-{\left(w{+}{\mathsf{C}}z\middle|\Sigma\,\square\right)_{\mathbf{H}}}\end{pmatrix},$

which satisfies the fluctuation-dissipation relation

${}_{\mathrm{GEN}}{}_{\mathrm{GEN}}^{*}=2{\mathbb{K}}_{\mathrm{GEN}}.$ (4.21)

######  Remark 4.11.

It is remarkable that the splitting
${\mathbb{D}}={{\mathbb{D}}_{\mathrm{sym}}}+{{\mathbb{D}}_{\mathrm{skw}}}$ is
compatible with the non-interaction conditions of GENERIC. Moreover, taking
into account the special structure of the differentials
${\mathrm{D}}{\mathcal{H}}_{\mathrm{GEN}}=({\mathrm{D}}{\mathcal{H}}_{\mathcal{A}}(z)+{\mathsf{C}}^{*}z,w{+}{\mathsf{C}}z,1)^{\top}$
and ${\mathrm{D}}{\mathcal{S}}_{\mathrm{GEN}}=(0,0,\beta)^{\top}$ and the
structure of ${\mathbb{K}}_{\mathrm{GEN}}$ we see that in the differential
equation (4.16b) only the sum
$({{\mathbb{D}}_{\mathrm{sym}}}{+}{{\mathbb{D}}_{\mathrm{skw}}})(w{+}{\mathsf{C}}z)$
appears. However, in the stochastic forcing we see only the symmetric part
${{\mathbb{D}}_{\mathrm{sym}}}$ through

$\Sigma\Sigma^{*}=\frac{1}{\beta}\big{(}{\mathbb{D}}{+}{\mathbb{D}}^{*}\big{)}=\frac{2}{\beta}\,{{\mathbb{D}}_{\mathrm{sym}}}.$

∎

######  Example 4.12 (Running example, part 4).

For the model in Examples (1.1) and (1.3), where we identify ${\mathbf{W}}$
with ${\mathbb{R}}^{3}$ we obtain a even more specific form of the GENERIC
Poisson and Onsager structure in in ${\mathbb{R}}^{(2n+4){\times}(2n+4)}$:

${\mathbb{J}}_{\mathrm{GEN}}=\left(\begin{array}[]{cccc}0&{\mathbb{I}}&0&0\\\
\\!\\!{-}{\mathbb{I}}&0&0&0\\\ 0&0&\\!\\!-{{\mathbb{D}}_{\mathrm{skw}}}&0\\\
0&0&0&0\end{array}\right)\ \text{
and}\quad{\mathbb{K}}_{\mathrm{GEN}}=\frac{1}{\beta}\left(\begin{array}[]{cccc}0&0&0&0\\\
0&0&0&0\\\
0&0&{{\mathbb{D}}_{\mathrm{sym}}}&\\!\\!\\!-{{\mathbb{D}}_{\mathrm{sym}}}(w{+}\mathsf{C}q)\\!\\!\\\
0&0&\\!\\!{-}\big{(}{{\mathbb{D}}_{\mathrm{sym}}}(w{+}\mathsf{C}q)\big{)}^{\\!\top}\\!\\!\\!&|w{+}\mathsf{C}q|^{2}_{{{\mathbb{D}}_{\mathrm{sym}}}}\end{array}\right)$

where $\mathsf{C}\in{\mathbb{R}}^{3{\times}n}$ and
${{\mathbb{D}}_{\mathrm{sym}}},\,{{\mathbb{D}}_{\mathrm{skw}}}\in{\mathbb{R}}^{3{\times}3}$
have the explicit form

${{\mathbb{D}}_{\mathrm{skw}}}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&\\!{-}\varsigma\\!\\!\\\
0&\varsigma&0\end{array}\right),\quad{{\mathbb{D}}_{\mathrm{sym}}}=\left(\begin{array}[]{ccc}\vartheta_{1}&0&0\\\
0&\vartheta_{2}&0\\\ 0&0&\vartheta_{2}\end{array}\right),\quad\text{and
}|\widehat{w}|^{2}_{{{\mathbb{D}}_{\mathrm{sym}}}}=\widehat{w}{\cdot}{{\mathbb{D}}_{\mathrm{sym}}}\widehat{w}.$

