+1}\in\Delta_{n}^{i+1,i+1}(\pi_{i+1}(r_{n}))}\underbrace{\left|\left|\sum_{x\in\Delta_{n}^{i+2,d}(r_{n}^{(i+1)})}g_{y_{1},\dots,y_{i+1},x}\right|\right|_{L_{p}}^{2}}_{\lesssim\left(2^{-n\gamma-n\frac{E_{i+1}}{2}}\lambda^{-\frac{E+E_{i+1}}{2}}\left|\left|\psi\right|\right|_{C_{c}^{r}}\right)^{2}}\Bigg{)}^{\frac{1}{2}}$
$\displaystyle\lesssim
2^{-n\gamma-n\frac{E_{i}}{2}}\lambda^{-\frac{E+E_{i}}{2}}\left|\left|\psi\right|\right|_{C_{c}^{r}}.$

This shows the claim. Therefore, $i=0$ implies

$\left|\left|h(P_{n}\psi_{x_{0}}^{\lambda})\right|\right|_{L_{p}}=\left|\left|\sum_{x\in\Delta_{n}}g_{x}\right|\right|_{L_{p}}\lesssim\sum_{r_{n}^{(0)}}\left|\left|\sum_{x\in\Delta_{n}^{1,d}(r_{n}^{(0)})}g_{x}\right|\right|_{L_{p}}\lesssim
2^{-n\gamma}\lambda^{-\frac{E}{2}}\left|\left|\psi\right|\right|_{C_{c}^{r}},$

which finishes the proof of the lemma. ∎

It is now immediate to show the last bound we need:

######  Lemma 36.

Let $(F_{x})$ be as in Theorem 22. Then it holds that for any $x_{0}$ in some
compact $K\subset\mathbb{R}^{d}$, $1\leq\lambda\leq 2^{-n}$ and $\psi\in
C_{c}^{r}$ with support in $[-\tilde{R},\tilde{R}]^{d}$ for some
$\tilde{R}>0$, we have:

$\left|\left|(f^{(n+1)}-f^{(n)})(P_{n}\psi_{x_{0}}^{\lambda})\right|\right|_{L_{p}}\lesssim
2^{-n\gamma}\lambda^{-\frac{E}{2}}\left|\left|\psi\right|\right|_{C_{c}^{r}}.$
(4.16)

There the constant in $\lesssim$ is allowed to depend on $K$ and $\tilde{R}$.

###### Proof.

Analogously to 4.7, one shows that

$\left|\left|(f^{(n+1)}-f^{(n)})(\phi_{x}^{n})\right|\right|_{L_{p}}\lesssim
2^{-n\gamma-n\frac{\left|S\right|-E}{2}}\,.$ (4.17)

Observe that $(f^{(n+1)}-f^{(n)})(\phi_{x}^{n})$ is
$\mathcal{F}^{(i)}_{\hat{S}_{i}(x)}$-measurable for
$\hat{S}_{i}(x):=\pi_{i}(x)+(2^{-ns_{i}}+2^{-(n+1)s_{i}})R$. This together
with 4.17 and 4.7 shows that $h=f^{(n+1)}-f^{(n)}$ fulfills the conditions of
Lemma 33, which immediately gives us 4.16.

∎

We proceed with the proof of properties 1 and 2. To do so, we first make the
additional assumption that $F_{x}$ is a random distribution with enough
regularity to invoke the second part of Lemma 62. Without this assumption, the
reconstructing sequence 4.1 does not converge, but it is still possible to
construct a distribution $f$ such that property 2 holds. This construction
will be shown in Lemma 38.

######  Lemma 37.

Let $(F_{x})$ be as in Theorem 22 and assume that for all
$x\in\mathbb{R}^{d}$, $F_{x}\in C^{\beta}(L_{p})$ with $\beta+\tilde{r}>0$.
Then for each $\psi\in C_{c}^{r}$, $f^{(n)}(\psi)$ converges against a random
variable $f(\psi)$ in $L_{p}(\Omega)$ and $f:C_{c}^{r}\to L_{p}(\Omega)$ is a
random distribution with stochastic dimension $e$ and fulfills 3.8 and 3.9.

###### Proof.

The idea of the proof is to show that
$(f^{(l)}-P_{l}F_{x})(\psi_{x}^{\lambda})$ converges to some random variable
in $L_{p}$ as $l\to\infty$ and that its $L_{p}$-norm is bound by
$\lambda^{\gamma-\frac{E}{2}}$, while the $L_{p}$\- norm of its conditional
expectation is bound by $\lambda^{\gamma}$. Since $P_{l}F_{x}\to F_{x}$ is
convergent thanks to the additional assumption and Lemma 62, it follows that
$f^{(l)}$ converges to some $f$, and that
$(f-F_{x})(\psi_{x}^{\lambda})=\lim_{l\to\infty}(f^{(l)}-P_{l}F_{x})(\psi_{x}^{\lambda})$
fulfills 3.8 and 3.9.

Let us fix a test function $\psi$ with support in $[\tilde{C},\infty)^{d}$ for
some $\tilde{C}>0$, an $x\in K$ as well as a $\lambda>0$. We choose $n$ in
such a way, that $B2^{-n}\leq\lambda\leq B2^{-n+1}$ for
$B:=\min_{i=1,\dots,d}\left(\frac{\tilde{C}}{2R}\right)^{-\frac{1}{s_{i}}}$.
For $l\geq n$, we decompose
$f^{(l)}=f^{(n)}+\sum_{m=n}^{l-1}(f^{(m+1)}-f^{(m)})$ as well as
$P_{l}\psi=P_{n}\psi+\sum_{m=n}^{l-1}\hat{P}_{m}\psi$ to get the decomposition

$\displaystyle f^{(l)}(\psi_{x}^{\lambda})-F_{x}(P_{l}\psi_{x}^{\lambda})$
$\displaystyle=\underbrace{(f^{(n)}-P_{n}F_{x})(\psi_{x}^{\lambda})}_{(I)}+\sum_{m=n}^{l-1}\underbrace{\hat{P}_{m}(f^{(m+1)}-f^{(m)}-F_{x})(\psi_{x}^{\lambda})}_{(II)_{m}}$
$\displaystyle\qquad+\sum_{m=n}^{l-1}\underbrace{P_{m}(f^{(m+1}-f^{(m)})(\psi_{x}^{\lambda})}_{(III)_{m}}\,,$
(4.18)

where we used $f^{(l)}(\psi_{x}^{\lambda})=f^{(l)}(P_{l}\psi_{x}^{\lambda})$,
which follows from $f^{(l)}\in V_{l}$ A.3. $(II)_{m}$ and $(III)_{m}$ are
bound by Lemma 32 and 36, so we only need to find a bound for $(I)$. Using
A.5, A.6 as well as stochastic coherence, the following holds:

$\displaystyle\left|\left|(I)\right|\right|_{L_{p}}$
$\displaystyle\leq\sum_{y\in\Delta_{n}}\left|\left|(F_{y}-F_{x})(\phi_{y}^{n})\right|\right|_{L_{p}}\left|\langle\phi_{y}^{n},\psi_{x}^{\lambda}\rangle\right|$
$\displaystyle\lesssim\lambda^{\gamma-\frac{E}{2}}\left|\left|\psi\right|\right|_{C_{c}^{r}}\,.$

Note that the positive support of $\psi$ together with our choice of $n$
implies
$\pi_{i}(y)\geq\pi_{i}(x)+\lambda^{s_{i}}\tilde{C}-2^{-ns_{i}}R\geq\pi_{i}(x)$
by our choice of $n$, which justifies our use of stochastic coherence.
Analogously

$\left|\left|E^{\mathcal{F}_{\pi_{i}(x)}^{(i)}}(I)\right|\right|_{L_{p}}\lesssim\lambda^{\gamma}\left|\left|\psi\right|\right|_{C_{c}^{r}}\,.$

We now have all inequalities we need and want to argue that
$(f^{(n)}-P_{n}F_{x})(\psi)$ is a Cauchy sequence. By Lemma 32 and 36 it holds
that for any $n\leq k\leq l$:

