# Error analysis of kernel EDMD for prediction and control
in the Koopman framework

Friedrich Philipp  F. Philipp Technische Universität Ilmenau, Optimization-
based Control Group, Institute for Mathematics, Ilmenau, Germany
<EMAIL_ADDRESS>,  Manuel Schaller  M. Schaller Technische
Universität Ilmenau, Optimization-based Control Group, Institute for
Mathematics, Ilmenau, Germany<EMAIL_ADDRESS>,  Karl Worthmann
K. Worthmann Optimization-based Control Group, Institute of Mathematics,
Technische Universität Ilmenau, Ilmenau, Germany<EMAIL_ADDRESS>,  Sebastian Peitz  S. Peitz Paderborn University, Department of Computer
Science, Data Science for Engineering, Germany<EMAIL_ADDRESS>and
Feliks Nüske  F. Nüske Max Planck Institute for Dynamics of Complex Technical
Systems, Magdeburg, Germany
and Department of Mathematics and Computer Science, Freie Universität Berlin,
Germany<EMAIL_ADDRESS>
###### Abstract.

Extended Dynamic Mode Decomposition (EDMD) is a popular data-driven method to
approximate the Koopman operator for deterministic and stochastic (control)
systems. This operator is linear and encompasses full information on the
(expected stochastic) dynamics. In this paper, we analyze a kernel-based EDMD
algorithm, known as kEDMD, where the dictionary consists of the canonical
kernel features at the data points. The latter are acquired by i.i.d. samples
from a user-defined and application-driven distribution on a compact set. We
prove bounds on the prediction error of the kEDMD estimator when sampling from
this (not necessarily ergodic) distribution. The error analysis is further
extended to control-affine systems, where the considered invariance of the
Reproducing Kernel Hilbert Space is significantly less restrictive in
comparison to invariance assumptions on an a-priori chosen dictionary.

##  1\. Introduction

Extended Dynamic Mode Decomposition (EDMD, see, e.g., [56]) and its
predecessor DMD [48] have received much attention in recent years. The reason
is the capability to predict so-called observables along the flow of dynamical
(control) systems in a data-driven manner. These observable functions are
quantities of interests, which can be used for prediction [20], computation
and analysis of key characteristics like attractors [27] or metastable sets
[35, 21, 34] as well as control [18], cf. the book [31] and the recent
overview article [7] for further applications. The particular appeal of the
EDMD approach is due to several reasons: First, the implementation of the
learning scheme is comparatively simple. Second, the resulting model is linear
and, last, the Koopman framework provides a mathematically-sound foundation.

Application of the EDMD method usually requires two steps: First, the linear
Koopman operator [23] (or its generator) is restricted to a subspace
$\mathcal{D}$ spanned by a finite set of observable functions. Second, the
projection of this finite-rank linear operator to $\mathcal{D}$, also called
compression, is approximated using samples, i.e., evaluations of the
observables along the flow of the dynamical system. The first step usually
requires the a-priori construction of a dictionary consisting of functions on
the state space such as, e.g., monomials, which inevitably leads to very large
dictionaries for high-dimensional state spaces. Consequently, the respective
data requirements to learn the compression of the Koopman operator grows
rapidly due to the curse of dimensionality. Data-informed dictionaries using
kernel methods have proven their potential to alleviate this issue. In kernel
EDMD (kEDMD), the dictionary is built up by kernel feature evaluations on the
data set [57, 22] which are contained in an infinite-dimensional ambient
Reproducing Kernel Hilbert Space (RKHS, see, e.g., [39, 49]). Thus, the
a-priori choice and tuning of the dictionary reduces to the mere choice of a
suitable kernel. As another advantage, one gains access to powerful tools from
the rich theory of RKHSs such as the kernel trick. For further recent works on
the use of reproducing kernels in the context of dynamical systems see also
[5, 12, 25, 26].

To derive data-driven approximations of Koopman operator or generator, two
fundamentally different sampling techniques are common. The first one is to
draw independent-and-identically-distributed (i.i.d.) initial values and to
propagate the flow by one time step. In this way, the dynamics is learned by
means of more and more i.i.d samples and results from, e.g., Monte-Carlo
integration, can be invoked. The second approach, which is very appealing from
a practical point of view, is to aqcuire data from one (or several) long
trajectories. This approach is called ergodic sampling and is widely applied
to systems stemming, for instance, from statistical physics problems, see [28,
29]. As these measurements are clearly not independent, the exploration of the
state space has to be ensured by means of properties of the dynamical system
such as ergodicity. Ergodic sampling is often mandated if the target measure
is a high-dimensional multi-modal distribution, where efficient algorithms to
generate i.i.d samples are unavailable. It is also appealing due to its ease
of implementation and its ability to reveal unknown ergodic measures and long-
term behavior of the system. However, i.i.d. sampling also has a particular
advantage: In contrast to ergodic sampling, where one usually obtains
characteristics of the behavior through the lens of the ergodic invariant
measure, e.g., on an attractor of the system, i.i.d. sampling allows for
learning the dynamics on any desired region of the state space.

For an a-priori chosen dictionary, i.e., the original EDMD approach, the
approximation error was analyzed in [37, 59] for systems governed by ordinary
and stochastic differential equations. However, for kEDMD, this analysis is so
far only available if some invariant measure exists on the whole state space,
see [42]. In this work, we complement these works by extending the results of
[42] on the kernel- and data-based approximation of the Koopman operator to a
much more flexible setting without (potentially restrictive) invariance
assumptions on the sets and/or measures under consideration. Our main tool is
a probabilistic error bound for (cross-)covariance operators in the Hilbert-
Schmidt norm using Hoeffding’s inequality. The non-invariance of the
considered subset of the state space leads to Koopman operators which no
longer map the function space into itself, but onto another space in general.
Thus, the Koopman operators do not form a semigroup as it is the case under
invariance assumptions. Hence, we generalize the definition of the Koopman
generator to establish our findings without modifying the kernel-based EDMD
algorithm.

To model control systems in the Koopman framework, a popular approach consists
of seeking a linear surrogate model in a lifted state space by means of
augmenting the state by the control, cf. [24]. Bi-linear models are an
alternative [55, 53]. This is motivated by the fact that the Koopman generator
inherits control affinity and can, thus, be constructed using Koopman
generators of autonomous systems. While loosing linearity of the data-driven
surrogate model, the bi-linear approach has proven to be favorable for
nonlinear systems with state-control couplings [40]. For EDMD, a respective
analysis of the approximation error for dynamical control systems can be
conducted by splitting the error into its two sources, cf. [37] for the
estimation error and [47] for the projection error. In these works, error
bounds for the generator approximation of autonomous systems were leveraged
exploiting the control-affine structure. In addition, for set-point
stabilization, an error bound proportional to the size of the state and the
control was established in [6]. On the one hand, these error bounds were
leveraged in [52] to design a robust controller of the bi-linear system using
techniques based on linear matrix inequalities. On the other hand, they were
exploited in [6] to rigorously ensure practical asymptotic stability of EDMD-
based Model Predictive Control using cost controllability [10] of the original
system. In this paper, we provide the first error bounds for the approximation
of control-affine systems using kernel-based EDMD. Whereas many existing
results concerning EDMD for control require the Koopman invariance of the
finite-dimensional linear span of the pre-defined dictionary [16, 6], we are
able to improve our approximation bounds by assuming that the Koopman operator
leaves the (usually infinite-dimensional) ambient RKHS invariant. In contrast
to the standard EDMD case with a finite-dimensional dictionary space, this
assumption appears to be much more naturally satisfied in many relevant cases.

##  2\. Summary of the key results

We briefly highlight our two main findings. Our first result considers error
bounds for kernel EDMD (kEDMD) of systems governed by a stochastic
differential equation

$dX_{t}=b(X_{t})\,dt+\sigma(X_{t})\,dW_{t}$

for initial values contained in a compact set $\mathcal{X}$. The Koopman
operator is defined by the conditional expectation of observables along the
flow, which is assumed to be contained in possibly unbounded set
$\mathcal{Y}$. More precisely, for an observable function
$\psi:\mathcal{Y}\to\mathbb{R}$ and for all $x\in\mathcal{X}$, we define

$\displaystyle(K^{t}\psi)(x)=\mathbb{E}\big{[}\psi(X_{t})|X_{0}=x\big{]}.$
(2.1)

We explicitly allow for $\sigma\equiv 0$ such that ordinary differential
equations are included. To approximate the Koopman operator from data, we
collect samples $(x_{k},y_{k})_{k=0}^{m-1}$, where $x_{k}\in\mathcal{X}$ is
distributed according to a chosen Borel measure $\mu$ on $\mathcal{Y}$ such
that $\mu(\mathcal{X})=1$ holds and $y_{k}\sim\rho_{t}(x_{k},\cdot)$ denotes
the corresponding successor state. Here, $\rho_{t}(x,\cdot)$ is the
probability distribution associated with the transition kernel of the process,
i.e., $\rho_{t}(x,A)=\mathbb{P}(X_{t}\in A\,|\,X_{0}=x)$. As we are interested
in approximations of the Koopman operator on Reproducing Kernel Hilbert Spaces
(RKHS) on $\mathcal{X}$ and $\mathcal{Y}$ denoted by
$\mathbb{H}_{\mathcal{X}}$ and $\mathbb{H}_{\mathcal{Y}}$, respectively, the
data matrices $\Psi(X)$ and $\Psi(Y)$ are then formed by kernel evaluations at
the data points, i.e.,

$\Psi(X)=\big{(}k(x_{i},x_{j})\big{)}_{i,j=0}^{m-1}\qquad\text{and}\qquad\Psi(Y)=\big{(}k(x_{i},y_{j})\big{)}_{i,j=0}^{m-1}.$

Correspondingly, the kEDMD estimator $\widehat{K}_{r}^{m,t}$ of the Koopman
operator $K^{t}$ on the function space $\mathcal{D}$ spanned by the kernel
features, i.e.,
$\mathcal{D}=\operatorname{span}\\{k(x_{0},\cdot),\ldots,k(x_{m-1},\cdot)\\}$,
is given by

$\displaystyle\widehat{K}_{r}^{m,t}:\mathcal{D}\to\mathcal{D}\quad\text{with\
matrix\
representation}\quad\widehat{M}^{m,t}_{r}=[\Psi(X)]^{\dagger}_{r}\Psi(Y)^{\top},$

where $[\Psi(X)]^{\dagger}_{r}$ is a pseudo-inverse of a rank-$r$ truncation
of $\Psi(X)$ to avoid ill-conditioning.

First main result: error bound for autonomous systems, cf. Theorem 4.2. Let an
error bound $\varepsilon>0$ and a probabilistic tolerance $\delta\in(0,1)$ be
given. Then, under suitable assumptions to be specified later, there is a
sufficient amount of data $m_{0}\in\mathbb{N}$ such that for all $m\geq
m_{0}$,

$\|P_{r}K^{t}-\widehat{K}_{r}^{m,t}\|_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X};\mu)}\,\leq\,c_{r}\varepsilon$ (2.2)

holds with probability $1-\delta$. Here, $c_{r}\geq 0$ is a constant depending
on the truncation order $r\in\mathbb{N}$, and $P_{r}$ is an orthogonal
projection onto the first $r$ components of the Mercer orthogonal basis of
$L^{2}(\mathcal{X};\mu)$.

Further, if the chosen RKHS is invariant under the Koopman flow, then, with
probability at least $1-\delta$,

$\|K^{t}-\widehat{K}_{r}^{m,t}\|_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X};\mu)}\,\leq\,\sqrt{\lambda_{r+1}}\,\|K^{t}\|_{\mathbb{H}_{\mathcal{Y}}\to\mathbb{H}_{\mathcal{X}}}+c_{r}\varepsilon$

with Mercer eigenvalues $\lambda_{r}\to 0$ for $r\to\infty$.

As a second result, we provide bounds for kEDMD-based predictions of control-
affine systems

$\displaystyle\dot{x}(t)=f(x(t))+\sum_{i=1}^{{n_{u}}}u_{i}(t)g_{i}(x(t))$
(2.3)

with a piecewise constant control function taking values in a compact set
$\mathcal{U}\subset\mathbb{R}^{n_{u}}$, resulting e.g. from sampling with
zero-order hold. For a constant control function $u(t)\equiv u\in\mathcal{U}$,
the counterpart of the Koopman operator (2.1) is given by

$\displaystyle(K_{u}^{t}\psi)(x^{0}):=\psi\big{(}x(t;x^{0},u)\big{)},$ (2.4)

where $x^{0}\in\mathcal{X}$ and $x(\cdot;x^{0},u)$ is the solution to (2.3)
with $x(0)=0$. Again, $\mathcal{Y}$ is chosen in a way such that
$x(t;x^{0},u)\in\mathcal{Y}$ for $x^{0}\in\mathcal{X}$. As the generator of
the Koopman operator (2.4) inherits the control affine structure of the
underlying control system (2.3), we show that the Koopman operator is
approximately control affine (for small times $t\geq 0$). Then, we define its
approximation by

$\displaystyle\widehat{K}^{m,t}_{r,u}:=\widehat{K}^{m,t}_{r}+\sum_{i=1}^{{n_{u}}}\frac{u_{i}}{\gamma_{i}}\big{(}\widehat{K}^{m,t}_{r,\gamma_{i}e_{i}}-\widehat{K}^{m,t}_{r}\big{)},$
(2.5)

where $\widehat{K}_{r,\gamma_{i}e_{i}}^{m,t}$, $\widehat{K}^{m,t}_{r}$ are
estimators of the Koopman operators corresponding to (2.3) for the constant
controls $u=\gamma_{i}e_{i}$, $i=1,\ldots,{n_{u}}$, and $u=0$, respectively,
using the kEDMD method as outlined above for autonomous systems.

Second main result: error bound for control systems, cf. Theorem 5.1. Let an
error bound $\varepsilon>0$, a probabilistic tolerance $\delta\in(0,1)$, and a
control $u\in\mathcal{U}\subset\mathbb{R}^{m}$, $\mathcal{U}$ compact be
given. Then, under suitable assumptions to be specified later, there is a
sufficient amount of data $m_{0}\in\mathbb{N}$ such that for all $m\geq
m_{0}$,

$\|P_{r}K_{u}^{t}-\widehat{K}_{r,u}^{m,t}\|_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X};\mu)}\,\leq\,c_{r}\varepsilon+t^{2}\cdot
C_{k}\widehat{\lambda}_{r}^{-1}.$

The first term resembles the right-hand side of (2.2) and the second term
stems from the approximate control-affinity of the Koopman operator.

The outline of this paper is as follows. In the following section, the reader
is introduced to the setting and fundamentals of this work, such as stochastic
differential equations, Reproducing Kernel Hilbert spaces, Koopman operators,
and kEDMD including the main assumptions. In Section 4 we present and prove
our main result, Theorem 4.2, on the approximation error on the kEDMD
estimator for autonomous systems. In Section 5 we first recall the affine
dependence of the Koopman generator on the control for control-affine systems
and show that the same holds approximately (for small sampling times) for the
Koopman operator, resulting in a natural estimator via an affine combination
of autonomous operators. Our second main result, Theorem 5.1, provides an
error bound on this estimator. Finally, Section 6 concludes the paper with an
exhibition of numerical results.

##  3\. Setting

In the following, we introduce the reader to the setting considered in this
paper which involves stochastic differential equations (Section 3.1), Koopman
operators (Section 3.3), symmetric positive definite kernels and their
associated reproducing kernel Hilbert spaces (Section 3.4), and the kEDMD
approximation method (Section 3.5). In Section 3.2 we provide our assumptions
and discuss them in detail.

###  3.1. Stochastic differential equations

Let a stochastic differential equation (SDE) with drift vector field
$b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and diffusion matrix field
$\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d\times d}$ be given, i.e.,

$dX_{t}=b(X_{t})\,dt+\sigma(X_{t})\,dW_{t},$ (3.1)

where $W_{t}$ is $d$-dimensional Brownian motion. We assume that both $b$ and
$\sigma$ are Lipschitz-continuous. Then [38, Theorem 5.2.1] guarantees the
existence of a unique solution $(X_{t})_{t\geq 0}$ to (3.1). The case
$\sigma\equiv 0$ will be referred to as “the deterministic case” as the SDE
then reduces to the ODE $\dot{x}=b(x)$.