Thus, the macroscopic ODE takes the explicit form

$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\left(\begin{array}[]{c}q\\\ p\\\ w\\\
e\end{array}\right)=\left(\begin{array}[]{c}p\\\
\\!\\!-{\mathrm{D}}V(q)-\mathsf{C}^{*}w\\!\\\
-{{\mathbb{D}}_{\mathrm{skw}}}(w{+}\mathsf{C}q)\\\
0\end{array}\right)+\left(\begin{array}[]{c}0\\\ 0\\\
\\!\\!{-}{{\mathbb{D}}_{\mathrm{sym}}}(w{+}\mathsf{C}q)\\!\\\
|w{+}\mathsf{C}q|^{2}_{{{\mathbb{D}}_{\mathrm{sym}}}}\end{array}\right)=\left(\begin{array}[]{c}p\\\
-{\mathrm{D}}V(q)-\mathsf{C}^{*}w\\\ -{\mathbb{D}}(w{+}\mathsf{C}q)\\\
|w{+}\mathsf{C}q|^{2}_{{{\mathbb{D}}_{\mathrm{sym}}}}\end{array}\right).$

∎

###  4.4 Interpretation of ${\mathcal{E}}_{\mathrm{GEN}}$,
${\mathbb{J}}_{\mathrm{GEN}}$, and ${\mathbb{K}}_{\mathrm{GEN}}$

##### Interpretation of the energy ${\mathcal{E}}_{\mathrm{GEN}}$.

The discussion in Section 4.3 gives a clear interpretation of the energy

${\mathcal{E}}_{\mathrm{GEN}}(z,w,e)={\mathcal{H}}_{\mathcal{A}}(z)+\left({\mathsf{C}}z\middle|w\right)_{\mathbf{H}}+\frac{1}{2}\|w\|_{\mathbf{H}}^{2}+e$

as a renormalized version of ${\mathcal{H}}_{\mathrm{total}}$:
$\frac{1}{2}\|w\|_{{\mathbf{H}}}^{2}$ represents the ‘visible’ part of the
heat-bath energy, $e$ represents changes in the ‘invisible’ part, and an
additional ‘infinite but constant’ component of the heat-bath energy has been
renormalized out. This interpretation also explains why
${\mathcal{E}}_{\mathrm{GEN}}$ should be preserved by the GENERIC SDE (4.19).

##### Interpretation of the Poisson operator ${\mathbb{J}}_{\mathrm{GEN}}$.

The Poisson operator ${\mathbb{J}}_{\mathrm{GEN}}$ is a combination of the
symplectic operator ${\mathbb{J}}_{{\mathcal{A}}}$ with the skew-symmetric
part ${{\mathbb{D}}_{\mathrm{skw}}}$ of the compression operator
${\mathbb{D}}$. This shows how a skew-symmetric component of the compression
dynamics should be considered as a conservative (Hamiltonian) component of the
evolution. Clearly ${\mathbb{J}}_{\mathrm{GEN}}(y)$ is skew-symmetric, and it
can be easily shown that the Jacobi identity (4.2) holds, because
${\mathbb{J}}_{\mathcal{A}}$ depends on $z$ but not on $(w,e)$.

It would be interesting to understand in what sense or in which cases the
formula (4.20g) for ${\mathbb{J}}_{\mathrm{GEN}}$ coincides with Öttinger’s
coarse-graining formula in [Ö05, Eq. (6.67+68)]:

${\mathbb{J}}(z,w,e)=\Big{\langle}{\mathrm{D}}\Pi(z,\eta){\mathbb{J}}_{\mathrm{micro}}(z,\eta){\mathrm{D}}\Pi(z,\eta)^{*}\Big{\rangle}_{(z,w,e)},$
(4.52)

where $\langle\cdot\rangle_{(z,w,e)}$ means averaging with respect to the
conditional equilibrium measure given the macro-state $(z,w,e)$.

In our case, omitting the energy variable $e$, the projection mapping reads

$\begin{pmatrix}z\\\
w\end{pmatrix}=\widetilde{\Pi}(z,\eta)=\begin{pmatrix}z\\\
{\mathbb{P}}\eta\end{pmatrix}.$

Hence, Öttinger’s formula would take the form

${\mathbb{J}}_{\text{macro}}(z,w)=\left\langle{\mathrm{D}}\widetilde{\Pi}(z,\eta)\begin{pmatrix}{\mathbb{J}}_{\mathcal{A}}(z)&0\\\
0&{\mathbb{J}}_{\mathcal{B}}\end{pmatrix}{\mathrm{D}}\widetilde{\Pi}(z,\eta)\right\rangle_{(z,w)}=\begin{pmatrix}{\mathbb{J}}_{\mathcal{A}}(z)&0\\\
0&\text{``}{\mathbb{P}}\,{\mathbb{J}}_{\mathcal{B}}{\mathbb{P}}\text{''}\end{pmatrix}.$

Here, averaging is not needed, because the integrand is constant. However, as
discussed in Remark LABEL:re:Dom.JB.bfW, the term
${\mathbb{P}}\,{\mathbb{J}}_{\mathcal{B}}{\mathbb{P}}$ is not well defined,
because ${\mathbf{W}}\cap\mathrm{dom}({\mathbb{J}}_{\mathcal{B}})=\\{0\\}$ by
construction.