$\displaystyle\sum_{m=k}^{l-1}\left|\left|(II)_{m}\right|\right|_{L_{p}}+\sum_{m=k}^{l-1}\left|\left|(III)_{m}\right|\right|_{L_{p}}$
$\displaystyle\lesssim\sum_{m=k}^{l-1}(2^{-m(\alpha+\tilde{r})}\lambda^{\gamma-\frac{E}{2}-\alpha-\tilde{r}}+2^{-m\gamma}\lambda^{-\frac{E}{2}})\left|\left|\psi\right|\right|_{C_{c}^{r}}$
$\displaystyle\lesssim(2^{-k(\alpha+\tilde{r})}\lambda^{\gamma-\frac{E}{2}-\alpha-\tilde{r}}+2^{-k\gamma}\lambda^{-\frac{E}{2}})\left|\left|\psi\right|\right|_{C_{c}^{r}}\,,$
(4.19)

where we used $\alpha+\tilde{r}>0$. For an arbitrary $\tilde{\psi}\in
C_{c}^{r}$, we choose $\lambda=1$ and $x\in\mathbb{R}^{d}$ such that
$\tilde{\psi}=\psi_{x}^{\lambda}$ for some $\psi$ with support in
$[\tilde{C},\infty)^{d}$. 4.19 than implies that
$f^{(n)}(\tilde{\psi})-F_{x}(P_{n}\tilde{\psi})$ is a Cauchy-sequence, and
thus converges in $L_{p}(\Omega)$. Since $F_{x}(P_{n}\tilde{\psi})\rightarrow
F_{x}(\tilde{\psi})$ in $L_{p}(\Omega)$, it follows that
$f^{(n)}(\tilde{\psi})$ converges to a limit $f(\tilde{\psi})$. To show that
$f$ is a random distribution, note that by construction, $f$ is linear in
$\tilde{\psi}$. Furthermore, the above inequalities show that for
$\lambda=1\approx 2^{-n}$:

$\left|\left|f(\psi)-F_{x}(\psi)\right|\right|_{L_{p}}=\left|\left|(I)\right|\right|_{L_{p}}+\sum_{m=n}^{\infty}\left|\left|(II)_{m}\right|\right|_{L_{p}}+\sum_{m=n}^{\infty}\left|\left|(III)_{m}\right|\right|_{L_{p}}\lesssim\left|\left|\psi\right|\right|_{C_{c}^{r}}.$

Since $F_{x}$ is already a random distribution, it follows that
$\left|\left|f(\psi)\right|\right|_{L_{p}}\lesssim\left|\left|\psi\right|\right|_{C_{c}^{r}}$,
which shows that $f$ is a random distribution.

To show that $f$ has stochastic dimension $e$, simply observe that
$f^{(n)}(\psi)=\sum_{x}F_{x}(\phi_{x}^{n})\langle\phi_{x}^{n},\psi\rangle$ is
$\mathcal{F}^{(i)}_{y+2^{-ns_{i}}R}$ measurable for any $\psi$ with support in
$\mathbb{R}^{i-1}\times(-\infty,y]\times\mathbb{R}^{d-i}$. The completeness
and right continuity of $\mathcal{F}^{(i)}$ now implies, that the limit
$f(\psi)$ is $\mathcal{F}^{(i)}_{y}$-measurable for $i=1,\dots,e$.

It remains to show 3.8 and 3.9. We use 4.19 and $2^{-n}\approx\lambda$ to get
for $l\to\infty$

$\displaystyle\left|\left|(f-F_{x})(\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\left(\lambda^{\gamma-\frac{E}{2}}+\sum_{m=n}^{\infty}2^{-m(\alpha+\tilde{r})}\lambda^{\gamma-\frac{E}{2}-\alpha\tilde{r}}+\sum_{m=n}^{\infty}2^{-m\gamma}\lambda^{-\frac{E}{2}}\right)\left|\left|\psi\right|\right|_{C_{c}^{r}}$
$\displaystyle\lesssim\lambda^{\gamma-\frac{E}{2}}\left|\left|\psi\right|\right|_{C_{c}^{r}}\,.$

Analogously, it follows that

$\left|\left|E^{\mathcal{F}_{\pi_{i}(x)}^{(i)}}(f-F_{x})(\psi_{x}^{\lambda})\right|\right|_{L_{p}}\lesssim\lambda^{\gamma}\left|\left|\psi\right|\right|_{C_{c}^{r}}\,.$

This finishes the proof. ∎

While the above proof needs $F_{x}\in C^{\beta}(L_{p})$ for sufficiently large
$\beta$ to get $f^{(n)}\to f$ for $n\to\infty$, a closer look at the proof
reveals that the property is not needed to get $f^{(n)}-P_{n}F_{x}$ to
converge to some random distribution $G_{x}$, which should formally be
$f-F_{x}$. So we can get rid of the assumption $F_{x}\in C^{\beta}(L_{p})$ by
setting

$f(\psi):=\lim_{n\to\infty}(f^{(n)}-P_{n}F_{x})(\psi)+F_{x}(\psi)\,.$

As it turns out, this gives us the reconstruction if we do not have $F_{x}\in
C^{\beta}(L_{p})$, as the next lemma shows.

######  Lemma 38.

Under the assumptions of Theorem 22, there is a random distribution $f$ on
$C_{c}^{r}$ with stochastic dimension $e$, such that 3.8 and 3.9 hold.

###### Proof.

By Remark 9, we get that for all $x\in\mathbb{R}^{d}$, $F_{x}\in
C^{-\left|S\right|-r}(L_{p})$. So if we restrict ourselves to the case
$\psi\in C^{q}_{c}$, $q+\left|S\right|-r>0$, we get that $F_{x}$ fulfills the
conditions of Lemma 37 and can thus be reconstructed to get a random
distribution $f(\psi)$ for $\psi\in C_{c}^{q}$. As stated above, the proof of
Lemma 37 shows that for any $\psi\in C_{c}^{r}$, $(f^{(l)}-P_{l}F_{x})(\psi)$
converges against some random $C_{c}^{r}$ distribution $\psi\mapsto
G_{x}(\psi)$ as $l\to\infty$.

Therefore, we set $f_{x}:=G_{x}+F_{x}$ and show that $f_{x}=:f$ is independent
of $x\in\mathbb{R}^{d}$ and thus a well-defined random $C_{c}^{r}$
distribution. This is a slight abuse of notation, as we defined $f(\psi)$
twice for $\psi\in C_{c}^{q}$. However, by the proof of Lemma 37, it is clear
that for all $\psi\in C_{c}^{q}$,
$f(\psi)=\lim_{l\to\infty}(f^{(l)}-P_{l}F_{x})(\psi)+F_{x}(\psi)=(G_{x}+F_{x})(\psi)=f_{x}(\psi)$,
justifying the notation.

Note that this would imply that
$(f-F_{x})(\psi)=G_{x}(\psi)=\lim_{l\to\infty}(f^{(l)}-P_{l}F_{x})(\psi)$ in
$L_{p}$, which means that $f$ has stochastic dimension $e$ and fulfills 3.8
and 3.9.

So it only remains to show that $f_{x}(\psi)$ is independent of $x$. Let
$x,y\in\mathbb{R}^{d}$. Then $f_{x}(\psi)=f_{y}(\psi)$ a.s. for all $\psi\in
C_{c}^{q}$, since $f_{x}(\psi)=f_{y}(\psi)=\lim_{n\to\infty}f^{(n)}(\psi)$ in
$L_{p}$ holds on $C_{c}^{q}$. Since $f_{x},f_{y}:C_{c}^{r}\to L_{p}(\Omega)$
are continuous and $C_{c}^{q}\subset C_{c}^{r}$ is dense, it follows that
$f_{x}=f_{y}$ a.s. ∎

It remains to prove the uniqueness of the reconstructed germ. We do so in the
next lemma, which finishes the proof of Theorem 22:

######  Lemma 39.

There is at most one random distribution $f$ (up to modification) with
stochastic dimension $e$, which fulfills 3.8 and 3.9.

