# Optimal parameter estimation for linear SPDEs from multiple measurements

Randolf Altmeyer, Anton Tiepner, Martin Wahl Center for Mathematical Sciences,
University of Cambridge, Wilberforce Road, CB3 0WB Cambridge, UK, Email:
<EMAIL_ADDRESS>of Mathematics, Aarhus University, Ny Munkegade 118,
DK-8000 Aarhus C, Denmark, Email<EMAIL_ADDRESS>für Mathematik,
Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany,
E-mail<EMAIL_ADDRESS>
###### Abstract

The coefficients in a general second order linear stochastic partial
differential equation (SPDE) are estimated from multiple spatially localised
measurements. Assuming that the spatial resolution tends to zero and the
number of measurements is non-decreasing, the rate of convergence for each
coefficient depends on the order of the parametrised differential operator and
is faster for higher order coefficients. Based on an explicit analysis of the
reproducing kernel Hilbert space of a general stochastic evolution equation, a
Gaussian lower bound scheme is introduced. As a result, minimax optimality of
the rates as well as sufficient and necessary conditions for consistent
estimation are established.

MSC 2010: 60H15, 60F05; Secondary: 62F12, 62F35.

Keywords: stochastic convection-diffusion equation, parameter estimation,
central limit theorem, minimax lower bound, reproducing kernel Hilbert space,
local measurements.

##  1 Introduction

Stochastic partial differential equations (SPDEs) form a flexible class of
models for space-time data. They combine phenomena such as diffusion and
transport that occur naturally in many processes, but also include random
forcing terms, which may arise from microscopic scaling limits or account for
model uncertainty. Quantifying the size of these different effects and testing
for their presence from data is an important step in model validation.

Suppose that $X=(X(t))_{0\leq t\leq T}$ solves the linear parabolic SPDE

$dX(t)=A_{\vartheta}X(t)dt+dW(t),\quad 0\leq t\leq T,$ (1.1)

on an open, bounded and smooth domain $\Lambda\subset\mathbb{R}^{d}$ with some
initial value $X_{0}$, a space-time white noise $dW$ and a second order
elliptic operator

$\displaystyle A_{\vartheta}=\sum_{i=1}^{p}\vartheta_{i}A_{i}+A_{0}$ (1.2)

satisfying zero Dirichlet boundary conditions. The $A_{i}$ are known
differential operators of differential order $n_{i}$ and we aim at recovering
the unknown parameter $\vartheta\in\operatorname{{\mathbb{R}}}^{p}$. A
prototypical example is

$\displaystyle
A_{\vartheta}=\vartheta_{1}\Delta+\vartheta_{2}(b\cdot\nabla)+\vartheta_{3},\qquad\vartheta\in(0,\infty)\times\operatorname{{\mathbb{R}}}\times(-\infty,0],$
(1.3)

with diffusivity, transport and reaction coefficients $\vartheta_{1}$,
$\vartheta_{2}$, $\vartheta_{3}$ and a known unit vector
$b\in\operatorname{{\mathbb{R}}}^{d}$. Equations such as (1.1) are also called
stochastic advection–diffusion equations and often serve as building blocks
for more complex models, with applications in different areas such as
neuroscience [47, 41, 50], biology [2, 1], spatial statistics [43, 35] and
data assimilation [36].

While the estimation of a scalar parameter in front of the highest order
operator $A_{i}$ is well studied in the literature [24, 28, 12, 13, 20], there
is little known about estimating the lower order coefficients or the full
multivariate parameter $\vartheta$. Relying on discrete space-time
observations $X(t_{k},x_{j})$ in case of $\eqref{eq:canonical:model}$ and in
dimension $d=1$, [9, 23, 44] have analysed power variations and contrast
estimators. For two parameters in front of operators $A_{1}$ and $A_{2}$, [37]
computed the maximum likelihood estimator from $M$ spectral measurements
$(\langle X(t),e_{j}\rangle)_{0\leq t\leq T}$, $j=1,\dots,M$, where the
$e_{j}$ are the eigenfunctions of $A_{\vartheta}$. This leads as
$M\rightarrow\infty$ to rates of convergence depending on the differential
order of the operators $A_{1}$, $A_{2}$, but is restricted to domains and
diagonalisable operators with known $e_{j}$, independent of $\vartheta$. In
particular, in the spectral approach there is no known estimator for the
transport coefficient $\vartheta_{2}$ in (1.3). Estimators for nonlinearities
or noise specifications are studied e.g. by [11, 22, 18, 8].

In contrast, we construct an estimator $\hat{\vartheta}_{\delta}$ of
$\vartheta$ on general domains and arbitrary possibly anisotropic
$A_{\vartheta}$ from local measurement processes $X_{\delta,k}=(\langle
X(t),K_{\delta,x_{k}}\rangle)_{0\leq t\leq T}$, $X^{A_{i}}_{\delta,k}=(\langle
X(t),A^{*}_{i}K_{\delta,x_{k}}\rangle)_{0\leq t\leq T}$ for $i=0,\dots,p$ and
locations $x_{1},\dots,x_{M}\in\Lambda$. The $K_{\delta,x_{k}}$, also known as
_point spread functions_ in optical systems [6, 7], are compactly supported
functions on subsets of $\Lambda$ with radius $\delta>0$ and centered at the
$x_{k}$. They are part of the observation scheme and describe the physical
limitation that point sources $X(t_{k},x_{j})$ typically can only be measured
up to a convolution with the point spread function. Local measurements were
introduced in a recent contribution by [4] to demonstrate that a nonparametric
diffusivity can already be identified at $x_{k}$ from the spatially localised
information $X_{\delta,k}$ as $\delta\rightarrow 0$ with $T>0$ fixed. See also
[3, 2] for robustness to semilinear equations and different noise
configurations besides space-time white noise.

Let us briefly describe our main contributions. Our first result extends the
estimator $\hat{\vartheta}_{\delta}$ and the CLT of [4] to obtain asymptotic
normality of

$(M^{1/2}\delta^{1-n_{i}}(\hat{\vartheta}_{\delta,i}-\vartheta_{i}))_{i=1}^{p},\quad\delta\rightarrow
0,$

with $M=M(\delta)$ measurements. In particular, this yields the convergence
rates $M^{1/2}\delta^{1-n_{i}}$ for $\vartheta_{i}$, with the best rate for
diffusivity terms and the worst rate for reaction terms. We then turn to the
problem of establishing optimality of these rates in case of (1.3). To achieve
this, we compute the reproducing kernel Hilbert space (RKHS) of the Gaussian
measures induced by the laws of $X$ and of the local measurements. These
results are used to derive minimax lower bounds, implying that the rates in
the CLT are indeed optimal. In addition, our lower bounds also provide
conditions under which consistent estimation is impossible. Combining these
conditions with our CLT, we deduce that for general point spread functions
$K_{\delta,x_{k}}$ with non-intersecting supports, consistent estimation is
possible if and only if $M^{1/2}\delta^{1-n_{i}}\rightarrow\infty$. In (1.3),
we have $n_{2}=1$ and $n_{3}=0$, meaning that consistent estimation of
$\vartheta_{2}$ requires necessarily $M\rightarrow\infty$, and $\vartheta_{3}$
cannot be estimated consistently unless $M^{1/2}\delta$ remains bounded from
below. In particular, $\vartheta_{3}$ cannot be estimated consistently in
$d=1$, while $d=2$ appears as an interesting boundary case where consistency
depends on regularity properties of point spread functions. Conceptually,
spectral measurements can be obtained from local measurements approximately by
a discrete Fourier transform and we indeed recover the rates of convergence of
[24] by taking $M$ of maximal size $\delta^{-d}$.

The proofs for the CLT and the lower bound are involved due to the complex
information geometry arising from non-standard spatial asymptotics
$\delta\rightarrow 0$ and the correlations between measurements at different
locations. For instance, we introduce a novel lower bound scheme for Gaussian
measures by relating the Hellinger distance of their laws to properties of
their RKHS. This is different from the lower bound approach of [4] for $M=1$
and paves the way to rigorous lower bounds for each coefficient and an
arbitrary number of measurements. One of our key results states that the RKHS
of the Gaussian measure induced by $X$ with $A_{\vartheta}=\Delta$ consists of
all $h\in L^{2}([0,T];L^{2}(\Lambda))$ with $\Delta h,h^{\prime}\in
L^{2}([0,T];L^{2}(\Lambda))$ and its squared RKHS norm is upper bounded by an
absolute constant times

$\displaystyle\|\Delta
h\|_{L^{2}([0,T];L^{2}(\Lambda))}^{2}+\|h\|_{L^{2}([0,T];L^{2}(\Lambda))}^{2}+\|h^{\prime}\|_{L^{2}([0,T];L^{2}(\Lambda))}^{2}.$

This formula generalises the finite-dimensional Ornstein-Uhlenbeck case [32],
and provides a route to obtain the RKHS of local measurements as linear
transformations of $X$. To the best of our knowledge this result has not been
stated before in the literature, and may be of independent interest, e.g. in
constructing Bayesian procedures with Gaussian process priors, cf. [48].

The paper is organised as follows. Section 2 deals with the local measurement
scheme, the construction of our estimator and the CLT. Section 3 presents the
lower bounds for the rates established in the CLT, while Section 4 addresses
the RKHS of $X$ and of the local measurements. Section 5 covers model
examples, applications to inference and the boundary case for estimating zero
order terms in $d=2$. All proofs are deferred to Section 6 and Appendix A.

###  1.1 Basic notation

Throughout, we work on a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{0\leq t\leq
T},\operatorname{{\mathbb{P}}})$. We write $a\lesssim b$ if $a\leq Cb$ for a
universal constant $C$ not depending on $\delta$, but possibly depending on
other quantities such as $T$ and $\Lambda$. Unless stated otherwise, all
limits are understood as $\delta\rightarrow 0$ with non-decreasing
$M=M(\delta)$ possibly depending on $\delta$.

The Euclidean inner product and distance of two vectors
$a,b\in\operatorname{{\mathbb{R}}}^{p}$ is denoted by $a\cdot b$ and $|b-a|$.
We write $\|\cdot\|_{\operatorname{op}}$ for the operator norm of a matrix.
For a multi-index $\alpha=(\alpha_{1},\dots,\alpha_{d})$ let $D^{\alpha}$
denote the $\alpha$-th weak partial derivative operator of order
$|\alpha|=\alpha_{1}+\dots+\alpha_{d}$. The gradient, divergence and Laplace
operators are denoted by $\nabla$, $\nabla\cdot$ and $\Delta$.

For an open set $U\subset\operatorname{{\mathbb{R}}}^{d}$, $L^{2}(U)$ is the
usual $L^{2}$-space with inner product
$\langle\cdot,\cdot\rangle=\langle\cdot,\cdot\rangle_{L^{2}(U)}$. Let
$H^{k}(U)$ denote the usual Sobolev spaces and let $H_{0}^{1}(U)$ be the
completion of the space of smooth compactly supported functions
$C_{c}^{\infty}(U)$ relative to the $H^{1}(U)$-norm.

For a Hilbert space $\mathcal{H}$, the space $L^{2}([0,T];\mathcal{H})$
consists of all measurable functions $h:[0,T]\rightarrow\mathcal{H}$ with
$\int_{0}^{T}\|h(t)\|_{\mathcal{H}}^{2}dt<\infty$. We write $\lVert
T\rVert_{\operatorname{HS}(\mathcal{H}_{1},\mathcal{H}_{2})}$ for the Hilbert-
Schmidt norm of a linear operator
$T:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ between two Hilbert spaces
$\mathcal{H}_{1},\mathcal{H}_{2}$.

##  2 Joint parameter estimation

###  2.1 Setup

For an unknown parameter
$\vartheta\in\Theta\subseteq\operatorname{{\mathbb{R}}}^{p}$ let the operator
$A_{\vartheta}$ be as in (1.2) and suppose for $i=0,\dots,p$ that
$A_{i}=\sum_{|\alpha|\leq 2}a^{(i)}_{\alpha}D^{\alpha}$ with known
$a^{(i)}_{\alpha}\in\operatorname{{\mathbb{R}}}$. Let
$n_{i}=\operatorname{ord}(A_{i})\in\\{0,1,2\\}$ be their differential orders.
We suppose that $A_{\vartheta}$ has domain $H^{1}_{0}(\Lambda)\cap
H^{2}(\Lambda)$ and is strongly elliptic, that is, for some $C_{\vartheta}>0$
and with $x^{\alpha}=x^{\alpha_{1}}_{1}\cdots x^{\alpha_{d}}_{d}$

$\displaystyle\sum_{|\alpha|=2}\left(\sum_{i=1}^{p}\vartheta_{i}a^{(i)}_{\alpha}+a^{(0)}_{\alpha}\right)x^{\alpha}>C_{\vartheta}|x|^{2},\quad
x\in\operatorname{{\mathbb{R}}}^{d}.$ (2.1)

The operator $A_{\vartheta}$ generates an analytic semigroup on
$L^{2}(\Lambda)$, denoted by $(S_{\vartheta}(t))_{t\geq 0}$. With an
$\mathcal{F}_{0}$-measurable initial value $X_{0}$ and a cylindrical Wiener
process $W$ on $L^{2}(\Lambda)$ define a process $X=(X(t))_{0\leq t\leq T}$ by

$\displaystyle
X(t)=S_{\vartheta}(t)X_{0}+\int_{0}^{t}S_{\vartheta}(t-t^{\prime})dW(t^{\prime}),\quad
0\leq t\leq T.$ (2.2)

Due to the low spatial regularity of $W$ this process is understood as a
random element with values in $L^{2}(\Lambda)\subset\mathcal{H}_{1}$ almost
surely for any larger Hilbert space $\mathcal{H}_{1}$ with an embedding
$\iota:L^{2}(\Lambda)\rightarrow\mathcal{H}_{1}$ such that
$\int_{0}^{t}\lVert\iota
S_{\vartheta}(t^{\prime})\rVert_{\operatorname{HS}(L^{2}(\Lambda),\mathcal{H}_{1})}dt^{\prime}<\infty$
[21, Remark 5.6]. Such an embedding always exists. For example,
$\mathcal{H}_{1}$ can be realised as a negative Sobolev space (see Section 6.4
below). Let $\mathcal{H}_{1}^{\prime}$ denote the dual space of
$\mathcal{H}_{1}$ with the associated dual pairing
$\langle\cdot,\cdot\rangle_{\mathcal{H}_{1}\times\mathcal{H}_{1}^{\prime}}$.
Let $(e_{k})_{k\geq 1}$ be an orthonormal basis of $L^{2}(\Lambda)$ and let
$\beta_{k}$ be independent scalar Brownian motions. Then, realising the Wiener
process as $W=\sum_{k\geq 1}e_{k}\beta_{k}$, we find for all
$z\in\mathcal{H}_{1}^{\prime}\subset L^{2}(\Lambda)$, $0\leq t\leq T$, that
(see, e.g. [34, Lemma 2.4.1 and Proposition 2.4.5])

$\displaystyle\langle
X(t)-S_{\vartheta}(t)X_{0},z\rangle_{\mathcal{H}_{1}\times\mathcal{H}_{1}^{\prime}}=\sum_{k\geq
1}\int_{0}^{t}\langle
S_{\vartheta}(t-t^{\prime})e_{k},z\rangle_{\mathcal{H}_{1}\times\mathcal{H}_{1}^{\prime}}d\beta_{k}(t^{\prime})$
$\displaystyle\quad\quad=\sum_{k\geq 1}\int_{0}^{t}\langle
S_{\vartheta}(t-t^{\prime})e_{k},z\rangle
d\beta_{k}(t^{\prime})=\int_{0}^{t}\langle
S^{*}_{\vartheta}(t-t^{\prime})z,dW(t^{\prime})\rangle.$

According to [4, Proposition 2.1] and [34, Lemma 2.4.2] this allows us to
extend the dual pairings $\langle
X(t),z\rangle_{\mathcal{H}_{1}\times\mathcal{H}_{1}^{\prime}}$ to a real-
valued Gaussian process $(\langle X(t),z\rangle)_{0\leq t\leq T,z\in
L^{2}(\Lambda)}$ (the notation $\langle\cdot,\cdot\rangle$ is used for
convenience and indicates that the process does not depend on the embedding
space $\mathcal{H}_{1}$). This process solves the SPDE (1.1) in the sense that
for all $z\in H^{1}_{0}(\Lambda)\cap H^{2}(\Lambda)$ and $0\leq t\leq T$

$\displaystyle\langle X(t),z\rangle=\langle X_{0},z\rangle+\int_{0}^{t}\langle
X(t^{\prime}),A_{\vartheta}^{*}z\rangle dt^{\prime}+\langle W(t),z\rangle,$
(2.3)

where $\langle W(t),z\rangle/\lVert z\rVert_{L^{2}(\Lambda)}$ is a scalar
Brownian motion, and where

$\displaystyle
A_{\vartheta}^{*}=\sum_{i=1}^{p}\vartheta_{i}A_{i}^{*}+A_{0}^{*},\quad
A_{i}^{*}=\sum_{|\alpha|\leq 2}(-1)^{|\alpha|}a_{\alpha}^{(i)}D^{\alpha},$

is the adjoint operator of $A_{\vartheta}$ with the same domain
$H^{1}_{0}(\Lambda)\cap H^{2}(\Lambda)$.

###  2.2 Local measurements, construction of the estimator

Introduce for $z\in L^{2}(\operatorname{{\mathbb{R}}}^{d})$ the scale and
shift operation

$\displaystyle z_{\delta,x}(y)$
$\displaystyle=\delta^{-d/2}z(\delta^{-1}(y-x)),\quad
x,y\in\Lambda,\quad\delta>0.$ (2.4)

Suppose that a function $K\in H^{2}(\operatorname{{\mathbb{R}}}^{d})$ with
compact support is fixed, and consider locations
$x_{1},\dots,x_{M}\in\Lambda$, $M\in\operatorname{{\mathbb{N}}}$, and a
resolution level $\delta>0$, which is small enough to ensure that the
functions $K_{\delta,x_{k}}$ are supported on $\Lambda$. Local measurements of
$X$ at the locations $x_{1},\dots,x_{M}$ at resolution $\delta$ correspond to
the continuously observed processes $X_{\delta},X^{A_{0}}_{\delta}\in
L^{2}([0,T];\operatorname{{\mathbb{R}}}^{M})$, $X^{A}_{\delta}\in
L^{2}([0,T];\operatorname{{\mathbb{R}}}^{p\times M})$, where for
$i=1,\dots,p$, $k=1,\dots,M$

$\displaystyle(X_{\delta})_{k}$ $\displaystyle=X_{\delta,k}=(\langle
X(t),K_{\delta,x_{k}}\rangle)_{0\leq t\leq T},$
$\displaystyle(X^{A_{0}}_{\delta})_{k}$
$\displaystyle=X_{\delta,k}^{A_{0}}=(\langle
X(t),A^{*}_{0}K_{\delta,x_{k}}\rangle)_{0\leq t\leq T},$
$\displaystyle(X^{A}_{\delta})_{ik}$
$\displaystyle=X^{A_{i}}_{\delta,k}=(\langle
X(t),A^{*}_{i}K_{\delta,x_{k}}\rangle)_{0\leq t\leq T}.$

Let $W_{k}(t)=\langle W(t),K_{\delta,x_{k}}\rangle/\lVert
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ be scalar Brownian motions.
According to (2.3), every local measurement is an Itô process with initial
value $X_{\delta,k}(0)=\langle X_{0},K_{\delta,x_{k}}\rangle$ and

$dX_{\delta,k}(t)=\left(\sum_{i=1}^{p}\vartheta_{i}X^{A_{i}}_{\delta,k}(t)+X^{A_{0}}_{\delta,k}(t)\right)dt+\lVert
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}dW_{k}(t).$ (2.5)

We construct an estimator for $\vartheta$ by a generalised likelihood
principle. Neither (2.5) nor the system of equations augmented with
$X^{A}_{\delta}$, $X^{A_{0}}_{\delta}$ are Markov processes, because the time
evolution at $x_{k}$ depends on the spatial structure of the whole process
$X$, and not only of $X_{\delta}$. Therefore standard results for estimating
the parameters $\vartheta_{i}$ from continuously observed diffusion processes
by the maximum likelihood estimator (e.g., [31]) do not apply here. Instead, a
general Girsanov theorem for multivariate Itô processes, cf. [33, Section
7.6], yields after ignoring conditional expectations, the initial value and
possible correlations between measurements the modified log-likelihood
function

$\begin{split}\ell_{\delta}(\vartheta)=\lVert
K\rVert^{-1}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\sum_{k=1}^{M}\Bigg{(}\int_{0}^{T}\left(\sum_{i=1}^{p}\vartheta_{i}X^{A_{i}}_{\delta,k}(t)+X^{A_{0}}_{\delta,k}(t)\right)dX_{\delta,k}(t)\\\
-\frac{1}{2}\int_{0}^{T}\left(\sum_{i=1}^{p}\vartheta_{i}X^{A_{i}}_{\delta,k}(t)+X^{A_{0}}_{\delta,k}(t)\right)^{2}\Bigg{)}dt.\end{split}$

Maximising $\ell_{\delta}(\vartheta)$ with respect to $\vartheta$ leads to the
estimator

$\displaystyle\hat{\vartheta}_{\delta}$
$\displaystyle=\mathcal{I}_{\delta}^{-1}\sum_{k=1}^{M}\left(\int_{0}^{T}X^{A}_{\delta,k}(t)dX_{\delta,k}(t)-\int_{0}^{T}X^{A}_{\delta,k}(t)X^{A_{0}}_{\delta,k}(t)dt\right),$
(2.6)

which we call _augmented MLE_ in analogy to [2, Section 4.1], with the
_observed Fisher information_

$\mathcal{I}_{\delta}=\sum_{k=1}^{M}\int_{0}^{T}X^{A}_{\delta,k}(t)X^{A}_{\delta,k}(t)^{\top}dt.$
(2.7)

###  2.3 A central limit theorem

We show now that the augmented MLE $\hat{\vartheta}_{\delta}$ satisfies a CLT
as $\delta\rightarrow 0$. Replacing $dX_{\delta,k}(t)$ in the definition of
the augmented MLE by the right hand side in (2.5) yields the basic
decomposition

$\displaystyle\hat{\vartheta}_{\delta}=\vartheta+\mathcal{I}_{\delta}^{-1}\lVert
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\mathcal{M}_{\delta}$ (2.8)

with the martingale term

$\displaystyle\mathcal{M}_{\delta}=\sum_{k=1}^{M}\left(\int_{0}^{T}X^{A}_{\delta,k}(t)dW_{k}(t)\right).$
(2.9)

If the Brownian motions $W_{k}$ are independent, then the matrix
$\mathcal{I}_{\delta}$ corresponds to the quadratic co-variation process of
$\mathcal{M}_{\delta}$ and we therefore expect
$\mathcal{I}_{\delta}^{-1/2}\mathcal{M}_{\delta}$ to follow approximately a
multivariate normal distribution. The rate at which the estimation error in
(2.8) vanishes corresponds to the speed at which the components of the
observed Fisher information diverge. Exploiting scaling properties of the
underlying semigroup we will see that this depends on the action of the
‘highest order’ operators

$\displaystyle\bar{A}_{i}^{*}=\sum_{|\alpha|=n_{i}}(-1)^{|\alpha|}a_{\alpha}^{(i)}D^{\alpha},\quad\bar{A}_{\vartheta}=\sum_{|\alpha|=2}\left(\sum_{i=1}^{p}\vartheta_{i}a^{(i)}_{\alpha}+a^{(0)}_{\alpha}\right)D^{\alpha},$
(2.10)

on the point spread functions $K_{\delta,x_{k}}$. Let us define a diagonal
matrix of scaling coefficients
$\rho_{\delta}\in\operatorname{{\mathbb{R}}}^{p\times p}$,

$\displaystyle(\rho_{\delta})_{ii}=M^{-1/2}\delta^{n_{i}-1}.$ (2.11)

We consider $\bar{A}_{\vartheta}$ on the full space with domain
$H^{2}(\operatorname{{\mathbb{R}}}^{d})$ and make the following structural
assumptions.