Thus this formula should be considered in a renormalized sense, i.e. be
approximating ${\mathbb{J}}_{\mathcal{B}}$ suitably. Because of the generator
property, we have the approximation property by a symmetric difference of
going forward and backward in time (which means extracting irreversible
information):

$\forall\,\eta\in\mathrm{dom}({\mathbb{J}}_{\mathcal{B}}):\quad\frac{1}{2h}\big{(}\mathrm{e}^{h{\mathbb{J}}_{\mathcal{B}}}-\mathrm{e}^{-h{\mathbb{J}}_{\mathcal{B}}}\big{)}\eta=\frac{1}{2h}\big{(}\mathrm{e}^{h{\mathbb{J}}_{\mathcal{B}}}{-}{\mathbb{I}}\big{)}\eta-\frac{1}{2h}\big{(}\mathrm{e}^{-h{\mathbb{J}}_{\mathcal{B}}}{-}{\mathbb{I}}\big{)}\eta\;\stackrel{{\scriptstyle
h\downarrow 0}}{{\longrightarrow}}\;{\mathbb{J}}_{\mathcal{B}}\eta.$

At the same time, by the Compression Property (1.5) we obtain a different
behavior when applying ${\mathbb{P}}$ from left and right (for $h\searrow 0$):

$\forall\,v\in{\mathbf{W}}:\quad{\mathbb{P}}\frac{1}{2h}\big{(}\mathrm{e}^{h{\mathbb{J}}_{\mathcal{B}}}-\mathrm{e}^{-h{\mathbb{J}}_{\mathcal{B}}}\big{)}{\mathbb{P}}\,v\overset{\text{comp}}{=}\frac{1}{2h}\big{(}\mathrm{e}^{-h{\mathbb{D}}}{-}{\mathbb{I}}\big{)}v-\frac{1}{2h}\big{(}\mathrm{e}^{-h{\mathbb{D}}^{*}}{-}{\mathbb{I}}\big{)}v\longrightarrow-{\mathbb{D}}_{\mathrm{skw}}v.$

Thus, although this approximation is rather special (because of the symmetry
with $\pm h$), we see that it is reasonable to replace the ill-defined
expression ${\mathbb{P}}\\!\;{\mathbb{J}}_{\mathcal{B}}{\mathbb{P}}$ by
$-{\mathbb{D}}_{\mathrm{skw}}$. From this point of view it may be reasonable
to interpret the definition (4.20g) of ${\mathbb{J}}_{\mathrm{GEN}}$ as a
version of (4.52).

##### Interpretation of the Onsager operator ${\mathbb{K}}_{\mathrm{GEN}}$.

The structure of the Onsager operator ${\mathbb{K}}_{\mathrm{GEN}}$ can be
understood in the following way. To start with, note that in the absence of
interaction with $z$ or $e$, the component $w=\widehat{\mathbb{P}}\eta$ decays
as $\mathrm{e}^{-t{\mathbb{D}}}w$, which implies that the norm
$\|w\|^{2}_{\mathbf{H}}$ then decays with rate
$2{{\mathbb{D}}_{\mathrm{sym}}}$:

$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\|w_{t}\|_{\mathbf{H}}^{2}=\left(w_{t}\middle|{\mathbb{D}}w_{t}\right)_{\mathbf{H}}+\left({\mathbb{D}}w_{t}\middle|w_{t}\right)_{\mathbf{H}}=2\left(w_{t}\middle|{{\mathbb{D}}_{\mathrm{sym}}}w_{t}\right)_{\mathbf{H}}.$

This explains the operator ${{\mathbb{D}}_{\mathrm{sym}}}$ in the
$(w,w)$-block of ${\mathbb{K}}_{\mathrm{GEN}}$. The $(w,e)$-component of
${\mathbb{K}}_{\mathrm{GEN}}$ then necessarily has the form
$-{{\mathbb{D}}_{\mathrm{sym}}}(w{+}{\mathsf{C}}z)$ because of the non-
interaction condition (4.3), and the remaining components of
${\mathbb{K}}_{\mathrm{GEN}}$ follow by symmetry.