###### Proof.

Let $f,g$ be two random distributions as in the above lemma, and consider
$h=f-g$. By Lemma 62, we have $h(\psi)=\lim_{n\to\infty}h(P_{n}\psi)$ in
$L_{p}$ for $\psi\in C^{q}_{c}$ for a sufficiently large $q$. Since
$C^{q}_{c}$ is dense in $C_{c}^{r}$, it suffices to show that $h(P_{n}\psi)$
converges to $0$ in $L_{p}$ for $\psi\in C_{c}^{q}$. For a test function
$\psi$ with support in $[\tilde{C},\tilde{R}]^{d}$ for some
$0<\tilde{C}<\tilde{R}$, we can apply 3.8 to get

$\displaystyle\left|\left|h(\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\leq\left|\left|(f-F_{x})(\psi_{x}^{\lambda})\right|\right|_{L_{p}}+\left|\left|(g-F_{x})(\psi_{x}^{\lambda})\right|\right|_{L_{p}}$
$\displaystyle\lesssim\lambda^{\gamma-\frac{E}{2}}\,,$

and anlaogously, using 3.9 and for $i=1\dots,e$:

$\left|\left|E^{\mathcal{F}^{(i)}_{\pi_{i}(x)}}h(\psi_{x}^{\lambda})\right|\right|_{L_{p}}\lesssim\lambda^{\gamma}\,.$

If we substitute $\psi_{x}^{\lambda}$ with $\phi_{x}^{n}$ and note that $h$
has stochastic dimension $e$, we see that $h$ fulfills the conditions of Lemma
33. It follows that $\left|\left|h(P_{n}\psi)\right|\right|_{L_{p}}\lesssim
2^{-n\gamma}$ goes to $0$, which finishes the proof. ∎

###  4.2 Proof of Theorem 30

The strategy for the prove of Theorem 30 is the same as for Theorem 22: For
$l\geq n$, we use the decomposition 4.18 to show that
$(f^{(n)}-P_{n}F_{x})(\psi)$ is a Cauchy sequence and thus convergent in
$L_{2}(\Omega)$. By choosing a smart $n$, this decomposition will also be used
to show the bounds 3.12 and 3.13.

The main difference in the proofs is the uniqueness part: For Theorem 22, we
took two distributions $f,g$ fulfilling 3.8 and 3.9 and analyzed $h=f-g$.
However, if one follows this approach in the covariance setting, the term
$h(\psi_{x}^{\lambda})h(\psi_{y}^{\lambda})$ contains mixed expressions
$E\left((f-F_{x})(\psi_{x})(g-F_{y})(\psi_{y})\right)$, which are hard to
bound from above. So we use a different tactic to show uniqueness for the
covariance setting:

######  Lemma 40.

Under the assumptions of Theorem 30, there is at most one (up to modification)
random distribution $f$, such that 3.12 and 3.13 hold.

###### Proof.

We will show that any $f$ fulfilling 3.12 and 3.13 needs to fulfill

$f(\psi)=\lim_{n\to\infty}f^{(n)}(\psi)\,$

for $\psi\in C_{c}^{q}$ for high enough $q$, and $f^{(n)}$ as in 4.1, where
the right-hand-side limit is in $L_{2}(\Omega)$. Thus, $f(\psi)$ is uniquely
defined for $\psi\in C_{c}^{q}$ and since $C_{c}^{q}\subset C_{c}^{r}$ is
dense, this shows the claim.

So assume $f$ fulfills 3.12 and 3.13. Note that
$f(\psi)=\lim_{n\to\infty}f(P_{n}\psi)$ in $L_{2}$ holds for $\psi\in
C_{c}^{q}$ by Lemma 62 for high enough $q$. Using 3.12, 3.13 and counting the
non-zero terms, we can formally calculate

$\displaystyle\left|\left|f(P_{n}\psi)-f^{(n)}(\psi)\right|\right|_{L_{2}}^{2}$
$\displaystyle=\left|\left|\sum_{x\in\Delta_{n}}(f-F_{x})(\phi_{x}^{n})\langle\phi_{x}^{n},\psi\rangle\right|\right|_{L_{2}}^{2}$
$\displaystyle\leq\sum_{x\in\Delta_{n}^{1,e}}\left(\sum_{y\in\Delta_{n}^{e+1,d}}\left|\left|(f-F_{(x,y)})(\phi_{(x,y)}^{n})\langle\phi_{(x,y)}^{n},\psi\rangle\right|\right|_{L_{2}}\right)^{2}$
$\displaystyle\qquad+\sum_{x_{1}\neq
x_{2}\in\Delta_{n}^{1,e}}\sum_{y_{1},y_{2}\in\Delta_{n}^{e+1,d}}E\left(\prod_{i=1}^{2}(f-F_{(x_{i},y_{i})})(\phi_{(x_{i},y_{i})}^{n})\langle\phi_{(x_{i},y_{i})}^{n},\psi\rangle\right)$
$\displaystyle\lesssim 2^{-2n\gamma}\xrightarrow{n\to\infty}0\,,$ (4.20)

where the calculation is formal at the moment since we applied 3.13 without
checking that $(f-F_{(x_{i},y_{i})})(\phi^{n}_{(x_{i},y_{i})})$ have disjoint
effective stochastic supports for $i=1,2$. Looking at said effective
stochastic support, we see that it is given by
$[\pi_{1}(x_{i}),\pi_{1}(x_{i})+2\cdot
2^{-ns_{1}}R]\times\dots\times[\pi_{e}(x_{i}),\pi_{e}(x_{i})+2\cdot
2^{-ns_{e}}R]$, so the mesh size of $\Delta_{n}^{1,e}$ of $2^{-ns_{i}}$ for
$i=1\dots,e$ is not big enough to ensure that the effective stochastic
supports are disjoint for all $x_{1}\neq x_{2}\in\Delta_{n}^{1,e}$.

However, comparing the mesh size $2^{-ns_{i}}$ with the size of the effective
stochastic support $2\cdot 2^{-ns_{i}}R$ in the direction $i=1,\dots,e$, we
see that for each $x_{1}\in\Delta_{n}^{1,e}$, only a constant number of
$x_{2}\in\Delta_{n}^{1,e}$ will cause an overlapping effective stochastic
support. So it suffices to introduce finitely many offsets $r^{(0)}_{n}$ and
split the mesh $\Delta_{n}^{1,e}$ into finitely many meshes
$\Delta_{n}^{1,e}(r_{n}^{(0)})$. 4.20 holds true if we replace
$\Delta_{n}^{1,e}$ with $\Delta_{n}^{1,e}(r_{n}^{(0)})$ and summing over all
$r_{n}^{(0)}$ only adds a constant factor to the $\lesssim$. It follows that
$\left|\left|f(P_{n}\psi)-f^{(n)}(\psi)\right|\right|$ vanishes for $\psi\in
C_{c}^{q}$, which shows the claim. ∎

With this, we can tackle the proof of Theorem 30. As the proof is very similar
to the proof of Theorem 22, we will skip over some of the calculations and
focus on highlighting the broader arguments.

###### Proof of Theorem 30.

We only show the proof under the additional assumption that for any
$x\in\mathbb{R}^{d}$, $F_{x}\in C^{\beta}(L_{2})$ for a $\beta\in\mathbb{R}$
such that $\beta+\tilde{r}>0$. The general case can be recovered from this
case just as before.

For $n\leq l$, recall our decomposition 4.18 for a test function $\psi$ with
support in $[\tilde{C},\tilde{R}]^{d}$ for some $0<\tilde{C}<\tilde{R}$,
$0<\lambda\leq 1$ and $n$ such that $B2^{-n}\leq\lambda\leq B2^{-n+1}$ for
$B=\min_{i=1,\dots,d}\left(\frac{\tilde{C}}{2R}\right)^{-\frac{1}{s_{i}}}$:

$\displaystyle f^{(l)}(\psi_{x}^{\lambda})-F_{x}(P_{l}\psi_{x}^{\lambda})$
$\displaystyle=\underbrace{(f^{(n)}-P_{n}F_{x})(\psi_{x}^{\lambda})}_{(I)_{n}}+\sum_{m=n}^{l-1}\underbrace{\hat{P}_{m}(f^{(m+1)}-f^{(m)}-F_{x})(\psi_{x}^{\lambda})}_{(II)_{m}}$
$\displaystyle\qquad+\sum_{m=n}^{l-1}\underbrace{P_{m}(f^{(m+1)}-f^{(m)})(\psi_{x}^{\lambda})}_{(III)_{m}}\,.$

The bound $\left|\left|(II)_{m}\right|\right|_{L_{2}}\lesssim
2^{-m(\alpha+\tilde{r})}\lambda^{\gamma-\frac{E}{2}-\alpha-\tilde{r}}\left|\left|\psi\right|\right|_{C_{c}^{r}}$
holds by the same argument as in Theorem 22, so we only need to bound
$\left|\left|(III)_{m}\right|\right|_{L_{p}}$ to show convergence. Note that
4.17 still holds, and one can use 3.11 to replace 4.7 with