######  Assumption H.

  1. (i)

The functions $\bar{A}^{*}_{i}K$ are linearly independent for all
$i=1,\dots,p$.

  2. (ii)

If $d\leq 2$, then $n_{i}>0$ for all $i=1,\dots,p$.

  3. (iii)

The locations $x_{k}$, $k=1,\dots,M$, belong to a fixed compact set
$\mathcal{J}\subset\Lambda$, which is independent of $\delta$ and $M$. There
exists $\delta^{\prime}>0$ such that
$\mathrm{supp}(K_{\delta,x_{k}})\cap\mathrm{supp}(K_{\delta,x_{l}})=\emptyset$
for $k\neq l$ and all $\delta\leq\delta^{\prime}$.

  4. (iv)

$\sup_{x\in\mathcal{J}}\int_{0}^{T}\operatorname{{\mathbb{E}}}[\langle
X_{0},S^{*}_{\vartheta}(t)A_{i}^{*}K_{\delta,x}\rangle^{2}]dt=o(\delta^{2-2n_{i}})$
for all $i=1,\dots,p$.

Assumption Assumption H(i) guarantees invertibility of the observed Fisher
information, for a proof see Section A.1.

######  Lemma 2.1.

Under Assumption Assumption H(i), $\mathcal{I}_{\delta}$ is
$\operatorname{{\mathbb{P}}}$-almost surely invertible.

The support condition in Assumption Assumption H(iii) is natural in view of
applications in microscopy. It guarantees that the Brownian motions $W_{k}$ in
(2.5) are independent as $\delta\rightarrow 0$, while the processes
$X_{\delta,k}$ are _not_ independent due to the infinite speed of propagation
in the solution to the deterministic PDE (1.1) without space-time white noise.
The support condition requires the $x_{k}$ to be separated by a Euclidean
distance of at least $C\delta$ for a fixed constant $C$, which means that $M$
grows at most as $M=O(\delta^{-d})$. The next lemma shows that Assumption
Assumption H(iv) on the initial value is satisfied in most relevant
situations. For a proof see again Section A.1.

######  Lemma 2.2.

Given Assumption Assumption H(ii), Assumption Assumption H(iv) holds for any
$X_{0}$ taking values in $L^{q}(\Lambda)$, $q>2$, and if
$\sum_{|\alpha|=0}\left(\sum_{i=1}^{p}\vartheta_{i}a^{(i)}_{\alpha}+a^{(0)}_{\alpha}\right)\leq
0$ also for the stationary initial condition
$X_{0}=\int_{-\infty}^{0}S_{\vartheta}(-t^{\prime})dW(t^{\prime})$.

We establish now the asymptotic behavior of the observed Fisher information
and a CLT for the augmented MLE as the resolution $\delta$ tends to zero.

######  Theorem 2.3.

Under Assumption Assumption H the matrix
$\Sigma_{\vartheta}\in\operatorname{{\mathbb{R}}}^{p\times p}$ with entries

$\displaystyle(\Sigma_{\vartheta})_{ij}=(T/2)\langle(-\bar{A}_{\vartheta})^{-1/2}\bar{A}_{i}^{*}K,(-\bar{A}_{\vartheta})^{-1/2}\bar{A}_{j}^{*}K\rangle_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$

is well-defined and invertible, and we have
$\rho_{\delta}\mathcal{I}_{\delta}\rho_{\delta}\stackrel{{\scriptstyle\operatorname{{\mathbb{P}}}}}{{\rightarrow}}\Sigma_{\vartheta}$
as $\delta\rightarrow 0$. Moreover, the augmented MLE satisfies the CLT

$(\rho_{\delta}\mathcal{I}_{\delta}\rho_{\delta})^{1/2}\rho_{\delta}^{-1}(\hat{\vartheta}_{\delta}-\vartheta)\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,\lVert
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}I_{p}),\quad\delta\rightarrow
0,$

or equivalently,

$(M^{1/2}\delta^{1-n_{i}}(\hat{\vartheta}_{\delta,i}-\vartheta_{i}))_{i=1}^{p}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,\lVert
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\Sigma^{-1}_{\vartheta}).$

Theorem 2.3 shows that parameters $\vartheta_{i}$ of an operator $A_{i}$ with
differential order $n_{i}$ can be estimated at the rate of convergence
$M^{1/2}\delta^{1-n_{i}}$. The asymptotic variances for two parameters
$\vartheta_{i}$, $\vartheta_{j}$ are independent if the leading order terms
$\bar{A}^{*}_{i}K$ and $\bar{A}^{*}_{j}K$ are orthogonal in the geometry
induced by
$\lVert(-\bar{A}_{\vartheta})^{-1/2}\cdot\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$.
The theorem generalises [4, Theorem 5.3] (in the parametric case and with the
identity operator for $B$) to the anisotropic setting with $M$ measurement
locations, without requiring their kernel condition $\int Kdx=0$. Concrete
examples and applications are studied in Section 5.

##  3 Optimality

In this section we show that the rates of convergence
$M^{1/2}\delta^{1-n_{i}}$ achieved by the augmented MLE for parameters
$\vartheta_{i}$ with respect to operators $A_{i}$ of order
$n_{i}=\operatorname{ord}(A_{i})$ are indeed optimal and cannot be improved in
our general setup.

The proof strategy (presented in Section 6.3) relies on a novel lower bound
scheme for Gaussian measures by relating the Hellinger distance of their laws
to properties of their RKHS. The Gaussian lower bound is then applied to one-
dimensional submodels $(\mathbb{P}_{\vartheta})_{\vartheta\in\Theta_{i}}$ with
$A_{\vartheta}$ from (1.3) assuming a sufficiently regular kernel function $K$
and a stationary initial condition to simplify some computations. Extensions
to general initial values $X_{0}$ are possible. Note that this strategy also
yields lower bounds for nonparametric models, e.g. for estimating the local
diffusivity as in [4].

######  Assumption L.

Suppose that $\operatorname{{\mathbb{P}}}_{\vartheta}$ corresponds to the law
of the stationary solution $X$ to the SPDE (1.1) and assume that the following
conditions hold:

  1. (i)

The kernel function satisfies $K=\Delta^{2}\tilde{K}$ with $\tilde{K}\in
C_{c}^{\infty}(\operatorname{{\mathbb{R}}}^{d})$.

  2. (ii)

The models are
$A_{\vartheta}=\vartheta_{1}\Delta+\vartheta_{2}(b\cdot\nabla)+\vartheta_{3}$
for $\vartheta\in\operatorname{{\mathbb{R}}}^{3}$, a fixed unit vector
$b\in\operatorname{{\mathbb{R}}}^{d}$, and where $\vartheta$ lies in one of
the parameter classes

$\displaystyle\Theta_{1}$
$\displaystyle=\\{\vartheta=(\vartheta_{1},0,0):\vartheta_{1}\geq 1\\},$
$\displaystyle\Theta_{2}$
$\displaystyle=\\{\vartheta=(1,\vartheta_{2},0):\vartheta_{2}\in[0,1]\\},$
$\displaystyle\Theta_{3}$
$\displaystyle=\\{\vartheta=(1,0,\vartheta_{3}):\vartheta_{3}\leq 0\\}.$ (3.1)

  3. (iii)

Let $x_{1},\dots,x_{M}$ be $\delta$-separated points in $\Lambda$, that is,
$|x_{k}-x_{l}|>\delta$ for all $1\leq k\neq l\leq M$. Moreover, suppose that
$\mathrm{supp}(K_{\delta,x_{k}})\subset\Lambda$ for all $k=1,\dots,M$ and
$\mathrm{supp}(K_{\delta,x_{k}})\cap\mathrm{supp}(K_{\delta,x_{l}})=\emptyset$
for all $1\leq k\neq l\leq M$.

The parameter classes $\Theta_{i}$ correspond to the cases of estimating the
diffusivity $\vartheta_{1}$, transport coefficient $\vartheta_{2}$ and
reaction coefficient $\vartheta_{3}$ in front of operators $A_{i}$ with
differential orders $n_{1}=2$, $n_{2}=1$, $n_{3}=0$.

We start with a non-asymptotic lower bound when only $X_{\delta}$ is observed.

######  Theorem 3.1.

Grant Assumption Assumption L with $M\geq 1$, $T\geq 1$ and $i\in\\{1,2,3\\}$.
Then there exist constants $c_{1},c_{2}>0$ depending only on $K$ and an
absolute constant $c_{3}>0$ such that the following assertions hold:

  * (i)

If $\delta^{n_{i}-1}/\sqrt{TM}<1$ and $\delta\leq c_{1}$, then

$\displaystyle\inf_{\hat{\vartheta}_{i}}\sup_{\begin{subarray}{c}\vartheta\in\Theta_{i}\\\
|\vartheta-(1,0,0)^{\top}|\leq
c_{2}\frac{\delta^{n_{i}-1}}{\sqrt{TM}}\end{subarray}}\operatorname{{\mathbb{P}}}_{\vartheta}\left(|\hat{\vartheta}_{i}-\vartheta_{i}|\geq\frac{c_{2}}{2}\frac{\delta^{n_{i}-1}}{\sqrt{TM}}\right)>c_{3}.$

  * (ii)

If $\delta^{n_{i}-1}/\sqrt{TM}\geq 1$ and $\delta\leq c_{1}$, then

$\displaystyle\inf_{\hat{\vartheta}_{i}}\sup_{\begin{subarray}{c}\vartheta\in\Theta_{i}\\\
|\vartheta-(1,0,0)^{\top}|\leq
c_{2}\end{subarray}}\operatorname{{\mathbb{P}}}_{\vartheta}\left(|\hat{\vartheta}_{i}-\vartheta_{i}|\geq
c_{2}/2\right)>c_{3}.$

In (i) and (ii), the infimum is taken over all real-valued estimators
$\hat{\vartheta}_{i}=\hat{\vartheta}_{i}(X_{\delta})$.

Several comments are in order for the above result. First, by Markov’s
inequality Theorem 3.1 also implies lower bounds for the squared risk. Second,
part (ii) detects settings under which consistent estimation is impossible.
For instance, if $i=2$, then consistent estimation is impossible for $T=1$
(resp. $T$ bounded) and $M=1$, that is, if only a single spatial measurement
is observed in a bounded time interval. A similar conclusion holds in the case
$i=3$, in which case consistent estimation is even impossible in a full
observation scheme with $M=\lceil c\delta^{-d}\rceil$ locations for $d\leq 2$
and $T$ bounded. Third, part (i) of Theorem 3.1 shows that the different rates
in our CLT are minimax optimal. In particular, it easily implies an asymptotic
minimax lower bound when $\delta\rightarrow 0$. A first important case is
$M=1$ and $i=1$ in which case Theorem 3.1 also follows from Proposition 5.12
in [4] and gives the rate of convergence $\delta$. Another important case is
given by the full measurement scheme $M=\lceil c\delta^{-d}\rceil$, in which
case we get the following corollary.

######  Corollary 3.2.

Grant Assumption Assumption L with $M=\lceil c\delta^{-d}\rceil$, $T\geq 1$
and $i\in\\{1,2,3\\}$. If $n_{i}-1+d/2>0$, then

$\liminf_{\delta\rightarrow
0}\inf_{\hat{\vartheta}_{i}}\sup_{|\vartheta-(1,0,0)^{\top}|\leq
c_{1}}\operatorname{{\mathbb{P}}}_{\vartheta}\left(\delta^{n_{i}-1-d/2}|\hat{\vartheta}_{i}-\vartheta_{i}|\geq
c_{2}\right)>0,$

where the infimum is taken over all real-valued estimators
$\hat{\vartheta}_{i}=\hat{\vartheta}_{i}(X_{\delta})$. On the other hand, if
$n_{i}-1+d/2\leq 0$, then consistent estimation is impossible.

Similar optimality results have been derived in [24] for the case of $M$
spectral measurements. Provided there exists an orthonormal basis of
eigenfunctions $(e_{j})_{j=1}^{\infty}$ of $A_{\vartheta}$ independent of
$\vartheta$ (e.g., in the case $i=1$ or $i=3$), it is possible to estimate
$\vartheta_{i}$ from $M$ spectral measurements $(\langle
X(t),e_{j}\rangle)_{0\leq t\leq T,1\leq j\leq M}$ with rates $M^{-\tau}$ or
$\log M$ if $\tau=n_{i}/d-1/d+1/2>0$ or $\tau=0$, respectively. Consistent
estimation fails to hold for $\tau<0$. While [24] obtained asymptotic
efficiency by combining Girsanov’s theorem with LAN techniques, these rates
can also be easily derived from our RKHS results (cf. Lemma A.5) combined with
a version of Lemma 6.9. For $\delta=cM^{-1/d}$ the rate in Corollary 3.2 and
Theorem 2.3 coincides with $M^{-\tau}$ if $\tau>0$, and $\tau=0$ is again a
boundary case. Regarding the latter case, we briefly discuss in Section 5 that
a non-negative point spread function achieves the $\log M$-rate when $i=3$ and
$d=2$.

Recall that the augmented MLE $\hat{\vartheta}_{\delta}$ depends also on the
measurements $X^{A}_{\delta}$. We show next that including them into the lower
bounds does not change the optimal rates of convergence.

######  Theorem 3.3.

Theorem 3.1 remains valid when the infimum is taken over all real-valued
estimators
$\hat{\vartheta}_{i}=\hat{\vartheta}_{i}(X_{\delta},X_{\delta}^{\Delta},X_{\delta}^{b\cdot\nabla})$,
provided that $K$, $\Delta K$ and $(b\cdot\nabla)K$ are independent and
Assumption Assumption L(i) holds for $K$, $\Delta K$ and $(b\cdot\nabla)K$.

##  4 The RKHS

A crucial ingredient for the proofs of the lower bounds in the last section is
a good understanding of the RKHS of the Gaussian measure induced by the law of
the observations when $A_{\vartheta}=\Delta$. For some background on the RKHS
of a Gaussian measure see Section 6.3.1.

We first derive the RKHS of the stochastic convolution (2.2) in a more general
setting. Suppose that $A_{\vartheta}$ is an (unbounded) negative self-adjoint
closed operator on a Hilbert space
$(\mathcal{H},\lVert\cdot\rVert_{\mathcal{H}})$ with domain
$\mathcal{D}(A_{\vartheta})\subset\mathcal{H}$ such that
$A_{\vartheta}e_{j}=-\lambda_{j}e_{j}$ for a sequence $(\lambda_{j})_{j\geq
1}$ of positive real numbers and an orthonormal basis $(e_{j})_{j\geq 1}$ of
$\mathcal{H}$, and such that $A_{\vartheta}$ generates a strongly continuous
semigroup $(S_{\vartheta}(t))_{t\geq 0}$ on $\mathcal{H}$. With a cylindrical
Wiener process $W$, consider the stationary stochastic convolution

$\displaystyle
X(t)=\int_{-\infty}^{t}S_{\vartheta}(t-t^{\prime})dW(t^{\prime}),\quad t\geq
0.$ (4.1)

As discussed after (2.2) the process $X=(X(t))_{0\leq t\leq T}$ is understood
as a random element with values in $\mathcal{H}\subset\mathcal{H}_{1}$ almost
surely for some larger Hilbert space $\mathcal{H}_{1}$.

Since the RKHS of a Gaussian measure depends only on its distribution, the
RKHS, as well as its norm, in the next theorem are independent of the
embedding space $\mathcal{H}_{1}$ (see, e.g., Exercise 2.6.5 in [19]).

######  Theorem 4.1.

Let $(H_{X},\|\cdot\|_{X})$ be the RKHS of the process $X$ in (4.1). Then

$\displaystyle H_{X}$ $\displaystyle=\\{h\in L^{2}([0,T];\mathcal{H}):h\text{
absolutely continuous, }A_{\vartheta}h,h^{\prime}\in
L^{2}([0,T];\mathcal{H})\\}$

and

$\displaystyle\|h\|_{X}^{2}=\|A_{\vartheta}h\|_{L^{2}([0,T];\mathcal{H})}^{2}+\|h^{\prime}\|_{L^{2}([0,T];\mathcal{H})}^{2}$
$\displaystyle\quad\quad+\|(-A_{\vartheta})^{1/2}h(0)\|^{2}_{\mathcal{H}}+\|(-A_{\vartheta})^{1/2}h(T)\|^{2}_{\mathcal{H}},\quad
h\in H_{X}.$

Moreover, we have

$\displaystyle\|h\|_{X}^{2}\leq
3\|A_{\vartheta}h\|_{L^{2}([0,T];\mathcal{H})}^{2}+\|h\|_{L^{2}([0,T];\mathcal{H})}^{2}+2\|h^{\prime}\|_{L^{2}([0,T];\mathcal{H})}^{2},\quad
h\in H_{X}.$

The theorem generalises the corresponding result for scalar Ornstein-Uhlenbeck
processes to the infinite dimensional process $X$, cf. Lemma 6.11 below. Next,
as in (2.3), consider the Gaussian process $(\langle
X(t),z\rangle_{\mathcal{H}})_{t\geq 0,z\in\mathcal{H}}$, where the ‘inner
product’ here corresponds to

$\displaystyle\langle X(t),z\rangle_{\mathcal{H}}=\int_{-\infty}^{t}\langle
S_{\vartheta}(t-t^{\prime})z,dW(t^{\prime})\rangle_{\mathcal{H}},$

satisfying (2.3) for
$z\in\mathcal{D}(A_{\vartheta}^{*})=\mathcal{D}(A_{\vartheta})$ by analogous
arguments. We study the RKHS of $(\langle X(t),z\rangle_{\mathcal{H}})_{0\leq
t\leq T}$ for finitely many $z$. We will deduce the result directly from
Theorem 4.1 by realising $\langle X,z\rangle_{\mathcal{H}}$ as a linear
transformation of $X$ by a bounded linear map $L$ from
$L^{2}([0,T];\mathcal{H}_{1})$ to $L^{2}([0,T])^{M}$ and using the machinery
described in Section 6.4. This restricts $z$ to lie in the dual space of
$\mathcal{H}_{1}$. Another proof presented in Appendix A.6.1 circumvents this
by an approximation argument.

######  Theorem 4.2.

For $K_{1},\dots,K_{M}\in\mathcal{D}(A_{\vartheta})$ and with $X$ in (4.1)
consider the process $X_{K}$ with $X_{K}(t)=(\langle
X(t),K_{k}\rangle_{\mathcal{H}})_{k=1}^{M}$. Suppose that the Gram matrix
$G=(\langle K_{k},K_{l}\rangle_{\mathcal{H}})_{1\leq k,l\leq M}$ is non-
singular. Then the RKHS $(H_{X_{K}},\|\cdot\|_{X_{K}})$ of $X_{K}$ satisfies
$H_{X_{K}}=H^{M}$, where

$\displaystyle H=\\{h\in L^{2}([0,T]):h\text{ absolutely
continuous},h^{\prime}\in L^{2}([0,T])\\},$

and

$\displaystyle\lVert h\rVert_{X_{K}}^{2}$
$\displaystyle\leq(3\|G^{-1}\|_{\operatorname{op}}^{2}\|G_{A_{\vartheta}}\|_{\operatorname{op}}+\|G^{-1}\|_{\operatorname{op}})\sum_{k=1}^{M}\lVert
h_{k}\rVert^{2}_{L^{2}([0,T])}$
$\displaystyle+2\|G^{-1}\|_{\operatorname{op}}\sum_{k=1}^{M}\lVert
h_{k}^{\prime}\rVert^{2}_{L^{2}([0,T])}$

for all $h=(h_{k})_{k=1}^{M}\in H_{X_{K}}$, where $G_{A_{\vartheta}}=(\langle
A_{\vartheta}K_{k},A_{\vartheta}K_{l}\rangle_{\mathcal{H}})_{1\leq k,l\leq
M}$.

In the specific case $A_{\vartheta}=\Delta$ and local measurements with
$K_{k}=K_{\delta,x_{k}}$ we obtain the following.

######  Corollary 4.3.

Let $(H_{X_{\delta}},\|\cdot\|_{X_{\delta}})$ be the RKHS of $X_{\delta}$ with
respect to $A_{\vartheta}=\Delta$, let $K\in
H^{2}(\operatorname{{\mathbb{R}}}^{d})$, $\lVert
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}=1$, satisfy Assumption
Assumption L(iii) and suppose $\delta^{2}\leq\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$, $T\geq 1$. Then
$H_{X_{\delta}}=H^{M}$ and for $h=(h_{k})_{k=1}^{M}\in H_{X_{\delta}}$

$\displaystyle\|h\|_{X_{\delta}}^{2}$ $\displaystyle\leq 4\frac{\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{4}}\sum_{k=1}^{M}\|h_{k}\|_{L^{2}([0,T])}^{2}+2\sum_{k=1}^{M}\|h_{k}^{\prime}\|_{L^{2}([0,T])}^{2}.$

Similar results hold for the RKHS of the jointly Gaussian process
$(X_{\delta},X^{A}_{\delta})$, see e.g. Corollary A.4 below.