In fact, in equation (4.16b) the drift term is generated by the
$(w,e)$-component of ${\mathbb{K}}_{\mathrm{GEN}}$, not the $(w,w)$-component,
since ${\mathbb{K}}_{\mathrm{GEN}}$ multiplies
${\mathrm{D}}{\mathcal{S}}_{\mathrm{GEN}}=(0,0,\beta)^{\top}$. This does not
change the interpretation above, since the non-interaction condition implies
that the same drift term also would be generated by an alternative entropy,
$(z,w,e)\mapsto-{\mathcal{H}}_{z,w}(z,w)$. We discuss the role of different
entropies in the next section.

###  4.5 Interpretation of ${\mathcal{S}}_{\mathrm{GEN}}$

For the interpretation of ${\mathcal{S}}_{\mathrm{GEN}}$ we start by
commenting on the historical development of ‘entropy’. Entropy is commonly
associated with a description of a system at two levels of aggregation, a
‘microscopic’ level and a ‘macroscopic’ level. Different authors, however,
have emphasized different ways of placing entropy in this context. Boltzmann’s
famous postulate

${\mathcal{S}}(x)=k\ln W(x)$ (4.53)

already illustrates this: the symbol $W(x)$ could either represent probability
_(Wahrscheinlichkeit)_ [Bre75, p. 44], or alternatively ‘the number of
microstates associated with the macrostate $x$’. Boltzmann presumably based
his discussion on equal probability of microstates, in which case these two
interpretations reduce to the same thing.

Einstein [Ein10] postulated that by rewriting (4.53) as

$\operatorname{Prob}(x)=\mathrm{e}^{\frac{1}{k}{\mathcal{S}}(x)},$ (4.54)

this expression could be abstracted away from the specific microscopic
situation, and could therefore hold also in situations without equal
probability of microstates; in fact, the probability distribution over
microstates only needs to be stationary under the microscopic dynamics. This
approach focuses on an interpretation of entropy as ‘logarithm of
probabilities’ rather than as ‘number of microstates’. We take the same view
here.

In continuous state spaces an expression such as (4.54) should be interpreted
as a density with respect to some fixed measure. We illustrate this on a
simple example: in the case of SDEs of the form

${\mathrm{d}}X_{t}=-K(X_{t})\nabla
V(X_{t})\;\\!\mathrm{d}t+\Sigma(X_{t})\;\\!\mathrm{d}B_{t},\qquad\Sigma\Sigma^{*}=2K,$

one can consider $-V$ to be an entropy in the sense above, because the flow
preserves the measure $\mathrm{e}^{-V(x)}{\mathrm{d}}x$. The appearance of the
Lebesgue measure ‘${{\mathrm{d}}x}$’ is explained by observing that this
Lebesgue measure is invariant under the Brownian motion $B_{t}$, and the
function $-V$ converts this invariant measure into
$\mathrm{e}^{-V(x)}{\mathrm{d}}x$ through the pair $(K,\Sigma)$ as described
by the Girsanov theorem. In this sense the entropy can be understood as
describing an exponential _modification_ of an underlying measure, and that is
the sense in which (4.54) should be interpreted.

This point of view provides the SDE (3.3) for $(z,w)$ (i.e. before extending
with $e$) with a natural entropy

${\mathcal{S}}(z,w):=\log\frac{{\mathrm{d}}\nu_{\beta}}{{\mathrm{d}}{\mathscr{L}}_{{\mathbf{Z}}{\times}{\mathbf{W}}}}(z,w)\stackrel{{\scriptstyle\eqref{eqdef:nu-
beta}}}{{=}}-\beta{\mathcal{H}}_{zw}(z,w)+\text{constant},$ (4.55)

where $\nu_{\beta}$ is given in (3.5) and
${\mathscr{L}}_{{\mathbf{Z}}{\times}{\mathbf{W}}}$ is the Lebesgue measure on
${\mathbf{Z}}{\times}{\mathbf{W}}$.

Below we follow this entropy ${\mathcal{S}}$ as it propagates into the
extended system with variables $(z,w,e)$, but that propagation is akin to
bookkeeping. The actual appearance of entropy has happened here, at the
coarse-graining step from variables $(z,\eta)$ to variables $(z,w)$.