The solution $(X_{t})_{t\geq 0}$ of the SDE (3.1) is a continuous time-
homogeneous Markov process, whose transition kernel is denoted by
$\rho_{t}:\mathbb{R}^{d}\times\mathcal{B}\to\mathbb{R}$, where $\mathcal{B}$
is the Borel $\sigma$-algebra on $\mathbb{R}^{d}$. Then $\rho_{t}(x,\cdot)$ is
a probability measure for all $x\in\mathbb{R}^{d}$, and for each
$A\in\mathcal{B}$ we have that $\rho_{t}(x,A)$ is a representative of the
conditional probability for $A$ containing $X_{t}$, given $X_{0}=x$, i.e.,

$\rho_{t}(x,A)=\mathbb{P}(X_{t}\in A|X_{0}=x)\quad\text{for
$\mathbb{P}^{X_{0}}$-a.e. $x\in\mathbb{R}^{d}$},$

where $\mathbb{P}^{X_{0}}$ denotes the law of $X_{0}$. Note that the law of
$X_{t}$ is determined by

$\displaystyle\mathbb{P}^{X_{t}}(A)=\int\rho_{t}(x,A)\,d\mathbb{P}^{X_{0}}(x),\qquad
A\in\mathcal{B}.$ (3.2)

It is well known [3, Section 1.3] that the transition kernel satisfies the so-
called Chapman-Kolmogorov equation

$\rho_{t+s}(x,A)=\int\rho_{s}(y,A)\,\rho_{t}(x,dy),\qquad
A\in\mathcal{B},\;x\in\mathbb{R}^{d}\;s,t\geq 0.$ (3.3)

As it is usual, we agree to write $\rho_{t}(x,dy)$ for
$d\rho_{t}(x,\cdot)(y)$.

###  3.2. Assumptions

In many works it is assumed that there is some set
$\mathcal{X}\subseteq\mathbb{R}^{d}$ which is invariant under the flow of the
SDE and that there is an invariant distribution $\pi$ on $\mathcal{X}$ such
that the solution process $X_{t}$ is stationary with respect to $\pi$, i.e.,
$\mathbb{P}^{X_{t}}=\pi$ for all $t\geq 0$. In view of (3.2), this is
equivalent to

$\int\rho_{t}(x,A)\,d\pi(x)=\pi(A),\qquad
A\in\mathcal{B}_{\mathcal{X}},\,t\geq 0.$

Here, $\mathcal{B}_{\mathcal{X}}$ denotes the Borel sigma algebra on
$\mathcal{X}$.

However, although the Krylov–Bogolyubov Theorem [11] ensures the existence of
such an invariant measure $\pi$ for a large class of SDEs, both the set
$\mathcal{X}$ as the measure $\pi$ are often unknown in practice; even if
$\pi$ is known, generating independent and identically distributed (i.i.d)
samples with respect to it remains, in general, a challenging problem.

In this paper, we relax both conditions in a way that the objects become both
more flexible and more accessible for applications:

  * •

$\mathcal{X}:=\overline{\Omega}$, where $\Omega\subset\mathbb{R}^{d}$ is a
bounded and connected open set;

  * •

$T>0$ is a finite time horizon;

  * •

$\mathcal{Y}\subset\mathbb{R}^{d}$ is a closed set such that
$\mathcal{X}\subset\mathcal{Y}$ and

$\displaystyle\forall t\geq 0\,\forall
x\in\mathcal{X}:\rho_{t}(x,\mathcal{Y})=1.$ (3.4)

  * •

$\mu$ is a finite Borel measure on $\mathcal{Y}$ such that
$\mu(\mathcal{X})=1$ and there exists a constant $L>0$ such that

$\displaystyle\int_{\mathcal{X}}\rho_{t}(x,A)\,d\mu(x)\,\leq\,L\cdot\mu(A),\quad
A\in\mathcal{B}_{\mathcal{Y}},\,t\in[0,T].$ (3.5)

The set $\mathcal{X}$ will later serve as the set, which initial data for the
process is sampled from. The assumption (3.4) then assures that a subsequent
sample after time $t\in(0,T]$ of the process remains in $\mathcal{Y}$ almost
surely. Clearly, $\mathcal{Y}=\mathbb{R}^{d}$ is a possible choice.

Let us discuss the above assumptions in the following remark.

######  Remark 3.1.

(a) In the case $\mathcal{Y}=\mathcal{X}$, the set $\mathcal{X}$ is invariant
under the stochastic flow of the SDE (3.1), i.e.,

$\forall t\geq 0\,\forall x\in\mathcal{X}:\rho_{t}(x,\mathcal{X})=1.$

A sufficient condition on the coefficients of the SDE (3.1) for the invariance
of $\mathcal{X}$ under the SDE flow is given by

$\displaystyle\sigma(x)$ $\displaystyle=0\quad\text{for all
$x\in\mathbb{R}^{d}\backslash\mathcal{X}$ and}$ (3.6a) $\displaystyle
b(x)^{\top}\nu(x)$ $\displaystyle\leq 0\quad\text{for all
$x\in\partial\mathcal{X}$ and all outer normal vectors
$\nu(x)\in\mathbb{R}^{n}$},$ (3.6b)

where an outer normal vector of $\mathcal{X}$ at $x\in\partial\mathcal{X}$ is
any vector $\nu\in\mathbb{R}^{d}$, $\nu\neq 0$, such that the open ball with
center $x+\nu$ and radius $\|\nu\|_{2}$ and $\mathcal{X}$ are disjoint111If
$\Omega$ has a $C^{1}$-boundary, the outer unit vector is unique. In case of
Lipschitz continuous boundary, as a consequence of Rademacher’s theorem [13,
Theorem 3.2], it is unique up to a set of Lebesgue measure zero on the
boundary.. This follows by an application of the Bony-Brezis (also known as
Nagumo) theorem, which is proven in [54, Theorem 10.XVI, p. 117f] for the
deterministic case under the tangent or interior pointing-condition (3.6b). In
the stochastic setting, continuity of solutions to the SDE ensures that any
trajectory leaving $\mathcal{X}$ has to cross the boundary. The condition
(3.6a) implies that the noise has no influence at the boundary ($\sigma$ is
continuous) and in the exterior of $\mathcal{X}$ such that the SDE reduces to
an ordinary differential equation. Hence, the proof in [54] for the
deterministic setting also applies to the SDE case.

(b) Note that (3.5) means that the push-forward measures $\nu_{t}$ of $\mu$
under the dynamics, defined on $\mathcal{Y}$ by
$\nu_{t}(A)=\int_{\mathcal{X}}\rho_{t}(x,A)\,d\mu(x)$, $t\in[0,T]$, are
uniformly Lipschitz-continuous w.r.t. $\mu$, i.e., $\nu_{t}(A)\leq
L\cdot\mu(A)$ for all $A\in\mathcal{B}_{\mathcal{Y}}$, cf. [41, Definition
3.1]. In consequence, $\mu$-integrable functions on $\mathcal{Y}$ are also
$\nu_{t}$-integrable, and for $\psi\in L^{1}(\mathcal{Y};\mu)$ we have

$\int_{\mathcal{X}}\int_{\mathcal{Y}}|\psi(y)|\,\rho_{t}(x,dy)\,d\mu(x)=\int_{\mathcal{Y}}|\psi|\,d\nu_{t}\,\leq\,L\int_{\mathcal{Y}}|\psi|\,d\mu.$
(3.7)

(c) In the deterministic case (i.e., $\sigma\equiv 0$), (3.4) means that
$F_{t}(x)\in\mathcal{Y}$ for all $t\in[0,T]$ and all $x\in\mathcal{X}$, where
$F_{t}$ denotes the flow of the ODE $\dot{x}=b(x)$. Moreover, the condition
(3.5) is equivalent to

$\displaystyle\mu(F_{t}^{-1}(A))\leq L\cdot\mu(A),\qquad
A\in\mathcal{B}_{\mathcal{Y}},\,t\in[0,T].$ (3.8)

In Section A we show that this condition is satisfied for a large class of
systems and measures.

###  3.3. Koopman operators

For $p\in[1,\infty]$, we let $\|\cdot\|_{p,\mathcal{Y}}$ denote the
$L^{p}(\mathcal{Y};\mu)$-norm. In particular, for $p<\infty$,

$\|\psi\|_{p,\mathcal{Y}}=\left(\int_{\mathcal{Y}}|\psi(y)|^{p}\,d\mu(y)\right)^{1/p}.$

Moreover, we define
$\langle\phi,\psi\rangle_{2,\mathcal{Y}}:=\int_{\mathcal{Y}}\phi(y)\overline{\psi(y)}\,d\mu(y)$
for $\phi,\psi\in L^{2}(\mathcal{Y};\mu)$. Similarly, we define the norm
$\|\cdot\|_{p,\mathcal{X}}$ on $L^{p}(\mathcal{X};\mu)$ and the scalar product
$\langle\cdot\,,\cdot\rangle_{2,\mathcal{X}}$ on $L^{2}(\mathcal{X};\mu)$,
respectively.

Let $B(\mathcal{Y})$ denote the set of all bounded Borel-measurable functions
on $\mathcal{Y}$. For $t\in[0,T]$, we define the Koopman operator $K^{t}$ on
$B(\mathcal{Y})$ associated with the SDE (3.1) by

$(K^{t}\psi)(x)=\int_{\mathcal{Y}}\psi(y)\,\rho_{t}(x,dy)=\mathbb{E}\big{[}\psi(X_{t})|X_{0}=x\big{]},\quad
x\in\mathcal{X},$

for $\psi\in B(\mathcal{Y})$.

######  Remark 3.2.

(a) For the definition of $K^{t}$ to carry over to spaces of equivalence
classes of functions that coincide a.e. on $\mathcal{Y}$, it must be ensured
that $K^{t}\psi_{1}=K^{t}\psi_{2}$ $\mu$-a.e. on $\mathcal{X}$ whenever
$\psi_{1}=\psi_{2}$ $\mu$-a.e. on $\mathcal{Y}$. This is equivalent to the
absolute continuity of the measure $\nu_{t}$ w.r.t. $\mu$ (see Remark 3.1),
which is a relaxation of (3.5). However, it is easy to see that $K^{t}$ maps
$L^{1}(\mu)$ (boundedly) into itself if and only if $\nu_{t}\leq L\mu$ for
some $L>0$, i.e., if (3.5) holds.

(b) Usually, the Koopman operator of a dynamical system on $\mathcal{X}$ is
considered as a linear operator from a space of functions on $\mathcal{X}$
into itself. The price we pay by relaxing the invariance condition on
$\mathcal{X}$ is that our version of the Koopman operator is a linear map
between two different spaces.

(c) Let us briefly consider the deterministic case. Here, we have
$\rho_{t}(x,A)=\delta_{F_{t}(x)}(A)$, and thus
$(K^{t}\psi)(x)=\int_{\mathcal{Y}}\psi(y)d\delta_{F_{t}(x)}(y)=\psi(F_{t}(x))$
for $t\in[0,T]$, cf. (3.4). This complies with the usual definiton of the
Koopman operator for deterministic dynamical systems as a composition
operator.

A proof of the following proposition can be found in the Appendix Section B.

######  Proposition 3.3.

Let $p\in[1,\infty]$. For every $t\geq 0$, the operator $K^{t}$ extends
uniquely to

  * •

a contraction from $L^{\infty}(\mathcal{Y};\mu)$ to $L^{p}(\mathcal{X};\mu)$;

  * •

a bounded operator from $L^{p}(\mathcal{Y};\mu)$ to $L^{p}(\mathcal{X};\mu)$.

If $p<\infty$, for $\psi\in L^{p}(\mathcal{Y};\mu)$ we have
$K^{t}\psi\to\psi|_{\mathcal{X}}$ in $L^{p}(\mathcal{X};\mu)$ as $t\to 0$.

######  Remark 3.4.

If $\mathcal{X}$ is invariant under the SDE flow (i.e.,
$\mathcal{Y}=\mathcal{X}$), the operators $K^{t}$ satisfy the semigroup
property $K^{t+s}=K^{t}K^{s}$ for $s,t\geq 0$, which is a simple consequence
of the Chapman-Kolmogorov equation (3.3). If $\mu$ is an invariant measure,
$(K^{t})_{t\geq 0}$ is a $C_{0}$-semigroup of contractions on222In what
follows, $L(\mathcal{H},\mathcal{K})$ denotes the set of all bounded linear
operators between Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. As usual, we
also set $L(\mathcal{H}):=L(\mathcal{H},\mathcal{H})$.
$L(L^{p}(\mathcal{X};\pi))$, where $p\in[1,\infty)$. However, if $\mu$ is not
invariant but at least satisfies (3.5) on each interval $[0,T]$ (with the
constants $L$ depending on $T$), the semigroup $(K^{t})_{t\geq 0}$ might not
be uniformly bounded. An example is given in Appendix C.

###  3.4. Reproducing kernel Hilbert spaces

In what follows, let $k:\mathcal{Y}\times\mathcal{Y}\to\mathbb{R}$ be a
continuous and bounded symmetric positive definite kernel, that is, we have
$k(x,y)=k(y,x)$ for all $x,y\in\mathcal{Y}$ and

$\sum_{i,j=1}^{m}k(x_{i},x_{j})c_{i}c_{j}\,\geq\,0$

for all choices of $x_{1},\ldots,x_{m}\in\mathcal{Y}$ and
$c_{1},\ldots,c_{m}\in\mathbb{R}$. It is well known that $k$ generates a so-
called reproducing kernel Hilbert space (RKHS) [2, 4, 39]
$(\mathbb{H}_{\mathcal{Y}},\langle\cdot\,,\cdot\rangle_{\mathbb{H}_{\mathcal{Y}}})$
of bounded continuous functions on $\mathcal{Y}$, such that for
$\psi\in\mathbb{H}_{\mathcal{Y}}$ the reproducing property

$\psi(x)=\langle\psi,\Phi_{\mathcal{Y}}(x)\rangle_{\mathbb{H}_{\mathcal{Y}}},\qquad
x\in\mathcal{Y},$ (3.9)

holds, where $\Phi_{\mathcal{Y}}:\mathcal{Y}\to\mathbb{H}_{\mathcal{Y}}$
denotes the so-called feature map corresponding to the kernel $k$, i.e.,

$\Phi_{\mathcal{Y}}(x)=k(x,\cdot),\qquad x\in\mathcal{Y}.$

In the sequel, we shall denote the norm on $\mathbb{H}_{\mathcal{Y}}$ by
$\|\cdot\|_{\mathbb{H}_{\mathcal{Y}}}$ and the kernel diagonal by $\varphi$:

$\varphi(x)=k(x,x),\qquad x\in\mathcal{Y}.$

Then for $x\in\mathcal{Y}$ we have

$\|\Phi_{\mathcal{Y}}(x)\|_{\mathbb{H}_{\mathcal{Y}}}^{2}=\langle\Phi_{\mathcal{Y}}(x),\Phi_{\mathcal{Y}}(x)\rangle_{\mathbb{H}_{\mathcal{Y}}}=\langle
k(x,\cdot),k(x,\cdot)\rangle_{\mathbb{H}_{\mathcal{Y}}}=k(x,x)=\varphi(x).$

We shall frequently make use of the following estimate:

$|k(x,y)|=|\langle\Phi_{\mathcal{Y}}(x),\Phi_{\mathcal{Y}}(y)\rangle_{\mathbb{H}_{\mathcal{Y}}}|\leq\|\Phi_{\mathcal{Y}}(x)\|_{\mathbb{H}_{\mathcal{Y}}}\|\Phi_{\mathcal{Y}}(y)\|_{\mathbb{H}_{\mathcal{Y}}}=\sqrt{\varphi(x)\varphi(y)}.$

For example, it shows that $\|k\|_{\infty}=\|\varphi\|_{\infty}$.

In contrast to other works, where the kernel is induced by an observation map
[12] or is specifically designed by taking samples of the dynamics into
account [5], we admit any continuous kernel on $\mathcal{Y}$ which satisfies
the following

Compatibility assumptions:

  1. (C1)

If $\psi\in L^{2}(\mathcal{Y};\mu)$ such that
$\int_{\mathcal{Y}}\\!\int_{\mathcal{Y}}k(x,y)\psi(x)\psi(y)\,d\mu(x)\,d\mu(y)=0$,
then $\psi=0$ $\mu$-a.e.

  2. (C2)

If $\psi\in\mathbb{H}_{\mathcal{Y}}$ such that $\psi(y)=0$ for $\mu$-a.e.
$y\in\mathcal{Y}$, then $\psi=0$.