##  5 Applications and extensions

###  5.1 Examples

Let us illustrate the main results in a few examples.

######  Example 5.1.

Suppose $A_{\vartheta}=\vartheta_{1}\Delta+\vartheta_{2}b\cdot\nabla+c$. This
corresponds to (1.2) with $A_{0}=c$, $A_{1}=\Delta$, $A_{2}=b\cdot\nabla$ for
$c\in\operatorname{{\mathbb{R}}}$ and a unit vector $b$, and with differential
orders $n_{1}=2$, $n_{2}=1$. A typical realisation of the solution $X$ in
$d=1$ can be seen in Figure 1(left). The ellipticity condition (2.1) holds if
$\vartheta_{1}>0$. We know from Theorem 3.1 and Proposition 5.4 that $c$
cannot be estimated consistently unless $d\geq 2$. For known $c$, the
augmented MLE $\hat{\vartheta}_{\delta}$ is a consistent estimator of
$\vartheta\in\operatorname{{\mathbb{R}}}^{2}$ by Theorem 2.3, attaining the
optimal rates of convergence $M^{1/2}\delta^{-1}$, $M^{1/2}$ for the
diffusivity and the transport terms, respectively according to the lower
bounds in Theorem 3.1. If we suppose for simplicity $\lVert
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}=1$, then the CLT holds with a
diagonal matrix

$\Sigma_{\vartheta}=\frac{T}{2\vartheta_{1}}\operatorname{diag}\left(\lVert\nabla
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})},\lVert(-\Delta)^{-1/2}(b\cdot\nabla)K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\right),$

implying that $\hat{\vartheta}_{\delta,1}$ and $\hat{\vartheta}_{\delta,2}$
are asymptotically independent. Interestingly, in $d=1$,
$\lVert(-\Delta)^{-1/2}(b\cdot\nabla)K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}=\lVert
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ and so the asymptotic
variance of $\hat{\vartheta}_{\delta,2}$ is independent of $K$.

Figure 1(right) presents root mean squared estimation errors in $d=1$ for
local measurements obtained from the data displayed in the left part of the
figure with $K(x)=\exp(-5/(1-\lVert x\rVert^{2}))\mathbf{1}(-1<x<1)$ and the
maximal choice of $M\asymp\delta^{-1}$. We see that the optimal rates of
convergence, and even the exact asymptotic variances (blue dashed lines) are
approached quickly as $\delta\rightarrow 0$. For comparison, we have included
in Figure 1(right) estimation errors also for an estimator
$\bar{\vartheta}_{\delta}$ without the correction factor depending on the
lower order ’nuisance operator’ $A_{0}$ in (2.6). We can see that this
introduces only a small bias, which is negligible as $\delta\rightarrow 0$.

######  Example 5.2.

Consider the operator
$A_{\vartheta}=\vartheta_{1}\Delta+\vartheta_{2}b\cdot\nabla+\vartheta_{3}$
from (1.3) such that $A_{0}=0$, $A_{1}=\Delta$, $A_{2}=b\cdot\nabla$ and
$A_{3}=1$, with differential orders $n_{1}=2$, $n_{2}=1$, $n_{3}=0$. If $d\geq
3$ and $M^{1/2}\delta\rightarrow\infty$, then the CLT in Theorem 2.3 applies
again with optimal rates of convergence as in the last example for
$\hat{\vartheta}_{\delta,1}$, $\hat{\vartheta}_{\delta,2}$ and $M^{1/2}\delta$
for the reaction term. Using integration by parts we find this time

$\Sigma_{\vartheta}=\frac{T}{2\vartheta_{1}}\begin{pmatrix}\lVert\nabla
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}&0&-1\\\
0&\lVert(-\Delta)^{-1/2}(b\cdot\nabla)K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}&0\\\
-1&0&\lVert(-\Delta)^{-1/2}K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\end{pmatrix}.$

######  Example 5.3.

The results of the last example apply with the same asymptotic distribution to
$A_{\vartheta}=\vartheta_{1}(\Delta+b^{\prime}\cdot\nabla+c^{\prime})+2\vartheta_{2}(b\cdot\nabla+c^{\prime\prime})+\vartheta_{3}$
for $b^{\prime}\in\operatorname{{\mathbb{R}}}^{d}$,
$c^{\prime},c^{\prime\prime}\in\operatorname{{\mathbb{R}}}$ with lower order
perturbations, because the leading-order operators $\bar{A}_{i}$ remain
unchanged.

Figure 1: (left) heat map for a typical realisation of $X(t,x)$ corresponding
to (1.1) in $d=1$ with domain $\Lambda=(0,1)$ in Example 5.1; (right)
$\log_{10}\log_{10}$ plot of the root mean squared errors for estimating
$\vartheta_{1}$ and $\vartheta_{2}$ in Example 5.1.

###  5.2 Statistical inference

The CLT in Theorem 2.3 easily yields statistical inference procedures. For
instance, if $\alpha\in(0,1)$, $q_{1-\alpha}$ is the $1-\alpha$ quantile of
the $\chi^{2}_{p}$-distribution and if $\lVert
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}=1$ for simplicity, then

$C_{\alpha}=\\{\vartheta\in\mathbb{R}^{p}:(\hat{\vartheta}_{\delta}-\vartheta)^{\top}\mathcal{I}_{\delta}(\hat{\vartheta}_{\delta}-\vartheta)\leq
q_{1-\alpha}\\}$

is an asymptotic joint $1-\alpha$ confidence set for the parameter
$\vartheta$.

Moreover, we can test for the presence of lower order terms. In Example 5.2,
with the $1-\alpha$ quantile $z_{1-\alpha}$ of the standard normal
distribution,

$\displaystyle\phi_{\alpha}=\mathbf{1}\left(\left|\left(TM\delta^{2}/(2\hat{\vartheta}_{\delta,1})\lVert(-\Delta)^{-1/2}K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\right)^{1/2}\hat{\vartheta}_{\delta,3}\right|>z_{1-\alpha}\right)$

is an asymptotic level $\alpha$ test for testing the null hypothesis
$H_{0}:\vartheta_{3}=0$ against the alternative $H_{1}:\vartheta_{3}<0$.

###  5.3 A boundary case: estimation in $d=2$

The augmented MLE generally does not satisfy the CLT in Theorem 2.3 in $d\leq
2$ for reaction terms $\vartheta_{i}$ with differential order $n_{i}=0$. We
show now that in $d=2$ and for the maximal choice of measurements with
$M=\lceil c\delta^{-2}\rceil$ the CLT can be recovered under integrability
conditions on $K$, but consistency is lost, while for non-negative $K$ a
logarithmic rate holds. This is consistent with results for the MLE from
spectral observations in $d=2$, cf. [24]. For a proof see Section A.2.

######  Proposition 5.4.

Suppose that $d=2$, $A_{\vartheta}=\Delta+\vartheta$ for
$\vartheta\in\operatorname{{\mathbb{R}}}$, $K\neq 0$, $X(0)=0$ and
$M\delta^{2}\rightarrow 1$ as $\delta\rightarrow 0$. Then the following holds:

  1. (i)

If $\int_{\operatorname{{\mathbb{R}}}^{2}}K(x)dx=0$,
$\int_{\operatorname{{\mathbb{R}}}^{2}}xK(x)dx=0$, then
$\hat{\vartheta}_{\delta}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(\vartheta,\lVert
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\Sigma^{-1}_{\vartheta})$
with
$\Sigma_{\vartheta}=(T/2)\lVert(-\Delta)^{-1/2}K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}^{2}$.

  2. (ii)

If $K\geq 0$, then
$\hat{\vartheta}_{\delta}=\vartheta+O_{\operatorname{{\mathbb{P}}}}(\log(\delta^{-1})^{-1/2})$.

##  6 Proofs

Let us first define some notation. For convenience in the proofs we write the
elliptic operator as $A_{\vartheta}=\nabla\cdot
a_{\vartheta}\nabla+b_{\vartheta}\cdot\nabla+c_{\vartheta}$ with a symmetric
positive definite matrix $a_{\vartheta}\in\operatorname{{\mathbb{R}}}^{d\times
d}$, $b_{\vartheta}\in\operatorname{{\mathbb{R}}}^{d}$,
$c_{\vartheta}\in\operatorname{{\mathbb{R}}}$ and let
$\tilde{A}_{\vartheta}=\nabla\cdot a_{\vartheta}\nabla$. The operators
$A_{\vartheta}$ and $\tilde{A}_{\vartheta}$ with domain
$H^{1}_{0}(\Lambda)\cap H^{2}(\Lambda)$ generate the analytic semigroups
$(S_{\vartheta}(t))_{t\geq 0}$ and $(\tilde{S}_{\vartheta}(t))_{t\geq 0}$. By
a standard-PDE result (see, e.g., [27, Example III.6.11] or [39, equation
(5.1)]) $A_{\vartheta}$ (and by [16, Example 2.1 in Section II.2]) its
generated semigroup are diagonalizable, which yields the useful
representations

$\displaystyle
A_{\vartheta}=U_{\vartheta}(\tilde{A}_{\vartheta}+\tilde{c}_{\vartheta})U_{\vartheta}^{-1},\quad
S_{\vartheta}(t)=e^{t\tilde{c}_{\vartheta}}U_{\vartheta}\tilde{S}_{\vartheta}(t)U_{\vartheta}^{-1}$
(6.1)

with the multiplication operator
$(U_{\vartheta}z)(x)=\exp\left(-\frac{1}{2}(a_{\vartheta}^{-1}b_{\vartheta})\cdot
x\right)z(x)$ and with
$\tilde{c}_{\vartheta}=c_{\vartheta}-\frac{1}{4}b_{\vartheta}\cdot(a_{\vartheta}^{-1}b_{\vartheta})$.
Below we will use $U_{\vartheta}$ to denote both the multiplication operator
and the function
$x\mapsto\exp\left(-\frac{1}{2}(a_{\vartheta}^{-1}b_{\vartheta})\cdot
x\right)$.

###  6.1 The rescaled semigroup

In this section we collect a number of results on the semigroup operators
$S_{\vartheta}(t)$ and their actions on localised functions
$z_{\delta,x}(\cdot)=\delta^{-d/2}z(\delta^{-1}(\cdot-x))$. Write
$\Lambda_{\delta,x}=\\{\delta^{-1}(y-x):y\in\Lambda\\}$,
$\Lambda_{0,x}=\mathbb{R}^{d}$ and introduce the operators with domains
$H_{0}^{1}(\Lambda_{\delta,x})\cap H^{2}(\Lambda_{\delta,x})$

$\displaystyle A_{\vartheta,\delta,x}$ $\displaystyle=\nabla\cdot
a_{\vartheta}\nabla+\delta
b_{\vartheta}\cdot\nabla+\delta^{2}c_{\vartheta},\quad\tilde{A}_{\vartheta,\delta,x}=\nabla\cdot
a_{\vartheta}\nabla,$ (6.2) $\displaystyle A_{i,\delta,x}$
$\displaystyle=\delta^{n_{i}}\sum_{|\alpha|\leq
n_{i}}\delta^{-|\alpha|}a^{(i)}_{\alpha}D^{\alpha}.$

The operators in (6.2) generate the analytic semigroups
$(S_{\vartheta,\delta,x}(t))_{t\geq 0}$ and
$(\tilde{S}_{\vartheta,\delta,x}(t))_{t\geq 0}$ on
$L^{2}(\Lambda_{\delta,x})$. Let $(\bar{S}_{\vartheta}(t))_{t\geq 0}$ be the
semigroup on $L^{2}(\operatorname{{\mathbb{R}}}^{d})$ generated by
$\bar{A}_{\vartheta}$. We also write $e^{t\Delta_{0}}=\bar{S}_{\vartheta}(t)$
when $a_{\vartheta}$ is the identity matrix.

######  Lemma 6.1.

Let $\delta^{\prime}\geq\delta\geq 0$, $x\in\Lambda$, $i=1,\dots,p$, $t\geq
0$.

  1. (i)

If $z\in H_{0}^{1}(\Lambda_{\delta,x})\cap H^{2}(\Lambda_{\delta,x})$, then
$A_{i}^{*}z_{\delta,x}=\delta^{-n_{i}}(A^{*}_{i,\delta,x}z)_{\delta,x}$,
$A^{*}_{\vartheta}z_{\delta,x}=\delta^{-2}(A^{*}_{\vartheta,\delta,x}z)_{\delta,x}$.

  2. (ii)

If $z\in H^{2}(\operatorname{{\mathbb{R}}}^{d})$ is compactly supported in
$\bigcap_{x\in\mathcal{J}}\Lambda_{\delta^{\prime},x}$, then
$\sup_{x\in\mathcal{J}}\lVert
A^{*}_{i,\delta,x}z-\bar{A}_{i}z\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\rightarrow
0$, $\sup_{x\in\mathcal{J}}\lVert
A^{*}_{\vartheta,\delta,x}z-\bar{A}_{\vartheta}z\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\rightarrow
0$ as $\delta\rightarrow 0$.

  3. (iii)

If $z\in L^{2}(\Lambda_{\delta,x})$ then
$S_{\vartheta}^{*}(t)z_{\delta,x}=(S_{\vartheta,\delta,x}^{*}(t\delta^{-2})z)_{\delta,x}$.

###### Proof.

Parts (i), (ii) are clear, part (iii) follows analogously to [4, Lemma 3.1]. ∎

The semigroup on the bounded domain $\Lambda_{\delta,x}$ is after zooming in
as $\delta\rightarrow 0$ intuitively close to the semigroup on
$\operatorname{{\mathbb{R}}}^{d}$. The next result states precisely in which
way this holds, uniformly in $x\in\mathcal{J}$.

######  Lemma 6.2.

Under Assumption Assumption H(iii) the following holds:

  * (i)

There exists $C>0$ such that if $z\in C_{c}(\mathbb{R}^{d})$ is supported in
$\bigcap_{x\in\mathcal{J}}\Lambda_{\delta,x}$ for some $\delta\geq 0$, then
for all $t\geq 0$

$\sup_{x\in\mathcal{J}}\left|(S^{*}_{\vartheta,\delta,x}(t)z)(y)\right|\leq
Ce^{\tilde{c}_{\vartheta}t\delta^{2}}(\bar{S}_{\vartheta}(t)|z|)(y),\quad
y\in\operatorname{{\mathbb{R}}}^{d}.$

  * (ii)

If $z\in L^{2}(\mathbb{R}^{d})$, then as $\delta\rightarrow 0$ for all $t>0$

$\sup_{x\in\mathcal{J}}\mathop{}\\!\mathbin{\lVert}S^{*}_{\vartheta,\delta,x}(t)(z|_{\Lambda_{\delta,x}})-\bar{S}_{\vartheta}(t)z\mathop{}\\!\mathbin{\rVert}_{L^{2}(\mathbb{R}^{d})}\rightarrow
0.$

###### Proof.

(i). Apply first the scaling in Lemma 6.1 in reverse order such that
$(S^{*}_{\vartheta,\delta,x}(t)z)_{\delta,x}=S^{*}_{\vartheta}(t\delta^{2})z_{\delta,x}$.
By (6.1) and direct computation, noting that the function $x\mapsto
U_{\vartheta}(x)$ is uniformly upper and lower bounded on $\Lambda$, we get

$\displaystyle|S^{*}_{\vartheta}(t\delta^{2})z_{\delta,x}|$
$\displaystyle=|U_{\vartheta}^{-1}\tilde{S}_{\vartheta}(t\delta^{2})\exp(t\delta^{2}\tilde{c}_{\vartheta})U_{\vartheta}z_{\delta,x}|\lesssim
e^{t\delta^{2}\tilde{c}_{\vartheta}}\tilde{S}_{\vartheta}(t\delta^{2})|z_{\delta,x}|.$

Applying again the $\delta$-scaling it is therefore enough to prove the claim
with respect to $\tilde{S}_{\vartheta}$ and with $|z|$ instead of $z$. By the
classical Feynman-Kac formulas (cf. [26, Chapter 4.4], the anisotropic case is
an easy generalisation, which can also be obtained by a change of variables
leading to a diagonal diffusivity matrix $a_{\vartheta}$, which corresponds to
$d$ scalar heat equations) we have with a process
$Y_{t}=y+\mathop{}\\!\mathbin{a}_{\vartheta}^{1/2}\tilde{W}_{t}$ and a
$d$-dimensional Brownian motion $\tilde{W}$, all defined on another
probability space with expectation and probability operators
$\tilde{\operatorname{{\mathbb{E}}}}_{y}$,
$\tilde{\operatorname{{\mathbb{P}}}}_{y}$, that
$(\bar{S}_{\vartheta}(t)z)(y)=\tilde{\mathbb{E}}_{y}[z(Y_{t})]$ and
$\tilde{S}_{\vartheta,\delta,x}(t)z(y)=\tilde{\mathbb{E}}_{y}\left[z(Y_{t})\mathbf{1}\left(t<\tau_{\delta,x}\right)\right]$
with the stopping times $\tau_{\delta,x}:=\inf\\{t\geq
0:Y_{t}\notin\Lambda_{\delta,x}\\}$. The claim follows now from

$\sup_{x\in\mathcal{J}}(\tilde{S}_{\vartheta,\delta,x}(t)|z|)(y)\leq\tilde{E}_{y}[|z(Y_{t})|]=(\bar{S}_{\vartheta}(t)|z|)(y),$

where the right hand side is
$L^{2}(\operatorname{{\mathbb{R}}}^{d})$-integrable.

(ii). By an approximation argument it is enough to consider $z\in
C_{c}(\bar{\Lambda})$ and $0<\delta\leq\delta^{\prime}$ such that $z$ is
supported in $\Lambda_{\delta^{\prime},x}$, hence $z|_{\Lambda_{\delta,x}}=z$
for all such $\delta$. Compactness of $\mathcal{J}$ according to Assumption
Assumption H(iii) guarantees for sufficiently small $\delta$ the existence of
a ball $B_{\rho\delta^{-1}}\subset\bigcap_{x\in\mathcal{J}}\Lambda_{\delta,x}$
with center $0$ and radius $\rho\delta^{-1}$ for some $\rho>0$. With this and
the representation formulas in (i), combined with the Cauchy-Schwarz
inequality, we have for all $y\in\operatorname{{\mathbb{R}}}^{d}$

$\displaystyle\sup_{x\in\mathcal{J}}|(\tilde{S}_{\vartheta,\delta,x}(t)z)(y)-(\bar{S}_{\vartheta}(t)z)(y)|^{2}=\sup_{x\in\mathcal{J}}|\tilde{\mathbb{E}}_{y}\left[z(Y_{t})\mathbf{1}(\tau_{\delta,x}\leq
t)\right]|^{2}$
$\displaystyle\leq\sup_{x\in\mathcal{J}}\tilde{\operatorname{{\mathbb{E}}}}_{y}[z^{2}(Y_{t})]\tilde{\mathbb{P}}_{y}(\tau_{\delta,x}\leq
t)\leq(\bar{S}_{\vartheta}(t)z^{2})(y)\tilde{\mathbb{P}}_{y}(\max_{0\leq s\leq
t}|Y_{s}|\geq\rho\delta^{-1})$
$\displaystyle\leq(\bar{S}_{\vartheta}(t)z^{2})(y)\tilde{\mathbb{P}}_{y}(\max_{0\leq
s\leq
t}|\tilde{W}_{s}|\geq\tilde{\rho}\delta^{-1})\lesssim(\bar{S}_{\vartheta}(t)z^{2})(y)(\delta
t^{1/2}e^{-\delta^{-2}t^{-1}})\rightarrow 0$ (6.3)

as $\delta\rightarrow 0$ for a modified constant $\tilde{\rho}$, concluding by
[26, equation (2.8.4)]. As a consequence of (6.1) and the $\delta$-scaling we
have
$(S^{*}_{\vartheta,\delta,x}(t)z)(y)=(U_{\vartheta,\delta,x}^{-1}\tilde{S}_{\vartheta,\delta,x}(t)e^{t\delta^{2}\tilde{c}_{\vartheta}}U_{\vartheta,\delta,x}z)(y)$
with
$U_{\vartheta,\delta,x}z(y)=\exp(-(a_{\vartheta}^{-1}b_{\vartheta})\cdot(\delta
y+x)/2)z(y)$. This converges to $z(y)$ uniformly in $x\in\mathcal{J}$ as
$\delta\rightarrow 0$, and we also have
$e^{t\delta^{2}\tilde{c}_{\vartheta}}\rightarrow 1$. With this conclude

$\displaystyle\sup_{x\in\mathcal{J}}|(S^{*}_{\vartheta,\delta,x}(t)z)(y)-(\bar{S}_{\vartheta}(t)z)(y)|$
$\displaystyle=\sup_{x\in\mathcal{J}}|(U_{\vartheta,\delta,x}^{-1}\tilde{S}_{\vartheta,\delta,x}(t)e^{t\delta^{2}\tilde{c}_{\vartheta}}U_{\vartheta,\delta,x}z)(y)-(\bar{S}_{\vartheta}(t)z)(y)|\rightarrow
0,$

by the uniform pointwise convergence in (6.3). From (i) we know that
$\sup_{0\leq\delta\leq\delta^{\prime}}\sup_{x\in\mathcal{J}}|(S^{*}_{\vartheta,\delta,x}(t)z)(y)|$
is $L^{2}(\operatorname{{\mathbb{R}}}^{d})$-integrable. Dominated convergence
yields the claim. ∎

We require frequently quantitative statements on the decay of the action of
the semigroup operators $S^{*}_{\vartheta,\delta,x}(t)$ as
$t\rightarrow\infty$ when applied to functions of a certain smoothness and
integrability. This is well-known for a fixed general analytic semigroups, but
is shown here to hold true regardless of $\delta$ and uniformly in
$x\in\mathcal{J}$.