While (4.55) gives a clear suggestion for an entropy for the variables
$(z,w)$, an indeterminacy arises when extending to variables $(z,w,e)$.
Indeed, after extension there is a multitude of invariant measures: for
instance, if we fix an energy level ${\mathcal{E}}_{0}\in{\mathbb{R}}$, then
under the extension map

$T^{{\mathcal{E}}_{0}}:(z,w)\mapsto\big{(}z,w,e:={\mathcal{E}}_{0}-{\mathcal{H}}_{zw}(z,w)\big{)}$

the measure $\nu_{\beta}$ is mapped to the measure

$\nu_{\beta}^{{\mathcal{E}}_{0}}:=T^{{\mathcal{E}}_{0}}_{\\#}\nu_{\beta},$

which is concentrated on the ‘${\mathcal{E}}_{0}$-energy-leaf’
$\\{(z,w,e)\in{\mathbf{Z}}{\times}{\mathbf{H}}{\times}{\mathbb{R}}:{\mathcal{E}}_{\mathrm{GEN}}(z,w,e)={\mathcal{E}}_{0}\\}$.
The measure $\nu_{\beta}^{{\mathcal{E}}_{0}}$ is invariant under the flow of
the extended SDE (4.16), for each choice of ${\mathcal{E}}_{0}$. In fact, for
any measure $\zeta\in{\mathcal{P}}({\mathbb{R}})$ on the set of energy values
${\mathcal{E}}_{0}$, the compound measure

$\nu_{\beta}^{\zeta}(A):=\int_{\mathbb{R}}\nu_{\beta}^{{\mathcal{E}}_{0}}(A)\,\zeta({\mathrm{d}}{\mathcal{E}}_{0})\qquad\text{for
}A\subset{\mathbf{Z}}{\times}{\mathbf{W}}{\times}{\mathbb{R}},$

has energy values ${\mathcal{E}}_{\mathrm{GEN}}$ distributed according to
$\zeta$ and is invariant under the SDE (4.16).

An equivalent way of observing the same indeterminacy is the following. If
$\nu_{\beta}^{\zeta}$ is one of the invariant measures that we just
constructed, then there is a family of other invariant measures of the form

$f({\mathcal{E}}_{\mathrm{GEN}}(z,w,e))\,\nu_{\beta}^{\zeta}({\mathrm{d}}z\;\\!\mathrm{d}w\;\\!\mathrm{d}e),\qquad\text{for
arbitrary }f\colon{\mathbb{R}}\to{\mathbb{R}}.$ (4.56)

Indeed,
$f{\circ}{\mathcal{E}}_{\mathrm{GEN}}\,\nu_{\beta}^{\zeta}=\nu_{\beta}^{\zeta_{f}}$
with
$\zeta_{f}({\mathrm{d}}\tilde{e})=f(\tilde{e})\zeta({\mathrm{d}}\tilde{e})$.

This non-uniqueness of the invariant measure generates a corresponding non-
uniqueness in the entropy. The family (4.56) generates the corresponding
family of ‘entropies’

${\mathcal{S}}_{g}(z,w,e):=-\beta{\mathcal{H}}_{zw}(z,w)+g({\mathcal{E}}_{\mathrm{GEN}}(z,w,e)),\quad\text{for
arbitrary }g\colon{\mathbb{R}}\to{\mathbb{R}}.$ (4.57)

The choice ${\mathcal{S}}_{\mathrm{GEN}}(z,w,e):=\beta e$ of (4.20c) is one
member of this family, corresponding to $g(\tilde{e})=\beta\tilde{e}$.

Although (4.57) shows that many functions could be called ‘entropy’, we claim
that ${\mathcal{S}}_{\mathrm{GEN}}(z,w,e):=e$ is the one that should appear in
a GENERIC structure, and we give two reasons for this. The first reason is
that the GENERIC structure requires
${\mathbb{J}}_{\mathrm{GEN}}{\mathrm{D}}{\mathcal{S}}_{\mathrm{GEN}}=0$ (see
(4.3)); within the family (4.57), only “$\beta e+\text{constant}$” satisfies
this. The second reason takes more time to explain, and we do this in the next
section.

###  4.6 Characterization of entropy through finite-dimensional approximations

In this section we give an independent explanation why the expression
${\mathcal{S}}(z,w,e)=\beta e$ should characterize the entropy in the GENERIC
SDE (4.19). The origin of this expression lies in the scaling of ‘surface
area’, and we now describe this in detail.