$\left|E\left(\prod_{i=1}^{2}(f^{(m+1)}-f^{(m)})(\phi_{x_{i}}^{m})\right)\right|\lesssim
2^{-2m\gamma-m\left|S\right|}\,,$ (4.21)

for $\left|x_{1}^{(e)}-x_{2}^{(e)}\right|\geq 2eR2^{-m}$ (where
$x_{i}^{(e)}=(\pi_{1}(x_{i}),\dots,\pi_{e}(x_{i}))$), since the stochastic
effective support of all terms in the sum
$(f^{(m+1)}-f^{(m)})(\phi_{x_{i}}^{m})=\sum_{k\in\Delta^{n+1}}a_{k}^{(n+1)}(F_{x+k}-F_{x})(\phi_{k+x}^{(n+1)})$
(see 4.6) are contained in
$[\pi_{1}(x_{i}),\pi_{1}(x_{i})+2R2^{-ms_{1}}]\times\dots\times[\pi_{e}(x_{i}),\pi_{e}(x_{i})+2R2^{-ms_{e}}]$.
Recall that $(III)_{m}$ is of the form

$(III)_{m}=\sum_{x\in\Delta_{m}^{1,e},y\in\Delta_{m}^{e+1,d}}g_{(x,y)}\,$

where
$g_{x}:=(f^{m+1}-f^{(m)})(\phi_{x}^{m})\langle\phi_{x}^{m},\psi_{x_{0}}^{\lambda}\rangle$.
Just as in the proof of Lemma 40, we do not have a satisfying upper bound for
$\left|E(g_{(x_{1},y_{1})}g_{(x_{2},y_{2})})\right|$ with $x_{1}\neq
x_{2}\in\Delta_{m}^{1,e},y_{1},y_{2}\in\Delta_{m}^{e+1,d}$, unless
$\left|x_{1}^{(e)}-x_{2}^{(e)}\right|$ is big enough to ensure disjoint
effective stochastic supports of the involved terms. This can be achieved by
splitting the net $\Delta_{n}$ into different nets $\Delta_{n}(r_{n}^{(0)})$,
as before. This will only add a constant factor to our calculations, so we
suppress this notation and write with a slight abuse of notation:

$\displaystyle\left|\left|\sum_{x\in\Delta_{m}^{1,e},y\in\Delta_{m}^{e+1,d}}g_{(x,y)}\right|\right|_{L_{2}}^{2}$
$\displaystyle\leq\sum_{x\in\Delta_{m}^{1,e}}\left(\sum_{y\in\Delta_{m}^{e+1,d}}\left|\left|g_{(x,y)}\right|\right|_{L_{2}}\right)^{2}+\sum_{x_{1}\neq
x_{2}\in\Delta_{m}^{1,e}}\sum_{y_{1},y_{2}\in\Delta_{m}^{e+1,d}}E(g_{(x_{1},y_{1})}g_{(x_{2},y_{2})})$
$\displaystyle\lesssim
2^{-2m\gamma}(\lambda^{-E}+1)\left|\left|\psi\right|\right|_{C_{c}^{r}}^{2}\,,$

where one uses the same bounds 4.17, 4.21 and A.5 while counting the order of
non-zero terms. This shows that

$\left|\left|(III)_{m}\right|\right|_{L_{2}}\lesssim
2^{-m\gamma}\lambda^{-\frac{E}{2}}\left|\left|\psi\right|\right|_{C_{c}^{r}}\,,$

which is sufficient to show that $f^{(n)}(\psi)$ is cauchy and thus
convergent. As before, the limit $f$ is a random distribution.

It remains to show 3.12 and 3.13. Note that 3.12 follows exactly as 3.8 in
Theorem 22, so we only need to focus on 3.13. To show this, we use again the
decomposition 4.18 and write
$(I)_{n}(\psi_{x}^{\lambda})=(f^{(n)}-P_{n}F_{x})(\psi_{x}^{\lambda})$. Before
we calculate
$\left|E((I)_{n}(\psi_{x_{1}}^{\lambda})(I)_{n}(\psi_{x_{2}}^{\lambda}))\right|$,
we need to check that there is no problem with the effective stochastic
supports of any involved term: It holds that

$(I)_{n}(\psi_{x}^{\lambda})=\sum_{y\in\Delta_{n}}(F_{y}-F_{x})(\phi_{y}^{n})\langle\phi_{y}^{n},\psi_{x}^{\lambda}\rangle\,,$
(4.22)

which has for all non-zero terms and for $j=1,\dots,e$:
$\pi_{j}(y)\geq\pi_{j}(x)$ and
$\pi_{j}(y)\leq\pi_{j}(x)+\tilde{R}\lambda^{s_{j}}$ by our choice of $n$. Thus
the effective stochastic support of $(F_{x}-F_{y})(\phi_{y}^{n})$ is a subset
of
$\prod_{j=1}^{e}[\pi_{j}(x),\pi_{j}(x)+\tilde{R}\lambda^{s_{j}}+2^{-ns_{j}}R]$.
Using again our choice of $n$, we get that
$2^{-ns_{j}}R\lesssim\lambda^{s_{i}}\tilde{C}/2\leq\lambda^{s_{i}}\tilde{R}$
and thus
$\pi_{j}(x)+\tilde{R}\lambda^{s_{j}}+2^{-ns_{j}}R\leq\pi_{j}(x)+2\tilde{R}\lambda^{s_{i}}$.
Thus, the stochastic effective support of all terms in the sum
$I_{n}(\psi_{x}^{\lambda})$ are contained in the one of
$(f-F_{x})(\psi_{x}^{\lambda})$. So as long as
$(f-F_{x_{i}})(\psi_{x_{i}}^{\lambda})$ has disjoined stochastic effective
supports for $i=1,2$, we can apply the coherence property 3.11 to all terms
appearing in $(I)_{n}(\psi_{x_{1}}^{\lambda})(I_{n})(\psi_{x_{2}}^{\lambda})$.
Similarly, it holds that the stochastic effective supports of terms in
$(II)_{m}(\psi_{x}^{\lambda})$ and $(III)_{m}(\psi_{x}^{\lambda})$ (where we
define $(II)_{m}(\psi_{x}^{\lambda}),(III)_{m}(\psi_{x}^{\lambda})$
analogously to $(I)_{n}(\psi_{x}^{\lambda})$) are contained in the one of
$(f-F_{x})(\psi_{x}^{\lambda})$, so there is never a problem applying 3.11.

So let $x_{1},x_{2}$ be chosen in such a way, that
$(f-F_{x_{i}})(\psi_{x_{i}}^{\lambda})$ has disjoined stochastic support. We
use 4.22 to calculate

$\displaystyle\left|E((I)_{n}(\psi_{x_{1}}^{\lambda})(I)_{n}(\psi_{x_{2}}^{\lambda}))\right|$
$\displaystyle\leq\sum_{y_{1},y_{2}\in\Delta_{n}}\left|E\left(\prod_{i=1}^{2}(F_{y_{i}}-F_{x_{i}})(\phi_{y_{i}}^{n})\langle\phi_{y_{i}}^{n},\psi_{x_{i}}^{\lambda}\rangle\right)\right|$
$\displaystyle\lesssim
2^{-2n\gamma-n\left|S\right|}\sum_{y_{1},y_{2}\in\Delta_{n}}\left|\prod_{i=1}^{2}\langle\phi_{y_{i}}^{n},\psi_{x_{i}}^{\lambda}\rangle\right|$
$\displaystyle\lesssim 2^{-2n\gamma}\approx\lambda^{2\gamma}\,,$

since the number of non-zero terms is of order
$2^{2n\left|S\right|}\lambda^{2\left|S\right|}$. More generally, by using the
identities

$\displaystyle(II)_{m}(\psi_{x}^{\lambda})$
$\displaystyle=\sum_{y\in\Delta_{m}}\sum_{z\in\Delta_{m+1}}\sum_{\hat{\phi}\in\Phi}(F_{z}-F_{x})(\phi_{z}^{m+1})\langle\phi_{z}^{m+1},\hat{\phi}_{y}^{m}\rangle\langle\hat{\phi}_{y}^{m},\psi_{x}^{\lambda}\rangle$
(4.23) $\displaystyle(III)_{m}(\psi_{x}^{\lambda})$
$\displaystyle=\sum_{y\in\Delta_{m}}(f^{(m+1)}-f^{(m)})(\phi_{y}^{m})\langle\phi_{y}^{m},\psi_{x}^{\lambda}\rangle$
$\displaystyle=\sum_{y\in\Delta_{m}}\sum_{k\in\Delta^{m+1}}a_{k}^{(m+1)}(F_{y+k}-F_{y})(\phi_{y+k}^{(m+1)})\langle\phi_{y}^{m},\psi_{x}^{\lambda}\rangle\,$
(4.24)

we see that $(I)_{n}(\psi_{x}^{\lambda})$ interacts with the respective other
terms by adding a factor