######  Lemma 6.3.

Let $0\leq\delta\leq 1$, $t>0$, $x\in\mathcal{J}$ and $1<p\leq\infty$.
Moreover, let $z\in L^{p}(\Lambda_{\delta,x})$ if $1<p<\infty$ and $z\in
C(\Lambda_{\delta,x})$ with $z=0$ on $\partial\Lambda_{\delta,x}$ if
$p=\infty$. Then it holds with implied constants not depending on $x$:

$\displaystyle\lVert
A^{*}_{\vartheta,\delta,x}S^{*}_{\vartheta,\delta,x}(t)z\rVert_{L^{p}(\Lambda_{\delta,x})}\lesssim
t^{-1}\lVert z\rVert_{L^{p}(\Lambda_{\delta,x})}.$

###### Proof.

Apply first the scaling in Lemma 6.1 in reverse order such that with
$1<p\leq\infty$

$\displaystyle\lVert
A^{*}_{\vartheta,\delta,x}S^{*}_{\vartheta,\delta,x}(t)z\rVert_{L^{p}(\Lambda_{\delta,x})}=\delta^{d(1/2-1/p)-2}\mathop{}\\!\mathbin{\lVert}{A^{*}_{\vartheta}S^{*}_{\vartheta}(t\delta^{-2})z_{\delta,x}}\mathop{}\\!\mathbin{\rVert}_{L^{p}(\Lambda)}.$

If $p<\infty$, by the semigroup property for analytic semigroups in [5,
Theorem V.2.1.3], the $L^{p}(\Lambda)$-norm is up to a constant upper bounded
by $(t\delta^{-2})^{-1}\lVert z_{\delta,x}\rVert_{L^{p}(\Lambda)}$, and the
claim follows. The same proof applies to $p=\infty$, noting that
$A^{*}_{\vartheta}$ generates an analytic semigroup on $\\{u\in
C(\Lambda),u=0$ on $\partial\Lambda\\}$, cf. [38, Theorem 7.3.7]. ∎

The proof for the next result relies on the Bessel-potential spaces
$H_{0}^{s,p}(\Lambda_{\delta,x})$, $1<p<\infty$,
$s\in\operatorname{{\mathbb{R}}}$, defined for $\delta>0$ as the domains of
the fractional weighted Dirichlet-Laplacian
$(-\tilde{A}_{\vartheta,\delta,x})^{s/2}$ of order $s/2$ on
$\Lambda_{\delta,x}$ with norms
$\lVert\cdot\rVert_{H^{s,p}(\Lambda_{\delta,x})}=\lVert(-\tilde{A}_{\vartheta,\delta,x})^{s/2}\cdot\rVert_{L^{p}(\Lambda_{\delta,x})}$,
see [15] for details and also Section 6.4 below.

######  Lemma 6.4.

Let $0<\delta\leq 1$, $t>0$, $1<p\leq 2$ and grant Assumption Assumption
H(iii). Let $z\in H_{0}^{s}(\mathbb{R}^{d})$, $s\geq 0$, be compactly
supported in $\bigcap_{x\in\mathcal{J}}\Lambda_{\delta,x}$, suppose that
$V_{\delta,x}:L^{p}(\Lambda_{\delta,x})\rightarrow
H_{0}^{-s,p}(\Lambda_{\delta,x})$ are bounded linear operators with $\lVert
V_{\delta,x}z\rVert_{H^{-s,p}(\Lambda_{\delta,x})}\leq
V_{\operatorname{op}}\lVert z\rVert_{L^{p}(\Lambda_{\delta,x})}$ for some
$V_{\operatorname{op}}$ independent of $\delta$, $x$. Then there exists a
universal constant $C>0$ such that for $1<p\leq 2$ and $\gamma=(1/p-1/2)d/2$

$\displaystyle\sup_{x\in\mathcal{J}}\mathop{}\\!\mathbin{\lVert}S^{*}_{\vartheta,\delta,x}(t)V_{\delta,x}z\mathop{}\\!\mathbin{\rVert}_{L^{2}(\Lambda_{\delta,x})}$
$\displaystyle\leq
Ce^{\tilde{c}_{\vartheta}t\delta^{2}}\sup_{x\in\mathcal{J}}\left(\lVert
V_{\delta,x}z\rVert_{L^{2}(\Lambda_{\delta,x})}\wedge(V_{\operatorname{op}}t^{-s/2-\gamma}\lVert
z\rVert_{L^{p}(\Lambda_{\delta,x})})\right).$

If $s=0$, then this holds also for $p=1$.

###### Proof.

Let us write $u=V_{\delta,x}z$ and recall the multiplication operators
$U_{\vartheta,\delta,x}$ from the proof of Proposition 6.2(ii). They are
bounded operators on $L^{2}(\Lambda_{\delta,x})$ uniformly in $\delta\geq 0$
and $x\in\mathcal{J}$ and thus by (6.1)

$\displaystyle\mathop{}\\!\mathbin{\lVert}S^{*}_{\vartheta,\delta,x}(t)u\mathop{}\\!\mathbin{\rVert}_{L^{2}(\Lambda_{\delta,x})}$
$\displaystyle\lesssim
e^{\tilde{c}_{\vartheta}t\delta^{2}}\mathop{}\\!\mathbin{\lVert}\tilde{S}_{\vartheta,\delta,x}(t)U_{\vartheta,\delta,x}u\mathop{}\\!\mathbin{\rVert}_{L^{2}(\Lambda_{\delta,x})}.$
(6.4)

Let first $s=0$ such that
$H_{0}^{-s,p}(\operatorname{{\mathbb{R}}}^{d})=L^{p}(\operatorname{{\mathbb{R}}}^{d})$.
Approximating $u$ by continuous and compactly supported functions we obtain by
Proposition 6.2(i), ellipticity of $a_{\vartheta}$ and hypercontractivity of
the heat kernel on $\mathbb{R}^{d}$ uniformly in $x\in\mathcal{J}$

$\displaystyle\mathop{}\\!\mathbin{\lVert}S^{*}_{\vartheta,\delta,x}(t)u\mathop{}\\!\mathbin{\rVert}_{L^{2}(\Lambda_{\delta,x})}$
$\displaystyle\lesssim
e^{\tilde{c}_{\vartheta}t\delta^{2}}\mathop{}\\!\mathbin{\lVert}e^{Ct\Delta}|U_{\vartheta,\delta,x}u|\mathop{}\\!\mathbin{\rVert}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$
$\displaystyle\lesssim e^{\tilde{c}_{\vartheta}t\delta^{2}}t^{-\gamma}\lVert
u\rVert_{L^{p}(\mathbb{R}^{d})}\lesssim
e^{\tilde{c}_{\vartheta}t\delta^{2}}t^{-\gamma}\lVert
z\rVert_{L^{p}(\operatorname{{\mathbb{R}}}^{d})}.$

This yields the result for $s=0$. These inequalities hold also for $p=1$, thus
proving the supplement of the statement. For $s>0$ and $p>1$ note first that
by [3, Proposition 17(i)] we have
$\lVert(-t\tilde{A}_{\vartheta,\delta,x})^{s/2}\tilde{S}_{\vartheta,\delta,x}(t)z\rVert_{L^{2}(\Lambda_{\delta,x})}\lesssim_{s}\lVert
z\rVert_{L^{2}(\Lambda_{\delta,x})}$. Inserting this and then the last display
with $u$ replaced by
$(-\tilde{A}_{\vartheta,\delta,x})^{-s/2}U_{\vartheta,\delta,x}u$ into (6.4)
we get

$\displaystyle\mathop{}\\!\mathbin{\lVert}S^{*}_{\vartheta,\delta,x}(t)u\mathop{}\\!\mathbin{\rVert}_{L^{2}(\Lambda_{\delta,x})}\lesssim
e^{\tilde{c}_{\vartheta}t\delta^{2}}\mathop{}\\!\mathbin{\lVert}{(-\tilde{A}_{\vartheta,\delta,x})^{s/2}\tilde{S}_{\vartheta,\delta,x}(t)(-\tilde{A}_{\vartheta,\delta,x})^{-s/2}U_{\vartheta,\delta,x}u}\mathop{}\\!\mathbin{\rVert}_{L^{2}(\Lambda_{\delta,x})}$
$\displaystyle\quad\lesssim
e^{\tilde{c}_{\vartheta}t\delta^{2}}t^{-s/2}\mathop{}\\!\mathbin{\lVert}{\tilde{S}_{\vartheta,\delta,x}(t/2)(-\tilde{A}_{\vartheta,\delta,x})^{-s/2}U_{\vartheta,\delta,x}u}\mathop{}\\!\mathbin{\rVert}_{L^{2}(\Lambda_{\delta,x})}$
$\displaystyle\quad\lesssim
e^{\tilde{c}_{\vartheta}t\delta^{2}}t^{-s/2-\gamma}\mathop{}\\!\mathbin{\lVert}{U_{\vartheta,\delta,x}u}\mathop{}\\!\mathbin{\rVert}_{H^{-s,p}(\Lambda_{\delta,x})},$

uniformly in $x\in\mathcal{J}$. Since the functions $U_{\vartheta,\delta,x}$
are smooth and uniformly bounded on $\operatorname{{\mathbb{R}}}^{d}$ for
$x\in\mathcal{J}$, they induce a family of multiplication operators on
$H^{s,p}_{0}(\operatorname{{\mathbb{R}}}^{d})$ for $s\geq 0$ with operator
norms uniformly bounded in $x\in\mathcal{J}$, cf. [45, Theorem 2.8.2]. By
duality and restriction this transfers to $H^{s,p}_{0}(\Lambda_{\delta,x})$
for general $s$ according to [45, Theorem 3.3.2]. Hence,

$\displaystyle\mathop{}\\!\mathbin{\lVert}S^{*}_{\vartheta,\delta,x}(t)u\mathop{}\\!\mathbin{\rVert}_{L^{2}(\Lambda_{\delta,x})}\lesssim
e^{\tilde{c}_{\vartheta}t\delta^{2}}t^{-s/2-\gamma}\mathop{}\\!\mathbin{\lVert}{u}\mathop{}\\!\mathbin{\rVert}_{H^{-s,p}(\Lambda_{\delta,x})}\lesssim
e^{\tilde{c}_{\vartheta}t\delta^{2}}t^{-s/2-\gamma}V_{\operatorname{op}}\lVert
z\rVert_{L^{p}(\Lambda_{\delta,x})}.$

∎

###  6.2 Proof of the central limit theorem

####  6.2.1 Covariance structure of multiple local measurements

######  Lemma 6.5.

  1. (i)

If $X_{0}=0$, then the Gaussian process from (2.3) has mean zero and
covariance function

$\displaystyle\operatorname{Cov}(\langle X(t),z\rangle,\langle
X(t^{\prime}),z^{\prime}\rangle)=\int_{0}^{t\wedge t^{\prime}}\langle
S^{*}_{\vartheta}(t-s)z,S^{*}_{\vartheta}(t^{\prime}-s)z^{\prime}\rangle ds.$

  2. (ii)

If $X_{0}$ is the stationary initial condition from Lemma 2.2, then the
Gaussian process from (2.3) has mean zero and covariance function

$\displaystyle\operatorname{Cov}(\langle X(t),z\rangle,\langle
X(t^{\prime}),z^{\prime}\rangle)=\int_{0}^{\infty}\langle
S^{*}_{\vartheta}(t+s)z,S^{*}_{\vartheta}(t^{\prime}+s)z^{\prime}\rangle ds.$

###### Proof.

Part (i) follows from (2.2) and Itô’s isometry [14, Proposition 4.28]. For
part (ii) we conclude in the same way from noting that the stationary solution
given by $\langle X(t),z\rangle=\int_{-\infty}^{t}\langle
S_{\vartheta}(t-s)z,dW(s)\rangle$ has mean zero. ∎

######  Lemma 6.6.

Grant Assumption Assumption H and let $X_{0}=0$. Introduce for
$z,z^{\prime}\in L^{2}(\operatorname{{\mathbb{R}}}^{d})$

$\displaystyle\Psi_{\vartheta}(z,z^{\prime})$
$\displaystyle=\int_{0}^{\infty}\langle\bar{S}_{\vartheta}(t^{\prime})z,\bar{S}_{\vartheta}(t^{\prime})z^{\prime}\rangle_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}dt^{\prime}.$

If $n_{i}+n_{j}>2-d$ for some $i,j=1,\dots,p$, then
$\Psi_{\vartheta}(\bar{A}_{i}^{*}K,\bar{A}_{j}^{*}K)$ is well-defined and we
have as $\delta\rightarrow 0$

$\displaystyle\delta^{-2+n_{i}+n_{j}}(MT)^{-1}\sum_{k=1}^{M}\int_{0}^{T}\operatorname{{\mathbb{E}}}\left[\langle
X(t),A_{i}^{*}K_{\delta,x_{k}}\rangle\langle
X(t),A_{j}^{*}K_{\delta,x_{k}}\rangle\right]dt\rightarrow\Psi_{\vartheta}(\bar{A}_{i}^{*}K,\bar{A}_{j}^{*}K).$

###### Proof.

Fix $i,j$ with $n_{i}+n_{j}>2-d$. Then, applying Lemma 6.5(i), the scaling
from Lemma 6.1 and changing variables give

$\displaystyle\delta^{-2+n_{i}+n_{j}}(MT)^{-1}\sum_{k=1}^{M}\int_{0}^{T}\operatorname{{\mathbb{E}}}\left[\langle
X(t),A_{i}^{*}K_{\delta,x_{k}}\rangle\langle
X(t),A_{j}^{*}K_{\delta,x_{k}}\rangle\right]dt=\int_{0}^{\infty}f_{\delta}(t^{\prime})dt^{\prime}$

with

$\displaystyle f_{\delta}(t^{\prime})=(MT)^{-1}\sum_{k=1}^{M}\langle
S^{*}_{\vartheta,\delta,x_{k}}(t^{\prime})A^{*}_{i,\delta,x_{k}}K,S^{*}_{\vartheta,\delta,x_{k}}(t^{\prime})A^{*}_{j,\delta,x_{k}}K\rangle_{L^{2}(\Lambda_{\delta,x_{k}})}\int_{0}^{T}\mathbf{1}_{\\{0\leq
t^{\prime}\leq t\delta^{-2}\\}}dt.$

Consider now the differential operators
$V_{\delta,x_{k}}=A^{*}_{i,\delta,x_{k}}$. If $D^{m}$ is a composition of $m$
partial differential operators, then Theorem 1.43 of [52] yields that $D^{m}$
is a bounded linear operator from $L^{p}(\Lambda)$ to $H_{0}^{-m,p}(\Lambda)$,
implying
$\mathop{}\\!\mathbin{\lVert}{D^{m}K_{\delta,x_{k}}}\mathop{}\\!\mathbin{\rVert}_{H^{-m,p}(\Lambda)}\lesssim\delta^{-m}\mathop{}\\!\mathbin{\lVert}{K_{\delta,x_{k}}}\mathop{}\\!\mathbin{\rVert}_{L^{p}(\Lambda)}$.
Since $(D^{m}K)_{\delta,x_{k}}=\delta^{m}D^{m}K_{\delta,x_{k}}$, changing
variables gives
$\mathop{}\\!\mathbin{\lVert}{D^{m}K}\mathop{}\\!\mathbin{\rVert}_{H^{-m,p}(\Lambda_{\delta,x_{k}})}\lesssim\mathop{}\\!\mathbin{\lVert}{K}\mathop{}\\!\mathbin{\rVert}_{L^{p}(\Lambda_{\delta,x_{k}})}$.
From this we find $\lVert
V_{\delta,x_{k}}K\rVert_{H^{-n_{i},p}(\Lambda_{\delta,x})}\leq\lVert
K\rVert_{L^{p}(\Lambda_{\delta,x_{k}})}$, $\lVert
V_{\delta,x_{k}}K\rVert_{L^{2}(\Lambda_{\delta,x_{k}})}\lesssim\lVert
K\rVert_{H^{n_{i}}(\operatorname{{\mathbb{R}}}^{d})}$, and Lemma 6.4 shows for
$0\leq t^{\prime}\leq T\delta^{-2}$, $\varepsilon>0$ and all sufficiently
small $\delta>0$

$\sup_{x\in\mathcal{J}}\lVert
S^{*}_{\vartheta,\delta,x}(t^{\prime})A^{*}_{i,\delta,x}K\rVert_{L^{2}(\Lambda_{\delta,x})}\lesssim
1\wedge(t^{\prime})^{-n_{i}/2-d/4+\varepsilon}.$ (6.5)

This allows us to infer by the Cauchy-Schwarz inequality
$|f_{\delta}(t^{\prime})|\lesssim
1\wedge(t^{\prime})^{-n_{i}/2-n_{j}/2-d/2+2\varepsilon}$. In particular,
taking $\varepsilon$ so small that $n_{i}+n_{j}>2-d-4\varepsilon$ yields
$\sup_{\delta>0}|f_{\delta}|\in L^{1}([0,\infty))$. Lemma 6.2(ii), Lemma
6.1(ii) and continuity of the $L^{2}$-scalar product show now pointwise for
all $t^{\prime}>0$ that
$f_{\delta}(t^{\prime})\rightarrow\langle\bar{S}_{\vartheta}(t^{\prime})\bar{A}^{*}_{i}K,\bar{S}_{\vartheta}(t^{\prime})\bar{A}^{*}_{j}K\rangle_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$.
The wanted convergence follows from the dominated convergence theorem, which
also shows that $\Psi_{\vartheta}(\bar{A}_{i}^{*}K,\bar{A}^{*}_{j}K)$ is well-
defined. ∎

######  Lemma 6.7.

Grant Assumption Assumption H and let $X_{0}=0$. If $n_{i}+n_{j}>2-d$ for
$i,j=1,\dots,p$, then
$\sup_{x\in\mathcal{J}}\operatorname{Var}\left(\int_{0}^{T}\langle
X(t),A^{*}_{i}K_{\delta,x}\rangle\langle X(t),A^{*}_{j}K_{\delta,x}\rangle
dt\right)=o(\delta^{4-2n_{i}-2n_{j}})$.

###### Proof.

Applying the scaling from Lemma 6.1 and using Wicks theorem [25, Theorem 1.28]
we have for $x\in\mathcal{J}$

$\displaystyle\delta^{2n_{i}+2n_{j}}\operatorname{Var}\left(\int_{0}^{T}\langle
X(t),A_{i}^{*}K_{\delta,x}\rangle\langle X(t),A_{j}^{*}K_{\delta,x}\rangle
dt\right)$ $\displaystyle\quad=\operatorname{Var}\left(\int_{0}^{T}\langle
X(t),(A^{*}_{i,\delta,x}K)_{\delta,x}\rangle\langle
X(t),(A^{*}_{j,\delta,x}K)_{\delta,x}\rangle dt\right)=V_{1}+V_{2}$

with

$\displaystyle V_{1}$
$\displaystyle=V_{\delta,x}(A^{*}_{i,\delta,x}K,A^{*}_{i,\delta,x}K,A^{*}_{j,\delta,x}K,A^{*}_{j,\delta,x}K),$
$\displaystyle V_{2}$
$\displaystyle=V_{\delta,x}(A^{*}_{i,\delta,x}K,A^{*}_{j,\delta,x}K,A^{*}_{j,\delta,x}K,A^{*}_{i,\delta,x}K),$

and where for $v,v^{\prime},z,z^{\prime}\in L^{2}(\Lambda_{\delta,x})$

$\displaystyle
V_{\delta,x}(v,v^{\prime},z,z^{\prime})=\int_{0}^{T}\int_{0}^{T}\operatorname{{\mathbb{E}}}[\langle
X(t),v_{\delta,x}\rangle\langle
X(t^{\prime}),v^{\prime}_{\delta,x}\rangle]\operatorname{{\mathbb{E}}}[\langle
X(t),z_{\delta,x}\rangle\langle
X(t^{\prime}),z^{\prime}_{\delta,x}\rangle]dt^{\prime}dt.$

We only upper bound $V_{1}$, the arguments for $V_{2}$ are similar. Set
$f_{i,j}(s,s^{\prime})=\langle
S^{*}_{\vartheta,\delta,x}(s+s^{\prime})A^{*}_{i,\delta,x}K,S^{*}_{\vartheta,\delta,x}(s^{\prime})A^{*}_{j,\delta,x}K\rangle_{L^{2}(\Lambda_{\delta,x})}$.
Using Lemma 6.5(i) and the scaling from Lemma 6.1 we have

$\displaystyle V_{1}=2\delta^{6}\int_{0}^{T}\int_{0}^{t\delta^{-2}}$
$\displaystyle\left(\int_{0}^{t\delta^{-2}-s}f_{i,i}(s,s^{\prime})ds^{\prime}\right)\left(\int_{0}^{t\delta^{-2}-s}f_{j,j}(s,s^{\prime\prime})ds^{\prime\prime}\right)dsdt,$

cf. [4, Proof of Proposition A.9]. From (6.5) and the Cauchy-Schwarz
inequality we infer for $\varepsilon>0$

$\displaystyle\sup_{x\in\mathcal{J}}\left|f_{i,i}(s,s^{\prime})f_{j,j}(s,s^{\prime\prime})\right|\lesssim$
$\displaystyle\left(1\wedge
s^{-n_{i}/2-n_{j}/2-d/2+2\varepsilon}\right)\left(1\wedge s^{\prime-
n_{i}/2-d/4+\varepsilon}\right)$ $\displaystyle\quad\cdot\left(1\wedge
s^{\prime\prime-n_{j}/2-d/4+\varepsilon}\right),$

which gives

$\displaystyle\sup_{x\in\mathcal{J}}\left|V_{1}\right|$
$\displaystyle\lesssim\delta^{6}\int_{0}^{T\delta^{-2}}\left(1\wedge
s^{-n_{i}/2-n_{j}/2-d/2+2\varepsilon}\right)ds\int_{0}^{T\delta^{-2}}\left(1\wedge
s^{\prime-n_{i}/2-d/4+\varepsilon}\right)ds^{\prime}$
$\displaystyle\quad\quad\cdot\int_{0}^{T\delta^{-2}}\left(1\wedge
s^{\prime\prime-n_{j}/2-d/4+\varepsilon}\right)ds^{\prime\prime}$
$\displaystyle\lesssim\delta^{6}\left(1\vee\delta^{n_{i}+n_{j}+d-2-4\varepsilon}\right)\left(1\vee\delta^{n_{i}+d/2-2-2\varepsilon}\right)\left(1\vee\delta^{n_{j}+d/2-2-2\varepsilon}\right).$

Without loss of generality let $n_{i}\leq n_{j}$. Taking $\varepsilon$ small
enough, we can ensure $1\vee\delta^{n_{i}+n_{j}+d-2-4\varepsilon}=1$, as
$n_{i}+n_{j}>2-d$. In $d\leq 2$ only the pairs
$(n_{i},n_{j})\in\\{(0,0),(0,1)\\}$ are excluded, and in every case the
claimed bound holds. The same applies to $d\geq 3$ for all pairs
$(n_{i},n_{j})$. ∎

####  6.2.2 Proof of Theorem 2.3

###### Proof.

We begin with the observed Fisher information. Suppose first $X_{0}=0$. Under
Assumption Assumption H we find that $n_{i}+n_{j}>2-d$ for all $i,j=1,\dots,p$
in all dimensions $d\geq 1$. It follows from Lemmas 6.6 and 6.7 that

$\displaystyle(\rho_{\delta}\mathcal{I}_{\delta}\rho_{\delta})_{ij}$
$\displaystyle=\delta^{-2+n_{i}+n_{j}}M^{-1}\sum_{k=1}^{M}\int_{0}^{T}\langle
X(t),A_{i}^{*}K_{\delta,x_{k}}\rangle\langle
X(t),A_{j}^{*}K_{\delta,x_{k}}\rangle dt$
$\displaystyle=T\Psi_{\vartheta}(\bar{A}^{*}_{i}K,\bar{A}^{*}_{j}K)+o_{\operatorname{{\mathbb{P}}}}(1)=(\Sigma_{\vartheta})_{ij}+o_{\operatorname{{\mathbb{P}}}}(1),$

concluding by
$\Psi_{\vartheta}(z,z^{\prime})=\frac{1}{2}\langle(-\bar{A}_{\vartheta})^{-1/2}z,(-\bar{A}_{\vartheta})^{-1/2}z^{\prime}\rangle_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$.
This yields for $X_{0}=0$ the wanted convergence

$\displaystyle\rho_{\delta}\mathcal{I}_{\delta}\rho_{\delta}\stackrel{{\scriptstyle\operatorname{{\mathbb{P}}}}}{{\rightarrow}}\Sigma_{\vartheta}.$
(6.6)

In order to extend this to the general $X_{0}$ from Assumption Assumption H,
let $\bar{X}$ be defined as $X$, but starting in $\bar{X}(0)=0$ such that for
$v\in L^{2}(\Lambda)$, $\langle
X(t),v\rangle=\langle\bar{X}(t),v\rangle+\langle
S_{\vartheta}(t)X_{0},v\rangle$. If $\bar{\mathcal{I}}_{\delta}$ is the
observed Fisher information corresponding to $\bar{X}$, then by the Cauchy-
Schwarz inequality, with $v_{i}=\sup_{k}\delta^{2n_{i}-2}\int_{0}^{T}\langle
X_{0},S^{*}_{\vartheta}(t)A^{*}_{i}K_{\delta,x_{k}}\rangle^{2}dt$,

$\displaystyle|(\rho_{\delta}\mathcal{I}_{\delta}\rho_{\delta})_{ij}-(\rho_{\delta}\bar{\mathcal{I}}_{\delta}\rho_{\delta})_{ij}|\lesssim(\rho_{\delta}\bar{\mathcal{I}}_{\delta}\rho_{\delta})_{ii}^{1/2}v_{j}^{1/2}+(\rho_{\delta}\bar{\mathcal{I}}_{\delta}\rho_{\delta})_{jj}^{1/2}v_{i}^{1/2}+v_{i}^{1/2}v_{j}^{1/2}.$

By the first part,
$(\rho_{\delta}\bar{\mathcal{I}}_{\delta}\rho_{\delta})_{ii}$ is bounded in
probability and Assumption Assumption H(iv) gives
$v_{i}=o_{\operatorname{{\mathbb{P}}}}(1)$ for all $i$. From this obtain again
(6.6).