Since the functions in $\mathbb{H}_{\mathcal{Y}}$ are continuous, note that
condition (C2) holds if the measure $\mu$ is absolutely continuous with
respect to Lebesgue measure with positive density. The condition allows us to
consider $\mathbb{H}_{\mathcal{Y}}$ as a subspace of $L^{2}(\mathcal{Y};\mu)$,
i.e., the inclusion operator $S_{\mathcal{Y}}:\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{Y};\mu)$, defined by

$S_{\mathcal{Y}}\psi:=[\psi],\quad\psi\in\mathbb{H}_{\mathcal{Y}},$

is injective. Here, $[\psi]$ denotes the equivalence class w.r.t. a.e.
equality. It is easily seen that $S_{\mathcal{Y}}$ is bounded with norm
$\|S_{\mathcal{Y}}\|\leq\|\varphi\|_{1,\mathcal{Y}}^{1/2}$. Hence,
$\mathbb{H}_{\mathcal{Y}}$ is continuously embedded in
$L^{2}(\mathcal{Y};\mu)$, i.e.,

$\|\psi\|_{2,\mathcal{Y}}\lesssim\|\psi\|_{\mathbb{H}_{\mathcal{Y}}},\quad\psi\in\mathbb{H}_{\mathcal{Y}}.$

Furthermore, the condition (C1) is equivalent to the density of
$\mathbb{H}_{\mathcal{Y}}$ in $L^{2}(\mathcal{Y};\mu)$, see [42]. Hence,
$\mathbb{H}_{\mathcal{Y}}$ is continuously and densely embedded in
$L^{2}(\mathcal{Y};\mu)$.

In fact, the embedding is even compact as $S_{\mathcal{Y}}$ is a Hilbert-
Schmidt operator with Hilbert-Schmidt norm
$\|S_{\mathcal{Y}}\|_{HS}^{2}=\|\varphi\|_{1,\mathcal{Y}}$, see [42]. For
this, recall that an operator $T\in L(\mathcal{H})$ on a Hilbert space
$\mathcal{H}$ is trace class if for some (and hence for each) orthonormal
basis (ONB) $(e_{j})_{j\in\mathbb{N}}$ of $\mathcal{H}$ we have
$\sum_{j=1}^{\infty}\langle(T^{*}T)^{1/2}e_{i},e_{i}\rangle<\infty$. An
operator $S\in L(\mathcal{H},\mathcal{K})$ between Hilbert spaces
$\mathcal{H}$ and $\mathcal{K}$ is said to be Hilbert-Schmidt [14, Chapter
III.9] if $S^{*}S$ is trace class, i.e.,
$\|S\|_{HS}^{2}:=\sum_{j=1}^{\infty}\|Se_{i}\|^{2}<\infty$ for some (and hence
for each) ONB $(e_{j})_{j\in\mathbb{N}}$ of $\mathcal{H}$. Trace-class and
Hilbert-Schmidt operators are compact operators. In particular, their spectrum
consists of discrete eigenvalues with the origin as the only accumulation
point. As $S_{\mathcal{Y}}$ is Hilbert-Schmidt, it follows that the covariance
operator

$C_{\mathcal{Y}}:=S_{\mathcal{Y}}^{*}S_{\mathcal{Y}}\,\in\,L(\mathbb{H}_{\mathcal{Y}})$

is trace-class. Mercer’s theorem shows the existence of a particular ONB
$(e_{j})_{j=1}^{\infty}$ of $L^{2}(\mathcal{Y};\mu)$ composed of
eigenfunctions of $S_{\mathcal{Y}}S_{\mathcal{Y}}^{*}$, which we shall
henceforth call the Mercer basis corresponding to the kernel $k$.

######  Theorem 3.5 (Mercer’s Theorem [45]).

There exists an ONB $(e_{j})_{j=1}^{\infty}$ of $L^{2}(\mathcal{Y};\mu)$
comprised of eigenfunctions of $S_{\mathcal{Y}}S_{\mathcal{Y}}^{*}$ with
corresponding eigenvalues $\lambda_{j}>0$ such that
$\sum_{j=1}^{\infty}\lambda_{j}=\|\varphi\|_{1,\mathcal{Y}}<\infty$.
Furthermore, $(f_{j})_{j=1}^{\infty}$ with $f_{j}=\sqrt{\lambda_{j}}e_{j}$
constitutes an ONB of $\mathbb{H}_{\mathcal{Y}}$ consisting of eigenfunctions
of $C_{\mathcal{Y}}$ with corresponding eigenvalues $\lambda_{j}$. Moreover,
for all $x,y\in\mathcal{Y}$,

$k(x,y)=\sum_{j}f_{j}(x)f_{j}(y)=\sum_{j}\lambda_{j}e_{j}(x)e_{j}(y),$

where the series converges absolutely.

The restriction $k_{\mathcal{X}}$ of the kernel $k$ to
$\mathcal{X}\times\mathcal{X}$ is certainly also a continuous and bounded
symmetric positive definite kernel. Hence, the discussion above applies with
$\mathcal{Y}$ replaced by $\mathcal{X}$, which introduces the symbols
$\mathbb{H}_{\mathcal{X}}$, $\Phi_{\mathcal{X}}$, $S_{\mathcal{X}}$, and
$C_{\mathcal{X}}$. Note that condition (C1) for $\mathcal{Y}$ implies (C1) for
$\mathcal{X}$. However, the analogous implication does not apply to (C2),
which we explicitly assume here for $\mathcal{X}$:

  1. (C2X)

If $\psi\in\mathbb{H}_{\mathcal{X}}$ such that $\psi(x)=0$ for $\mu$-a.e.
$x\in\mathcal{X}$, then $\psi=0$ on $\mathcal{X}$.

Then all the statements above also hold for $\mathcal{Y}$ replaced by
$\mathcal{X}$. Again, (C2X) holds if the measure $\mu$ is absolutely
continuous with respect to Lebesgue measure with positive density. By [39,
Corollary 5.8],

$\mathbb{H}_{\mathcal{X}}=\\{\psi|_{\mathcal{X}}:\psi\in\mathbb{H}_{\mathcal{Y}}\\}$

and

$\|\psi\|_{\mathbb{H}_{\mathcal{X}}}=\inf\\{\|\eta\|_{\mathbb{H}_{\mathcal{Y}}}:\eta|_{\mathcal{X}}=\psi\\}.$

Covariance and cross-covariance operators. For any $x\in\mathcal{X}$ and
$y\in\mathcal{Y}$ define the rank-one operator
$\Phi(x)\otimes\Phi(y):\mathbb{H}_{\mathcal{Y}}\to\mathbb{H}_{\mathcal{X}}$ by

$[\Phi(x)\otimes\Phi(y)]\,\psi:=\langle\psi,\Phi(y)\rangle_{\mathbb{H}_{\mathcal{Y}}}\Phi(x)=\psi(y)\Phi(x),\quad\psi\in\mathbb{H}_{\mathcal{Y}}.$

Clearly, if $x\in\mathcal{X}$, then
$\Phi(x)\otimes\Phi(x):\mathbb{H}_{\mathcal{X}}\to\mathbb{H}_{\mathcal{X}}$.
The covariance operator $C_{\mathcal{X}}=S_{\mathcal{X}}^{*}S_{\mathcal{X}}$
then has the representation

$C_{\mathcal{X}}=\int_{\mathcal{X}}\Phi(x)\otimes\Phi(x)\,d\mu(x).$

For $t\geq 0$ and $\psi\in\mathbb{H}_{\mathcal{Y}}$ we further define the
cross-covariance operator
$C_{\mathcal{X}\mathcal{Y}}^{t}:\mathbb{H}_{\mathcal{Y}}\to\mathbb{H}_{\mathcal{X}}$
by

$C_{\mathcal{X}\mathcal{Y}}^{t}:=\int_{\mathcal{X}}\int_{\mathcal{Y}}\Phi(x)\otimes\Phi(y)\,\rho_{t}(x,dy)\,d\mu(x)=S_{\mathcal{X}}^{*}K^{t}S_{\mathcal{Y}}.$

As the product of the two Hilbert-Schmidt operators $S_{\mathcal{X}}^{*}K^{t}$
and $S_{\mathcal{Y}}$, the operator $C_{\mathcal{X}\mathcal{Y}}^{t}$ is trace
class for all $t\geq 0$ (cf. [19, p. 521]). Note that for $t=0$ we have
$C_{\mathcal{X}\mathcal{Y}}^{t}\psi=S_{\mathcal{X}}^{*}S_{\mathcal{X}}\psi|_{\mathcal{X}}=C_{\mathcal{X}}\psi|_{\mathcal{X}}$.
Moreover, for all $\eta\in\mathbb{H}_{\mathcal{X}}$ and
$\psi\in\mathbb{H}_{\mathcal{Y}}$ we have

$\displaystyle\langle\eta,C_{\mathcal{X}\mathcal{Y}}^{t}\psi\rangle_{\mathbb{H}_{\mathcal{X}}}=\langle\eta,K^{t}\psi\rangle_{2,\mathcal{X}}.$
(3.10)

###  3.5. Data samples and Kernel EDMD

Next, we briefly recall the Koopman approximation method called Extended
Dynamic Mode Decomposition (EDMD), where certain (usually continuous)
functions from $L^{2}(\mathcal{Y};\mu)$ are evaluated on a set of sampled
data.

######  Assumption 3.6.

The data $(x_{k},y_{k})$, $k=0,\ldots,m-1$, are drawn i.i.d. from the joint
distribution

$d\mu_{0,t}(x,y):=\rho_{t}(x,dy)\,d\mu(x)$

on $\mathcal{X}\times\mathcal{Y}$, where the time-step $t>0$ is fixed. This
then implies that the $x_{k}$ are sampled i.i.d. with respect to $\mu$ and
$y_{k}$ is drawn dependently of $x_{k}$ from the probability distribution
$\rho_{t}(x_{k},\,\cdot\,)$.

The EDMD Algorithm [56]. In EDMD, one chooses functions in
$L^{2}(\mathcal{Y};\mu)$ from a so-called dictionary
$\mathcal{D}=\\{\psi_{1},\ldots,\psi_{N}\\}$ spanning the finite-dimensional
space $\mathbb{V}=\operatorname{span}\\{\psi_{1},\ldots,\psi_{N}\\}\subset
L^{2}(\mathcal{Y};\mu)$ with the goal of approximating the compression
$P_{\mathbb{V}}K^{t}|_{\mathbb{V}}$ of $K^{t}$ to $\mathbb{V}$. Here,
$P_{\mathbb{V}}$ denotes the orthogonal projection onto the space $\mathbb{V}$
in $L^{2}(\mathcal{Y};\mu)$. The matrix representation of the compression
operator with respect to the basis $\mathcal{D}$ of $\mathbb{V}$ is then given
by $C^{-1}A$, where $C=(\langle\psi_{i},\psi_{j}\rangle_{\mu})_{i,j=1}^{N}$ is
the so-called mass matrix and
$A=(\langle\psi_{i},K^{t}\psi_{j}\rangle_{\mu})_{i,j=1}^{N}$ is called the
stiffness matrix. In order to define empirical estimators for $C$ and $A$, we
replace the scalar products in the matrices’ entries by sums of
$\psi_{i}(x_{k})\psi_{j}(x_{k})$ and $\psi_{i}(x_{k})\psi_{j}(y_{k})$,
respectively, which leads to the matrices

$\widehat{C}_{m}=\tfrac{1}{m}\Psi(X)\Psi(X)^{\top}\qquad\text{and}\qquad\widehat{A}_{m}=\tfrac{1}{m}\Psi(X)\Psi(Y)^{\top},$

where $\Psi=[\psi_{1},\ldots,\psi_{N}]^{\top}$ and

$\Psi(X)=[\Psi(x_{0}),\ldots,\Psi(x_{m-1})]\qquad\text{and}\qquad\Psi(Y)=[\Psi(y_{0}),\ldots,\Psi(y_{m-1})].$

Hence, the EDMD approximation of $P_{\mathbb{V}}K^{t}|_{\mathbb{V}}$ is given
by the estimator $\widehat{M}^{m,t}=\widehat{C}_{m}^{-1}\widehat{A}_{m}$. In
[37] it was proved that $\widehat{M}^{m,t}$ indeed converges in probability to
$C^{-1}A$ as $m\to\infty$, and bounds on the accuracy were provided.

The kEDMD Algorithm [57, 22]. In contrast to traditional EDMD, in kernel EDMD
(kEDMD) the dictionary is not chosen a priori, but is given by the features
sampled on the data points, that is,
$\mathcal{D}=\\{\Phi(x_{0}),\ldots,\Phi(x_{m-1})\\}$. The data matrices
$\Psi(X)$ and $\Psi(Y)$ are then easily seen to be given by

$\Psi(X)=\big{(}k(x_{i},x_{j})\big{)}_{i,j=0}^{m-1}\qquad\text{and}\qquad\Psi(Y)=\big{(}k(x_{i},y_{j})\big{)}_{i,j=0}^{m-1}.$

Hence, since $\Psi(X)$ is symmetric,

$\widehat{C}_{m}^{-1}\widehat{A}_{m}=\big{[}\tfrac{1}{m}\Psi(X)\Psi(X)^{\top}\big{]}^{-1}\big{[}\tfrac{1}{m}\Psi(X)\Psi(Y)^{\top}\big{]}=\Psi(X)^{-1}\Psi(Y)^{\top}.$

However, since the eigenvalues of $\Psi(X)$ typically decay fast, in practice
one often replaces the inverse of the ill-conditioned matrix $\Psi(X)$ by a
pseudo-inverse of a rank-$r$ ($r<m$) eigendecomposition truncation of
$\Psi(X)$. Hence, the kEDMD estimator for the Koopman operator compression
becomes

$\displaystyle\widehat{M}^{m,t}_{r}=[\Psi(X)]_{r}^{\dagger}\Psi(Y)^{\top}.$
(3.11)

Now, the empirical estimators for the operators $C_{\mathcal{X}}$ and
$C_{\mathcal{X}\mathcal{Y}}^{t}$ are given by

$\widehat{C}_{\mathcal{X}}^{m}=\frac{1}{m}\sum_{k=0}^{m-1}\Phi(x_{k})\otimes\Phi(x_{k})\qquad\text{and}\qquad\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}=\frac{1}{m}\sum_{k=0}^{m-1}\Phi(x_{k})\otimes\Phi(y_{k}).$

We have
$\widehat{C}_{\mathcal{X}}^{m}:\mathbb{H}_{\mathcal{X}}\to\mathbb{H}_{\mathcal{X}}$
and
$\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}:\mathbb{H}_{\mathcal{Y}}\to\mathbb{H}_{\mathcal{X}}$.
In fact, both operators map to
$\mathbb{V}=\operatorname{span}\mathcal{D}\subset\mathbb{H}_{\mathcal{X}}$.
Since

$\displaystyle\widehat{C}_{\mathcal{X}}^{m}\Phi(x_{j})$
$\displaystyle=\frac{1}{m}\sum_{k=0}^{m-1}[\Phi(x_{k})\otimes\Phi(x_{k})]\Phi(x_{j})=\frac{1}{m}\sum_{k=0}^{m-1}k(x_{j},x_{k})\Phi(x_{k}),$
$\displaystyle\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}\Phi(x_{j})$
$\displaystyle=\frac{1}{m}\sum_{k=0}^{m-1}[\Phi(x_{k})\otimes\Phi(y_{k})]\Phi(x_{j})=\frac{1}{m}\sum_{k=0}^{m-1}k(x_{j},y_{k})\Phi(x_{k}),$

the matrix representations of $\widehat{C}_{\mathcal{X}}^{m}|_{\mathbb{V}}$
and $\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}|_{\mathbb{V}}$ with respect to
the basis $\mathcal{D}=\\{\Phi(x_{0}),\ldots,\Phi(x_{m-1})\\}$ are given by
$\frac{1}{m}\Psi(X)$ and $\frac{1}{m}\Psi(Y)^{\top}$, respectively. Thus, the
kEDMD estimator matrix $\widehat{M}^{m,t}_{r}$ in (3.11) represents the
operator
$[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}|_{\mathbb{V}}$.
In the next section, we shall give probabilistic bounds on the
$\|\cdot\|_{\mathbb{H}_{\mathcal{Y}}\to L^{2}(\mathcal{X};\mu)}$-error between
$P_{r}K^{t}$ and

$\widehat{K}_{r}^{m,t}:=[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t},$
(3.12)

which then serves as a quantification of the prediction error made by kEDMD.
Here, $P_{r}$ denotes the orthogonal projection in $L^{2}(\mathcal{X};\mu)$
onto the span of eigenvectors corresponding to the largest $r$ eigenvalues of
$C_{\mathcal{X}}$.