$\displaystyle\left|E((I)_{n}(\psi_{x_{1}}^{\lambda})(\bigstar_{1}))\right|$
$\displaystyle=\left|E\left(\left(\sum_{y_{1}\in\Delta_{n}}(F_{y_{1}}-F_{x_{1}})(\phi_{y_{1}}^{n})\langle\phi_{y_{1}}^{n},\psi_{x_{1}}^{\lambda}\rangle\right)\cdot(\bigstar_{2})\right)\right|$
(4.25) $\displaystyle\lesssim\lambda^{\gamma}(\bigstar_{3})\,,$

where $(\bigstar_{1})$ is given by $(I)_{n}(\psi_{x_{2}}^{\lambda})$
$(II)_{m}(\psi_{x_{2}}^{\lambda})$ or $(III)_{m}(\psi_{x_{2}}^{\lambda})$,
$(\bigstar_{2})$ is given by 4.22, 4.23 or 4.24, respectively and
$(\bigstar_{3})$ is given by $\lambda^{a}2^{-mb}$ for certain rates $a,b$
depending on $(\bigstar_{1})$. A similar analysis leads to the bounds

$\displaystyle\left|E((II)_{m}(\psi_{x_{1}}^{\lambda})(\bigstar_{1}))\right|$
$\displaystyle=\left|E\left(\left(\sum_{y_{1}\in\Delta_{m}}\sum_{z_{1}\in\Delta_{m+1}}\sum_{\hat{\phi}\in\Phi}(F_{z_{1}}-F_{x_{1}})(\phi_{z_{1}}^{m+1})\langle\phi_{z_{1}}^{m+1},\hat{\phi}_{y_{1}}^{m}\rangle\langle\hat{\phi}_{y_{1}}^{m},\psi_{x_{1}}^{\lambda}\rangle\right)\cdot(\bigstar_{2})\right)\right|$
(4.26) $\displaystyle\lesssim
2^{-m(\alpha+\tilde{r})}\lambda^{\gamma-\alpha-\tilde{r}}(\bigstar_{3})$

as well as

$\displaystyle\left|E((III)_{m}(\psi_{x_{1}}^{\lambda})(\bigstar_{1}))\right|$
$\displaystyle=\left|E\left(\left(\sum_{y_{1}\in\Delta_{m}}\sum_{k_{1}\in\Delta^{m+1}}a_{k_{1}}^{(m+1)}(F_{y_{1}+k_{1}}-F_{y_{1}})(\phi_{y_{1}+k_{1}}^{(m+1)})\langle\phi_{y_{1}}^{m},\psi_{x_{1}}^{\lambda}\rangle\right)\cdot(\bigstar_{2})\right)\right|$
(4.27) $\displaystyle\lesssim 2^{-m\gamma}(\bigstar_{3})\,.$

Combining these gives us the bounds

$\displaystyle\left|E((II)_{l}(\psi_{x_{1}}^{\lambda})(II)_{k}(\psi_{x_{2}}^{\lambda}))\right|$
$\displaystyle\lesssim
2^{-(k+l)(\alpha+\tilde{r})}\lambda^{2\gamma-2(\alpha+\tilde{r})}$
$\displaystyle\left|E((III)_{k}(\psi_{x_{1}}^{\lambda})(III)_{l}(\psi_{x_{2}}^{\lambda}))\right|$
$\displaystyle\lesssim 2^{-(k+l)\gamma}$
$\displaystyle\left|E((I)_{n}(\psi_{x_{1}}^{\lambda})(II)_{m}(\psi_{x_{2}}^{\lambda}))\right|$
$\displaystyle\lesssim
2^{-m(\alpha+\tilde{r})}\lambda^{2\gamma-(\alpha+\tilde{r})}$
$\displaystyle\left|E((I)_{n}(\psi_{x_{1}}^{\lambda})(III)_{m}(\psi_{x_{2}}^{\lambda}))\right|$
$\displaystyle\lesssim 2^{-m\gamma}\lambda^{\gamma}$
$\displaystyle\left|E((II)_{k}(\psi_{x_{1}}^{\lambda})(III)_{l}(\psi_{x_{2}}^{\lambda}))\right|$
$\displaystyle\lesssim
2^{-k(\alpha+\tilde{r})-l\gamma}\lambda^{\gamma-(\alpha+\tilde{r})}\,.$

Note that $\gamma>0,\alpha+\tilde{r}>0$ implies that all of the above terms
are summable in $k,l,m$, respectively. Summing those expressions over their
respective indices in the decomposition 4.18 and using $2^{-n}\cong\lambda$
shows

$\displaystyle\left|E\left(\prod_{i=1}^{2}(f-F_{x_{i}})(\psi_{x_{i}}^{\lambda})\right)\right|$
$\displaystyle\lesssim\lambda^{2\gamma}\,,$

which finishes the proof. ∎

######  Remark 41.

A surprising detail of this proof is that the existence part only makes use of
3.11 for $\psi_{1}=\psi_{2}$ and $\epsilon_{1}=\epsilon_{2}$, while the
general case was used to show 3.13. This implies that the reconstruction is
possible under the weaker condition that 3.11 holds only for
$\psi_{1}=\psi_{2}$ and $\epsilon_{1}=\epsilon_{2}$, but in this case, we do
not know how to uniquely characterize the limit.

Last but not least, let us proof Corollary 31

###### Proof of Corollary 31.

This proof is essentially identical to the proof of 3.13 in Theorem 30. The
only difference is that we use two natural numbers $n_{1},n_{2}$ such that
$B2^{-n_{i}}\leq\lambda_{i}\leq B2^{-n_{i}+1}$ for $i=1,2$, and decompose
$(f-F_{x})((\psi_{i})_{x_{i}}^{\lambda_{i}})$ with 4.18 with the respective
$n_{i}$. One observes that the calculations 4.25, 4.26 and 4.27 still hold if
we add the corresponding indexes $i$ everywhere. This directly gives us

$\left|E\left(\prod_{i=1}^{2}(f-F_{x_{i}})((\psi_{i})_{x_{i}}^{\lambda_{i}})\right)\right|\lesssim\lambda_{1}^{\gamma}\lambda_{2}^{\gamma}\,.$

∎

##  5 Stochastic reconstruction is stochastic sewing in 1 dimension

As we discussed in the introduction, sewing and reconstruction are closely
related to one another. It is well known [BZ22],[Zor22], that for a two-
parameter process $(A(s,t))_{s,t\in[0,T]}$ which fulfills the conditions of
the classical sewing lemma, the germ given by

$F_{s}(t)=\frac{\partial}{\partial t}A(s,t)$

fulfills the conditions of the reconstruction theorem (as formulated in
[CZ20]). Here, $\frac{\partial}{\partial t}$ should be seen as a
distributional derivative, i.e.

$F_{s}(\psi)=-\int_{-\infty}^{\infty}A(s,t)\psi^{\prime}(t)dt$

for any $\psi\in C_{c}^{r}$.

######  Remark 42.

We should note that the classical sewing lemma defines $A(s,t)$ only on
$\Delta:=\\{(s,t)\in[0,T]^{2}\leavevmode\nobreak\ |\leavevmode\nobreak\ s\leq
t\\}$ and assumes $A(s,s)=0$ for all $s$. In this case, we can extend $A$
anti-symmetrically by setting $A(t,s):=-A(s,t)$ for all $s\leq t$.

Using the above germ $(F_{s})_{s\in[0,T]}$ and under the assumptions of the
sewing lemma, it holds that $(F_{s})_{s\in\mathbb{R}}$ can be reconstructed to
get a distribution $f$ and that $f=\frac{\partial}{\partial t}I$ is the
distributional derivative of the process $(I(t))_{t\in[0,T]}$ gained from the
sewing lemma.

Thus, sewing can indeed be seen as the one-dimensional case of reconstruction.
As a result of [Bra19], Lemma 3.10, we can reverse the time derivative and
regain $I(t)$ as “$f(1_{[0,t]})$”, which is of course only rigorous as the
function $z$ from [Bra19], Lemma 3.10. This allows one to construct the
process $I$ from the reconstruction $f$.