Regarding the invertibility of $\Sigma_{\vartheta}$, let
$\lambda\in\operatorname{{\mathbb{R}}}^{p}$ such that

$\displaystyle
0=\sum_{i,j=1}^{p}\lambda_{i}\lambda_{j}(\Sigma_{\vartheta})_{ij}$
$\displaystyle=T\Psi_{\vartheta}\left(\sum_{i=1}^{p}\lambda_{i}\bar{A}^{*}_{i}K,\sum_{i=1}^{p}\lambda_{i}\bar{A}^{*}_{i}K\right).$

By the definition of $\Psi_{\vartheta}$ this implies
$\bar{S}_{\vartheta}(t)(\sum_{i=1}^{p}\lambda_{i}\bar{A}^{*}_{i}K)=0$ for all
$t\geq 0$ and thus $\sum_{i=1}^{p}\lambda_{i}\bar{A}^{*}_{i}K=0$. Since the
functions $\bar{A}^{*}_{i}K$ are linearly independent by Assumption Assumption
H(i), conclude that $\Sigma_{\vartheta}$ is invertible.

We proceed next to the proof of the CLT. The augmented MLE and the statement
of the limit theorem remain unchanged when $K$ is multiplied by a scalar
factor. We can therefore assume without loss of generality that $\lVert
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}=1$. By the basic error
decomposition (2.8) and because $\Sigma_{\vartheta}$ is invertible, this means

$\displaystyle(\rho_{\delta}\mathcal{I}_{\delta}\rho_{\delta})^{1/2}\rho_{\delta}^{-1}(\hat{\vartheta}_{\delta}-\vartheta)=(\rho_{\delta}\mathcal{I}_{\delta}\rho_{\delta})^{-1/2}\Sigma_{\vartheta}^{1/2}(\Sigma_{\vartheta}^{-1/2}\rho_{\delta}\mathcal{M}_{\delta}).$
(6.7)

Note that $\mathcal{M}_{\delta}=\mathcal{M}_{\delta}(T)$ corresponds to a
$p$-dimensional continuous and square integrable martingale
$(\mathcal{M}_{\delta}(t))_{0\leq t\leq T}$ with respect to the filtration
$(\mathcal{F}_{t})_{0\leq t\leq T}$ evaluated at $t=T$. In view of the support
Assumption Assumption H(i) let $\delta\leq\delta^{\prime}$ such that for
$k,k^{\prime}$ with the Kronecker delta $\delta_{k,k^{\prime}}$

$\operatorname{{\mathbb{E}}}[W_{k}(t)W_{k^{\prime}}(t)]=\operatorname{{\mathbb{E}}}[\langle
W(t),K_{\delta,x_{k}}\rangle\langle
W(t),K_{\delta,x_{k^{\prime}}}\rangle]=\langle
K_{\delta,x_{k}},K_{\delta,x_{k^{\prime}}}\rangle=\delta_{k,k^{\prime}}.$

This means that the Brownian motions $W_{k}$ and $W_{k^{\prime}}$ are
independent for $k\neq k^{\prime}$ and thus their quadratic co-variation
process at $t$ is $[W_{k},W_{k^{\prime}}]=t\delta_{k,k^{\prime}}$. From this
infer that the quadratic co-variation process of the martingale
$(\mathcal{M}_{\delta}(t))_{0\leq t\leq T}$ at $t=T$ for
$\delta\leq\delta^{\prime}$ is equal to

$\displaystyle[\mathcal{M}_{\delta}]_{T}=\sum_{k,k^{\prime}=1}^{M}\int_{0}^{T}X^{A}_{\delta,k}(t)X^{A}_{\delta,k^{\prime}}(t)d[W_{k},W_{k^{\prime}}]_{t}=\mathcal{I}_{\delta}.$

Theorem A.1 now implies
$\Sigma_{\vartheta}^{-1/2}\rho_{\delta}\mathcal{M}_{\delta}\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,I_{p})$. Conclude in (6.7) by (6.6) and
Slutsky’s lemma. ∎

###  6.3 Proof of Theorem 3.1

In this section, we give the main steps of the proof of Theorem 3.1. The RKHS
computations and the proofs of two key lemmas are deferred to Section 6.4 and
the appendix.

####  6.3.1 Gaussian minimax lower bounds

Let $(\mathbb{P}_{\vartheta})_{\vartheta\in\Theta}$ be a family of probability
measures defined on the same measurable space with a parameter set
$\Theta\subseteq\mathbb{R}^{p}$. For $\vartheta^{0},\vartheta^{1}\in\Theta$,
the (squared) Hellinger distance between $\mathbb{P}_{\vartheta^{0}}$ and
$\mathbb{P}_{\vartheta^{1}}$ is defined by
$H^{2}(\mathbb{P}_{\vartheta^{0}},\mathbb{P}_{\vartheta^{1}})=\int(\sqrt{\mathbb{P}_{\vartheta^{0}}}-\sqrt{\mathbb{P}_{\vartheta^{1}}})^{2}$
(see, e.g. [46, Definition 2.3]). Moreover, if
$\vartheta^{0},\vartheta^{1}\in\Theta$ satisfy

$\displaystyle\
H^{2}(\mathbb{P}_{\vartheta^{0}},\mathbb{P}_{\vartheta^{1}})\leq 1,$ (6.8)

then we have the lower bound

$\displaystyle\inf_{\hat{\vartheta}}\max_{\vartheta\in\\{\vartheta^{0},\vartheta^{1}\\}}\mathbb{P}_{\vartheta}\left(|\hat{\vartheta}-\vartheta|\geq\frac{|\vartheta^{0}-\vartheta^{1}|}{2}\right)\geq\frac{1}{4}\frac{2-\sqrt{3}}{4}=:c_{3},$
(6.9)

where the infimum is taken over all $\operatorname{{\mathbb{R}}}^{p}$-valued
estimators $\hat{\vartheta}$ and $|\cdot|$ denotes the Euclidean norm. For a
proof of this lower bound, see [46, Theorem 2.2(ii)].

Next, let $\mathbb{P}_{\vartheta^{0}}$ and $\mathbb{P}_{\vartheta^{1}}$ be two
Gaussian measures defined on a separable Hilbert space $\mathcal{H}$ with
expectation zero and positive self-adjoint trace-class covariance operators
$C_{\vartheta^{0}}$ and $C_{\vartheta^{1}}$, respectively. By the spectral
theorem, there exist (strictly) positive eigenvalues $(\sigma_{j}^{2})_{j\geq
1}$ and an associated orthonormal system of eigenvectors $(u_{j})_{j\geq 1}$
such that $C_{\vartheta^{0}}=\sum_{j\geq 1}\sigma_{j}^{2}(u_{j}\otimes
u_{j})$. Given the Gaussian measure $\mathbb{P}_{\vartheta^{0}}$, we can
associate the so-called kernel or RKHS
$(H_{\vartheta^{0}},\|\cdot\|_{H_{\vartheta^{0}}})$ of
$\mathbb{P}_{\vartheta^{0}}$ given by

$\displaystyle
H_{\vartheta^{0}}=\\{h\in\mathcal{H}:\|h\|_{H_{\vartheta^{0}}}<\infty\\},\qquad\|h\|_{H_{\vartheta^{0}}}^{2}=\sum_{j\geq
1}\frac{\langle u_{j},h\rangle_{\mathcal{H}}^{2}}{\sigma_{j}^{2}}$ (6.10)

(see, e.g., [32, Example 4.2] and also [32, Chapters 4.1 and 4.3] and [19,
Chapter 3.6] for other characterizations of the RKHS of a Gaussian measure or
process). Alternatively, we have
$H_{\vartheta^{0}}=C_{\vartheta^{0}}^{1/2}\mathcal{H}$ and
$\|h\|_{H_{\vartheta^{0}}}=\lVert C_{\vartheta^{0}}^{-1/2}h\rVert$ for $h\in
H_{\vartheta^{0}}$. A useful tool to compute the RKHS is the fact that the
RKHS behaves well under linear transformation. More precisely, if
$L:\mathcal{H}\rightarrow\mathcal{H}^{\prime}$ is a bounded linear operator
between Hilbert spaces, then the image measure
$Q_{\vartheta^{0}}=\mathbb{P}_{\vartheta^{0}}\circ L^{-1}$ is a centered
Gaussian measure having RKHS $L(H_{\vartheta^{0}})$ with norm $\lVert
h\rVert_{L(H_{\vartheta^{0}})}=\inf\\{\lVert f\rVert_{\mathcal{H}}:f\in
L^{-1}h\\}$ (see Proposition 4.1 in [32] and also Chapter 3.6 in [19]).

Finally, combining (6.8) with the RKHS machinery, we get the following lower
bound.

######  Lemma 6.8.

In the above Gaussian setting, suppose that $(u_{j})_{j\geq 1}$ is an
orthonormal basis of $\mathcal{H}$ and that

$\displaystyle\sum_{j\geq
1}\sigma_{j}^{-2}\|(C_{\vartheta^{1}}-C_{\vartheta^{0}})u_{j}\|_{H_{\vartheta^{0}}}^{2}\leq
1/2.$ (6.11)

Then the lower bound in (6.9) holds, that is

$\displaystyle\inf_{\hat{\vartheta}}\max_{\vartheta\in\\{\vartheta^{0},\vartheta^{1}\\}}\mathbb{P}_{\vartheta}\left(|\hat{\vartheta}-\vartheta|\geq\frac{|\vartheta^{0}-\vartheta^{1}|}{2}\right)\geq
c_{3}.$

Lemma 6.8 is a consequence of the proof of the Feldman-Hajek theorem [14,
Theorem 2.25] in combination with basic properties of the Hellinger distance
and the minimax risk. A proof is given in Appendix A.3.

####  6.3.2 Proof of Theorem 3.1

Our goal is to apply Lemma 6.8 and Corollary 4.3 to the Gaussian process
$X_{\delta}$ under Assumption Assumption L. We assume without loss of
generality that $\lVert K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}=1$.
We choose $\vartheta^{0}=(1,0,0)$ and
$\vartheta^{1}\in\Theta_{1}\cup\Theta_{2}\cup\Theta_{3}$, meaning that the
null model is $A_{\vartheta^{0}}=\Delta$ and the alternatives are
$A_{\vartheta^{1}}=\vartheta^{1}_{1}\Delta+\vartheta^{1}_{2}(b\cdot\nabla)+\vartheta^{1}_{3}$
for $\vartheta^{1}\in\operatorname{{\mathbb{R}}}^{3}$, where $\vartheta^{1}$
lies in one of the parameter classes $\Theta_{1}$, $\Theta_{2}$ or
$\Theta_{3}$. For $\vartheta\in\\{\vartheta^{0},\vartheta^{1}\\}$, let
$\operatorname{{\mathbb{P}}}_{\vartheta,\delta}$ be the law of $X_{\delta}$ on
$\mathcal{H}=L^{2}([0,T])^{M}$, let $C_{\vartheta,\delta}$ be its covariance
operator, and let
$(H_{\vartheta,\delta},\lVert\cdot\rVert_{H_{\vartheta,\delta}})$ be the
associated RKHS. For $(f_{k})_{k=1}^{M}\in\mathcal{H}$, we have
$C_{\vartheta,\delta}(f_{k})_{k=1}^{M}=(\sum_{l=1}^{M}C_{\vartheta,\delta,k,l}f_{l})_{k=1}^{M}$
with (cross-)covariance operators defined by

$\displaystyle C_{\vartheta,\delta,k,l}:L^{2}([0,T])\rightarrow L^{2}([0,T]),$
$\displaystyle
C_{\vartheta,\delta,k,l}f_{l}(t)=\operatorname{{\mathbb{E}}}_{\vartheta}[\langle
X_{\delta,l},f_{l}\rangle_{L^{2}([0,T])}X_{\delta,k}(t)],\qquad 0\leq t\leq T$

(see, e.g., Appendix A.4 for more background on bounded linear operator on
$\mathcal{H}$). Because of the stationary of $X_{\delta}$ under Assumption
Assumption L, we have

$\displaystyle C_{\vartheta,\delta,k,l}f_{l}(t)$
$\displaystyle=\int_{0}^{t}c_{\vartheta,\delta,k,l}(t-t^{\prime})f_{l}(t^{\prime})\,dt^{\prime}+\int_{t}^{T}c_{\vartheta,\delta,l,k}(t^{\prime}-t)f_{l}(t^{\prime})\,dt^{\prime},\quad
0\leq t\leq T$

with covariance kernels

$c_{\vartheta,\delta,k,l}(t)=\operatorname{{\mathbb{E}}}_{\vartheta}[X_{\delta,k}(t)X_{\delta,l}(0)],\qquad
0\leq t\leq T.$

Following the notation of Section 6.3.1, let $(\sigma_{j}^{2})_{j\geq 1}$ be
the strictly positive eigenvalues of $C_{\vartheta^{0},\delta}$ and let
$(u_{j})_{j\geq 1}$ with $u_{j}=(u_{j,k})_{k=1}^{m}\in\mathcal{H}$ be a
corresponding orthonormal system of eigenvectors. By Corollary 4.3, we have
$H_{\vartheta^{0},\delta}=H^{M}$ as sets. Since $H^{M}$ is dense in
$\mathcal{H}$, $(u_{j})_{j\geq 1}$ forms an orthonormal basis of
$\mathcal{H}$. This means that the first assumption of Lemma 6.8 is satisfied.
To verify the second assumption in (6.11), we will use the bound for the RKHS
norm in Corollary 4.3 to turn the left-hand side of (6.11) into a more
accessible expression.

######  Lemma 6.9.

In the above setting, we have

$\displaystyle\sum_{j=1}^{\infty}\sigma_{j}^{-2}\lVert(C_{\vartheta^{0},\delta}-C_{\vartheta^{1},\delta})u_{j}\rVert^{2}_{H_{\vartheta^{0},\delta}}$
$\displaystyle\leq cT\sum_{k,l=1}^{M}\Big{(}\frac{\lVert\Delta
K\rVert^{4}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{8}}\lVert
c_{\vartheta^{0},\delta,k,l}-c_{\vartheta^{1},\delta,k,l}\rVert^{2}_{L^{2}([0,T])}+\lVert
c_{\vartheta^{0},\delta,k,l}^{\prime\prime}-c_{\vartheta^{1},\delta,k,l}^{\prime\prime}\rVert^{2}_{L^{2}([0,T])}\Big{)}$

for all $\delta^{2}\leq\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ and all $T\geq 1$, where
$c>0$ is an absolute constant.

The proof of Lemma 6.9 can be found in Appendix A.4. Moreover, combining Lemma
6.5(ii) with perturbation arguments for semigroups, we prove the following
upper bound in Section A.5.

######  Lemma 6.10.

In the above setting let
$\vartheta^{1}=(\vartheta_{1},\vartheta_{2},\vartheta_{3})\in\Theta_{1}\cup\Theta_{2}\cup\Theta_{3}$
with $M\geq 1$. Then there exists a constant $c>0$, depending only on $K$ such
that

$\displaystyle\sum_{k,l=1}^{M}\left(\delta^{-8}\lVert
c_{\vartheta^{0},\delta,k,l}-c_{\vartheta^{1},\delta,k,l}\rVert^{2}_{L^{2}([0,T])}+\lVert
c_{\vartheta^{0},\delta,k,l}^{\prime\prime}-c_{\vartheta^{1},\delta,k,l}^{\prime\prime}\rVert^{2}_{L^{2}([0,T])}\right)$
$\displaystyle\qquad\leq
cM(\delta^{-2}(1-\vartheta_{1})^{2}+\vartheta_{2}^{2}+\delta^{2}\vartheta_{3}^{2}).$

Choosing consecutively

$\displaystyle\vartheta^{1}$
$\displaystyle=(\vartheta_{1},0,0)\in\Theta_{1},\qquad\vartheta_{1}=1+c_{2}\frac{\delta}{\sqrt{TM}},$
$\displaystyle\vartheta^{1}$
$\displaystyle=(1,\vartheta_{2},0)\in\Theta_{2},\qquad\vartheta_{2}=c_{2}\frac{1}{\sqrt{TM}},$
$\displaystyle\vartheta^{1}$
$\displaystyle=(1,0,\vartheta_{3})\in\Theta_{3},\qquad\vartheta_{3}=c_{2}\min\Big{(}1,\frac{\delta^{-1}}{\sqrt{TM}}\Big{)},$

claims (i) and (ii) of Theorem 3.1 follow from Lemma 6.8 in combination with
Lemmas 6.9 and 6.10. ∎

###  6.4 RKHS computations

The proofs of the RKHS results from Section 4 are achieved by basic operations
on RKHS, in particular under linear transformation (see, e.g., Section 6.3.1
and [32, Chapter 4] or [49, Chapter 12]).

Recall the stationary process $X$ in (4.1) and that
$A_{\vartheta}e_{j}=-\lambda_{j}e_{j}$. The cylindrical Wiener process can be
realised as $W=\sum_{j\geq 1}e_{j}\beta_{j}$ for independent scalar Brownian
motions $\beta_{j}$ and we obtain the decomposition

$\displaystyle X(t)=\sum_{j\geq
1}\int_{-\infty}^{t}e^{-\lambda_{j}(t-t^{\prime})}d\beta_{j}(t^{\prime})e_{j}=\sum_{j\geq
1}Y_{j}(t)e_{j},$ (6.12)

with independent stationary Ornstein-Uhlenbeck processes $Y_{j}$ satisfying

$\displaystyle dY_{j}(t)=-\lambda_{j}Y_{j}(t)dt+d\beta_{j}(t).$ (6.13)

For a sequence $(\mu_{j})$ of non-decreasing, positive real numbers, take
$\mathcal{H}_{1}$ to be the closure of $\mathcal{H}$ under the norm

$\displaystyle\lVert z\rVert^{2}_{\mathcal{H}_{1}}=\sum_{j\geq
1}\frac{1}{\mu_{j}^{2}}\langle z,e_{j}\rangle^{2}_{\mathcal{H}},$

such that $\mathcal{H}$ is continuously embedded in $\mathcal{H}_{1}$. If for
$0\leq t\leq T$

$\displaystyle\int_{-\infty}^{t}\lVert
S_{\vartheta}(t^{\prime})\rVert^{2}_{\operatorname{HS}(\mathcal{H},\mathcal{H}_{1})}dt^{\prime}$
$\displaystyle=\sum_{j\geq 1}\int_{-\infty}^{t}\lVert
S_{\vartheta}(t^{\prime})e_{j}\rVert^{2}_{\mathcal{H}_{1}}dt^{\prime}$
$\displaystyle=\sum_{j\geq
1}\frac{1}{\mu_{j}^{2}}\int_{-\infty}^{t}e^{-2\lambda_{j}t^{\prime}}dt^{\prime}<\infty,$
(6.14)

then we conclude by [14, Theorem 5.2] that the law of $X$ induces a Gaussian
measure on $L^{2}([0,T];\mathcal{H}_{1})$. A first universal choice is given
by $\mu_{j}=j$ for all $j\geq 1$. Moreover, if $A_{\vartheta}$ is a
differential operator, then Weyl’s law [42, Lemma 2.3] says that the
$\lambda_{j}$ are positive real numbers of the order $j^{2/d}$, meaning that
the choice $\mu_{j}=\lambda_{j}^{s/2}$ is possible whenever $s\geq 0$ and
$s+1>d/2$. In this case, $\mathcal{H}_{1}$ corresponds to a Sobolev space of
negative order $-s$ induced by the eigensequence $(\lambda_{j},e_{j})_{j\geq
1}$.