##  4\. Bound on the Koopman prediction error

Before we present our main result, let us bound the error made by estimating
the operators $C_{\mathcal{X}}$ and $C_{\mathcal{X}\mathcal{Y}}$ by their
empirical estimators $\widehat{C}_{\mathcal{X}}^{m}$ and
$\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}$. Since $C_{\mathcal{X}}$ and
$C_{\mathcal{X}\mathcal{Y}}$ are Hilbert-Schmidt, we may quantify the error in
the Hilbert-Schmidt norm.

######  Proposition 4.1.

For every $t\geq 0$, the following probabilistic bounds on the Hilbert-Schmidt
estimation errors holds:

$\displaystyle\mathbb{P}\big{(}\|C_{\mathcal{X}\mathcal{Y}}^{t}-\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}\|_{HS}>\varepsilon\big{)}$
$\displaystyle\leq 2\,e^{-\frac{m\varepsilon^{2}}{8\|k\|_{\infty}^{2}}}$
$\displaystyle\mathbb{P}\big{(}\|C_{\mathcal{X}}-\widehat{C}_{\mathcal{X}}^{m}\|_{HS}>\varepsilon\big{)}$
$\displaystyle\leq 2\,e^{-\frac{m\varepsilon^{2}}{8\|k\|_{\infty}^{2}}}.$

###### Proof.

For $x\in\mathcal{X}$ and $y\in\mathcal{Y}$ let us abbreviate
$C_{x}:=\Phi(x)\otimes\Phi(x)$ and $C_{xy}:=\Phi(x)\otimes\Phi(y)$. If
$(f_{i})\subset\mathbb{H}_{\mathcal{Y}}$ denotes the Mercer ONB of
$\mathbb{H}_{\mathcal{Y}}$ corresponding to $k$ on
$\mathcal{Y}\times\mathcal{Y}$, for $x,x^{\prime}\in\mathcal{X}$ and
$y,y^{\prime}\in\mathcal{Y}$ we have

$\langle C_{xy},C_{x^{\prime}y^{\prime}}\rangle_{HS}=\sum_{i}\langle
C_{xy}f_{i},C_{x^{\prime}y^{\prime}}f_{i}\rangle_{\mathbb{H}_{\mathcal{X}}}=\sum_{i}f_{i}(y)f_{i}(y^{\prime})k(x,x^{\prime})=k(x,x^{\prime})k(y,y^{\prime}).$

This proves $\|C_{xy}\|_{HS}^{2}=\varphi(x)\varphi(y)$. Moreover, it yields

$\displaystyle\|C_{\mathcal{X}\mathcal{Y}}^{t}\|_{HS}^{2}$
$\displaystyle=\left\|\int C_{xy}\,d\mu_{0,t}(x,y)\right\|_{HS}^{2}=\int\int
k(x,x^{\prime})k(y,y^{\prime})\,d\mu_{0,t}(x,y)\,d\mu_{0,t}(x^{\prime},y^{\prime})\,\leq\,\|k\|_{\infty}^{2}.$

Hence, since the $C_{x_{k},y_{k}}$ are independent,
$\mathbb{E}[C_{\mathcal{X}\mathcal{Y}}^{t}-C_{x_{k},y_{k}}]=0$, and

$\|C_{\mathcal{X}\mathcal{Y}}^{t}-C_{x_{k},y_{k}}\|_{HS}\leq\|C_{\mathcal{X}\mathcal{Y}}^{t}\|_{HS}+\|C_{x_{k},y_{k}}\|_{HS}\leq
2\|k\|_{\infty},$

we may apply Hoeffding’s inequality for Hilbert space-valued random variables
[43, Theorem 3.5] (see also [32, Theorem A.5.2]) to
$C_{\mathcal{X}\mathcal{Y}}^{t}-C_{x_{k},y_{k}}$ and obtain

$\mathbb{P}\big{(}\|C_{\mathcal{X}\mathcal{Y}}^{t}-\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}\|_{HS}>\varepsilon\big{)}=\mathbb{P}\left(\left\|\frac{1}{m}\sum_{k=0}^{m-1}(C_{\mathcal{X}\mathcal{Y}}^{t}-C_{x_{k},y_{k}})\right\|_{HS}>\varepsilon\right)\,\leq\,2e^{-\frac{m\varepsilon^{2}}{8\|k\|_{\infty}^{2}}}.$

The proof of the second claim follows analogous lines. ∎

Let $(e_{j})$ be the Mercer orthonormal basis of $L^{2}(\mathcal{X};\mu)$
corresponding to the kernel $k$ and let
$\lambda_{j}=\|S_{\mathcal{X}}S_{\mathcal{X}}^{*}e_{j}\|_{2,\mathcal{X}}$ as
well as $f_{j}:=\sqrt{\lambda_{j}}e_{j}\in\mathbb{H}_{\mathcal{X}}$ (cf.
Theorem 3.5). We arrange the Mercer eigenvalues $\lambda_{j}$ of
$C_{\mathcal{X}}$ in a non-increasing way, i.e.,

$\lambda_{1}\geq\lambda_{2}\geq\ldots$

and denote by $P_{r}$ the orthogonal projection in $L^{2}(\mathcal{X};\mu)$
onto $\operatorname{span}\\{e_{1},\ldots,e_{r}\\}$.

######  Theorem 4.2.

Fix an arbitrary $r\in\mathbb{N}$ and assume that the first $r+1$ eigenvalues
$\lambda_{j}$ of $C_{\mathcal{X}}$ are simple, i.e.,
$\lambda_{j+1}<\lambda_{j}$ for all $j=1,\ldots,r$, and let

$\delta_{r}=\min_{j=1,\ldots,r}\frac{\lambda_{j}-\lambda_{j+1}}{2}\qquad\text{and}\quad
c_{r}=\frac{1}{\sqrt{\lambda_{r}}}+\frac{r+1}{\delta_{r}\lambda_{r}}(1+\|\varphi\|_{1,\mathcal{Y}})\|\varphi\|_{1,\mathcal{X}}^{1/2}.$
(4.1)

Further, let $\varepsilon\in(0,\delta_{r})$ and $\delta\in(0,1)$ be arbitrary,
fix some
$m\geq\max\\{r,\frac{8\|k\|_{\infty}^{2}\ln(4/\delta)}{\varepsilon^{2}}\\}$
and define $\widehat{K}_{r}^{m,t}$ as in (3.12). Then with probability at
least $1-\delta$, we have that

$\|P_{r}K^{t}-\widehat{K}_{r}^{m,t}\|_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X};\mu)}\,\leq\,c_{r}\varepsilon.$

If, furthermore,
$K^{t}\mathbb{H}_{\mathcal{Y}}\subset\mathbb{H}_{\mathcal{X}}$, then, with
probability at least $1-\delta$,

$\|K^{t}-\widehat{K}_{r}^{m,t}\|_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X};\mu)}\,\leq\,\sqrt{\lambda_{r+1}}\,\|K^{t}\|_{\mathbb{H}_{\mathcal{Y}}\to\mathbb{H}_{\mathcal{X}}}+c_{r}\varepsilon.$
(4.2)

###### Proof of Theorem 4.2.

Let $\widehat{\lambda}_{1}\geq\ldots\geq\widehat{\lambda}_{r}$ denote the
largest $r$ eigenvalues of $\widehat{C}_{\mathcal{X}}^{m}$ in descending order
and let $\widehat{e}_{1},\ldots,\widehat{e}_{r}$ be corresponding
eigenfunctions, respectively, such that
$\|\widehat{e}_{j}\|_{\mathbb{H}_{\mathcal{X}}}=\widehat{\lambda}_{j}^{-1/2}$
for $j=1,\ldots,m$. If we set
$\widehat{f}_{j}=\widehat{\lambda}_{j}^{1/2}\cdot\widehat{e}_{j}$, then for
$\psi\in\mathbb{H}_{\mathcal{Y}}$ we have

$\widehat{K}_{r}^{m,t}\psi=[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}\psi=\sum_{j=1}^{r}\widehat{\lambda}_{j}^{-1}\langle\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}\psi,\widehat{f}_{j}\rangle_{\mathbb{H}_{\mathcal{X}}}\widehat{f}_{j}=\sum_{j=1}^{r}\langle\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}\psi,\widehat{e}_{j}\rangle_{\mathbb{H}_{\mathcal{X}}}\widehat{e}_{j},$

which estimates (see (3.10))

$\sum_{j=1}^{r}\langle
C_{\mathcal{X}\mathcal{Y}}^{t}\psi,e_{j}\rangle_{\mathbb{H}_{\mathcal{X}}}e_{j}=\sum_{j=1}^{r}\langle
K^{t}\psi,e_{j}\rangle_{2,\mathcal{X}}e_{j}=P_{r}K^{t}\psi.$

Hence, we have to estimate the deviation of
$\sum_{j=1}^{r}\langle\widehat{C}_{\mathcal{X}\mathcal{Y}}^{m,t}\psi,\widehat{e}_{j}\rangle_{\mathbb{H}_{\mathcal{X}}}\widehat{e}_{j}$
from $\sum_{j=1}^{r}\langle
C_{\mathcal{X}\mathcal{Y}}^{t}\psi,e_{j}\rangle_{\mathbb{H}_{\mathcal{X}}}e_{j}$,
which was done in [42] for the special case of
$\mathcal{X}=\mathcal{Y}=\mathbb{R}^{d}$. The proof for the present case
follows analogous lines. Also the estimate in the case
$K^{t}\mathbb{H}_{\mathcal{Y}}\subset\mathbb{H}_{\mathcal{X}}$ can be proved
by following the lines of the proof of the counterpart in [42]. ∎

######  Remark 4.3.

The assumption $K^{t}\mathbb{H}_{\mathcal{Y}}\subset\mathbb{H}_{\mathcal{X}}$
is not too exotic. In fact, we prove in Section D in the Appendix that the
condition holds for the Ornstein-Uhlenbeck process, if
$\mathcal{Y}=\mathbb{R}$, $\mathcal{X}\subset\mathbb{R}$ is any compact set
with non-empty interior, and $k$ is a Gaussian RBF kernel.

##  5\. Extension to deterministic control-affine systems

In this part, we extend the error estimates to deterministic control systems
of the form

$\displaystyle\dot{x}(t)=f(x(t))+\sum_{i=1}^{{n_{u}}}u_{i}(t)g_{i}(x(t)),\qquad
t\in[0,T],$ (5.1)

with vector fields $f:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$g_{i}:\mathbb{R}^{d}\to\mathbb{R}^{d}$,
$i\in[1:{n_{u}}]:=\\{1,\ldots,{n_{u}}\\}$, ${n_{u}}\in\mathbb{N}$, and a
prediction horizon $T>0$. We let $\mathcal{X}$ and $\mathcal{Y}$ be compact
sets in $\mathbb{R}^{d}$ and choose specifically

$\mathrm{d}\mu(x)=\frac{1}{|\mathcal{X}|}\,\mathrm{d}x,$

where $|\mathcal{X}|$ denotes the Lebesgue measure of $\mathcal{X}$ on
$\mathbb{R}^{d}$. We furthermore assume that the restrictions of the vector
fields $f$ and $g_{1},\ldots,g_{n_{u}}$ to $\mathcal{Y}$ are $C^{1}$.
Admissible controls $u$ in (5.1) are measurable functions
$u:[0,T]\to\mathcal{U}$, where $\mathcal{U}\subset\mathbb{R}^{n_{u}}$ is a
compact, convex set satisfying $0\in\operatorname{int}(\mathcal{U})$. To avoid
excessive notation, we assume that for all $x^{0}\in\mathcal{X}$ and all
admissible controls $u:[0,T]\to\mathcal{U}$,

  * •

$[0,T]$ is contained in the existence interval for the solution
$x(\,\cdot\,;x^{0},u)$ to (5.1) and

  * •

$x(t;x^{0},u)\in\mathcal{Y}$ for each $t\in[0,T]$.

If the first assumption is satisfied and $\mathcal{X}$ is contained in the
interior of $\mathcal{Y}$, then by continuity of the flow in time and the
compactness of both $\mathcal{X}$ and $\mathcal{U}$, there exists a time
horizon $T_{0}$ such that the second assumption is satisfied for all
$T\in(0,T_{0})$.

Here, we consider sampled-data systems with Zero-order Hold (ZoH), i.e., for a
given, typically small sampling period $t>0$, $u(s)\equiv u$ on $[kt,(k+1)t)$,
$k\in\mathbb{N}_{0}$. This system class is well established in systems and
control, see, e.g., [8, 33] and the references therein. Sampled-data systems
can be used, e.g., for output-feedback stabilization of nonlinear systems [30]
or in the redesign of continuous-time control signals [17]. In particular,
there are many results showing that system-theoretic properties like
stabilizability, or even cost controllability, are preserved under sampling
[58]. The latter even is inherited by the EDMD-based surrogate model in the
Koopman framework as recently shown in [6] analyzing EDMD-based Model
Predictive Control. Hence, we also restrict our consideration to sampled-data
systems knowing that the analysis on one sampling period suffices to obtain
similar results based on the concatenation of the control signal for multiple
sampling periods.

For a constant control $u\in\mathcal{U}$ we let $K_{u}^{t}$ be the Koopman
operator for the autonomous system (5.1), that is,

$\displaystyle(K_{u}^{t}\psi)(x^{0}):=\psi\big{(}x(t;x^{0},u)\big{)},\qquad
x^{0}\in\mathcal{X},\,\psi\in L^{2}(\mathcal{Y}).$

In order to control the system by piecewise constant controls, in principle,
one could compute the kEDMD estimator for $K_{u}^{t}$ for every applied
control $u$, which, however, would require a lot of computation time in each
step. As this is infeasible in practice, we circumvent this issue by
approximating $K_{u}^{t}$ by a linear combination of ${n_{u}}+1$ Koopman
operators $K_{u_{i}}^{t}$ with quadratic precision. In this course, we
leverage the affine representation of the Koopman generator.