Given all this, it should come as little surprise, that the statement stays
true in the stochastic case: indeed, the stochastic reconstruction theorem is
nothing else than the distributional version of Khoa Lê’s stochastic sewing
Lemma in one dimension. We recall said stochastic sewing lemma [Lê20]:

######  Theorem 43 (stochastic sewing lemma).

Let $2\leq p<\infty$, and let $(A(s,t))_{s,t\in[0,T]}$ be a two parameter
process in $L_{p}(\Omega)$, which is adapted to some complete, right
continuous filtration $\mathcal{F}_{t}$ in the sense, that
$A(s,t)\in\mathcal{F}_{t}$ for all $0\leq s\leq t\leq T$. Suppose that there
is a $\gamma>0$, such that for all $0\leq s\leq u\leq t\leq T$, it holds that

$\displaystyle\left|\left|A(s,t)-A(s,u)-A(u,t)\right|\right|_{L_{p}}$
$\displaystyle\lesssim\left|t-s\right|^{\frac{1}{2}+\gamma}$
$\displaystyle\left|\left|E^{\mathcal{F}_{s}}(A(s,t)-A(s,u)-A(u,t))\right|\right|_{L_{p}}$
$\displaystyle\lesssim\left|t-s\right|^{1+\gamma}.$

Then, there is a unique (up to modifications) process $(I(t))_{t\in[0,T]}$ in
$L_{p}(\Omega)$, which is adapted to $\mathcal{F}_{t}$, has $I(0)=0$ and
fulfills for all $0\leq s\leq t\leq T$

$\displaystyle\left|\left|I(t)-I(s)-A(s,t)\right|\right|_{L_{p}}$
$\displaystyle\lesssim\left|t-s\right|^{\frac{1}{2}+\gamma}$
$\displaystyle\left|\left|E^{\mathcal{F}_{s}}(I(t)-I(s)-A(s,t))\right|\right|_{L_{p}}$
$\displaystyle\lesssim\left|t-s\right|^{1+\gamma}.$

Note that we can easily extend $A(s,t)$ to be a process over $\mathbb{R}^{2}$,
by setting $A(s,t)=A(0,t)$ for $s\leq 0,t\in[0,T]$ and $A(s,t)=A(s,T)$ for
$t\geq T,s\in[0,T]$. Should both parameter $s,t\notin[0,T]$, we simply set
$A(s,t)=A(0,T)$ if $s\leq 0$ and $t\geq T$ and $A(s,t)=0$, if both are less
than $0$ or greater than $T$. This leads to a process $I(t)$, which is the
same for $t\in[0,T]$ and constant outside of this interval.

We want to set $F_{s}(t)=\frac{\partial}{\partial t}A(s,t)$, as above. To do
so, we need to assume that almost surely $A(s,\cdot)$ is locally integrable.
In practice, most candidates for $A$ are almost surely continuous or at least
piecewise continuous, so this additional assumption is reasonable to make.
Under this assumption, we define the germ

$F_{s}(\psi)=-\int_{-\infty}^{\infty}A(s,t)\psi^{\prime}(t)dt$

as before. Note that $\mathcal{F}_{t}$ fulfills our assumptions of
completeness and right-continuity by assumption, and is trivially orthogonal
as we only have one filtration. It follows that the germ $(F_{s})$ has
stochastic dimension $e=d=1$. We can now show the following:

######  Theorem 44.

Let $A(s,t)$ fulfill the assumptions of the stochastic sewing lemma, and
assume $A(s,\cdot)$ is in $L_{1}^{loc}(\mathbb{R})$ almost surely. Then,
$(F_{s})_{s\in\mathbb{R}}$ is stochastically $\gamma$-coherent with stochastic
dimension $1$. Let $I(t)$ be the process we get from the sewing lemma from
$A(s,t)$, and let $f$ be the distribution gained from Theorem 22 applied to
$(F_{x})$. It holds that

$f(\psi)=-\int_{-\infty}^{\infty}I(t)\psi^{\prime}(t)dt,$

for all $\psi\in C_{c}^{r}$ a.s.

###### Proof.

Let us check the coherence property: Observe that

$\displaystyle(F_{s}-F_{u})(\psi)=\int_{-\infty}^{\infty}(A(s,t)-A(s,u)-A(u,t))\psi^{\prime}(t)dt,$

where we used that $A(s,u)\int_{-\infty}^{\infty}\psi^{\prime}(t)=0$ since
$\psi$ is only compactly supported. It follows that for all $s\leq u$ and
$\psi$ with compact, positive support

$\displaystyle\left|\left|(F_{s}-F_{u})(\psi_{u}^{\lambda})\right|\right|_{L_{2}}$
$\displaystyle\leq\sup_{t\in\mathrm{supp}(\psi_{u}^{\lambda})}\left|\left|A(s,t)-A(s,u)-A(u,t)\right|\right|_{L_{2}}\int_{-\infty}^{\infty}\left|(\psi_{u}^{\lambda})^{\prime}(t)\right|dt$
$\displaystyle\lesssim(\left|u-s\right|+\lambda)^{\frac{1}{2}+\gamma}\lambda^{-1},$

and analogously

$\left|\left|E^{\mathcal{F}_{s}}(F_{s}-F_{u})(\psi_{u}^{\lambda})\right|\right|_{L_{2}}\lesssim(\left|u-s\right|+\lambda)^{1+\gamma}\lambda^{-1}.$

Thus, $(F_{s})_{s\in\mathbb{R}}$ is indeed stochastically $\gamma$-coherent
with $\alpha=-1$ and there exists a reconstruction $f$. It remains to show
that $f=\frac{\partial}{\partial t}I$. To this end, we calculate

$\displaystyle\left(F_{s}-\frac{\partial}{\partial
t}I\right)(\psi)=-\int_{-\infty}^{\infty}(A(s,t)-(I(t)-I(s)))\psi^{\prime}(t)dt,$

where we again used that $I(s)\int\psi^{\prime}(t)dt=0$. It follows that for
all test functions with strictly positive support

$\displaystyle\left|\left|\left(F_{s}-\frac{\partial}{\partial
t}I\right)(\psi_{s}^{\lambda})\right|\right|_{L_{2}}$
$\displaystyle\leq\sup_{t\in\mathrm{supp}(\psi_{s}^{\lambda})}\left|\left|A(s,t)-(I(t)-I(s))\right|\right|_{L_{2}}\int_{-\infty}^{\infty}\left|(\psi_{s}^{\lambda})^{\prime}(t)\right|dt$
$\displaystyle\lesssim\lambda^{-\frac{1}{2}+\gamma},$

and analogously,

$\left|\left|E^{\mathcal{F}_{s}}\left(F_{s}-\frac{\partial}{\partial
t}I\right)(\psi_{s}^{\lambda})\right|\right|_{L_{2}}\lesssim\lambda^{\gamma}.$

By the uniqueness of the reconstruction, this concludes the proof. ∎

##  6 Gaussian Martingale Measure

As an application of stochastic reconstruction, we show that integration
against Gaussian martingale measures can be seen as a product of a process in
$C^{\alpha}$ and a distribution in $C^{\beta}$, similar to the young product
between distributions presented in the introduction. The martingale properties
of the measure will allow us to do this reconstruction up to
$\alpha+\beta>-\frac{1}{2}$, in comparison to the classical assumption
$\alpha+\beta>0$.

We begin by introducing the notation of martingale measures, which can be
found in [Dal+09] or [Wal86].

###  6.1 Gaussian martingale measures and Walsh-type Integration

Loosely speaking, a Gaussian martingale measure is a Gaussian family
$(W_{t}(A))_{t\geq 0,A\in\mathcal{A}_{K}}$ for some subset
$\mathcal{A}_{K}\subset B(\mathbb{R}^{d})$, such that

  * •

$W_{t}(A)$ is a martingale in $t$, and

  * •

$W_{t}(A)$ is a measure in $A$ in the sense that $W_{t}(\emptyset)=0$ and for
all disjoint sets $(A_{n})_{n\in\mathbb{N}}$,
$W_{t}\left(\bigcup_{n}A_{n}\right)=\sum_{n}W_{t}(A_{n})$ (where both
equations hold almost surely).

To get a Gaussian measure (or a Gaussian family in general), we need to
clarify its covariance. This is given through the notion of a covariance
measure:

######  Definition 45.

Let $K$ be a symmetric, positive definite and $\sigma$-finite signed measure
on $(\mathbb{R}^{d}\times\mathbb{R}^{d},B(\mathbb{R}^{d})\otimes
B(\mathbb{R}^{d}))$. Then, $K$ is called a _covariance measure_ , if there is
a symmetric, positive definite, and $\sigma$-finite measure $\left|K\right|$,
such that for all $A,B\in B(\mathbb{R}^{d})$:

$\left|K(A\times B)\right|\leq\left|K\right|(A\times B).$

We set

$\left|\left|f\right|\right|_{K}^{2}:=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}f(x)f(y)K(dx,dy),$

and define $\left|\left|f\right|\right|_{\left|K\right|}$ analogously. We
further set $\mathcal{A}_{K}:=\\{A\in B(\mathbb{R}^{d})\leavevmode\nobreak\
|\leavevmode\nobreak\ K(A\times A)<\infty\\}$. This allows us to formally
define the Gaussian martingale measure $W$ as follows:

######  Definition 46.

Let $K$ be a covariance measure. A _Gaussian martingale measure_ is a family
of centered Gaussian variables

$(W_{t}(A)\leavevmode\nobreak\ |\leavevmode\nobreak\ t\geq
0,A\in\mathcal{A}_{k}),$

such that

  1. i)

$W_{0}(A)=0$.