####  6.4.1 RKHS of an Ornstein-Uhlenbeck process

We start by computing the RKHS $(H_{Y_{j}},\lVert\cdot\rVert_{Y_{j}})$ of the
processes $Y_{j}$. We show that the RKHS is, as a set, independent of $j$ and
equal to

$\displaystyle H=\\{h\in L^{2}([0,T]):h\text{ absolutely
continuous},h^{\prime}\in L^{2}([0,T])\\}$

from Theorem 4.2, while the corresponding norm depends on the parameter
$\lambda_{j}$.

######  Lemma 6.11.

For every $j\geq 1$ we have $H_{Y_{j}}=H$ and

$\displaystyle\lVert h\rVert^{2}_{Y_{j}}$ $\displaystyle=\lambda_{j}^{2}\lVert
h\rVert^{2}_{L^{2}([0,T])}+\lambda_{j}(h^{2}(T)+h^{2}(0))+\lVert
h^{\prime}\rVert^{2}_{L^{2}([0,T])}.$ (6.15)

###### Proof.

By Example 4.4 in [32], a scalar Brownian motion $(\beta(t))_{0\leq t\leq T}$
starting in zero has RKHS $H_{\beta}=\\{h:h(0)=0,h\text{ absolutely
continuous, }h,h^{\prime}\in L^{2}([0,T])\\}$ with norm $\lVert
h\rVert_{H_{\beta}}^{2}=\int_{0}^{T}(h^{\prime}(t))^{2}\,dt$. Moreover, the
Brownian motion $(\bar{\beta}(t))_{0\leq t\leq T}$ with
$\bar{\beta}(t)=X_{0}+\beta(t)$, where $X_{0}$ is a standard Gaussian random
variable independent of $(\beta(t))_{0\leq t\leq T}$ has RKHS

$H_{\bar{\beta}}=\\{\alpha+h:\alpha\in\operatorname{{\mathbb{R}}},h\in
H_{\beta}\\}=H,\quad\lVert
h\rVert_{H_{\bar{\beta}}}^{2}=\int_{0}^{T}(h^{\prime}(t))^{2}\,dt+h^{2}(0),$

as can be seen from Proposition 4.1 in [32] or Example 12.28 in [49]. To
compute the RKHS of $Y_{j}$ we now proceed similarly as in Example 4.8 in
[32]. Define the bounded linear map
$L:L^{2}([0,e^{2\lambda_{j}T}-1])\rightarrow L^{2}([0,T])$,
$(Lf)(t)=(2\lambda_{j})^{-1/2}e^{-\lambda_{j}t}f(e^{2\lambda_{j}t}-1)$. Then
we have $L\bar{\beta}=Y_{j}$ in distribution and $L$ is bijective with inverse
$L^{-1}h(s)=\sqrt{2\lambda_{j}(s+1)}h((2\lambda_{j})^{-1}\log(s+1))$, $0\leq
s\leq e^{2\lambda_{j}T}-1$. By Proposition 4.1 in [32] (see also the
discussion after (6.10)), we conclude that
$H_{Y_{j}}=L(H_{\bar{\beta}})=L(H)=H$ with

$\displaystyle\|h\|_{Y_{j}}^{2}$
$\displaystyle=\|L^{-1}h\|_{H_{\bar{\beta}}}^{2}$
$\displaystyle=\int_{0}^{e^{2\lambda_{j}T}-1}\Big{(}\frac{d}{ds}\sqrt{2\lambda_{j}(s+1)}h\Big{(}\frac{1}{2\lambda_{j}}\log(s+1)\Big{)}\Big{)}^{2}\,ds+2\lambda_{j}h^{2}(0)$
$\displaystyle=\int_{0}^{T}(\lambda_{j}h(t)+h^{\prime}(t))^{2}\,dt+2\lambda_{j}h^{2}(0)$
$\displaystyle=\lambda_{j}^{2}\int_{0}^{T}h^{2}(t)\,dt+\lambda_{j}(h^{2}(T)+h^{2}(0))+\int_{0}^{T}(h^{\prime}(t))^{2}\,dt.$

This completes the proof. ∎

####  6.4.2 RKHS of the SPDE

We compute next the RKHS of the process $X$. Let us start with the following
series representation, which is independent of $\mathcal{H}_{1}$.

######  Lemma 6.12.

The RKHS $(H_{X},\|\cdot\|_{X})$ of the process $X$ in (4.1) satisfies

$H_{X}=\Big{\\{}h=\sum_{j\geq 1}h_{j}e_{j}:h_{j}\in
H,\|h\|_{X}<\infty\Big{\\}}\quad\text{and}\quad\|h\|_{X}^{2}=\sum_{j\geq
1}\|h_{j}\|_{Y_{j}}^{2}.$

###### Proof.

For simplicity, choose $\mu_{j}=j$ for all $j\geq 1$. Since $X(t)=\sum_{j\geq
1}j^{-1}Y_{j}(t)\tilde{e}_{j}$ with orthonormal basis $\tilde{e}_{j}=je_{j}$
of $\mathcal{H}_{1}$, the covariance operator $C_{X}$ of $X$ is isomorphic to
$\bigoplus_{j\geq 1}j^{-2}C_{Y_{j}}$ with $C_{Y_{j}}:L^{2}([0,T])\rightarrow
L^{2}([0,T])$ being the covariance operator of $Y_{j}$. Hence, using the
definition of the RKHS given after (6.10), the RKHS of $X$ consists of all
elements of the form

$\displaystyle h=C_{X}^{1/2}f=\sum_{j\geq
1}j^{-1}(C_{Y_{j}}^{1/2}f_{j})\tilde{e}_{j}=\sum_{j\geq
1}(C_{Y_{j}}^{1/2}f_{j})e_{j}$

with $f=\sum_{j\geq 1}f_{j}\tilde{e}_{j}\in L^{2}([0,T];\mathcal{H}_{1})$ and
we have

$\displaystyle\|h\|_{X}^{2}$
$\displaystyle=\|C_{X}^{-1/2}h\|_{L^{2}([0,T];\mathcal{H}_{1})}^{2}$
$\displaystyle=\|f\|_{L^{2}([0,T];\mathcal{H}_{1})}^{2}=\int_{0}^{T}\|f(t)\|_{\mathcal{H}_{1}}^{2}dt=\sum_{j\geq
1}\|f_{j}\|_{L^{2}([0,T])}^{2}<\infty.$ (6.16)

Using Lemma 6.11, we can write $h=\sum_{j\geq 1}h_{j}e_{j}$ with
$h_{j}=C_{Y_{j}}^{1/2}f_{j}\in H$ and

$\|h_{j}\|_{Y_{j}}^{2}=\|C_{Y_{j}}^{-1/2}h_{j}\|_{L^{2}([0,T])}^{2}=\|f_{j}\|_{L^{2}([0,T])}^{2}.$

Inserting this into (6.4.2), we conclude that

$\displaystyle\|h\|_{X}^{2}=\sum_{j\geq
1}\|f_{j}\|_{L^{2}([0,T])}^{2}=\sum_{j\geq 1}\|h_{j}\|_{Y_{j}}^{2},$

which completes the proof. ∎

###### Proof of Theorem 4.1.

In the proof we write

$\displaystyle\tilde{H}_{X}$ $\displaystyle=\\{h\in
L^{2}([0,T];\mathcal{H}):A_{\vartheta}h,h^{\prime}\in
L^{2}([0,T];\mathcal{H})\\}$

and

$\displaystyle\|h\|_{\tilde{H}_{X}}^{2}$
$\displaystyle=\|A_{\vartheta}h\|_{L^{2}([0,T];\mathcal{H})}^{2}+\|h^{\prime}\|_{L^{2}([0,T];\mathcal{H})}^{2}$
$\displaystyle\quad\quad+\|(-A_{\vartheta})^{1/2}h(0)\|^{2}_{\mathcal{H}}+\|(-A_{\vartheta})^{1/2}h(T)\|^{2}_{\mathcal{H}}.$

By the RKHS computations in Lemma 6.12 it remains to check that
$H_{X}=\tilde{H}_{X}$ and $\lVert h\rVert_{X}=\lVert h\rVert_{\tilde{H}_{X}}$
for all $h\in H_{X}$. First, let $h=\sum_{j\geq 1}h_{j}e_{j}\in H_{X}$. Then
$h$ is absolute continuous and we have

$\displaystyle h^{\prime}$ $\displaystyle=\sum_{j\geq 1}h_{j}^{\prime}e_{j}\in
L^{2}([0,T];\mathcal{H}),$ $\displaystyle A_{\vartheta}h$
$\displaystyle=-\sum_{j\geq 1}\lambda_{j}h_{j}e_{j}\in
L^{2}([0,T];\mathcal{H}).$ (6.17)

Hence, $h\in\tilde{H}_{X}$ and therefore $H_{X}\subseteq\tilde{H}_{X}$. To see
the second claim in (6.17), set $h_{m}=\sum_{j=1}^{m}h_{j}e_{j}$ and
$g_{m}=-\sum_{j=1}^{m}\lambda_{j}h_{j}e_{j}$ for $m\geq 1$. Then, $h_{m}(t)$
and $g_{m}(t)$ are in $\mathcal{H}$ for all $t\in[0,T]$ and we have
$A_{\vartheta}h_{m}=g_{m}$ because $(\lambda_{j},e_{j})_{j\geq 1}$ is an
eigensequence of $-A_{\vartheta}$. Moreover $h_{m}(t)\rightarrow h(t)$ and
$A_{\vartheta}h_{m}=g_{m}(t)\rightarrow g(t)=\sum_{j\geq
1}\lambda_{j}h_{j}(t)e_{j}$ for a.e. $t$. Since $A_{\vartheta}$ is closed, we
conclude that $A_{\vartheta}h(t)=g(t)$ for a.e. $t$. Next, let
$h\in\tilde{H}_{X}$. Since $(-A_{\vartheta})^{1/2}$ is self-adjoint, we have
for all $0\leq t^{\prime}\leq t\leq T$ (cf. the proof of [17, Theorem 5.9.3])

$\displaystyle 2\lVert(-A_{\vartheta})^{1/2}h(t)\rVert^{2}_{\mathcal{H}}$
$\displaystyle=2\langle(-A_{\vartheta})^{1/2}h(t),(-A_{\vartheta})^{1/2}h(t)\rangle_{\mathcal{H}}=2\langle(-A_{\vartheta})h(t),h(t)\rangle_{\mathcal{H}}$
$\displaystyle\leq 2\lVert A_{\vartheta}h(t)\rVert_{\mathcal{H}}\lVert
h(t)\rVert_{\mathcal{H}}\leq\lVert
A_{\vartheta}h(t)\rVert_{\mathcal{H}}^{2}+\lVert h(t)\rVert_{\mathcal{H}}^{2}$

and

$\displaystyle\frac{d}{dt}\lVert(-A_{\vartheta})^{1/2}h(t)\rVert^{2}_{\mathcal{H}}$
$\displaystyle=2\langle(-A_{\vartheta})^{1/2}h(t),(-A_{\vartheta})^{1/2}h^{\prime}(t)\rangle_{\mathcal{H}}=2\langle(-A_{\vartheta})h(t),h^{\prime}(t)\rangle_{\mathcal{H}},$

where the absolute value of the latter term is bounded by $\lVert
A_{\vartheta}h(t)\rVert_{\mathcal{H}}^{2}+\lVert
h^{\prime}(t)\rVert_{\mathcal{H}}^{2}$. Letting $t_{0}\in[0,T]$ such that
$\lVert(-A_{\vartheta})^{1/2}h(t_{0})\rVert^{2}_{\mathcal{H}}=T^{-1}\int_{0}^{T}\lVert(-A_{\vartheta})^{1/2}h(t)\rVert^{2}_{\mathcal{H}}dt$,
we deduce that

$\displaystyle\lVert(-A_{\vartheta})^{1/2}h(0)\rVert^{2}_{\mathcal{H}}+\lVert(-A_{\vartheta})^{1/2}h(T)\rVert^{2}_{\mathcal{H}}$
$\displaystyle=\Big{(}\int_{t_{0}}^{0}+\int_{t_{0}}^{T}\Big{)}\frac{d}{dt}\lVert(-A_{\vartheta})^{1/2}h(t)\rVert^{2}_{\mathcal{H}}dt+2T^{-1}\int_{0}^{T}\lVert(-A_{\vartheta})^{1/2}h(t)\rVert^{2}_{\mathcal{H}}dt$
$\displaystyle\leq 2\int_{0}^{T}\lVert
A_{\vartheta}h^{\prime}(t)\rVert_{\mathcal{H}}^{2}dt+\int_{0}^{T}\lVert
h(t)\rVert_{\mathcal{H}}^{2}dt+\int_{0}^{T}\lVert
h(t)\rVert_{\mathcal{H}}^{2}dt,$

where we also used that $T\geq 1$. The latter can be written as

$\displaystyle\lVert(-A_{\vartheta})^{1/2}h(0)\rVert^{2}_{\mathcal{H}}+\lVert(-A_{\vartheta})^{1/2}h(T)\rVert^{2}_{\mathcal{H}}$
$\displaystyle\leq 2\lVert
A_{\vartheta}h\rVert^{2}_{L^{2}([0,T];\mathcal{H})}+\lVert
h\rVert^{2}_{L^{2}([0,T];\mathcal{H})}+\lVert
h^{\prime}\rVert^{2}_{L^{2}([0,T];\mathcal{H})},$ (6.18)

so that $\lVert h\rVert_{\tilde{H}_{X}}<\infty$. Moreover, writing
$h=\sum_{j\geq 1}h_{j}e_{j}$, we have $(-A_{\vartheta})^{1/2}h(t)=\sum_{j\geq
1}\lambda_{j}^{1/2}h_{j}(t)e_{j}$ and the relations in (6.17) continue to
hold, as can be seen from the identities
$\langle(-A_{\vartheta})^{1/2}h(t),e_{j}\rangle=\langle
h(t),(-A_{\vartheta})^{1/2}e_{j}\rangle=\lambda_{j}^{1/2}h_{j}(t)$, $\langle
A_{\vartheta}h(t),e_{j}\rangle=\lambda_{j}h_{j}(t)$ and $\langle
h^{\prime},e_{j}\rangle=\langle h,e_{j}\rangle^{\prime}=h_{j}^{\prime}$ (for
the latter, see also [34, Proposition A.22]). It follows that

$\displaystyle\lVert h\rVert_{X}^{2}=\sum_{j\geq 1}(\lambda_{j}^{2}\lVert
h_{j}\rVert^{2}_{L^{2}([0,T])}+\lambda_{j}(h_{j}^{2}(T)+h_{j}^{2}(0))+\lVert
h^{\prime}_{j}\rVert^{2}_{L^{2}([0,T])})=\|h\|_{\tilde{H}_{X}}^{2}<\infty.$

Hence, $h\in H_{X}$ and therefore $\tilde{H}_{X}\subseteq H_{X}$. Since we
have also shown that the norms are equal, this completes the proof. The upper
bound for the RKHS norm follows from inserting (6.18). ∎

####  6.4.3 RKHS of multiple measurements

In this section, we deduce Theorem 4.2 from Theorem 4.1. This requires the
$K_{1},\dots,K_{M}$ to lie in the dual space of $\mathcal{H}_{1}$. In the case
$A_{\vartheta}=\Delta$, this requires the $K_{1},\dots,K_{M}$ to lie in a
Sobolev space of order $s>d/2-1$ (see the beginning of Section 6.4). In
Appendix A.6.1, we give a second, slightly more technical proof based on an
approximation argument, which provides the claim under the weaker assumption
$K_{1},\dots,K_{M}\in\mathcal{D}(A_{\vartheta})$.

###### First proof of Theorem 4.2.

For a sequence $(\mu_{j})$ of non-decreasing, positive real numbers, take

$H_{\mu}=\\{f\in\mathcal{H}:\lVert f\rVert_{H_{\mu}}^{2}=\sum_{j\geq
1}\mu_{j}^{2}\langle f,e_{j}\rangle^{2}_{\mathcal{H}}<\infty\\},$

and take $\mathcal{H}_{1}=\mathcal{H}_{\mu}^{\prime}$ to be the closure of
$\mathcal{H}$ under the norm

$\displaystyle\lVert z\rVert^{2}_{\mathcal{H}_{\mu}^{\prime}}=\sum_{j\geq
1}\frac{1}{\mu_{j}^{2}}\langle z,e_{j}\rangle^{2}_{\mathcal{H}}.$

Then, $\mathcal{H}_{\mu}$ is continuously embedded in $\mathcal{H}$ and
$\mathcal{H}_{\mu}^{\prime}$ is indeed the dual of $\mathcal{H}_{\mu}$.
Moreover, we can extend $\langle f,g\rangle=\langle f,g\rangle_{\mathcal{H}}$
to pairs $f\in\mathcal{H}_{\mu}$ and $g\in\mathcal{H}_{\mu}^{\prime}$ and we
have the (generalised) Cauchy-Schwarz inequality

$\displaystyle|\langle f,g\rangle|\leq\lVert f\rVert_{\mathcal{H}_{\mu}}\lVert
g\rVert_{\mathcal{H}_{\mu}^{\prime}}.$ (6.19)

We choose the sequence $(\mu_{j})$ such that (6.4) holds, meaning that $X$ can
be considered as a Gaussian random variable in
$L^{2}([0,T];\mathcal{H}_{\mu}^{\prime})$. For
$K_{1},\dots,K_{M}\in\mathcal{H}_{\mu}$, consider now the linear map

$\displaystyle L:L^{2}([0,T];\mathcal{H}_{\mu}^{\prime})\rightarrow
L^{2}([0,T])^{M},$ $\displaystyle Lf(t)=(\langle
K_{k},f(t)\rangle)_{k=1}^{M},t\in[0,T].$

Then, $LX=X_{K}$ in distribution. Using (6.19), it is easy to see that $L$ is
a bounded operator with norm bounded by $(\sum_{k=1}^{M}\lVert
K_{k}\rVert_{\mathcal{H}_{\mu}}^{2})^{1/2}$:

$\displaystyle\sum_{k=1}^{M}\int_{0}^{T}\langle K_{k},f(t)\rangle^{2}dt$
$\displaystyle\leq\Big{(}\sum_{k=1}^{M}\lVert
K_{k}\rVert_{\mathcal{H}_{\mu}}^{2}\Big{)}\lVert
f\rVert_{L^{2}([0,T];\mathcal{H}_{\mu}^{\prime})}^{2}.$

Next, we show that $L(H_{X})=H^{M}$. First, for $(h_{k})_{k=1}^{M}\in H^{M}$,
the function

$\displaystyle f=\sum_{k,l=1}^{M}G^{-1}_{k,l}K_{k}h_{l}\in
H_{X}\quad\text{satisfies}\quad Lf=(h_{k})_{k=1}^{M}.$ (6.20)

Hence $H^{M}\subseteq L(H_{X})$. To see the reverse inclusion, let $f\in
H_{X}$. Set $(h_{k})_{k=1}^{M}=Lf$ such that $h_{k}(t)=\langle
K_{k},f(t)\rangle$. By the definition of $H_{X}$ and properties of the Bochner
integral (see, e.g., [34, Proposition A.22]), the $h_{k}$ are absolutely
continuous with derivatives $h_{k}^{\prime}(t)=\langle
K_{k},f^{\prime}(t)\rangle$, and we have

$\displaystyle\int_{0}^{T}(h^{\prime}(t))^{2}dt\leq\lVert
K_{k}\rVert_{\mathcal{H}}^{2}\int_{0}^{T}\lVert
f^{\prime}(t)\rVert_{\mathcal{H}}^{2}dt=\lVert
K_{k}\rVert_{\mathcal{H}}^{2}\lVert
f^{\prime}\rVert_{L^{2}([0,T];\mathcal{H})}^{2}<\infty.$

We get $h_{k}\in H$ for all $k=1,\dots,M$. Hence, $L(H_{X})\subseteq H^{M}$
and therefore $L(H_{X})=H^{M}$. It remains to prove the bound for the norm.
Using (6.20) and the behavior of the RKHS under linear transformation (see
Proposition 4.1 in [32]), we have

$\displaystyle\lVert(h_{k})_{k=1}^{M}\rVert_{X_{K}}\leq\lVert\sum_{k,l=1}^{M}G^{-1}_{k,l}K_{k}h_{l}\rVert_{X}$
$\displaystyle\leq
3\|\sum_{k,l=1}^{M}G^{-1}_{k,l}A_{\vartheta}K_{k}h_{l}\|_{L^{2}([0,T];L^{2}(\Lambda))}^{2}+\|\sum_{k,l=1}^{M}G^{-1}_{k,l}K_{k}h_{l}\|_{L^{2}([0,T];L^{2}(\Lambda))}^{2}$
$\displaystyle+2\|\sum_{k,l=1}^{M}G^{-1}_{k,l}K_{k}h_{l}^{\prime}\|_{L^{2}([0,T];L^{2}(\Lambda))}^{2}.$

Using the definition of $G_{A_{\vartheta}}$, the last display becomes

$\displaystyle\lVert(h_{k})_{k=1}^{M}\rVert_{X_{K}}$ $\displaystyle\leq
3\int_{0}^{T}\sum_{k,l=1}^{M}(G^{-1}G_{A_{\vartheta}}G^{-1})_{kl}h_{k}(t)h_{l}(t)\,dt+\int_{0}^{T}\sum_{k,l=1}^{M}(G^{-1})_{kl}h_{k}(t)h_{l}(t)\,dt$
$\displaystyle+2\int_{0}^{T}\sum_{k,l=1}^{M}(G^{-1})_{kl}h_{k}^{\prime}(t)h_{l}^{\prime}(t)\,dt,$

and the claim follows from standard results for the operator norm of symmetric
matrices. ∎

###### Proof of Corollary 4.3.

Since the Laplace operator is negative and self-adjoint, the stochastic
convolution (4.1) is just the weak solution in (2.2) and
$\mathcal{H}=L^{2}(\Lambda)$. If
$(K_{k})_{k=1}^{M}=(K_{\delta,x_{k}})_{k=1}^{M}$ with $\lVert
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}=1$, then $K_{1},\dots,K_{M}$
have disjoint supports and satisfy the assumptions of Theorem 4.2 with
$G=I_{M}$ being the identity matrix and $G_{\Delta}$ being a diagonal matrix
with $(G_{\Delta})_{kk}=\lVert\Delta
K_{\delta,x_{k}}\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$. By
construction and the Cauchy-Schwarz inequality, we have $\lVert
K_{\delta,x_{k}}\rVert=1$ and $\lVert\Delta
K_{\delta,x_{k}}\rVert\leq\delta^{-2}\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$. From Theorem 4.2, we obtain
the RKHS $H_{X_{K}}=H^{M}$ of $X_{K}$ with the claimed upper bound on its
norm, where we also used that $\delta^{2}\leq\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ by assumption. ∎

##  Appendix A Additional proofs

###  A.1 Additional proofs from Section 2

The proof of invertibility of the observed Fisher information is classical
when the solution process is a multivariate Ornstein-Uhlenbeck process [29],
but requires a different proof for the Itô processes $X_{\delta,k}^{A}$.