The Koopman generator $\mathcal{L}^{u}$ associated with (5.1) is given by

$\displaystyle\mathcal{L}^{u}\psi=\lim_{t\to
0}\frac{K_{u}^{t}\psi-\psi|_{\mathcal{X}}}{t}$ (5.2)

for $\psi\in\operatorname{dom}\mathcal{L}^{u}$, where the limit is taken in
$L^{2}(\mathcal{X})$, and $\operatorname{dom}\mathcal{L}^{u}$ consists of
those functions $\psi\in L^{2}(\mathcal{Y})$ for which this $L^{2}$-limit
exists. It is easy to see that
$C^{1}(\mathcal{Y})\subset\operatorname{dom}\mathcal{L}^{u}$ for each
$u\in\mathcal{U}$ such that for $\psi\in C^{1}(\mathcal{Y})$ and
$x^{0}\in\mathcal{X}$,

$\displaystyle(\mathcal{L}^{u}\psi)(x^{0})$ $\displaystyle=\lim_{t\to
0}\tfrac{1}{t}\big{(}(K_{u}^{t}\psi)(x^{0})-\psi(x^{0})\big{)}=\frac{\mathrm{d}}{\mathrm{d}t}(K_{u}^{t}\psi)(x^{0})\Big{|}_{t=0}=\frac{\mathrm{d}}{\mathrm{d}t}\psi(x(t;x^{0},u))\Big{|}_{t=0}$
$\displaystyle=\nabla\psi(x^{0})^{\top}\Big{(}f(x^{0})+\sum_{i=1}^{{n_{u}}}u_{i}g_{i}(x^{0})\Big{)}.$
(5.3)

Let $\\{e_{i}\\}_{i=1}^{{n_{u}}}$ denote the standard basis of
$\mathbb{R}^{n_{u}}$ and let

$\gamma_{i}:=\inf\\{\gamma>0:\gamma e_{i}\notin\mathcal{U}\\},\qquad
i=1,\ldots,{n_{u}}.$

Then the $\gamma_{i}$ are positive numbers thanks to
$0\in\operatorname{int}(\mathcal{U})$, and we have
$\gamma_{i}e_{i}\in\partial\mathcal{U}\subset\mathcal{U}$. Now, if
$u=\sum_{i=1}^{n_{u}}u_{i}e_{i}$ is the coordinate representation of the
vector $u\in\mathcal{U}$, the representation (5.3) reveals that, on
$C^{1}(\mathcal{Y})$,

$\displaystyle\mathcal{L}^{u}=\mathcal{L}^{0}+\sum_{i=1}^{{n_{u}}}\frac{u_{i}}{\gamma_{i}}\left(\mathcal{L}^{\gamma_{i}e_{i}}-\mathcal{L}^{0}\right).$
(5.4)

To transfer the error bounds from autonomous systems derived in this work to
control systems, we will show that the restriction of the Koopman operator to
$C^{2}(\mathcal{Y})$ is also affine in the control—at least approximately for
small $t>0$. For this, set

$f_{u}(x):=f(x)+\sum_{i=1}^{{n_{u}}}u_{i}g_{i}(x)$

and let $\psi\in C^{2}(\mathcal{Y})$, $u\in\mathcal{U}$, and
$x^{0}\in\mathcal{X}$. We apply Taylor’s theorem to $\psi\circ
x(\,\cdot\,;x^{0},u)$ at $t_{0}=0$ and observe that

$\displaystyle(K_{u}^{t}\psi-\psi|_{\mathcal{X}})(x^{0})$
$\displaystyle=\psi(x(t;x^{0},u))-\psi(x(0;x^{0},u))$
$\displaystyle=t\nabla\psi(x^{0})\cdot f_{u}(x^{0})+\frac{1}{2}r(\xi)t^{2},$

for some $\xi\in[0,t]$, where, abbreviating $x(s)=x(s;x^{0},u)$,

$\displaystyle
r(s)=f_{u}(x(s))^{\top}\nabla^{2}\psi(x(s))f_{u}(x(s))+\nabla\psi(x(s))^{\top}Df_{u}(x(s))f_{u}(x(s)).$

By compactness of $\mathcal{U}$, $\mathcal{X}$, and $\mathcal{Y}$ and the
continuity of all functions involved, we can bound the remainder term
uniformly in $u\in\mathcal{U}$, $x^{0}\in\mathcal{X}$ and $s\in[0,T]$. More
specifically,

$\displaystyle\max_{s\in[0,T]}|r(s)|\leq
c(f,g_{1},\ldots,g_{n_{u}})\cdot\|\psi\|_{C^{2}(\mathcal{Y})},$ (5.5)

where
$\|\psi\|_{C^{2}(\mathcal{Y})}:=\|\psi\|_{C(\mathcal{Y})}+\|\nabla\psi\|_{C(\mathcal{Y})^{d}}+\|\nabla^{2}\psi\|_{C(\mathcal{Y})^{d\times
d}}$. Thus,

$\displaystyle K_{u}^{t}\psi=\psi+t\mathcal{L}^{u}\psi+\mathcal{O}(t^{2})$
(5.6)

where the constant in $\mathcal{O}(t^{2})$ as defined in (5.5) can be chosen
independently of $u$.

Invoking control-affinity of the generator (5.4), we thus get

$\displaystyle
K_{u}^{t}\psi=K^{t}\psi+\sum_{i=1}^{{n_{u}}}\frac{u_{i}}{\gamma_{i}}\left(K^{t}_{\gamma_{i}e_{i}}-K^{t}\right)\psi+O(t^{2}),$
(5.7)

which means that the Koopman operator is control affine up to second order in
time. This is Equation (3.14) in [40], where we stress however that the
remainder term here depends on the chosen dictionary function.

The error bound (4.2) obtained in Theorem 4.2 can now be transferred to the
controlled case. Thus, for $u\in\mathcal{U}$ we define the estimator

$\displaystyle\widehat{K}^{m,t}_{r,u}:=\widehat{K}^{m,t}_{r}+\sum_{i=1}^{{n_{u}}}\frac{u_{i}}{\gamma_{i}}\big{(}\widehat{K}^{m,t}_{r,\gamma_{i}e_{i}}-\widehat{K}^{m,t}_{r}\big{)},$

where the $\widehat{K}_{r,\gamma_{i}e_{i}}^{m,t}$ are defined as in (3.12) for
the system $\dot{x}=f(x)+\gamma_{i}g_{i}(x)$.

######  Theorem 5.1.

Let the assumptions in Theorem 4.2 be satisfied. For fixed $r\in\mathbb{N}$,
let $\delta_{r}$ and $c_{r}$ be as in (4.1). Further, let
$\varepsilon\in(0,\delta_{r})$ and $\delta\in(0,1)$ be arbitrary and fix some
$m\geq\max\\{r,\frac{8\|k\|_{\infty}^{2}\ln(4(n_{u}+1)/\delta)}{\varepsilon^{2}}\\}$.
Then, for each $\psi\in\mathbb{H}_{\mathcal{Y}}\cap C^{2}(\mathcal{X})$, there
is a constant $c_{\psi}\geq 0$ such that with probability at least $1-\delta$
for all $u\in\mathcal{U}$ we have

$\displaystyle\|P_{r}K_{u}^{t}\psi-\widehat{K}_{r,u}^{m,t}\psi\|_{L^{2}(\mathcal{X};\mu)}\,\leq\,(1+2\|\Gamma^{-1}u\|_{1})c_{r}\varepsilon\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}+c_{\psi}t^{2}.$
(5.8)

where $\Gamma=\operatorname{diag}_{i=1,\ldots,n_{u}}\gamma_{i}$. In the case
where $K_{u}^{t}\mathbb{H}_{\mathcal{Y}}\subset\mathbb{H}_{\mathcal{X}}$ for
all
$u\in\mathcal{U}_{\partial}:=\\{0,\gamma_{1}e_{1},\ldots,\gamma_{n_{u}}e_{n_{u}}\\}$,
we obtain

$\displaystyle\|K_{u}^{t}\psi-\widehat{K}_{r,u}^{m,t}\psi\|_{L^{2}(\mathcal{X};\mu)}\,\leq\,(1+2\|\Gamma^{-1}u\|_{1})(\sqrt{\lambda_{r+1}}M+c_{r}\varepsilon)\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}+c_{\psi}t^{2},$
(5.9)

where
$M=\max\\{\|K_{u}^{t}\|_{\mathbb{H}_{\mathcal{Y}}\to\mathbb{H}_{\mathcal{X}}}:u\in\mathcal{U}_{\partial}\\}$.

If the kernel $k$ enjoys $C^{2}$-regularity, for
$m\geq\max\\{r,\frac{8\|k\|_{\infty}^{2}\ln(4/\delta)}{\varepsilon^{2}}\\}$
data points we have with probability at least $1-\delta$ that

$\|P_{r}K_{u}^{t}-\widehat{K}_{r,u}^{m,t}\|_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X};\mu)}\,\leq\,c_{r}\varepsilon+t^{2}\cdot
C_{k}\widehat{\lambda}_{r}^{-1}$

with a constant $C_{k}$ depending only on the kernel $k$. Here,
$\widehat{\lambda}_{r}$ denotes the $r$-th largest eigenvalue of the operator
$\widehat{C}_{\mathcal{X}}^{m}$. Again, if
$K_{u}^{t}\mathbb{H}_{\mathcal{Y}}\subset\mathbb{H}_{\mathcal{X}}$ for all
$u\in\mathcal{U}_{\partial}$, then

$\|K_{u}^{t}-\widehat{K}_{r,u}^{m,t}\|_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X};\mu)}\,\leq\,\sqrt{\lambda_{r+1}}M+c_{r}\varepsilon+t^{2}\cdot
C_{k}\widehat{\lambda}_{r}^{-1}.$

###### Proof.

Using the approximate control-affinity (5.7), we compute

$\displaystyle(P_{r}K_{u}^{t}-\widehat{K}_{r,u}^{m,t})\psi=(P_{r}K^{t}-\widehat{K}_{r}^{m,t})\psi+\sum_{i=1}^{n_{u}}\frac{u_{i}}{\gamma_{i}}\left(P_{r}K^{t}_{\gamma_{i}e_{i}}-\widehat{K}^{m,t}_{r,\gamma_{i}e_{i}}-P_{r}K^{t}+\widehat{K}^{m,t}_{r}\right)\psi+t^{2}\phi,$

where $\|\phi\|_{L^{\infty}(\mathcal{X})}\leq c_{\psi}$ for all choices of
$u\in\mathcal{U}$ and $t\in[0,T]$. By the choice of $m$, the respective bounds
in Theorem 4.2 on $\|P_{r}K^{t}-\widehat{K}_{r}^{m,t}\|$ and
$\|P_{r}K^{t}_{\gamma_{i}e_{i}}-\widehat{K}^{m,t}_{r,\gamma_{i}e_{i}}\|$,
$i\in\\{1,\ldots,{n_{u}}\\}$, hold with probability at least
$1-\delta(n_{u}+1)^{-1}$ each. Hence, the probability for all $n_{u}+1$ bounds
to hold exceeds $1-\delta$, cf. [37, Lemma 22]. In that event,

$\displaystyle\Big{\|}(P_{r}K^{t}-\widehat{K}_{r}^{m,t})\psi+\sum_{i=1}^{n_{u}}\frac{u_{i}}{\gamma_{i}}\left(P_{r}K^{t}_{\gamma_{i}e_{i}}-\widehat{K}^{m,t}_{r,\gamma_{i}e_{i}}-P_{r}K^{t}+\widehat{K}^{m,t}_{r}\right)\psi\Big{\|}_{L^{2}(\mathcal{X})}$
$\displaystyle\leq\|P_{r}K^{t}-\widehat{K}_{r}^{m,t}\|_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X})}\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}$
$\displaystyle\qquad+\Big{(}\max_{i\in\\{1,\ldots,{n_{u}}\\}}\big{\|}P_{r}K^{t}_{\gamma_{i}e_{i}}-\widehat{K}^{m,t}_{r,\gamma_{i}e_{i}}\big{\|}_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X})}+\|P_{r}K^{t}-\widehat{K}_{r}^{m,t}\|_{\mathbb{H}_{\mathcal{Y}}\to
L^{2}(\mathcal{X})}\Big{)}\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}\sum_{i=1}^{n_{u}}\frac{|u_{i}|}{\gamma_{i}}$
$\displaystyle\leq\big{(}1+2\|\Gamma^{-1}u\|_{1}\big{)}c_{r}\varepsilon\|\psi\|_{\mathbb{H}_{\mathcal{Y}}},$

which proves the given estimate. The proof for the case
$K_{u}^{t}\mathbb{H}_{\mathcal{Y}}\subset\mathbb{H}_{\mathcal{X}}$ for all
$u\in\mathcal{U}_{\partial}$ follows analogous lines, making use of (4.2).

Now, let $k\in C^{2}$ and denote the flow of $\dot{x}=f_{v}(x)$ by $F_{v}$,
$v\in\mathcal{U}$. Also, let $F:=F_{0}$. In addition to the random variables
$y_{k}=F^{t}(x_{k})$, $k=0,\ldots,m-1$, let $y_{k,v}=F_{v}^{t}(x_{k})$,
$i=1,\ldots,{n_{u}}$. Moreover, let
$\widehat{C}^{m,t}_{\mathcal{X}\mathcal{Y},v}$ be the empirical estimator of
the cross-covariance operator $C^{t}_{\mathcal{X}\mathcal{Y},v}$ for the
system $\dot{x}=f_{v}(x)$. Then we have

$\displaystyle\widehat{K}^{m,t}_{r,u}$
$\displaystyle=\widehat{K}^{m,t}_{r}+\sum_{i=1}^{{n_{u}}}\frac{u_{i}}{\gamma_{i}}\big{(}\widehat{K}^{m,t}_{r,\gamma_{i}e_{i}}-\widehat{K}^{m,t}_{r}\big{)}=[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\Big{[}\widehat{C}^{m,t}_{\mathcal{X}\mathcal{Y}}+\sum_{i=1}^{{n_{u}}}\frac{u_{i}}{\gamma_{i}}\big{(}\widehat{C}^{m,t}_{\mathcal{X}\mathcal{Y},\gamma_{i}e_{i}}-\widehat{C}^{m,t}_{\mathcal{X}\mathcal{Y}}\big{)}\Big{]}$
$\displaystyle=[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\,\frac{1}{m}\sum_{k=0}^{m-1}\Phi(x_{k})\otimes\Bigg{[}\Phi(F^{t}(x_{k}))+\sum_{i=1}^{{n_{u}}}\frac{u_{i}}{\gamma_{i}}\big{(}\Phi(F_{\gamma_{i}e_{i}}^{t}(x_{k}))-\Phi(F^{t}(x_{k}))\big{)}\Bigg{]}$
$\displaystyle=[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\,\frac{1}{m}\sum_{k=0}^{m-1}\Phi(x_{k})\otimes\Bigg{[}\mathbb{K}^{t}\Phi(x_{k})+\sum_{i=1}^{{n_{u}}}\frac{u_{i}}{\gamma_{i}}\big{(}\mathbb{K}_{\gamma_{i}e_{i}}^{t}\Phi(x_{k})-\mathbb{K}^{t}\Phi(x_{k})\big{)}\Bigg{]},$

where $\mathbb{K}_{v}^{t}:L^{2}(\mathcal{Y},\mathbb{H}_{\mathcal{Y}})\to
L^{2}(\mathcal{X},\mathbb{H}_{\mathcal{Y}})$ for $v\in\mathcal{U}$ is defined
by $\mathbb{K}_{v}^{t}\Psi(x)=\Psi(F_{v}^{t}(x))$, $\Psi\in
L^{2}(\mathcal{Y},\mathbb{H}_{\mathcal{Y}})$, and
$\mathbb{K}^{t}:=\mathbb{K}^{t}_{0}$. As $\Phi\in C^{2}$, we can, analogously
as in the discussion preceding the theorem, see that

$\mathbb{K}^{t}\Phi+\sum_{i=1}^{{n_{u}}}\frac{u_{i}}{\gamma_{i}}\left(\mathbb{K}^{t}_{\gamma_{i}e_{i}}-\mathbb{K}^{t}\right)\Phi=\mathbb{K}_{u}^{t}\Phi+t^{2}\Psi_{u,t}$

with some $\Psi_{u,t}\in L^{2}(\mathcal{X},\mathbb{H}_{\mathcal{Y}})$ such
that
$C_{\Phi}=\sup\\{\|\Psi_{u,t}\|_{L^{\infty}(\mathcal{X},\mathbb{H}_{\mathcal{Y}})}:u\in\mathcal{U},\,t\in[0,T]\\}<\infty$.
Hence, we obtain

$\displaystyle\widehat{K}^{m,t}_{r,u}$
$\displaystyle=[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\,\frac{1}{m}\sum_{k=0}^{m-1}\Phi(x_{k})\otimes\big{[}\Phi(F_{u}^{t}(x_{k}))+t^{2}\Psi_{u,t}(x_{k})\big{]}$
$\displaystyle=[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\Big{[}\widehat{C}^{m,t}_{\mathcal{X}\mathcal{Y},u}+\frac{t^{2}}{m}\sum_{k=0}^{m-1}\Phi(x_{k})\otimes\Psi_{u,t}(x_{k})\Big{]}.$

Note that
$[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\widehat{C}^{m,t}_{\mathcal{X}\mathcal{Y},u}$
is the estimator for $K^{t}_{u}$ as defined in (3.12). Here, we shall denote
it by $\widetilde{K}^{m,t}_{r,u}$. Now, if $\psi\in\mathbb{H}_{\mathcal{Y}}$,
then

$\displaystyle\|P_{r}K^{t}\psi-\widehat{K}^{m,t}_{r,u}\psi\|_{L^{2}(\mathcal{X})}$
$\displaystyle\leq\|P_{r}K^{t}\psi-\widetilde{K}^{m,t}_{r,u}\psi\|_{L^{2}(\mathcal{X})}+\|\widetilde{K}^{m,t}_{r,u}\psi-\widehat{K}^{m,t}_{r,u}\psi\|_{L^{2}(\mathcal{X})}$
$\displaystyle\leq
c_{r}\varepsilon\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}+\frac{t^{2}}{m}\Bigg{\|}\sum_{k=0}^{m-1}[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}[\Phi(x_{k})\otimes\Psi_{u,t}(x_{k})]\psi\Bigg{\|}_{L^{2}(\mathcal{X})}$
$\displaystyle\leq
c_{r}\varepsilon\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}+\frac{t^{2}}{m}\sum_{k=0}^{m-1}|\langle\psi,\Psi_{u,t}(x_{k})\rangle_{\mathbb{H}_{\mathcal{Y}}}|\cdot\|[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\Phi(x_{k})\|_{L^{2}(\mathcal{X})}$
$\displaystyle\leq
c_{r}\varepsilon\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}+\|\varphi\|_{1,\mathcal{X}}^{1/2}\cdot\frac{t^{2}}{m}\sum_{k=0}^{m-1}|\langle\psi,\Psi_{u,t}(x_{k})\rangle_{\mathbb{H}_{\mathcal{Y}}}|\|[\widehat{C}_{\mathcal{X}}^{m}]_{r}^{\dagger}\Phi(x_{k})\|_{\mathbb{H}_{\mathcal{X}}}$
$\displaystyle\leq
c_{r}\varepsilon\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}+\|k\|_{\infty}^{1/2}\cdot\frac{t^{2}}{m}\sum_{k=0}^{m-1}\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}\|\Psi_{u,t}(x_{k})\|_{\mathbb{H}_{\mathcal{Y}}}\widehat{\lambda}_{r}^{-1}\sqrt{\varphi(x_{k})}$
$\displaystyle\leq
c_{r}\varepsilon\|\psi\|_{\mathbb{H}_{\mathcal{Y}}}+\widehat{\lambda}_{r}^{-1}C_{\Phi}\|k\|_{\infty}\cdot
t^{2}\|\psi\|_{\mathbb{H}_{\mathcal{Y}}},$

which proves the claim with $C_{k}=C_{\Phi}\|k\|_{\infty}$. ∎

##  6\. Numerical results

In this part, we present two numerical examples. First we consider an
autonomous stochastic system to illustrate the results of Section 4. Second,
we present numerical results for a controlled deterministic system to showcase
the bounds deduced in Section 5.