  2. ii)

For all $A\in\mathcal{A}_{K}$, $(W_{t}(A))_{t\geq 0}$ is a continuous
martingale with respect to the filtration $\sigma(W_{s}(A)\leavevmode\nobreak\
|\leavevmode\nobreak\ 0\leq s\leq t,A\in\mathcal{A}_{K})$.

  3. iii)

Almost surely, $W_{t}(\emptyset)=0$ and
$W_{t}\left(\bigcup_{n}A_{n}\right)=\sum_{n}W_{t}(A_{n})$ holds for all
disjoint sequences $(A_{n})_{n\in\mathbb{N}}$.

  4. iv)

Its covariance is given by

$E(W_{s}(A)W_{t}(B))=(s\land t)K(A\times B).$

######  Remark 47.

One can show that iii) is redundant in the sense that it can be derived from
the other three properties. Since we decided not to present this result, we
included the property in the definition.

We also denote the closure of the filtration of $W_{t}$ as

$\mathcal{F}_{t}:=\overline{\sigma(W_{s}(A)\leavevmode\nobreak\
|\leavevmode\nobreak\ 0\leq s\leq t,A\in\mathcal{A}_{K})}\,,$

which is complete, right continuous, and, since we only have one filtration,
fulfills the orthogonality assumption. The most common example of such a
measure is space-time white noise, which is a Gaussian martingale measure with
covariance measure $K^{WN}(dx,dy)=dx\,\delta_{x}(dy)$, which should be read as

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}f(x)g(y)K^{WN}(dx,dy)=\int_{\mathbb{R}^{d}}f(x)g(x)dx\,.$

The construction of the integral over such martingale measure is rather
straight-forward: We call processes of the form

$H(s,x)=\sum_{k=1}^{n-1}\sum_{l=1}^{L_{k}}h_{k,l}1_{(t_{k},t_{k+1}]}(s)1_{A_{k,l}}(x)$

for $\mathcal{F}_{t_{k}}$ measurable random variables $h_{k,l}\in
L_{\infty}(\Omega)$ and sets $A_{k,l}\in\mathcal{A}_{K}$ _elementary
processes_ , and define the integral over such processes by

$\int_{0}^{\infty}\int_{\mathbb{R}^{d}}H(s,x)W(ds,dx):=\sum_{k=1}^{n-1}\sum_{l=1}^{L_{k}}h_{k,l}(W_{t_{k+1}}(A_{k,l})-W_{t_{k}}(A_{k,l})).$

We want to extend this definition to a certain $L_{2}$ space. To do this, let
$\mathcal{P}$ be the predictable $\sigma$-algebra, which is generated by the
elementary processes. We set

$L_{2}(W):=\left\\{H\text{
predictable},E\left(\int_{0}^{\infty}\left|\left|H(s,\cdot)\right|\right|_{\left|K\right|}^{2}ds\right)<\infty\right\\}.$

The following result can be found in [Wal86]:

######  Lemma 48.

The set of elementary processes is dense in $L_{2}(W)$.

Our notion of an integral extends to the space $L_{2}(W)$, as the following
theorem states:

######  Theorem 49 (Walsh Integral).

The above-given integral can be extended uniquely to a continuous, linear map

$\displaystyle L_{2}(W)$ $\displaystyle\rightarrow M^{2}_{0}$ $\displaystyle
H$ $\displaystyle\mapsto\int_{0}^{t}\int_{\mathbb{R}^{d}}H(s,x)W(ds,dx),$

where $M^{2}_{0}$ is the set of square-integrable martingales starting from 0.

The proof can be found in [Wal86].

By a simple approximation argument, one can show that the _Itô-isometry_ holds
for all $H\in L_{2}(W)$ and $t\geq 0$:

$E\left(\left(\int_{0}^{t}\int_{\mathbb{R}^{d}}H(s,x)W(ds,dx)\right)^{2}\right)=E\left(\int_{0}^{t}\left|\left|H(s,\cdot)\right|\right|_{K}^{2}ds\right).$

With this, we can tackle the main result of this section:

###  6.2 Reconstruction of the Walsh-type Integral

Let $W$ be a Gaussian martingale measure, and let $X(t,x)$ be a predictable
process. We want to show, that the distribution associated with the Walsh-type
integral

$\psi\mapsto\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\psi(s,x)X(s,x)W(ds,dx)\,,$

can be constructed with the stochastic reconstruction theorem. Note that the
integral is a martingale in time $t$, making our stochastic dimension $e=1$.

We need a local approximation for the integral, which we get by the constant
approximating $X(s,x)\approx X(t,y)$ for $(s,x)$ close to $(t,y)$. This
motivates using the germ $F_{t,y}=X(t,y)dW$, i.e.

$F_{t,y}(\psi):=X(t,y)\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\psi(s,x)W(ds,dx).$

######  Remark 50.

While this allows us to formally define the germ on any test function, it is
only really useful on test functions with a support on the right-hand-side of
$t$, so that $\tau\mapsto X(t,y)\int_{0}^{\tau}\int_{R^{d}}\psi(s,x)W(ds,dx)$
is a martingale. Our left-sided version of the reconstruction theorem mirrors
this property.

For simplicity, we restrict ourselves to $L_{2}(\Omega)$, but one can always
use Gaussianity to extend our result into $L_{p}$ spaces for more general $p$.
The regularity of $X$ is governed by its Hölder-continuity, i.e. we assume
$X\in C^{\alpha}(L_{2}(\Omega))$ for some $\alpha>0$, where we use the
classical scaling $s=(1,1,\dots,1)$. It is however not clear a priori how to
measure the regularity of $W$. To this end, let $K$ be the covariance measure
of $W$. We say that $K$ has a scaling $\delta>0$ if it holds that

$K(\lambda dx,\lambda dy)=\lambda^{\delta}K(dx,dy).$

The following lemma shows, that $W$ as a random distribution is
$-d+\frac{1}{2}+\frac{\delta}{2}$-Hölder continuous, if its covariance measure
has the scaling $\delta$:

######  Lemma 51.

Let $K$ have scaling $\delta$, as above. It holds that:

$\int_{0}^{\infty}\left|\left|\phi_{t,y}^{\lambda}(s,\cdot)\right|\right|_{K}^{2}ds\lesssim\lambda^{2\alpha}$

for $\alpha=-d-\frac{1}{2}+\frac{\delta}{2}$. We say that $K$ is of
homogeneity $\alpha.$

###### Proof.

A straightforward calculation shows:

$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\phi_{t,y}^{\lambda}(s,x_{1})$
$\displaystyle\phi_{t,y}^{\lambda}(s,x_{2})K(dx_{1},dx_{2})ds$
$\displaystyle=\lambda^{-2d-2}\int_{0}^{\infty}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\phi_{t,y}\left(\frac{s}{\lambda},\frac{x_{1}}{\lambda}\right)\phi_{t,y}\left(\frac{s}{\lambda},\frac{x_{2}}{\lambda}\right)K(dx_{1},dx_{2})ds$
$\displaystyle=\lambda^{-2d-2}\int_{0}^{\infty}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\phi_{\frac{t}{\lambda},\frac{y}{\lambda}}(r,v_{1})\phi_{\frac{t}{\lambda},\frac{y}{\lambda}}(r,v_{2})K(\lambda
dv_{1},\lambda dv_{2})\lambda dr$
$\displaystyle=\lambda^{-2d-1+\delta}\int_{0}^{\infty}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\phi_{\frac{t}{\lambda},\frac{y}{\lambda}}(r,v_{1})\phi_{\frac{t}{\lambda},\frac{y}{\lambda}}(r,v_{2})K(dv_{1},dv_{2})dr$
$\displaystyle=\lambda^{-2d-1+\delta}\int_{0}^{\infty}\left|\left|\phi_{\frac{t}{\lambda},\frac{y}{\lambda}}(r,\cdot)\right|\right|_{K}^{2}dr$
$\displaystyle=\lambda^{-2d-1+\delta}\int_{0}^{\infty}\left|\left|\phi(s,\cdot)\right|\right|_{K}^{2}ds.$

∎

######  Remark 52.

Using the Itô-isometry, this implies that

$\left|\left|\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\psi(s,x)W(ds,dx)\right|\right|_{L_{2}}=\left(\int_{0}^{\infty}\left|\left|\phi_{t,y}^{\lambda}(s,\cdot)\right|\right|_{K}^{2}ds\right)^{\frac{1}{2}}\lesssim\lambda^{\alpha}.$

Thus, the distribution
$\psi\mapsto\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}\psi(s,x)W(ds,dx)$ is
indeed in $C^{\alpha}(L_{2}(\Omega))$.