###### Proof of Lemma 2.1.

It is enough to show that the first summand with $k=1$ in the definition of
the observed Fisher information is $\operatorname{{\mathbb{P}}}$-almost surely
positive definite. By a density argument we can assume without loss of
generality that $K\in C^{\infty}_{c}(\operatorname{{\mathbb{R}}}^{d})$. Define
a symmetric matrix $\beta\in\operatorname{{\mathbb{R}}}^{p\times p}$,
$\beta_{ij}=\langle
A^{*}_{i}K_{\delta,x_{1}},A^{*}_{j}K_{\delta,x_{1}}\rangle$ and suppose for
$\lambda\in\operatorname{{\mathbb{R}}}^{p}$ that

$0=\sum_{i,j=1}^{p}\lambda_{i}\lambda_{j}\beta_{ij}=\lVert\sum_{i=1}^{p}\lambda_{i}A_{i}^{*}K_{\delta,x_{1}}\rVert^{2}.$

According to Assumption Assumption H(i) the functions $\bar{A}_{i}^{*}K$ are
linearly independent, which implies the same for the functions $A_{i}^{*}K$.
This yields $\lambda=0$ and so $\beta$ is invertible. It follows that

$dX_{\delta,1}^{A}(t)=(\langle
X(t),A^{*}_{\vartheta}A^{*}_{i}K_{\delta,x_{1}}\rangle)_{i=1}^{p}dt+\beta^{1/2}d\bar{W}(t)$

with a $p$-dimensional Brownian motion $\bar{W}(t)=\beta^{-1/2}(\langle
W(t),A_{i}^{*}K_{\delta,x_{1}}\rangle)_{i=1}^{p}$. Then
$Y=\beta^{-1/2}X^{A}_{\delta,1}$ satisfies $dY(t)=\alpha(t)dt+d\bar{W}(t)$ for
some $p$-dimensional Gaussian process $\alpha$. Invertibility of
$\int_{0}^{T}X_{\delta,1}^{A}(t)X_{\delta,1}^{A}(t)^{\top}dt$ is equivalent to
the invertibility of $\int_{0}^{T}Y(t)Y(t)^{\top}dt$. Applying first the
innovation theorem, cf. [33, Theorem 7.18], componentwise and then the
Girsanov theorem for multivariate diffusions, this is further equivalent to
the $\operatorname{{\mathbb{P}}}$-almost sure invertibility of
$\int_{0}^{T}\bar{W}(t)\bar{W}(t)^{\top}dt$. The result is now obtained from
noting that the determinant of the $p\times p$ dimensional random matrix
$(\bar{W}(t_{1}),\dots,\bar{W}(t_{p}))$ is
$\operatorname{{\mathbb{P}}}$-almost surely not zero for any pairwise
different time points $t_{1},\dots,t_{p}$, because $\bar{W}$ has independent
increments. ∎

###### Proof of Lemma 2.2.

It is enough to prove the claim for
$A^{*}_{i}\in\\{1,\partial_{j},\partial^{2}_{jk}\\}$ with
$n_{i}\in\\{0,1,2\\}$. Let $u_{i}=\delta^{-n_{i}}(v_{i})_{\delta,x}$ for
$v_{i}=A^{*}_{i,\delta,x}K$. Suppose first $X_{0}\in L^{p}(\Lambda)$. The
scaling in Lemma 6.1, the Hölder inequality and Lemma 6.4 applied to
$\delta=1$, $s=0$, yield for $1/p+1/q=1$

$\displaystyle\sup_{x\in\mathcal{J}}\langle
X_{0},S^{*}_{\vartheta}(t)u_{i}\rangle^{2}\lesssim\lVert
S_{\vartheta}(t)X_{0}\rVert_{L^{p}(\Lambda)}^{2}\sup_{x\in\mathcal{J}}\lVert
u_{i}\rVert^{2}_{L^{q}(\Lambda)}\lesssim\delta^{d(1-2/p)-2n_{i}}\lVert
X_{0}\rVert_{L^{p}(\Lambda)}^{2},$

The same Lemmas applied to $s=n_{i}$ also show for
$\varepsilon,\varepsilon^{\prime}>0$

$\displaystyle\sup_{x\in\mathcal{J}}\int^{T}_{\varepsilon^{\prime}}\langle
X_{0},S^{*}_{\vartheta}(t)u_{i}\rangle^{2}dt\leq\lVert
X_{0}\rVert^{2}\sup_{x\in\mathcal{J}}\int^{T}_{\varepsilon^{\prime}}\lVert
S^{*}_{\vartheta}(t)u_{i}\rVert^{2}dt$
$\displaystyle\quad\lesssim\delta^{-2n_{i}}\sup_{x\in\mathcal{J}}\int^{T}_{\varepsilon^{\prime}}\lVert
S^{*}_{\vartheta,\delta,x}(t\delta^{-2})v_{i}\rVert^{2}_{L^{2}(\Lambda_{\delta,x})}dt\lesssim\delta^{-2n_{i}}\int_{\varepsilon^{\prime}}^{T}(t\delta^{-2})^{-n_{i}-d/2+\varepsilon}dt.$

Assumption Assumption H(ii) implies $1-n_{k}-d/2<0$, and so the last line is
of order
$O((\varepsilon^{\prime})^{1-n_{i}-d/2+\varepsilon}\delta^{d-2\varepsilon})$.
After splitting up the integral we conclude

$\displaystyle\sup_{x\in\mathcal{J}}\int^{T}_{0}\langle
X_{0},S^{*}_{\vartheta}(t)u_{i}\rangle^{2}dt\lesssim\varepsilon^{\prime}\delta^{d(1-2/p)-2n_{i}}+(\varepsilon^{\prime})^{1-n_{i}-d/2+\varepsilon}\delta^{d-2\varepsilon}.$

Choosing
$\varepsilon^{\prime}=\delta^{\frac{2n_{i}+2d/p-2\varepsilon}{n_{i}+d/2-\varepsilon}}$
yields the order $O(\delta^{h(p)-\varepsilon^{\prime\prime}})$ with the
function $h(p)=d(1-2/p)+2(n_{i}+d/p)/(n_{i}+d/2)-2n_{i}$ for any
$\varepsilon^{\prime\prime}>0$. We get $h(2)=2-2n_{i}$ and $h^{\prime}(p)>0$.
From this obtain the claim when $X_{0}\in L^{p}(\Lambda)$.

Let now $X_{0}=\int_{-\infty}^{0}S_{\vartheta}(-t^{\prime})dW(t^{\prime})$ and
observe that
$c_{\vartheta}=\sum_{|\alpha|=0}\left(\sum_{i=1}^{p}\vartheta_{i}a^{(i)}_{\alpha}+a^{(0)}_{\alpha}\right)\leq
0$. By Itô’s isometry, the $\delta$-scaling and changing variables we get

$\displaystyle\operatorname{{\mathbb{E}}}[\langle
X_{0},S^{*}_{\vartheta}(t)u_{i}\rangle^{2}]$
$\displaystyle=\int_{0}^{\infty}\lVert
S^{*}_{\vartheta}(t^{\prime}+t)u_{i}\rVert^{2}dt^{\prime}$
$\displaystyle=\delta^{2-2n_{i}}\int_{0}^{\infty}\lVert
S_{\vartheta,\delta,x}^{*}(t^{\prime}+t\delta^{-2})v_{i}\rVert^{2}_{L^{2}(\Lambda_{\delta,x})}dt^{\prime}$

By Lemma 6.4 and $\tilde{c}_{\vartheta}\leq 0$ the integral is uniformly
bounded in $x\in\mathcal{J}$ and $0\leq t\leq T$ and converges to zero by
dominated convergence, because the integrand does so as $\delta\rightarrow 0$.
From this obtain the claim in the stationary case. ∎

######  Theorem A.1.

Let $M_{\delta}=(M_{\delta}(t))_{t\geq 0}$ be a family of continuous
$p$-dimensional square integrable martingales with respect to the filtered
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t}),\mathbb{P})$, with
$M_{\delta}(0)=0$ and with quadratic covariation processes
$([M_{\delta}]_{t})_{t\geq 0}$. If $T>0$ is such that

$[M_{\delta}]_{T}\stackrel{{\scriptstyle\operatorname{{\mathbb{P}}}}}{{\rightarrow}}I_{p},\quad\delta\rightarrow
0,$

then we have the convergence in distribution

$M_{\delta}(T)\stackrel{{\scriptstyle
d}}{{\rightarrow}}\mathcal{N}(0,I_{p}),\quad\delta\rightarrow 0.$

###### Proof.

For $x\in\mathbb{R}^{p}$ the process $Y_{\delta}(t)=x^{\top}M_{\delta}(t)x$
defines a one dimensional continuous martingale with respect to
$(\mathcal{F}_{t})$ with $Y_{\delta}(0)=0$ and with quadratic variation

$[Y_{\delta}]_{T}\stackrel{{\scriptstyle\mathbb{P}}}{{\rightarrow}}x^{\top}x,\quad\delta\rightarrow
0.$

An application of the Dambis-Dubins-Schwarz theorem ([26, Theorem 3.4.6])
shows $Y_{\delta}(t)=w_{\delta}([Y_{\delta}]_{t})$ with scalar Brownian
motions $(w_{\delta}(t))_{t\geq 0}$, which are possibly defined on an
extension of the underlying probability space. From the last display Slutsky’s
lemma implies the joint weak convergence
$(w_{\delta},[Y_{\delta}]_{T})\stackrel{{\scriptstyle
d}}{{\rightarrow}}(w_{0},x^{\top}x)$ on the product Borel sigma algebra of
$C([0,\infty))\times\operatorname{{\mathbb{R}}}$, where $C([0,\infty))$ is
endowed with the uniform topology on compact subsets of $[0,\infty)$, and
where $w_{0}$ is another scalar Brownian motion. The continuous mapping
theorem with respect to $(f,t)\mapsto\phi(f,t)=f(t)$ yields then the result,
noting that $w_{0}(x^{\top}x)$ has distribution $\mathcal{N}(0,x^{\top}x)$. ∎

###  A.2 Proof of Proposition 5.4

###### Proof.

Note first that $A_{\vartheta}=\Delta+\vartheta$ corresponds to $A_{1}=1$,
$A_{0}=\Delta$ and with observed Fisher information
$\mathcal{I}_{\delta}=\sum_{k=1}^{M}\int_{0}^{T}\langle
X(t),K_{\delta,x_{k}}\rangle^{2}dt$. In particular, Assumptions Assumption
H(i),(iii) and (iv) hold. Let us now proceed to the proof of the proposition.

(i). The claimed asymptotic normality of the augmented MLE follows from the
same proof as in Theorem 2.3 once we know that

$\displaystyle
M^{-1}\delta^{-2}\mathcal{I}_{\delta}\stackrel{{\scriptstyle\operatorname{{\mathbb{P}}}}}{{\rightarrow}}\frac{T}{2}\lVert(-\Delta)^{-1/2}K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}.$
(A.1)

The conditions $\int_{\operatorname{{\mathbb{R}}}^{2}}K(x)dx=0$,
$\int_{\operatorname{{\mathbb{R}}}^{2}}xK(x)dx=0$ imply the existence of a
compactly supported function $\tilde{K}\in
H^{4}(\operatorname{{\mathbb{R}}}^{2})$ such that $K=\Delta\tilde{K}$ [4,
Lemma A.5(iii)]. Lemma 6.4 therefore implies for any $\varepsilon>0$, $0<t\leq
T\delta^{-2}$ and in dimension $d=2$ that

$\displaystyle\sup_{x\in\mathcal{J}}\lVert
S^{*}_{\vartheta,\delta,x}(t)K\rVert_{L^{2}(\Lambda_{\delta,x})}\lesssim
1\wedge t^{-3/2+\varepsilon}.$

Substituting this for (6.5), the proofs of Lemmas 6.6 and 6.7 apply and allow
us to deduce

$\displaystyle
M^{-1}\delta^{-2}\mathcal{I}_{\delta}=M^{-1}\delta^{-2}\operatorname{{\mathbb{E}}}\mathcal{I}_{\delta}+o_{\operatorname{{\mathbb{P}}}}(1)\stackrel{{\scriptstyle\operatorname{{\mathbb{P}}}}}{{\rightarrow}}T\int_{0}^{\infty}\lVert\bar{S}_{\vartheta}(t^{\prime})\Delta\tilde{K}\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}dt^{\prime},$

which equals the right hand side in (A.1).

(ii). As in the proof of Theorem 2.3 we can suppose that $\lVert
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}=1$. Recall the basic
decomposition (2.8)
$\hat{\vartheta}_{\delta}=\vartheta+\mathcal{I}_{\delta}^{-1}\mathcal{M}_{\delta}$
and from the proof of Theorem 2.3 that
$\mathcal{M}_{\delta}=\mathcal{M}_{\delta}(T)$ for a square integrable
martingale $(\mathcal{M}_{\delta}(t))_{0\leq t\leq T}$, whose quadratic
variation at $t=T$ coincides with $\mathcal{I}_{\delta}$. We show below

$\displaystyle\log(\delta^{-1})^{-1}\mathcal{I}_{\delta}=O_{\operatorname{{\mathbb{P}}}}(1),\quad(\log(\delta^{-1})^{-1}\mathcal{I}_{\delta})^{-1}=O_{\operatorname{{\mathbb{P}}}}(1).$
(A.2)

A well-known result about tail properties of square integrable martingales
(e.g., [51, 3.8]) therefore implies
$\mathcal{M}_{\delta}=O_{\operatorname{{\mathbb{P}}}}(\log(\delta^{-1})^{1/2})$,
and we conclude from the basic decomposition that
$\hat{\vartheta}_{\delta}=\vartheta+O_{\operatorname{{\mathbb{P}}}}(\log(\delta^{-1})^{-1/2})$
as claimed.

For (A.2) it is enough to show that
$\mathcal{I}_{\delta}/\operatorname{{\mathbb{E}}}\mathcal{I}_{\delta}\stackrel{{\scriptstyle\operatorname{{\mathbb{P}}}}}{{\rightarrow}}1$
and $\log(\delta^{-1})\operatorname{{\mathbb{E}}}\mathcal{I}_{\delta}\asymp
1$, which in turn holds if for some $c,C>0$, independent of $\delta$,

$c\leq\log(\delta^{-1})^{-1}\operatorname{{\mathbb{E}}}\mathcal{I}_{\delta}\leq
C,\quad\log(\delta^{-1})^{-2}\operatorname{Var}(\mathcal{I}_{\delta})=o(1).$
(A.3)

As in the proofs of Lemmas 6.6, 6.7 and using their notation we compute

$\displaystyle\operatorname{{\mathbb{E}}}\mathcal{I}_{\delta}$
$\displaystyle\leq
M\delta^{2}\int_{0}^{T}\int_{0}^{t\delta^{-2}}\sup_{x\in\mathcal{J}}\lVert
S^{*}_{\vartheta,\delta,x}(t^{\prime})K\rVert_{L^{2}(\Lambda_{\delta,x})}^{2}dt^{\prime}dt,$
$\displaystyle\operatorname{Var}(\mathcal{I}_{\delta})$ $\displaystyle\lesssim
M^{2}\sup_{x\in\mathcal{J}}\operatorname{Var}\left(\int_{0}^{T}\langle
X(t),K_{\delta,x}\rangle^{2}dt\right)$
$\displaystyle=M^{2}\sup_{x\in\mathcal{J}}4\delta^{6}\int_{0}^{T}\int_{0}^{t\delta^{-2}}\left(\int_{0}^{t\delta^{-2}-s}f_{1,1}(s,s^{\prime})ds^{\prime}\right)^{2}dsdt.$

By the supplement in Lemma 6.4 we find in $d=2$ that

$\displaystyle\sup_{x\in\mathcal{J}}\lVert
S^{*}_{\vartheta,\delta,x}(t)K\rVert_{L^{2}(\Lambda_{\delta,x})}\lesssim
1\wedge t^{-1/2},\quad t\geq 0,\delta\geq 0.$ (A.4)

Plugging this into the last display and using $M\delta^{2}\lesssim 1$ provides
us with

$\displaystyle\operatorname{{\mathbb{E}}}\mathcal{I}_{\delta}$
$\displaystyle\lesssim\int_{0}^{T}\int_{0}^{t\delta^{-2}}(1\wedge
s^{-1})dsdt\lesssim\int_{0}^{T}\log(t\delta^{-2})dt\lesssim\log(\delta^{-1}),$
$\displaystyle\operatorname{Var}(\mathcal{I}_{\delta})$ $\displaystyle\lesssim
M^{2}\delta^{6}\left(\int_{0}^{T\delta^{-2}}(1\wedge(t^{\prime})^{-1})dt^{\prime}\right)\left(\int_{0}^{T\delta^{-2}}(1\wedge
t^{-1/2})dt\right)^{2}\lesssim\log(\delta^{-1}).$

We are thus left with showing
$\operatorname{{\mathbb{E}}}[\mathcal{I}_{\delta}]\gtrsim\log(\delta^{-1})$.
First, note that $\lVert
S^{*}_{\vartheta,\delta,x}(t)K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}\geq
e^{-T|\vartheta|}\lVert\tilde{S}_{\vartheta,\delta,x}(t)K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}$
and decompose

$\displaystyle\lVert\tilde{S}_{\vartheta,\delta,x}(t)K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}=\langle\tilde{S}_{\vartheta,\delta,x}(t)K,\tilde{S}_{\vartheta,\delta,x}(t)K\rangle_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}$
$\displaystyle\quad=\lVert
e^{t\Delta}K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}+\langle
e^{t\Delta}K+\tilde{S}_{\vartheta,\delta,x}(t)K,\tilde{S}_{\vartheta,\delta,x}(t)K-e^{t\Delta}K\rangle_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}.$

By (6.3) and recalling $K\geq 0$, the inner product here is uniformly in
$x\in\mathcal{J}$ up to a universal constant upper bounded by

$\displaystyle\langle
e^{t\Delta}K,(e^{t\Delta}K^{2})^{1/2}\rangle_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}(\delta
t^{1/2}e^{-\delta^{-2}t^{-1}})^{1/2}\leq\lVert
e^{t\Delta}K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}\lVert
e^{t\Delta}K^{2}\rVert_{L^{1}(\operatorname{{\mathbb{R}}}^{2})}\delta
t^{1/2},$

concluding by the Cauchy-Schwarz inequality and $\sup_{x\geq 0}xe^{-x}\lesssim
1$ in the last inequality. Since $\lVert
e^{t\Delta}K^{2}\rVert_{L^{1}(\operatorname{{\mathbb{R}}}^{2})}=\lVert
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}$ and using (A.4), it thus
follows for some $C>0$ that

$\displaystyle\lVert\tilde{S}_{\vartheta,\delta,x}(t)K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}\geq\lVert
e^{t\Delta}K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}-C(1\wedge
t^{-1/2})\delta t^{1/2}.$

Hence, using $M\delta^{2}\gtrsim 1$,

$\displaystyle\operatorname{{\mathbb{E}}}\mathcal{I}_{\delta}$
$\displaystyle\quad\geq\int_{0}^{\delta^{-1}}\lVert
e^{t\Delta}K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}^{2}dt-\int_{0}^{\delta^{-1}}C\delta
dt=\int_{0}^{\delta^{-1}}\lVert
e^{t\Delta}K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}^{2}dt-C.$

Suppose without loss of generality that $\operatorname{supp}(K)$ is contained
in the unit ball $B_{1}(0)$. Writing $e^{t\Delta}K=q_{t}*K$ as convolution
with the heat kernel $q_{t}(x)=(4\pi t)^{-1}\exp(-|x|^{2}/(4t))$ we have

$\displaystyle\lVert
e^{t\Delta}K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}^{2}=\int_{\operatorname{{\mathbb{R}}}^{2}}\left(\int_{B_{1}(0)}q_{t}(y-x)K(x)dx\right)^{2}dy.$

The heat kernel $q_{t}(x)$ is decreasing as $|x|\rightarrow\infty$. Hence, for
$x\in B_{1}(0),$ we bound $q_{t}(y-x)\geq q_{t}(y+y/|y|)$ for any
$y\in\operatorname{{\mathbb{R}}}^{2}\setminus\\{0\\}$. Plugging this into the
preceding display yields by $K\geq 0$

$\displaystyle\lVert
e^{t\Delta}K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{2})}^{2}$
$\displaystyle\geq\int_{\operatorname{{\mathbb{R}}}^{2}}\left(\int_{B_{1}(0)}q_{t}(y+y/|y|)K(x)dx\right)^{2}dy$
$\displaystyle=\lVert
K\rVert_{L^{1}(\operatorname{{\mathbb{R}}}^{2})}^{2}\int_{\operatorname{{\mathbb{R}}}^{2}}q_{t}(y+y/|y|)^{2}dy\gtrsim
t^{-1}.$

In all, we conclude that
$\operatorname{{\mathbb{E}}}\mathcal{I}_{\delta}\gtrsim
C^{\prime}\int_{0}^{\delta^{-1}}t^{-1}dt-C$ for $C^{\prime}>0$, implying the
wanted lower bound in (A.3). ∎

###  A.3 Proof of Lemma 6.8

By definition, we have

$\displaystyle\sum_{j\geq
1}\sigma_{j}^{-2}\|(C_{\vartheta^{1}}-C_{\vartheta^{0}})u_{j}\|_{H_{\vartheta^{0}}}^{2}=\sum_{j,k\geq
1}\sigma^{-2}_{j}\sigma^{-2}_{k}\langle
u_{j},(C_{\vartheta^{1}}-C_{\vartheta^{0}})u_{k}\rangle^{2}.$