###  6.1. Estimation Error for Ornstein-Uhlenbeck Process

We consider a one-dimensional Ornstein-Uhlenbeck (OU) process obeying the
stochastic differential equation

$dX_{t}=-\alpha X_{t}+\sqrt{2\beta^{-1}}dW_{t},$ (6.1)

with $\alpha=1$ and $\beta=2$ in the numerical tests. The initial domain is
chosen as $\mathcal{X}=[-1.5,1.5]$, the final domain is simply the entire real
line $\mathcal{Y}=\mathbb{R}$. The measure $\mu$ is chosen as the invariant
distribution of the OU process on $\mathcal{Y}$, which is normal with mean
zero and standard deviation $(\alpha\beta)^{-1/2}$. We then use the
restriction of $\mu$ to the finite interval $[-1.5,1.5]$ to generate initial
conditions for the SDE (6.1).

Mirroring the analysis in [42], we consider the approximation of the cross-
covariance operator $C^{t}_{\mathcal{X}\mathcal{Y}}$ at lag time $t=0.05$,
corresponding to five elementary time steps at integration time step
$\Delta_{t}=0.01$. We use the Gaussian radial basis function kernel

$k(x,y)=\exp\Big{[}-\frac{(x-y)^{2}}{\sigma^{2}}\Big{]},$

for bandwidths $\sigma\in\\{0.05,0.1,0.5\\}$. Setting the confidence level to
$1-\delta=0.9$, we use Proposition 4.1 to bound the minimal guaranteed error
that is achieved with this confidence, for a range of data sizes $m$ between
$m=20$ and $m=5\cdot 10^{4}$. As a comparison, we estimate the actual incurred
error for fifty independent collections of $m$ pairs $(x_{k},y_{k})$ each, and
plot the $(1-\delta$)-percentile for this error as a function of $m$. To do
so, we first observe that the Hilbert-Schmidt norm of the empirical covariance
operator $\widehat{C}^{t,m}_{\mathcal{X}\mathcal{Y}}$ can be calculated using
the Mercer features and eigenvalues of the Gaussian kernel:

$\|\widehat{C}^{t,m}_{\mathcal{X}\mathcal{Y}}\|^{2}_{\mathrm{HS}}=\sum_{i,j=1}^{\infty}\lambda_{i}\lambda_{j}(M^{m}_{ij})^{2},$
(6.2)

using the following matrix elements for the Mercer features:

$M^{m}_{ij}:=\frac{1}{m}\sum_{l=1}^{m}e_{i}(x_{l})e_{j}(y_{l}).$ (6.3)

In practice, we truncate the infinite sum in (6.2) after
$i_{\max}=j_{\max}=30$ terms. We then first generate ten large reference
collections $(x_{l},y_{l}),1\leq l\leq m_{\max}$, where $m_{\max}=10^{7}$, and
average the matrix elements $M^{m_{\max}}_{ij}$ over these large reference
samples. The actual error for $m<m_{\max}$ data pairs $(x_{k},y_{k}),1\leq
k\leq m$ is then estimated by the formula

$\|C^{t}_{\mathcal{X}\mathcal{Y}}-\widehat{C}^{t,m}_{\mathcal{X}\mathcal{Y}}\|^{2}_{\mathrm{HS}}\approx\|\widehat{C}^{t,m_{\max}}_{\mathcal{X}\mathcal{Y}}-\widehat{C}^{t,m}_{\mathcal{X}\mathcal{Y}}\|^{2}_{\mathrm{HS}}=\|\widehat{C}^{t,m_{\max}}_{\mathcal{X}\mathcal{Y}}\|^{2}_{\mathrm{HS}}-2\left\langle\widehat{C}^{t,m_{\max}}_{\mathcal{X}\mathcal{Y}},\,\widehat{C}^{t,m}_{\mathcal{X}\mathcal{Y}}\right\rangle_{\mathrm{HS}}+\|\widehat{C}^{t,m}_{\mathcal{X}\mathcal{Y}}\|^{2}_{\mathrm{HS}}.$

The Hilbert-Schmidt inner product can be computed by the formula

$\left\langle\widehat{C}^{t,m_{\max}}_{\mathcal{X}\mathcal{Y}},\,\widehat{C}^{t,m}_{\mathcal{X}\mathcal{Y}}\right\rangle_{\mathrm{HS}}=\sum_{i,j=1}^{\infty}\lambda_{i}\lambda_{j}M^{\max}_{ij}M^{m}_{ij}.$

The results are shown in Figure 1. On average, the bounds from Proposition 4.1
over-estimate the actual incurred error by about one order of magnitude.

Figure 1. Ornstein-Uhlenbeck process: Error bound from Prop. 4.1 (circles)
versus $(1-\delta)$-percentile of the actual estimation error estimated from
fifty independent trials (squares), for three different bandwidths $\sigma$
indicated by different colors and linestyles.

###  6.2. Duffing Oscillator

To conclude this study, we illustrate that kernel EDMD, combined with state-
of-the art model validation and low-rank approximation techniques, enables
accurate prediction of a non-linear control system with very few necessary
design choices on the modeler’s end. We study the two-dimensional Duffing
oscillator, governed by the ordinary differential equation

$\frac{\mathrm{d}}{\mathrm{d}t}\begin{bmatrix}z_{1}(t)\\\
z_{2}(t)\end{bmatrix}=\begin{bmatrix}z_{2}\\\ -\alpha z_{1}u-2\beta
z_{1}^{3}\end{bmatrix},$ (6.4)

which has been used as a model system in numerous previous studies. We set
$\alpha=-1$, $\beta=1$. By $z(t;z^{0},u)$, we denote the state at time $t\geq
0$ emanating from the initial condition $z^{0}$ at time zero and the control
$u:\mathbb{R}^{\geq 0}\to\mathbb{R}$. We generate $m=10^{4}$ uniformly sampled
initial conditions $\\{x_{k}\\}_{k=1}^{m}$ from the square
$\mathcal{X}=[-1.5,1.5]^{2}$. Next, for fixed control inputs
$\bar{u}\in\\{0,1\\}$, we compute the successor state
$y_{k}^{\bar{u}}=z(t;x_{k},\bar{u}),k=1,\ldots,m$ for $t=0.025$ via numerical
integration with time step $\Delta_{t}=0.005$. The data pairs
$\\{(x_{k},y_{k}^{\bar{u}})\\}_{k=1}^{m}$ serve as training data for the kEDMD
algorithm corresponding to the flow with constant control
$\bar{u}\in\\{0,1\\}$. As shown in Figure 2, the time-shifted points
$y_{k}^{\bar{u}}$ can indeed exit the initial domain $\mathcal{X}$, such that
invariance of $\mathcal{X}$ is not given. This in particular underlines the
necessity of incorporating an additional set $\mathcal{Y}$ as done in the
previous analysis.

Figure 2. Left: initial training data $x_{k}$ (blue) and corresponding time
shifted data $y_{k}$ (red) for fixed input $\bar{u}=0$. The black square
indicates the initial domain $\mathcal{X}=[-1.5,1.5]^{2}$. Right: Evaluation
of the state variable $x_{k}$ on the data $y_{k}^{\bar{u}}$, approximated in
terms of the RFF basis at $\sigma=1.0$ and $p=500$.

As reproducing kernel, we again select the Gaussian RBF kernel. To reduce the
computational effort, we approximate the resulting feature maps using random
Fourier features (RFFs) [44]. It was shown in [36] that this is equivalent to
applying plain EDMD with a randomly selected basis set of complex exponentials

$\psi_{i}(x)=e^{i\omega_{i}^{\top}x},\quad\ 1\leq i\leq p\in\mathbb{N},$

where $\omega_{i}\in\mathbb{R}^{d},\,1\leq i\leq p$, are random frequencies
drawn from the spectral measure associated with the kernel. In our case, the
spectral measure is a normal distribution with standard deviation
$\sigma^{-1}$, i.e. the inverse of the kernel bandwidth. Using the feature
matrices $\mathbf{M},\mathbf{M}_{t}^{\bar{u}}\in\mathbb{C}^{p\times m}$
defined by

$\displaystyle\mathbf{M}_{i,k}=e^{i\omega_{i}^{\top}x_{k}},\quad(\mathbf{M}_{t}^{\bar{u}})_{i,k}=e^{i\omega_{i}^{\top}y_{k}^{\bar{u}}},\quad
1\leq i\leq p,\ 1\leq k\leq m,$

the RFF Koopman matrix corresponding to the flow with constant control
$\bar{u}\in\\{0,1\\}$ amounts to

$\mathbf{K}_{\bar{u}}=\left(\mathbf{M}\mathbf{M}^{\rm H}+\gamma{\rm
Id}\right)^{-1}\mathbf{M}(\mathbf{M}_{t}^{\bar{u}})^{\rm H},$

where $\gamma>0$ is a regularization parameter.

We begin by tuning the bandwidth $\sigma>0$, the feature size
$p\in\mathbb{N}$, and the tolerance $\gamma>0$. To this end, we draw a test
ensemble of $100$ random initial conditions, from which we generate
simulations of the ODE (6.4) until time $T_{\rm test}=20t=0.5$. We also use
the learned Koopman models to predict the system state over the same time
horizon, starting from the same initial conditions, and compute the relative
mean-squared error between the true and predicted states over all time steps.
Note that state prediction with the Koopman operator requires an approximation
of the state variables $z_{1}$ and $z_{2}$ as linear combinations of the RFF
basis functions $\psi_{i}$. These approximations are computed by a least-
squares fit along with each Koopman model. An example is shown in Figure 2.

The results of the model validation are shown for different inputs
$u:[0,T_{\mathrm{test}}]\to\mathbb{R}$ and different hyper-parameters in
Figure 3. We conclude that the prediction error generally decreases with
regularization if $\gamma\geq 10^{-5}$, but seems to destabilize if the
regularization is decreased further. We therefore identify $\sigma=1.0$,
$p=500$ and $\gamma=10^{-5}$ as suitable model parameters.

Figure 3. Duffing oscillator: Mean-squared prediction error of the trained
Koopman model for $\bar{u}=0.0$ (left) and $\bar{u}=1.0$ (right), tested on
$100$ new trajectories of length $T_{\rm test}=20t=0.5$. The error is shown as
a function of the regularization strength $\gamma>0$ for different feature
sizes $p\in\mathbb{N}$ (dots, x-es, squares) and bandwiths
$\sigma\in\\{0.5,1.0,1.5\\}$ (blue, green, red).

We then evaluate the long-time prediction performance of the bi-linear Koopman
surrogate model

$\mathbf{K}_{u(t)}\mathbf{w}=\left[\mathbf{K}_{0}+(\mathbf{K}_{1}-\mathbf{K}_{0})u(t)\right]\mathbf{w}$
(6.5)

for the lifted state $\mathbf{w}\in\mathbb{C}^{p}$, generating $50$
trajectories for (a) constant control input $u\equiv 0$; (b) constant control
input $u\equiv 1$; (c) time-dependent control input $u(t)=\cos(t)$. To obtain
predictions corresponding to the real-valued system (6.4), we project the
propagated complex-valued lifted state onto its real part. The time horizon
covered by these trajectories is $T_{\rm long}=500t=12.5$, and we use the
project-and-lift scheme [15] every 25 steps to improve performance. Figure 4
shows the mean relative error over the real part of these trajectories as a
function of simulation time. We also show exemplary trajectories for each of
the three scenarios in the same figure, and observe excellent performance of
the kernel-based bi-linear Koopman model.

Figure 4. Upper left: Mean-squared prediction error for the Duffing oscillator
predicted by the bi-linear Koopman model (6.5) for different control inputs,
evaluated on $50$ new trajectories of length $T_{\rm long}=500t=12.5$. The
black dashed line indicates the time horizon $T_{\rm test}$ used for model
validation in Figure 3. The remaining panels show exemplary trajectories of
the same length $T_{\rm long}$ for different control inputs. Red dots mark the
predicted solution, the true ODE solution is shown in black. The initial value
is highlighted in green.

##  7\. Supplements

Acknowledgments. F. Philipp was funded by the Carl Zeiss Foundation within the
project DeepTurb–Deep Learning in and from Turbulence. He was further
supported by the free state of Thuringia and the German Federal Ministry of
Education and Research (BMBF) within the project THInKI–Thüringer
Hochschulinitiative für KI im Studium. K. Worthmann gratefully acknowledges
funding by the German Research Foundation (DFG grant WO 2056/14-1, project no
507037103).

Data Availability. Codes and data required for reproducing the numerical
results in this study are available at
https://doi.org/10.5281/zenodo.10378153.