With this result, we can reconstruct the Walsh integral:

######  Theorem 53.

Let $K$ be of homogeneity $\alpha$, and let $X\in C^{\beta}(L_{2}(\Omega))$
for some $\beta\in(0,1)$. Let $(F_{t,y})$ be as above. If
$\alpha+\beta>-\frac{1}{2}$, then $(F_{t,y})$ fulfills the assumption of
Theorem 22, and thus there exists a distribution $f$ fulfilling 3.8 and 3.9.
It further holds, that $f$ is given by the Walsh-type integral, i.e.

$f(\psi)=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}X(s,x)\psi(s,x)W(ds,dx),$ (6.1)

for all $\psi\in C_{c}^{r}$ a.s., $r=\max(\lfloor-\alpha\rfloor+1,1)$.

######  Remark 54.

Observe that by Lemma 51, Theorem 53 automatically applies to all $K$ with a
scaling $\delta$, such that

$\beta+\frac{\delta}{2}>d.$

######  Remark 55.

Looking at our example of white noise, we get that $K^{WN}(\lambda dx,\lambda
dy)=(\lambda dx)\delta_{x}(\lambda dy)=\lambda^{d}K(dx,dy)$, which implies
that $K^{WN}$ has scaling $\delta=d$ and thus, white noise has stochastic
Hölder regularity $-\frac{d+1}{2}$. Therefore, we would need
$\beta>\frac{d}{2}$ to apply the above theorem, which is a highly sub-optimal
requirement. Indeed, it is well known that $\beta>0$ suffices to define
integration against space-time white noise [Qua12]. The reason our result is
much worse is that we only use the martingale properties in the time
direction, i.e. we use stochastic dimension $e=1$, whereas white noise has
additional martingale properties in all spatial directions and should have
stochastic dimension $e=d+1$. Theorem 53 is much better suited to deal with
cases of colored noise, which only have martingale properties in the time
direction. The case of space-time white noise will be discussed at length in
Section 7.

###### Proof.

We first show that the assumptions of Theorem 22 are fulfilled: Let $t\leq s$,
and let $\psi$ be a test function with support in $[0,1]\times[-1,1]^{d}$. It
holds that:

$\displaystyle E^{\mathcal{F}_{t}}$
$\displaystyle(F_{t,y}-F_{s,x})(\psi_{s,x}^{\lambda})$
$\displaystyle=E^{\mathcal{F}_{t}}\left((X(t,y)-X(s,x))E^{\mathcal{F}_{s}}\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\psi_{s,x}^{\lambda}(u,v)W(du,dv)\right)=0,$

since $\int_{0}^{\tau}\int_{\mathbb{R}^{d}}\psi_{s,x}^{\lambda}(u,v)W(du,dv)$
is a martingale in $\tau$ and $s$ is on the edge of the support of
$\psi_{s,x}^{\lambda}$. It remains to show, that

$\left|\left|(F_{t,y}-F_{s,x})(\psi_{s,x}^{\lambda})\right|\right|_{L_{2}}\lesssim\lambda^{\tilde{\alpha}}(\left|(s,x)-(t,y)\right|+\lambda)^{\gamma-\frac{1}{2}-\tilde{\alpha}}$

for some $\tilde{\alpha}\in\mathbb{R},\gamma>0$. We chose
$\gamma=\alpha+\beta+\frac{1}{2}$ and observe the following:

$\displaystyle\left|\left|(F_{t,y}-F_{s,x})(\psi_{s,x}^{\lambda})\right|\right|_{L_{2}}^{2}$
$\displaystyle=\left|\left|(X(t,y)-X(s,x))\int_{0}^{\infty}\int_{\mathbb{R}^{d}}\psi_{s,x}^{\lambda}(u,v)W(du,dv)\right|\right|_{L_{2}}^{2}$
$\displaystyle\leq\int_{0}^{\infty}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\underbrace{\left|\left|(X(t,y)-X(s,x))\right|\right|_{L_{2}}^{2}}_{\lesssim\left|(t,y)-(s,x)\right|^{2\beta}}\psi_{s,x}^{\lambda}(u,v)\psi_{s,x}^{\lambda}(u,w)K(dv,dw)du$
$\displaystyle\lesssim\left|(t,y)-(s,x)\right|^{2\beta}\int_{0}^{\infty}\left|\left|\psi_{s,x}^{\lambda}(u,\cdot)\right|\right|_{K}^{2}du$
$\displaystyle\lesssim\left|(t,y)-(s,x)\right|^{2\beta}\lambda^{2\alpha}.$
(6.2)

Thus, using $\tilde{\alpha}=\alpha$, it follows that

$\displaystyle\left|\left|(F_{t,y}-F_{s,x})(\psi_{s,x}^{\lambda})\right|\right|_{L_{2}}\lesssim\lambda^{\tilde{\alpha}}(\left|(t,y)-(s,x)\right|+\lambda)^{\alpha+\beta-\tilde{\alpha}}=\lambda^{\tilde{\alpha}}(\left|(t,y)-(s,x)\right|+\lambda)^{\gamma-\frac{1}{2}-\tilde{\alpha}}.$

This shows that there is a unique reconstruction $f$. To find $r$, note that
the proof of Theorem 22 requires $\tilde{r}=r$ to be strictly greater than the
absolute value of $\tilde{\alpha}=\alpha$, and
$r>-\gamma+\frac{1}{2}>+\frac{1}{2}$. It follows that
$r=\max(\lfloor-\alpha\rfloor+1,1)$.

It remains to show 6.1. To this end, assume that the support of $\psi$ is in
$[\tilde{C},\tilde{R}]\times[-1,1]^{d}$ for some $0<\tilde{C}<\tilde{R}$. We
need to show that

$\tilde{f}(\psi):=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}X(s,x)\psi(s,x)W(ds,dx)$

fulfills 3.8 and 3.9. Note that 3.9 is again obvious due to the martingale
properties of
$\int_{0}^{t}\int_{\mathbb{R}^{d}}X(s,x)\psi(s,x)W(ds,dx)$, so it remains to
show 3.8. Using the same calculation as in 6.2, we see that

$\displaystyle\left|\left|(\tilde{f}-F_{t,y})(\psi_{t,y}^{\lambda})\right|\right|_{L_{2}}$
$\displaystyle=\left|\left|\int_{0}^{\infty}\int_{\mathbb{R}^{d}}(X(s,x)-X(t,y))\psi_{t,y}^{\lambda}(s,x)W(ds,dx)\right|\right|_{L_{2}}$
$\displaystyle\lesssim\sup_{(s,x)\in\mathrm{supp}(\psi_{t,y}^{\lambda})}\underbrace{\left|\left|X(s,x)-X(t,y)\right|\right|_{L_{2}}}_{\lesssim\lambda^{\beta}}\int_{0}^{\infty}\left|\left|\psi_{s,x}^{\lambda}\right|\right|_{K}^{2}ds$
$\displaystyle\lesssim\lambda^{\alpha+\beta}=\lambda^{\gamma-\frac{1}{2}}.$

Thus, $\tilde{f}=f$ up to modifications, which finishes the proof. ∎

##  7 Integration against white noise

Let us take a closer look at space-time white noise, which is given by the
Gaussian martingale measure $W^{WN}$ characterized by its covariance measure
$K^{WN}(dx,dy)=dx\,\delta_{x}(dy)$. As mentioned above, it has scaling
$\delta=d$ and does not fall into the setting of Theorem 53. This is not at
all surprising, as the previous section assumes a stochastic dimension of
$e=1$, whereas space-time white noise has martingale properties in all of its
directions and should be treated as $e=d+1$. This makes integration against
white noise an easy example of a stochastic reconstruction of a non-trivial
stochastic dimension, which is our main motivation for discussing it in this
paper.

Let us first discuss the construction of space-time white noise: Treating it