This interpretation of entropy as ‘surface area’ can be found in some form or
other in many treatments of the connection between statistical mechanics and
thermodynamics (see e.g. [Cha87, Sec. 3.2], [Cho94, p. 69], [Ber97, Sec. 1.5],
or [FrS01, (2.1.6)]). While the particular dependence of the surface area on
$n$, $\beta$, and $e$ given by Lemma 4.13 below must be generally known, we
could not find a reference, and therefore we provide the details here. The
same holds for the full-space equivalence-of-ensembles result of Lemma 4.14.

We choose $(\Xi_{n})_{n\geq 1}$ to be a particular increasing sequence of
$n$-dimensional subspaces of the orthogonal complement
${\mathbf{W}}^{\perp}\subset{\mathbf{H}}$, such that

${\mathbf{W}}^{\perp}=\overline{\operatorname{span}\bigcup_{n\geq
1}\Xi_{n}}^{\|\cdot\|_{\mathbf{H}}}.$

The spaces $\Xi_{n}$ are in the ‘invisible’ part of the heat bath, since
${\mathbb{P}}\Xi_{n}=0$ and ${\mathbb{Q}}\Xi_{n}=\Xi_{n}$. We describe this
choice in more detail in Appendix B.2.

Fix $\beta>0$ and $e\in{\mathbb{R}}$, and for each $n$ let $\zeta_{\beta,n,e}$
be the (non-normalized) measure on $\Xi_{n}$ defined by

$\zeta_{\beta,n,e}({\mathrm{d}}\xi):=\bm{\updelta}\left[\frac{1}{2}\|\xi\|_{\mathbf{H}}^{2}-\frac{n}{2\beta}-e\right](\;\\!\mathrm{d}\xi),$
(4.58)

where $\bm{\updelta}$ is the measure that is heuristically defined by (see
Appendix B.2)

$\int_{\Xi_{n}}\varphi(\xi)\bm{\updelta}\left[f(\xi)-e\right]({\mathrm{d}}\xi):=\lim_{h\downarrow
0}\frac{1}{h}\int_{\Xi_{n}}\varphi(\xi)\mathds{1}\big{\\{}e\leq
f(\xi)<e+h\big{\\}}\;\\!\mathrm{d}\xi\qquad\text{for all }\varphi\in
C_{c}(\Xi_{n}).$

The total mass of the measure
$\bm{\updelta}\left[f(\xi)-e\right]=\bm{\updelta}\left[f(\xi)-e\right]({\mathrm{d}}\xi)$
can be interpreted as the area of the surface $f^{-1}(e)\subset\Xi_{n}$
(weighted by $|\nabla f|$; see Appendix B.2).

In the thermodynamical or statistical-mechanical literature the measure
$\zeta_{\beta,n,e}$ would be called ‘microcanonical’. If the expression
$\frac{1}{2}\|\xi\|_{\mathbf{H}}^{2}$ is interpreted as an energy, as in the
case of this paper, then this measure is concentrated on an ‘energy shell’ at
energy level $n/2\beta+e$. This particular choice of energy level is geared
towards the limit $n\to\infty$: this choice causes each of the $n$ degrees of
freedom in $\Xi_{n}$ to have approximately energy level $1/2\beta$.

The following two properties of this sequence of microcanonical measures
probably are known (see e.g. [Cho94, p. 69] for a calculation similar to the
first one), but for neither we could find a statement that suits our purposes;
for completeness we give a proof in Appendix B.2.

######  Lemma 4.13.

Fix $\beta>0$ and $e\in{\mathbb{R}}$.

  1. 1.

There exists $C(\beta,n)$, converging to $\infty$ as $n\to\infty$, such that

${\mathsf{Z}}_{\beta,n,e}:=\int_{\Xi_{n}}\zeta_{\beta,n,e}({\mathrm{d}}x)$

has the asymptotic behavior

$\log{\mathsf{Z}}_{\beta,n,e}-C(\beta,n)\;\xrightarrow{n\to\infty}\;\beta e.$
(4.59)

  2. 2.

We have the narrow convergence of measures on ${\mathbf{X}}$

$\frac{1}{{\mathsf{Z}}_{\beta,n,e}}\zeta_{\beta,n,e}\relbar\joinrel\rightharpoonup\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}\qquad\text{as
}n\to\infty.$ (4.60)

The first statement above shows that for fixed $\beta$ and $e$,

$\text{surface area scales as}\quad\mathrm{e}^{C(\beta,n)+\beta
e}\qquad\text{as $n\to\infty$.}$

Note how the largest contribution $C(\beta,n)$ diverges as $n\to\infty$, and
the dependence on $e$ remains as an $O(1)$ contribution inside the
exponential.