## References

  * [1] H. Amann, Ordinary differential equations, An introduction to nonlinear analysis, de Gruyter, Berlin, New York, 1990.
  * [2] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.
  * [3] D. Bakry, I. Gentil, and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren der mathematischen Wissenschaften, Springer International Publishing Switzerland, 2014.
  * [4] A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer Academic Publishers, 2004.
  * [5] P. Bevanda, M. Beier, A. Lederer, S. Sosnowski, E. Hüllermeier, S. Hirche, Koopman kernel regression, Preprint, arXiv:2305.16215.
  * [6] L. Bold, L. Grüne, M. Schaller, and K. Worthmann, Practical asymptotic stability of data-driven model predictive control using extended DMD, preprint arXiv:2308.00296 (2023).
  * [7] S. L. Brunton, M. Budisic, E. Kaiser, and J. N. Kutz, Modern Koopman theory for dynamical systems, SIAM Rev. 64(2) (2022), 229–340.
  * [8] B. Castillo, S. Di Gennaro, S. Monaco, and D. Normand-Cyrot, On regulation under sampling, IEEE Transactions on Automatic Control 42 (1997), 864–868.
  * [9] Y. Chen and U. Vaidya, Sample complexity for non-linear stochastic systems, 2019 American Control Conference (ACC), Philadelphia, USA, July 10–12, 2019, pp. 3526–3531.
  * [10] J.-M. Coron and L. Grüne and K. Worthmann, Model predictive control, cost controllability, and homogeneity, SIAM Journal on Control and Optimization 58(5) (2020), 2979–2996.
  * [11] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Note Series, Cambridge University Press, 1996.
  * [12] S. Das, D. Giannakis, and J. Slawinska, Reproducing kernel Hilbert space compactification of unitary evolution groups, Appl. Comput. Harmon. Anal. 54 (2021), 75–136.
  * [13] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Taylor & Francis Group, CRC Press, 2015.
  * [14] I.C. Gohberg and M.G. Krein, Introduction To The Theory of Linear Nonselfadjoint Operators, volume 18 of Translations of Mathematical Monographs. American Mathematical Society, 1969.
  * [15] P. van Goor, R. Mahony, M. Schaller, K. Worthmann, Reprojection methods for Koopman-based modelling and prediction, preprint arXiv:2307.16188, 2023.
  * [16] D. Goswami and D.A. Paley, Bilinearization, reachability, and optimal control of control-affine nonlinear systems: A Koopman spectral approach, IEEE Transactions on Automatic Control 67(6) (2021), 2715–2728.
  * [17] L. Grüne and K. Worthmann, Sampled-data redesign for nonlinear multi-input systems, Geometric Control And Nonsmooth Analysis: In Honor of the 73rd Birthday of H. Hermes and of the 71st Birthday of R.T. Rockafellar, World Scientific (2008), 206–227.
  * [18] E. Kaiser, J.N. Kutz, and S.L. Brunton, Data-driven discovery of Koopman eigenfunctions for control, Mach. Learn. Sci. Technol. 2 (2021), 035023.
  * [19] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin Heidelberg, 1995.
  * [20] S. Klus, P. Koltai, and C. Schütte, On the numerical approximation of the Perron–Frobenius and Koopman operator, J. Comput. Dyn. 1 (2016), 51–79.
  * [21] S. Klus, F. Nüske, P. Koltai, H. Wu, I. Kevrekidis, C. Schütte, and F. Noé, Data-driven model reduction and transfer operator approximation, J. Nonlinear Sci. 28(3) (2018), 985–1010.
  * [22] S. Klus, I. Schuster, K. Muandet, Eigendecompositions of transfer operators in reproducing kernel Hilbert spaces, J. Nonlinear Sci. 30 (2020), 283–315.
  * [23] B.O. Koopman, Hamiltonian Systems and transformations in Hilbert space, Proc. Natl. Acad. Sci. 5 (1931), 315–318.
  * [24] M. Korda and I. Mezić, Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control, Automatica 93 (2018), 149–160.
  * [25] V. Kostic, P. Novelli, A. Maurer, C. Ciliberto, L. Rosasco, and M. Pontil, Learning dynamical systems via Koopman operator regression in reproducing kernel Hilbert spaces, Adv. Neural. Inf. Process. Syst. 35 (2022), 4017-4031.
  * [26] V. Kostic, K. Lounici, P. Novelli, and M. Pontil, Sharp spectral rates for Koopman operator learning, Preprint, arXiv:2302.02004, 2023.
  * [27] V. Kühner, What can Koopmanism do for attractors in dynamical systems?, The Journal of Analysis, 29 (2021), 449–471.
  * [28] T. Lelièvre, M. Rousset and G. Stoltz, Free Energy Computations, Impertial College Press (2010)
  * [29] T. Lelièvre and G. Stoltz, Partial differential equations and stochastic methods in molecular dynamics, Acta Numer. 25 (2016), 681–880.
  * [30] Z. Li and J. Zhao, Output feedback stabilization for a general class of nonlinear systems via sampled-data control, International Journal of Robust and Nonlinear Control 28:7 (2018), 2853—2867.
  * [31] A. Mauroy, Y. Susuki, and I. Mezić, Koopman operator in systems and control, Springer International Publishing, Berlin, 2020.
  * [32] M. Mollenhauer, On the Statistical Approximation of Conditional Expectation Operators, Dissertation, Freie Universität Berlin, 2021.
  * [33] D. Nešić and A.R. Teel, Sampled-data control of nonlinear systems: an overview of recent results, In: Moheimani, S.R. (eds) Perspectives in robust control. Lecture Notes in Control and Information Sciences 268. Springer, London (2007), 221–239.
  * [34] F. Nüske, P. Gelß, S. Klus, and C. Clementi, Tensor-based computation of metastable and coherent sets, Physica D 427 (2021), 133018.
  * [35] F. Nüske, B.G. Keller, G. Pérez-Hernández, A.S.J.S. Mey, and F. Noé, Variational approach to molecular kinetics, J. Chem. Theory Comput. 10 (2014), 1739–1752.
  * [36] F. Nüske and S. Klus, Efficient approximation of molecular kinetics using random Fourier features, J. Chem. Phys. 159 (7): 074105
  * [37] F. Nüske, S. Peitz, F. Philipp, M. Schaller, and K. Worthmann, Finite-data error bounds for Koopman-based prediction and control, J. Nonlinear Sci. 33 (2023), 1–34.
  * [38] B. Øksendal, Stochastic Differential Equations, An Introduction with Applications, Fifth Edition, Corrected Printing, Springer-Verlag Heidelberg New York, 2000.
  * [39] V.I. Paulsen and M. Raghupathi, An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge Studies in Advanced Mathematics 152, Cambridge University Press, 2016.
  * [40] S. Peitz, S.E. Otto, and C.W. Rowley, Data-driven model predictive control using interpolated Koopman generators, SIAM Journal on Applied Dynamical Systems, 19(3), 2162–2193 (2020).
  * [41] F. Philipp, Bessel orbits of normal operators, J. Math. Anal. Appl. 448 (2017), 767–785.
  * [42] F. Philipp, M. Schaller, K. Worthmann, S. Peitz and F. Nüske, Error bounds for kernel-based approximations of the Koopman operator. Submitted, preprint arXiv:2301.08637 (2023).
  * [43] I. Pinelis, Optimum bounds for the distributions of martingales in Banach spaces, The Annals of Probability 22 (1994), 1679–1706.
  * [44] A. Rahimi and B. Recht, Random features for large-scale kernel machines, Advances in Neural Information Processing Systems 20 (2007)
  * [45] F. Riesz and B. Nagy, Functional Analysis, Blackie & Son Ltd., Glasgow, Bombay, Toronto, 1955.
  * [46] W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, Inc., 1987.
  * [47] M. Schaller, K. Worthmann, F. Philipp, S. Peitz, and F. Nüske, Towards reliable data-based optimal and predictive control using extended DMD, to appear in IFAC PapersOnLine, arXiv preprint arXiv:2202.09084.
  * [48] P.J. Schmid, Dynamic mode decomposition of numerical and experimental data, Journal of fluid mechanics, 656, 5-28, 2010.
  * [49] I. Steinwart and A. Christmann, Support Vector Machines, Springer Science+Business Media, LLC, 2008.
  * [50] I. Steinwart, D. Hush, and C. Scovel, An explicit description of the Reproducing Kernel Hilbert Spaces of Gaussian RBF kernels, IEEE Transactions on Information Theory 52 (2006), 4635–4643.
  * [51] R. Strässer, J. Berberich, and F. Allgöwer, Robust data-driven control for nonlinear systems using the Koopman operator, preprint arXiv:2304.03519, 2023.
  * [52] R. Strässer, M. Schaller, K. Worthmann, J. Berberich, F. Allgöwer Koopman-based feedback design with stability guarantees, preprint arXiv:2312.01441, 2023.
  * [53] A. Surana, Koopman operator based observer synthesis for control-affine nonlinear systems, In 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 6492–6499. IEEE, 2016.
  * [54] W. Walter, Ordinary differential equations, Springer Science + Business Media, 1998.
  * [55] M.O. Williams, M.S. Hemati, S. Dawson, I.G. Kevrekidis, and C. W. Rowley, Extending data-driven Koopman analysis to actuated systems, IFAC-PapersOnLine 49, no. 18 (2016), 704–709.
  * [56] M.O. Williams, I.G. Kevrekidis, and C.W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci. 25 (2015), 1307–1346.
  * [57] M.O. Williams, C.W. Rowley, and I. Kevrekidis, A kernel-based method for data-driven Koopman spectral analysis, J. Comput. Dyn. 2(2) (2015), 247–265.
  * [58] K. Worthmann, M. Reble, L. Grüne, and F. Allgöwer, The role of sampling for stability and performance in unconstrained nonlinear model predictive control, SIAM Journal on Control and Optimization 52:1 (2014), 581–605.
  * [59] C. Zhang and E. Zuazua, A quantitative analysis of Koopman operator methods for system identification and predictions, Comptes Rendus. Mécanique, Online first (2023), 1–31.

##  Appendix A The condition (3.5) for deterministic systems

Let $\mathcal{X}=\overline{\Omega}$, where $\Omega\subset\mathbb{R}^{d}$ is
open and bounded. We consider the flow
$F:[0,T]\times\mathcal{X}\to\mathbb{R}^{d}$ of a given autonomous ODE
$\dot{x}=b(x)$. Let $\mathcal{Y}\subset\mathbb{R}^{d}$ such that

$\mathcal{Z}:=F([0,T]\times\mathcal{X})$

is contained in $\mathcal{Y}$. In this section, we provide conditions on the
flow and a finite measure $\mu$ on $\mathcal{Y}$ which guarantee that
condition (3.8) holds, i.e.,

$\mu(F_{t}^{-1}(A))\,\leq\,L\cdot\mu(A)\quad\forall
A\in\mathcal{B}(\mathcal{Y}),\,\forall t\in[0,T].$

First of all, we note that it is sufficient to prove the above for
$\mathcal{Y}$ replaced by $\mathcal{Z}$. Indeed, then for
$A\in\mathcal{B}(\mathcal{Y})$ we have

$\mu(F_{t}^{-1}(A))=\mu(F_{t}^{-1}(A\cap\mathcal{Z}))\leq
L\cdot\mu(A\cap\mathcal{Z})\leq L\cdot\mu(A).$

######  Proposition A.1.

The condition (3.8) is satisfied whenever the following conditions are met:

  * •

The flow $F:[0,\widetilde{T}]\times\mathcal{X}\to\mathbb{R}^{d}$ of the ODE
$\dot{x}=b(x)$ is defined for some $\widetilde{T}>T$ and is $C^{1}$.

  * •

$d\mu(x)=\varphi(x)\,dx$ with a positive density $\varphi\in
L^{\infty}(\mathcal{X})$ satisfying $1/\varphi\in L^{\infty}(\mathcal{Z})$.

Note that the first assumption is satisfied if $b\in C^{1}(\mathcal{Y})$, cf.
[1, Theorem 10.3].

###### Proof of Proposition A.1.

First of all, we show that

$\displaystyle\delta:=\inf\big{\\{}\det
F_{t}^{\prime}(x):x\in\mathcal{X},\,t\in[0,T]\big{\\}}>0.$ (A.1)

Since $F_{0}(z)=z$ for all $z\in\mathcal{Z}$, we have
$F_{0}^{\prime}(z)=I_{d}$ for all $z\in\mathcal{Z}$. Let
$t_{1}:=\widetilde{T}-T$. By assumption, the flow
$F:[0,t_{1}]\times\mathcal{Z}\to\mathbb{R}^{d}$ is defined on $\mathcal{Z}$
and is $C^{1}$. Hence, there exists some $t_{0}\in(0,t_{1})$ such that $\det
F_{t}^{\prime}(z)\geq\frac{1}{2}$ for all $t\in[0,t_{0}]$ and all
$z\in\mathcal{Z}$. Choose $t_{0}$ such that $T=nt_{0}$ for some
$n\in\mathbb{N}$. We prove by induction that for all $k\in\\{0,\ldots,n-1\\}$
we have

$\displaystyle\forall\,t\in[0,t_{0}]\,\forall x\in\mathcal{X}:\det
F_{kt_{0}+t}^{\prime}(x)\geq 2^{-k-1}.$ (A.2)

Then (A.1) holds with $\delta=2^{-n}$. We have proved (A.2) already for $k=0$.
So, let (A.2) hold for some $k<n-1$, and let $t\in[0,t_{0}]$ and
$x\in\mathcal{X}$. Then

$\det F_{(k+1)t_{0}+t}^{\prime}(x)=\det D_{x}F_{t_{0}}(F_{kt_{0}+t}(x))=\det
F_{t_{0}}^{\prime}(F_{kt_{0}+t}(x))\cdot\det
F_{kt_{0}+t}^{\prime}(x)\,\geq\,\tfrac{1}{2}\cdot\tfrac{1}{2^{k+1}}=\tfrac{1}{2^{k+2}},$

which concludes the proof of (A.1).

In order to prove (3.8), we may restrict the flow to
$F:[0,T]\times\Omega\to\mathbb{R}^{d}$ as $\mu(\partial\Omega)=0$. Let
$t\in[0,T]$, set $\mathcal{Z}_{t}:=F_{t}(\Omega)$, and let
$U\subset\mathbb{R}^{d}$ be open. Note that $\mathcal{Z}_{t}$ is open due to
the invariance of domain theorem333or simply the inverse function theorem
thanks to (A.1). Hence, also $U_{t}=U\cap\mathcal{Z}_{t}$ is open. By the
change of variables formula we have

$\displaystyle\mu(F_{t}^{-1}(U))$
$\displaystyle=\mu(F_{t}^{-1}(U_{t}))=\int_{F_{t}^{-1}(U_{t})}\varphi(x)\,dx=\int_{U_{t}}\varphi(F_{t}^{-1}(y))|\det
DF_{t}^{-1}(y)|\,dy$
$\displaystyle=\int_{U_{t}}\frac{\varphi(F_{t}^{-1}(y))}{|\det
F_{t}^{\prime}(F_{t}^{-1}(y))|\varphi(y)}\varphi(y)\,dy\,\leq\,\frac{\|\varphi\|_{\infty}\|\frac{1}{\varphi}\|_{\infty}}{\delta}\mu(U_{t})\leq\frac{\|\varphi\|_{\infty}\|\frac{1}{\varphi}\|_{\infty}}{\delta}\mu(U).$

The claim now follows from the outer regularity of the measures $\mu$ and
$\mu\circ F_{t}^{-1}$. ∎

##  Appendix B Proof of Proposition 3.3

Let $\psi\in B(\mathcal{Y})$. Then
$|(K^{t}\psi)(x)|\leq\int_{\mathcal{Y}}|\psi(y)|\,\rho_{t}(x,dy)\leq\|\psi\|_{\infty}$,
hence $K^{t}\psi\in B(\mathcal{X})$ with
$\|K^{t}\psi\|_{\infty}\leq\|\psi\|_{\infty}$ and
$\|K^{t}\psi\|_{p}\leq\|\psi\|_{\infty}$. Let now $1\leq p<\infty$. Then, by
Hölder’s inequality and (3.7),

$\int_{\mathcal{X}}|(K^{t}\psi)(x)|^{p}\,d\mu(x)\leq\int_{\mathcal{X}}\int_{\mathcal{Y}}|\psi(y)|^{p}\,\rho_{t}(x,dy)\,d\mu(x)\leq
L\int_{\mathcal{Y}}|\psi(y)|^{p}\,d\mu(y),$

and hence $\|K^{t}\psi\|_{p,\mathcal{X}}\leq L^{1/p}\|\psi\|_{p,\mathcal{Y}}$.

Let $\psi\in C_{c}(\mathcal{Y})$ (the set of continuous functions on
$\mathcal{Y}$ with compact support). For fixed $x\in\mathcal{X}$, denote the
stochastic solution process of the SDE (3.1) with initial value $x$ by
$X_{t}^{x}$. Since $X_{t}^{x}(\omega)$ is continuous in $t$ a.s. (see [38,
Theorem 5.2.1]) and $X_{t}^{x}\in\mathcal{Y}$ a.s., we have
$\psi(X_{t}^{x}(\omega))\to\psi(X_{0}^{x}(\omega))=\psi(x)$ as $t\to 0$ a.s.
Hence, by dominated convergence,

$K^{t}\psi(x)=\mathbb{E}[\psi(X_{t}^{x})]=\int\psi(X_{t}^{x}(\omega))\,d\mathbb{P}(\omega)\,\to\,\psi(x)$

as $t\to 0$. By the first part of this proof,

$|K^{t}\psi(x)-\psi(x)|\,\leq\,|K^{t}\psi(x)|+|\psi(x)|\leq
2\|\psi\|_{\infty},\qquad x\in\mathcal{X},$

hence, dominated convergence implies that
$\|K^{t}\psi-\psi|_{\mathcal{X}}\|_{p,\mathcal{X}}\to 0$ as $t\to 0$.