Combining this with (6.11) and the fact that $(u_{j})_{j\geq 1}$ is an
orthonormal basis of $\mathcal{H}$, the infinite matrix $(\langle
u_{j},(C_{\vartheta^{1}}-C_{\vartheta^{0}})u_{k}\rangle/(\sigma_{j}\sigma_{k}))_{j,k=1}^{\infty}$
defines an Hilbert-Schmidt operator $S$ on $\mathcal{H}$. Let
$(v_{j},\mu_{j})_{j\geq 1}$ be an eigensequence of $S$ with $(v_{j})_{j\geq
1}$ being an orthonormal basis of $\mathcal{H}$. Since
$\|S\|_{\operatorname{HS}(\mathcal{H})}^{2}\leq 1/2$ by (6.11), we have
$\tau_{j}:=\mu_{j}+1\in[1-2^{-1/2},1+2^{-1/2}]$ for all $j\geq 1$. Now, let
$\\{\xi_{j}\\}_{j\geq 1}$ be a sequence of independent standard Gaussian
random variables. Then the series

$\displaystyle\sum_{j\geq
1}\xi_{j}C_{\vartheta^{0}}^{1/2}v_{j}\qquad\text{and}\qquad\sum_{j\geq
1}\sqrt{\tau_{j}}\xi_{j}C_{\vartheta^{0}}^{1/2}v_{j}$

converge a.s. and their laws coincide with those of
$\mathbb{P}_{\vartheta^{0}}$ and $\mathbb{P}_{\vartheta^{1}}$, respectively
(see, e.g., [14, Proof of Theorem 2.25] or [30, Pages 166-167]). By standard
properties of the Hellinger distance (see, e.g., Equation (A.4) in [40]), we
have

$\displaystyle\mathcal{H}^{2}\Big{(}\bigotimes_{j\geq
1}\mathcal{N}(0,1),\bigotimes_{j\geq 1}\mathcal{N}(0,\tau_{j})\Big{)}$
$\displaystyle\leq\sum_{j\geq
1}H^{2}(\mathcal{N}(0,1),\mathcal{N}(0,\tau_{j}))$ $\displaystyle\leq
2\sum_{j\geq
1}(\tau_{j}-1)^{2}=2\|S\|_{\operatorname{HS}(\mathcal{H})}^{2}\leq 1.$ (A.5)

Moreover, defining $Q_{\vartheta^{0}}=\bigotimes_{j\geq 1}\mathcal{N}(0,1)$,
$Q_{\vartheta^{1}}=\bigotimes_{j\geq 1}\mathcal{N}(0,\tau_{j})$ and the
measurable map
$\mathcal{T}:\operatorname{{\mathbb{R}}}^{\infty}\rightarrow\mathcal{H}$ by
$\mathcal{T}(\\{\alpha_{j}\\})=\sum_{j\geq
1}\alpha_{j}C_{\vartheta^{0}}^{1/2}v_{j}$ if the limit exists and
$T(\\{\alpha_{j}\\})=0$ otherwise, the image measures satisfy
$Q_{\vartheta^{0}}\circ\mathcal{T}^{-1}=\mathbb{P}_{\vartheta^{0}}$ and
$Q_{\vartheta_{1}}\circ\mathcal{T}^{-1}=\mathbb{P}_{\vartheta_{1}}$. Finally,
by the transformation formula, the minimax risk in (6.9) can be written as
$\inf_{\hat{\vartheta}}\max_{\vartheta\in\\{\vartheta^{0},\vartheta^{1}\\}}Q_{\vartheta}(|\hat{\vartheta}\circ\mathcal{T}-\vartheta|\geq|\vartheta^{0}-\vartheta^{1}|/2)$,
where the infimum is taken over all measurable functions from $\mathcal{H}$ to
$\operatorname{{\mathbb{R}}}^{p}$. Allowing for general estimators depending
on the whole coefficent vector in $\operatorname{{\mathbb{R}}}^{\infty}$, the
claim follows from (A.5) and (6.9) applied to the product measures
$Q_{\vartheta^{0}}$ and $Q_{\vartheta^{1}}$.∎

###  A.4 Proof of Lemma 6.9

Let us recall some simple facts on the space $\mathcal{H}=L^{2}([0,T])^{M}$
and a bounded linear operator $I:\mathcal{H}\rightarrow\mathcal{H}$. First,
$\mathcal{H}$ is a Hilbert space equipped with the inner product
$\langle(f_{k})_{k=1}^{M},(g_{k})_{k=1}^{M}\rangle=\sum_{j=1}^{M}\langle
f_{j},g_{j}\rangle_{L^{2}([0,T])}$. Second, $I$ can be represented by linear
operators $I_{k,l}:L^{2}([0,T])\rightarrow L^{2}([0,T])$ such that
$I(f_{k})_{k=1}^{M}=(\sum_{l=1}^{M}I_{k,l}f_{l})_{k=1}^{M}$. Finally, $I$ is a
Hilbert-Schmidt operator if and only if all $I_{jk}$ are Hilbert-Schmidt
operators and we have

$\displaystyle\|I\|_{\operatorname{HS}(\mathcal{H})}^{2}=\sum_{k,l=1}^{M}\|I_{k,l}\|_{\operatorname{HS}(L^{2}([0,T]))}^{2},$

where $\lVert\cdot\rVert_{HS(L^{2}([0,T]))}$ denotes the Hilbert-Schmidt norm
on $L^{2}([0,T])$. Recall also that $(\sigma_{j}^{2})_{j\geq 1}$ are the
strictly positive eigenvalues of $C_{\vartheta^{0},\delta}$ and that
$(u_{j})_{j\geq 1}$ with $u_{j}=(u_{j,k})_{k=1}^{m}\in\mathcal{H}$ is a
corresponding orthonormal basis of eigenvectors. We first prove a more general
version of Lemma 6.9.

######  Lemma A.2.

Grant Assumption Assumption L. Consider an integral operator
$I=(I_{k,l})_{k,l=1}^{M}:\mathcal{H}\rightarrow\mathcal{H}$,
$I_{k,l}f(t)=\int_{0}^{t}\kappa_{k,l}(t-t^{\prime})f(t^{\prime})dt^{\prime}+\int_{t}^{T}\kappa_{l,k}(t^{\prime}-t)f(t^{\prime})dt^{\prime}$
with square integrable and twice continuously differentiable functions
$\kappa_{k,l}$ satisfying $\kappa_{k,l}(0)=\kappa_{l,k}(0)$ and
$\kappa^{\prime}_{k,l}(0)=-\kappa^{\prime}_{l,k}(0)$ for all $1\leq k,l\leq
M$. Then we have

$\displaystyle\sum_{j=1}^{\infty}\sigma_{j}^{-2}\lVert
Iu_{j}\rVert^{2}_{H_{\vartheta^{0},\delta}}$
$\displaystyle\leq\sum_{k,l=1}^{M}\left(240T\frac{\lVert\Delta
K\rVert^{4}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{8}}\lVert\kappa_{k,l}\rVert^{2}_{L^{2}([0,T])}+200T\lVert\kappa_{k,l}^{\prime\prime}\rVert^{2}_{L^{2}([0,T])}\right)$

for all $\delta^{2}\leq\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ and all $T\geq 1$.

###### Proof of Lemma A.2.

We divide the proof into the cases of single and multiple measurements.

##### Case $M=1$.

If $M=1$, then we consider an integral operator $I:L^{2}([0,T])\rightarrow
L^{2}([0,T])$,
$If(t)=\int_{0}^{T}\kappa(|t-t^{\prime}|)f(t^{\prime})dt^{\prime}$ with some
square integrable and twice continuously differentiable function $\kappa$
satisfying $\kappa^{\prime}(0)=0$. Define the operators

$\displaystyle I^{\prime}f(t)$
$\displaystyle=\int_{0}^{T}\operatorname{sign}(t-t^{\prime})\kappa^{\prime}(|t-t^{\prime}|)f(t^{\prime})dt^{\prime},$
$\displaystyle I^{\prime\prime}f(t)$
$\displaystyle=\int_{0}^{T}\kappa^{\prime\prime}(|t-t^{\prime}|)f(t^{\prime})dt^{\prime}.$

We show first

$\displaystyle(If)^{\prime}(t)=I^{\prime}f(t),\quad(If)^{\prime\prime}(t)=I^{\prime\prime}f(t).$
(A.6)

Indeed, after splitting up the integral defining $If(t)$ it follows from the
chain rule that

$\displaystyle(If)^{\prime}(t)$
$\displaystyle=\left(\int_{0}^{t}\kappa(t-t^{\prime})f(t^{\prime})dt^{\prime}+\int_{t}^{T}\kappa(t^{\prime}-t)f(t^{\prime})dt^{\prime}\right)^{\prime}$
$\displaystyle=\kappa(0)f(t)+\int_{0}^{t}\kappa^{\prime}(t-t^{\prime})f(t^{\prime})dt^{\prime}-\kappa(0)f(t)-\int_{t}^{T}\kappa^{\prime}(t^{\prime}-t)f(t^{\prime})dt^{\prime},$
$\displaystyle(If)^{\prime\prime}(t)$
$\displaystyle=\kappa^{\prime}(0)f(t)+\int_{0}^{t}\kappa^{\prime\prime}(t-t^{\prime})f(t^{\prime})dt^{\prime}+\kappa^{\prime}(0)f(t)+\int_{t}^{T}\kappa^{\prime\prime}(t^{\prime}-t)f(t^{\prime})dt^{\prime},$

from which (A.6) follows by inserting the assumption $\kappa^{\prime}(0)=0$.
Thus, Corollary 4.3 (applied with $M=1$) and (A.6) yield for all $j\geq 1$

$\displaystyle\lVert Iu_{j}\rVert^{2}_{H_{\vartheta^{0},\delta}}$
$\displaystyle\quad\leq 4\delta^{-4}\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\lVert
Iu_{j}\rVert^{2}_{L^{2}([0,T])}+2\lVert(Iu_{j})^{\prime}\rVert^{2}_{L^{2}([0,T])}$
$\displaystyle\quad=4\delta^{-4}\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\lVert
Iu_{j}\rVert^{2}_{L^{2}([0,T])}+2\lVert
I^{\prime}u_{j}\rVert^{2}_{L^{2}([0,T])}$

for all $\delta^{2}\leq\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ and all $T\geq 1$. By
construction, $I$ is symmetric, while $I^{\prime}$ is anti-symmetric, implying
that $\langle Iu_{j},u_{j^{\prime}}\rangle_{L^{2}([0,T])}^{2}=\langle
u_{j},Iu_{j^{\prime}}\rangle_{L^{2}([0,T])}^{2}$ and $\langle
I^{\prime}u_{j},u_{j^{\prime}}\rangle_{L^{2}([0,T])}^{2}=\langle
u_{j},I^{\prime}u_{j^{\prime}}\rangle_{L^{2}([0,T])}^{2}$ for all
$j,j^{\prime}\geq 1$. Combining this with Parseval’s identity, we get

$\displaystyle\lVert Iu_{j}\rVert^{2}_{H_{\vartheta^{0},\delta}}$
$\displaystyle\leq\quad\sum_{j^{\prime}=1}^{\infty}\big{(}4\delta^{-4}\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\langle
u_{j},Iu_{j^{\prime}}\rangle^{2}_{L^{2}([0,T])}+2\langle
u_{j},I^{\prime}u_{j^{\prime}}\rangle^{2}_{L^{2}([0,T])}\big{)}.$

Multiplying the right-hand side with $\sigma_{j}^{-2}$ and summing over $j\geq
1$ yields

$\displaystyle\sum_{j^{\prime}=1}^{\infty}\big{(}4\delta^{-4}\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\lVert
Iu_{j^{\prime}}\rVert^{2}_{H_{\vartheta^{0},\delta}}+2\lVert
I^{\prime}u_{j^{\prime}}\rVert^{2}_{H_{\vartheta^{0},\delta}}\big{)},$

as can be seen from (6.10). Applying again Corollary 4.3 and the definition of
the Hilbert-Schmidt norm, we arrive at

$\displaystyle\sum_{j=1}^{\infty}\sigma_{j}^{-2}\lVert
Iu_{j}\rVert^{2}_{H_{\vartheta^{0},\delta}}$ $\displaystyle\leq
16\frac{\lVert\Delta
K\rVert^{4}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{8}}\lVert
I\rVert^{2}_{\operatorname{HS}(L^{2}([0,T]))}$
$\displaystyle+16\frac{\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{4}}\lVert
I^{\prime}\rVert^{2}_{\operatorname{HS}(L^{2}([0,T]))}+4\lVert
I^{\prime\prime}\rVert^{2}_{\operatorname{HS}(L^{2}([0,T]))}$

for all $\delta^{2}\leq\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ and all $T\geq 1$. Inserting

$\displaystyle\lVert
I\rVert^{2}_{\operatorname{HS}(L^{2}([0,T]))}=\int_{0}^{T}\int_{0}^{T}\kappa^{2}(|t-t^{\prime}|)dtdt^{\prime}\leq
2T\lVert\kappa\rVert^{2}_{L^{2}([0,T])},$ $\displaystyle\lVert
I^{\prime}\rVert^{2}_{\operatorname{HS}(L^{2}([0,T]))}\leq
2T\lVert\kappa^{\prime}\rVert^{2}_{L^{2}([0,T])},\quad\lVert
I^{\prime\prime}\rVert^{2}_{\operatorname{HS}(L^{2}([0,T]))}\leq
2T\lVert\kappa^{\prime\prime}\rVert^{2}_{L^{2}([0,T])},$ (A.7)

we get

$\displaystyle\sum_{j}^{\infty}\sigma_{j}^{-2}\lVert
Iu_{j}\rVert^{2}_{H_{\vartheta^{0},\delta}}$ $\displaystyle\leq
32T\frac{\lVert\Delta
K\rVert^{4}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{8}}\lVert\kappa\rVert^{2}_{L^{2}([0,T])}$
$\displaystyle+32T\frac{\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{4}}\lVert\kappa^{\prime}\rVert^{2}_{L^{2}([0,T])}+8T\lVert\kappa^{\prime\prime}\rVert^{2}_{L^{2}([0,T])}$
(A.8)

for all $\delta^{2}\leq\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ and all $T\geq 1$. The claim
now follows from an interpolation inequality (see, e.g., [10]). To get precise
constants with respect to $T$, we give a self-contained argument. By partial
integration and the fact that $\kappa^{\prime}(0)=0$, we have

$\displaystyle\int_{0}^{T}(\kappa^{\prime}(t))^{2}\,dt=-\int_{0}^{T}\kappa^{\prime\prime}(t)\kappa(t)\,dt+\kappa^{\prime}(T)\kappa(T).$

Let $t_{0}\in[0,T]$ such that $\kappa(t_{0})=T^{-1}\int_{0}^{T}\kappa(t)\,dt$.
Then, by the Cauchy-Schwarz inequality, we have

$\displaystyle\kappa^{2}(T)$
$\displaystyle=2\int_{t_{0}}^{T}\kappa^{\prime}(t)\kappa(t)\,dt+\Big{(}T^{-1}\int_{0}^{T}\kappa(t)\,dt\Big{)}^{2}$
$\displaystyle\leq
2\|\kappa\|_{L^{2}([0,T])}\|\kappa^{\prime}\|_{L^{2}([0,T])}+T^{-1}\|\kappa\|_{L^{2}([0,T])}^{2}$

and similarly

$\displaystyle(\kappa^{\prime}(T))^{2}$
$\displaystyle=(\kappa^{\prime}(T))^{2}-(\kappa^{\prime}(0))^{2}$
$\displaystyle=\int_{0}^{T}2\kappa^{\prime\prime}(t)\kappa^{\prime}(t)\,dt\leq
2\|\kappa^{\prime}\|_{L^{2}([0,T])}\|\kappa^{\prime\prime}\|_{L^{2}([0,T])}.$

Combining these estimates, using also the Cauchy-Schwarz inequality, the fact
that $T\geq 1$ and the inequality $2ab\leq\varepsilon
a^{2}+\varepsilon^{-1}b^{2}$, $\varepsilon>0$,
$a,b\in\operatorname{{\mathbb{R}}}$ consecutively with
$\varepsilon\in\\{1/4,1/\sqrt{2},1\\}$, we get

$\displaystyle\|\kappa^{\prime}\|_{L^{2}([0,T])}^{2}$
$\displaystyle\leq\|\kappa\|_{L^{2}([0,T])}\|\kappa^{\prime\prime}\|_{L^{2}([0,T])}$
$\displaystyle+2\sqrt{\|\kappa\|_{L^{2}([0,T])}\|\kappa^{\prime\prime}\|_{L^{2}([0,T])}}\|\kappa^{\prime}\|_{L^{2}([0,T])}$
$\displaystyle+\sqrt{2}\sqrt{\|\kappa^{\prime}\|_{L^{2}([0,T])}\|\kappa^{\prime\prime}\|_{L^{2}([0,T])}}\|\kappa\|_{L^{2}([0,T])}$
$\displaystyle\leq\|\kappa\|_{L^{2}([0,T])}\|\kappa^{\prime\prime}\|_{L^{2}([0,T])}$
$\displaystyle+4\|\kappa\|_{L^{2}([0,T])}\|\kappa^{\prime\prime}\|_{L^{2}([0,T])}+\|\kappa^{\prime}\|_{L^{2}([0,T])}^{2}/4$
$\displaystyle+\|\kappa^{\prime}\|_{L^{2}([0,T])}\|\kappa\|_{L^{2}([0,T])}/2+\|\kappa\|_{L^{2}([0,T])}\|\kappa^{\prime\prime}\|_{L^{2}([0,T])}$
$\displaystyle\leq
6\|\kappa\|_{L^{2}([0,T])}\|\kappa^{\prime\prime}\|_{L^{2}([0,T])}+\|\kappa^{\prime}\|_{L^{2}([0,T])}^{2}/2+\|\kappa\|_{L^{2}([0,T])}^{2}/4$

and thus

$\displaystyle\|\kappa^{\prime}\|_{L^{2}([0,T])}^{2}\leq
12\|\kappa\|_{L^{2}([0,T])}\|\kappa^{\prime\prime}\|_{L^{2}([0,T])}+\|\kappa\|_{L^{2}([0,T])}^{2}/2.$

Using again the inequality $2ab\leq a^{2}+b^{2}$,
$a,b\in\operatorname{{\mathbb{R}}}$, we conclude that

$\displaystyle 32T\frac{\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{4}}\lVert\kappa^{\prime}\rVert^{2}_{L^{2}([0,T])}$
$\displaystyle\leq 192T\frac{\lVert\Delta
K\rVert^{4}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{8}}\|\kappa\|_{L^{2}([0,T])}^{2}+192T\lVert\kappa^{\prime\prime}\rVert^{2}_{L^{2}([0,T])}+16T\frac{\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{4}}\lVert\kappa\rVert^{2}_{L^{2}([0,T])}$
$\displaystyle\leq 208T\frac{\lVert\Delta
K\rVert^{4}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}}{\delta^{8}}\|\kappa\|_{L^{2}([0,T])}^{2}+192T\lVert\kappa^{\prime\prime}\rVert^{2}_{L^{2}([0,T])}$

where we used the inequality $\delta^{2}\leq\lVert\Delta
K\rVert_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}$ in the last step. Inserting
this into (A.8), the claim follows.

##### Case $M>1$.

We now extend the result to the general case $M>1$. Define the operators
$I^{\prime}=(I^{\prime}_{k,l})_{k,l=1}^{M}$ and
$I^{\prime\prime}=(I^{\prime\prime}_{k,l})_{k,l=1}^{M}$ by

$\displaystyle I_{k,l}^{\prime}f(t)$
$\displaystyle=\int_{0}^{t}\kappa_{k,l}^{\prime}(t-t^{\prime})f(t^{\prime})dt^{\prime}-\int_{t}^{T}\kappa_{l,k}^{\prime}(t^{\prime}-t)f(t^{\prime})dt^{\prime},$
$\displaystyle I_{k,l}^{\prime\prime}f(t)$
$\displaystyle=\int_{0}^{t}\kappa_{k,l}^{\prime\prime}(t-t^{\prime})f(t^{\prime})dt^{\prime}+\int_{t}^{T}\kappa_{l,k}^{\prime\prime}(t^{\prime}-t)f(t^{\prime})dt^{\prime}.$

Using that $\kappa_{k,l}(0)=\kappa_{l,k}(0)$ and
$\kappa^{\prime}_{k,l}(0)=-\kappa^{\prime}_{l,k}(0)$, we have

$\displaystyle(I_{k,l}f_{l})^{\prime}(t)=I_{k,l}^{\prime}f_{l}(t),\quad(I_{k,l}f_{l})^{\prime\prime}(t)=I_{k,l}^{\prime\prime}f_{l}(t),$

as can be seen by proceeding similarly as in the case $M=1$. Hence, we get

$\displaystyle(I((f_{k})_{k=1}^{M})^{\prime}=I^{\prime}(f_{k})_{k=1}^{M}\quad\text{
and
}\quad(I((f_{k})_{k=1}^{M})^{\prime\prime}=I^{\prime\prime}(f_{k})_{k=1}^{M}.$

Thus, Corollary 4.3 again yields for all $j\geq 1$

$\displaystyle\lVert
I(u_{j,k})_{k=1}^{M}\rVert^{2}_{H_{\vartheta^{0},\delta}}$ $\displaystyle\leq
4\delta^{-4}\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\lVert
I(u_{j,k})_{k=1}^{M}\rVert^{2}_{L^{2}([0,T])^{M}}+2\lVert
I^{\prime}(u_{j,k})_{k=1}^{M}\rVert^{2}_{L^{2}([0,T])^{M}}.$

Next, by construction, we have $(I_{k,l})^{*}=I_{l,k}$ and
$(I^{\prime}_{k,l})^{*}=-I^{\prime}_{l,k}$, implying that $I$ is symmetric,
while $I^{\prime}$ is anti-symmetric. Combining this with Parseval’s identity,
we get

$\displaystyle\lVert
I(u_{j,k})_{k=1}^{M}\rVert^{2}_{H_{\vartheta^{0},\delta}}$ $\displaystyle\leq
4\delta^{-4}\lVert\Delta
K\rVert^{2}_{L^{2}(\operatorname{{\mathbb{R}}}^{d})}\sum_{j^{\prime}=1}^{\infty}\langle(u_{j,k})_{k=1}^{M},I(u_{j^{\prime},k})_{k=1}^{M}\rangle^{2}_{L^{2}([0,T])^{M}}$
$\displaystyle+2\sum_{j^{\prime}=1}^{\infty}\langle(u_{j,k})_{k=1}^{M},I^{\prime}(u_{j^{\prime},k})_{k=1}^{M}\rangle^{2}_{L^{2}([0,T])^{M}}.$

Multiplying this with $\sigma_{j}^{-2}$ and summing over $j$ yields

$\displaystyle\sum_{j=1}^{\infty}\sigma_{j}^{-2}\lVert