The second statement above is a version of the ‘equivalence of ensembles’: it
can be interpreted as stating that in the large-$n$ limit the microcanonical
‘ensemble’ $\zeta_{\beta,n,e}$ is similar to the ‘canonical ensemble’
described by the Gaussian measure $\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$
(see Lemma LABEL:l:equivalence-of-ensembles). It differs from the usual
treatment of equivalence-of-ensemble results in that it describes the behavior
of the whole measure, not of a finite-marginal reduction. Note that under the
limit Gaussian measure $\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$ each scalar
degree of freedom has variance $1/\beta$, or equivalently energy $1/2\beta$,
thus matching the scaling choice described above.

For the discussion here it is relevant that the dependence on $e$ in
$\zeta_{\beta,n,e}$ is lost in the convergence (4.60) of the measures
themselves: the limit is independent of $e$. At the same time, the value of
$e$ _can_ be detected by following the $e$-dependence of the partition
function ${\mathsf{Z}}_{\beta,n,e}$, as described by (4.59).

We now apply this observation to the microscopic system of Section 2. We set
${\mathbf{H}}_{n}:={\mathbf{W}}+\Xi_{n}$; note that
${\mathbb{P}}{\mathbf{H}}_{n}={\mathbf{W}}$ and
${\mathbb{Q}}{\mathbf{H}}_{n}=\Xi_{n}$.

Similarly to the construction above we fix an ‘energy level’
${\mathcal{E}}_{0}$ and define the probability measure $\mu_{\beta,n}$ on the
subspace ${\mathbf{Z}}{\times}{\mathbf{H}}_{n}$ by

$\mu_{\beta,n}({\mathrm{d}}z\;\\!\mathrm{d}\eta):=\frac{1}{{\mathsf{Z}}_{\beta,n,{\mathcal{E}}_{0}}}\bm{\updelta}\left[{\mathcal{H}}_{\mathrm{total}}(z,\eta)-\frac{n}{2\beta}-{\mathcal{E}}_{0}\right]({\mathrm{d}}z\;\\!\mathrm{d}\eta).$

By extension we can also consider $\mu_{\beta,n}$ as a probability measure on
${\mathbf{Z}}{\times}{\mathbf{X}}$. In contrast to $\zeta_{\beta,n,e}$ above
we choose to normalize this measure.

The choice of energy level $n/2\beta+{\mathcal{E}}_{0}$ is very similar to the
one above: in the limit $n\to\infty$ the bulk of the energy will be located in
the $n$ degrees of freedom in the invisible part $\Xi_{n}$ of the heat bath,
with approximately energy $1/2\beta$ per degree of freedom. Indeed, we have an
equivalence-of-ensembles statement for the full system that mirrors that of
the heat bath in Lemma 4.13:

######  Lemma 4.14.

As $n\to\infty$ the measure $\mu_{\beta,n}$ converges weakly on
${\mathbf{Z}}{\times}{\mathbf{X}}$ to $\mu_{\beta}$.

Again, only the $O(n)$ scaling of the energy level
${\mathcal{E}}_{0}+n/2\beta$ is visible in the limit $\mu_{\beta}$; the $O(1)$
part given by the choice of ${\mathcal{E}}_{0}$ can not be recovered from the
limit measure $\mu_{\beta}$. The proof is given Appendix B.

We next map this microcanonical measure $\mu_{\beta,n}$ to a new state space
in which we can track the behavior of $z$, $w$, and $e$ independently from the
rest of the heat bath. Defining the closure in ${\mathbf{X}}$ of the $\Xi_{n}$
as

$\Xi^{\mathbf{X}}:=\overline{\bigcup_{n\geq
1}\Xi_{n}}^{\|\cdot\|_{\mathbf{X}}},$

we define the ‘decomposition’ map

$\displaystyle T_{n}$
$\displaystyle\colon{\mathbf{Z}}{\times}{\mathbf{H}}_{n}\longrightarrow{\mathbf{Z}}{\times}{\mathbf{W}}{\times}\Xi^{\mathbf{X}}{\times}{\mathbb{R}},$
$\displaystyle(z,\eta)\longmapsto\left(z,\,w:={\mathbb{P}}\eta,\,\xi={\mathbb{Q}}\eta,\,e:=\frac{1}{2}\|\xi\|^{2}_{\mathbf{H}}-\frac{n}{2\beta}\right),$

and the resulting measure on
${\mathbf{Z}}{\times}{\mathbf{W}}{\times}\Xi^{\mathbf{X}}{\times}{\mathbb{R}}$,

$\mu_{\beta,n,e}:=(T_{n})_{\\#}\mu_{\beta,n}.$ (4.61)

The map $T_{n}$ makes explicit the orthogonal split
$\eta=w+\xi={\mathbb{P}}\eta+{\mathbb{Q}}\eta$, and in addition exposes the
renormalized energy variable $\|\xi\|^{2}/2-n/2\beta$.