If $\psi\in L^{p}(\mathcal{Y};\mu)$ and $\varepsilon>0$, there exists $\eta\in
C_{c}(\mathcal{Y})$ such that
$\|\psi-\eta\|_{p,\mathcal{Y}}<\frac{\varepsilon}{2}(1+L^{1/p})^{-1}$, see
[46, Theorem 3.14]. Further, choose $\delta>0$ such that
$\|K^{t}\eta-\eta|_{\mathcal{X}}\|_{p,\mathcal{X}}<\frac{\varepsilon}{2}$ for
$t<\delta$. Then

$\|K^{t}\psi-\psi|_{\mathcal{X}}\|_{p,\mathcal{X}}\leq\|K^{t}(\psi-\eta)\|_{p,\mathcal{X}}+\|K^{t}\eta-\eta|_{\mathcal{X}}\|_{p,\mathcal{X}}+\|\eta-\psi\|_{p,\mathcal{Y}}<\varepsilon$

for $t<\delta$, which proves the claim.∎

##  Appendix C An Example: Lack of contractivity of the Koopman semigroup

Consider the deterministic case with the scalar ODE

$\displaystyle\dot{x}=x(2-x)\quad\text{on $\mathcal{X}=[1,2]$}.$

The solution of this ODE is given by

$\displaystyle x(t;x_{0})=\frac{2e^{2t}x_{0}}{2-x_{0}+e^{2t}x_{0}},\quad
x_{0}\in\mathcal{X},$

which is non-decreasing and approaches $2$ as $t\to\infty$ for any
$x_{0}\in\mathcal{X}$. In particular, $\mathcal{X}$ is invariant under the
flow of the ODE. Since $\mathcal{X}$ is compact, the convergence towards the
equilibrium $x=2$ is uniform in $x_{0}\in\mathcal{X}$. We will use this fact
to show that the Koopman semigroup is not a bounded semigroup on
$L^{2}(\mathcal{X})$, i.e., it does not satisfy
$\|K^{t}\|_{L(L^{2}(\mathcal{X}))}\leq M$ for some $M\geq 0$ and all $t\geq
0$. To show this, it suffices to find $\psi\in L^{2}(1,2)$ such that for any
$M\geq 0$, there is $t\geq 0$ such that
$\|K^{t}\psi\|_{L^{2}(\mathcal{X})}\geq M\|\psi\|_{L^{2}(\mathcal{X})}$.

Let $\psi(x):=\frac{1}{(2-x)^{1/4}}$. Then $\psi\in L^{2}(\mathcal{X})$ as

$\displaystyle\int_{1}^{2}\psi^{2}(x)\,\mathrm{d}x=\int_{1}^{2}\frac{1}{\sqrt{2-x}}\,\mathrm{d}x=\int_{0}^{1}\frac{1}{\sqrt{x}}\,\mathrm{d}x=2\sqrt{x}\Big{|}_{0}^{1}=2<\infty.$

Further, we have that $\psi(x)\to\infty$ as $x\to 2$ and $\psi$ is non-
decreasing such that

$\displaystyle\forall M\geq 0\,\,\exists\,\varepsilon>0:\,\psi(x)\geq
M\qquad\forall x\in(2-\varepsilon,2).$ (C.1)

Moreover, as the flow is non-decreasing towards $2$:

$\displaystyle\forall\varepsilon\geq 0\,\,\exists\,t^{*}\geq 0\,\,\forall
t\geq t^{*}\,\,\forall
x_{0}\in\mathcal{X}\backslash\\{2\\}:\,x(t;x_{0})\in(2-\varepsilon,2).$ (C.2)

Hence, for given $M$, choose $\varepsilon$ as in (C.1) and correspondingly
$t^{*}$ as in (C.2). Then $x(t;x^{0})\in(2-\varepsilon,2)$ for all $t\geq
t^{*}$ and $x_{0}\in\mathcal{X}\backslash\\{2\\}$, hence

$\displaystyle\forall t\geq
t^{*}:\quad\|K^{t}\psi\|^{2}_{L^{2}(\mathcal{X})}=\int_{1}^{2}|\psi(x(t;x_{0}))|^{2}\,\mathrm{d}x_{0}\geq
M^{2}=\tfrac{1}{2}M^{2}\|\psi\|_{L^{2}(\mathcal{X})}^{2}.$

##  Appendix D On the condition
$K^{t}\mathbb{H}_{\mathcal{Y}}\subset\mathbb{H}_{\mathcal{X}}$ in case of the
Gaussian kernel and the Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck (OU) process on $X=\mathbb{R}$ is the solution of the
SDE $dX_{t}=-\alpha X_{t}\,dt+dW_{t}$, where $\alpha>0$. The invariant measure
$\mu$ and the Markov transition kernel $\rho_{t}$, $t>0$, are known and given
by

$\displaystyle d\mu(x)=\sqrt{\frac{\alpha}{\pi}}\cdot e^{-\alpha
x^{2}}\,dx\qquad\text{and}\qquad\rho_{t}(x,dy)=\sqrt{\frac{c_{t}}{\pi}}\cdot\exp\Big{[}-c_{t}(y-e^{-\alpha
t}x)^{2}\Big{]}\,dy,$

where

$c_{t}=\frac{\alpha}{1-e^{-2\alpha t}}.$

We consider the Gaussian radial basis function (RBF) kernels on $\mathbb{R}$
with bandwidth $\sigma>0$, i.e.,

$k^{\sigma}(x,y)=\exp\left[-\frac{(x-y)^{2}}{\sigma^{2}}\right].$

For $y\in\mathbb{R}$ and a kernel $k$ on $\mathbb{R}$ set $k_{y}(x):=k(x,y)$,
$x\in\mathbb{R}$.

Let $X\subset\mathbb{R}$ be any set. By $\mathbb{H}_{\sigma}(X)$ we denote the
RKHS generated by the kernel $k^{\sigma}_{X}=k^{\sigma}|_{X\times X}$ on $X$.
Hilbert space norm and scalar product on $\mathbb{H}_{\sigma}(X)$ will be
denoted by $\|\cdot\|_{\sigma,X}$ and
$\langle\cdot\,,\,\cdot\rangle_{\sigma,X}$, respectively. For two positive
definite kernels on $X$ we write $k_{1}\preceq k_{2}$ if

$\sum_{i,j=1}^{n}\alpha_{i}\alpha_{j}k_{1}(x_{i},x_{j})\leq\sum_{i,j=1}^{n}\alpha_{i}\alpha_{j}k_{2}(x_{i},x_{j})$

for any choice of $n\in\mathbb{N}$ and $\alpha_{j}\in\mathbb{R}$, $x_{j}\in
X$, $j=1,\ldots,n$. We also write $V\lhook\joinrel\xrightarrow{\hskip
2.84544ptc\hskip 2.84544pt}W$ for two normed vector spaces $V$ and $W$ if
$V\subset W$ is continuously embedded in $W$.

######  Lemma D.1.

Let $X\subset\mathbb{R}$ and $0<\sigma_{1}<\sigma_{2}$. Then we have
$\mathbb{H}_{\sigma_{2}}(X)\\!\lhook\joinrel\xrightarrow{\hskip
2.84544ptc\hskip 2.84544pt}\mathbb{H}_{\sigma_{1}}(X)$ with

$\|\psi\|_{\sigma_{1},X}\leq\sqrt{\tfrac{\sigma_{2}}{\sigma_{1}}}\cdot\|\psi\|_{\sigma_{2},X},\qquad\psi\in\mathbb{H}_{\sigma_{2}}(X),$

and
$k_{X}^{\sigma_{2}}\preceq\frac{\sigma_{2}}{\sigma_{1}}k_{X}^{\sigma_{1}}$.

###### Proof.

The first claim is Corollary 6 in [50]. The second follows from Aronszajn’s
inclusion theorem [39, Theorem 5.1]. ∎

######  Theorem D.2.

Let $\mathcal{Y}=\mathbb{R}$ and let $\mathcal{X}\subset\mathbb{R}$ be any
compact set with non-empty interior. Then for each $t\geq 0$ and all
$\alpha,\sigma>0$ we have

$K^{t}\mathbb{H}_{\sigma}(\mathcal{Y})\subset\mathbb{H}_{\sigma}(\mathcal{X})\qquad\text{with}\qquad\big{\|}K^{t}\big{\|}_{\mathbb{H}_{\sigma}(\mathcal{Y})\to\mathbb{H}_{\sigma}(\mathcal{X})}\leq
e^{\frac{\alpha}{2}t}.$

###### Proof.

First of all, it follows from [50, Corollaries 4&5] that to each
$\psi\in\mathbb{H}_{\sigma}(\mathcal{X})$ there exists a unique extension
$\tilde{\psi}\in\mathbb{H}_{\sigma}(\mathcal{Y})$ such that
$\langle\tilde{\psi},\tilde{\phi}\rangle_{\sigma,\mathcal{Y}}=\langle\psi,\phi\rangle_{\sigma,\mathcal{X}}$
for all $\psi,\phi\in\mathbb{H}_{\sigma}(\mathcal{X})$. Conversely, we have

$\displaystyle\mathbb{H}_{\sigma}(\mathcal{X})=\\{\psi|_{\mathcal{X}}:\psi\in\mathbb{H}_{\sigma}(\mathcal{Y})\\}\qquad\text{and}\qquad\langle\psi,\phi\rangle_{\sigma,\mathcal{Y}}=\langle\psi|_{\mathcal{X}},\phi|_{\mathcal{X}}\rangle_{\sigma,\mathcal{X}},\qquad\psi,\phi\in\mathbb{H}_{\sigma}(\mathcal{Y}).$
(D.1)

For $t=0$ the claim is obviously true. Hence, suppose that $t>0$. For
$z\in\mathbb{R}$ let us compute $K^{t}k^{\sigma}_{z}$. For this, we define

$\tau=\sqrt{\frac{c_{t}\sigma^{2}}{1+c_{t}\sigma^{2}}},\qquad\text{and}\qquad\nu=\frac{e^{\alpha
t}}{\tau}\sigma\,>\,\frac{\sigma}{\tau}>\sigma.$

Since for $\sigma_{1},\sigma_{2}>0$ and $z,w\in\mathbb{R}$ we have

$\int_{-\infty}^{\infty}k_{z}^{\sigma_{1}}(x)\,k_{w}^{\sigma_{2}}(x)\,dx=\sqrt{\pi}\cdot\frac{\sigma_{1}\sigma_{2}}{\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}}\cdot\exp\left(-\frac{(z-w)^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}\right),$

with $\sigma_{t}=1/\sqrt{c_{t}}$ we obtain for $x\in\mathcal{X}$

$\displaystyle(K^{t}k_{z}^{\sigma})(x)$
$\displaystyle=\int_{-\infty}^{\infty}k_{z}^{\sigma}(y)\,\rho_{t}(x,dy)=\sqrt{\frac{c_{t}}{\pi}}\int_{-\infty}^{\infty}k_{z}^{\sigma}(y)\exp\Big{[}-c_{t}(y-e^{-\alpha
t}x)^{2}\Big{]}\,dy$
$\displaystyle=\sqrt{\frac{c_{t}}{\pi}}\int_{-\infty}^{\infty}k_{z}^{\sigma}(y)k_{e^{-\alpha
t}x}^{\sigma_{t}}(y)\,dy=\sqrt{c_{t}}\cdot\frac{\sigma\sigma_{t}}{\sqrt{\sigma^{2}+\sigma_{t}^{2}}}\cdot\exp\left(-\frac{(z-e^{-\alpha
t}x)^{2}}{\sigma^{2}+\sigma_{t}^{2}}\right)$
$\displaystyle=\frac{\sigma}{\sqrt{\sigma^{2}+1/c_{t}}}\cdot\exp\left(-\frac{(e^{\alpha
t}z-x)^{2}}{e^{2\alpha t}(\sigma^{2}+1/c_{t})}\right)=\tau\cdot
k^{\nu}_{e^{\alpha t}z}(x).$

Since $\nu>\sigma$, it follows from (D.1) and Lemma D.1 that

$K^{t}k_{z}^{\sigma}=\tau\cdot k^{\nu}_{e^{\alpha
t}z}|_{\mathcal{X}}\in\mathbb{H}_{\nu}(\mathcal{X})\subset\mathbb{H}_{\sigma}(\mathcal{X}).$

Note that

$k^{\nu}(e^{\alpha t}x,e^{\alpha t}y)=\exp\left(-\frac{e^{2\alpha
t}(x-y)^{2}}{\nu^{2}}\right)=\exp\left(-\frac{\tau^{2}(x-y)^{2}}{\sigma^{2}}\right)=k^{\sigma/\tau}(x,y).$

Now, let $n\in\mathbb{N}$ and $\alpha_{j},x_{j}\in\mathbb{R}$, $j=1,\ldots,n$,
be arbitrary and let
$\psi=\sum_{j=1}^{n}\alpha_{j}k^{\sigma}_{x_{j}}\in\mathbb{H}_{\sigma}(\mathcal{Y})$.
Then, again by (D.1) and Lemma D.1,

$\displaystyle\|K^{t}\psi\|_{\sigma,\mathcal{X}}^{2}$
$\displaystyle\leq\frac{\nu}{\sigma}\|K^{t}\psi\|_{\nu,\mathcal{X}}^{2}=\frac{\nu}{\sigma}\Bigg{\|}\sum_{j=1}^{n}\alpha_{j}K^{t}k^{\sigma}_{x_{j}}\Bigg{\|}_{\nu,\mathcal{X}}^{2}=\frac{\nu\tau^{2}}{\sigma}\Bigg{\|}\sum_{j=1}^{n}\alpha_{j}k^{\nu}_{e^{\alpha
t}x_{j}}\Bigg{\|}_{\nu,\mathcal{Y}}^{2}$
$\displaystyle=\frac{\nu\tau^{2}}{\sigma}\sum_{i,j=1}^{n}\alpha_{i}\alpha_{j}k^{\nu}(e^{\alpha
t}x_{i},e^{\alpha
t}x_{j})=\frac{\nu\tau^{2}}{\sigma}\sum_{i,j=1}^{n}\alpha_{i}\alpha_{j}k^{\sigma/\tau}(x_{i},x_{j})$
$\displaystyle\leq\frac{\nu\tau^{2}}{\sigma}\cdot\frac{1}{\tau}\sum_{i,j=1}^{n}\alpha_{i}\alpha_{j}k^{\sigma}(x_{i},x_{j})=\frac{\nu\tau}{\sigma}\|\psi\|_{\sigma,\mathcal{Y}}^{2}=e^{\alpha
t}\|\psi\|_{\sigma,\mathcal{Y}}^{2}.$

This shows that $K^{t}$ maps
$\mathbb{H}_{0,\sigma}(\mathcal{Y}):=\operatorname{span}\\{k^{\sigma}_{y}:y\in\mathcal{Y}\\}\subset\mathbb{H}_{\sigma}(\mathcal{Y})$
boundedly into $\mathbb{H}_{\sigma}(\mathcal{X})$. Since
$\mathbb{H}_{0,\sigma}(\mathcal{Y})$ is dense in
$\mathbb{H}_{\sigma}(\mathcal{Y})$, it follows that
$K^{t}|_{\mathbb{H}_{0,\sigma}}$ extends to a bounded operator
$T:\mathbb{H}_{\sigma}(\mathcal{Y})\to\mathbb{H}_{\sigma}(\mathcal{X})$. In
order to see that $T\psi=K^{t}\psi$ for
$\psi\in\mathbb{H}_{\sigma}(\mathcal{Y})$, let
$(\psi_{n})\subset\mathbb{H}_{0,\sigma}(\mathcal{Y})$ such that
$\psi_{n}\to\psi$ in $\mathbb{H}_{\sigma}(\mathcal{Y})$. Then
$K^{t}\psi_{n}=T\psi_{n}\to T\psi$ in $\mathbb{H}_{\sigma}(\mathcal{X})$.
Since $\mathbb{H}_{\sigma}(\mathcal{Y})\lhook\joinrel\xrightarrow{\hskip
2.84544ptc\hskip 2.84544pt}L^{2}_{\mu}(\mathcal{Y})$, we have
$\psi_{n}\to\psi$ in $L^{2}_{\mu}(\mathcal{Y})$ and thus $K^{t}\psi_{n}\to
K^{t}\psi$ in $L^{2}_{\mu}(\mathcal{X})$. Also, $K^{t}\psi_{n}\to T\psi$ in
$L^{2}_{\mu}(\mathcal{X})$. Hence, $K^{t}\psi=T\psi$ $\mu$-a.e. on
$\mathbb{R}$. But as both $K^{t}\psi$ and $T\psi$ are continuous and $\mu$ is
absolutely continuous w.r.t. Lebesgue measure with a positive density, we
conclude that $K^{t}\psi=T\psi\in\mathbb{H}_{\sigma}$. ∎

## Author Affiliations