Since for $\eta\in{\mathbf{H}}_{n}$ we have

${\mathcal{H}}_{\mathrm{total}}(z,\eta)={\mathcal{H}}_{zw}(z,{\mathbb{P}}\eta)+\frac{1}{2}\|{\mathbb{Q}}\eta\|_{{\mathbf{H}}}^{2},$

the properties of the ‘delta function’ $\bm{\updelta}$ allow us to rewrite
$\mu_{\beta,n,e}$ as (see Lemma B.2)

$\mu_{\beta,n,e}({\mathrm{d}}z\;\\!\mathrm{d}w\;\\!\mathrm{d}\xi\;\\!\mathrm{d}e)=\frac{1}{{\mathsf{Z}}_{\beta,n,{\mathcal{E}}_{0}}}\bm{\updelta}\Big{[}{\mathcal{H}}_{zw}(z,w)-{\mathcal{E}}_{0}+e\Big{]}({\mathrm{d}}z\;\\!\mathrm{d}w)\,\bm{\updelta}\left[\frac{1}{2}\|\xi\|_{\mathbf{H}}^{2}-\frac{n}{2\beta}-e\right]({\mathrm{d}}\xi)\,\;\\!\mathrm{d}e.$
(4.62)

By Lemma 4.13 the final $\bm{\updelta}$-measure has the asymptotic behavior

$\mathrm{e}^{-C(\beta,n)}\bm{\updelta}\left[\frac{1}{2}\|\xi\|_{\mathbf{H}}^{2}-\frac{n}{2\beta}-e\right]({\mathrm{d}}\xi)\
\relbar\joinrel\rightharpoonup\ \mathrm{e}^{\beta
e}\,(\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta})({\mathrm{d}}\xi)\qquad\text{as
}n\to\infty.$

Combining the factor $\mathrm{e}^{-C(\beta,n)}$ with
${\mathsf{Z}}_{\beta,n,{\mathcal{E}}_{0}}$ we thus have proved

######  Lemma 4.15.

As $n\to\infty$ we have the narrow convergence on
${\mathbf{Z}}{\times}{\mathbf{W}}{\times}\Xi^{\mathbf{X}}{\times}{\mathbb{R}}$,

$\displaystyle\mu_{\beta,n,e}\relbar\joinrel\rightharpoonup\mu_{\beta,e},$
(4.63a)
$\displaystyle\mu_{\beta,e}({\mathrm{d}}z\;\\!\mathrm{d}w\;\\!\mathrm{d}\xi\;\\!\mathrm{d}e):=\frac{1}{{\mathsf{Z}}_{\beta,{\mathcal{E}}_{0}}}\,\mathrm{e}^{\beta
e}\,\bm{\updelta}\big{[}{\mathcal{H}}_{zw}(z,w)-{\mathcal{E}}_{0}+e\big{]}({\mathrm{d}}z\;\\!\mathrm{d}w)\;\\!\mathrm{d}e\;(\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta})({\mathrm{d}}\xi).$
(4.63b)

We now use this convergence to understand the interpretation of
${\mathcal{S}}_{\mathrm{GEN}}(z,w,e)=\beta e+\mathrm{constant}$. The
microcanonical measure $\mu_{\beta,n,e}$, written in the form (4.62), can be
interpreted as the product of two surface measures, the first for System A and
the second one for the heat bath. Each surface measure is calculated at some
energy level parametrized by $e\in{\mathbb{R}}$, but neither are normalized;
the normalization happens at the level of the total product, not the
individual factors. This gives rise to a trade-off: as a function of $e$, the
surface area of the heat bath at energy level ‘$c_{1}+e$’ is multiplied by the
surface area of System A at energy level ‘$c_{2}-e$’.

From the discussion above we know that for large $n$ the surface area of the
heat bath at energy level $c_{1}+e$ has the exponential dependence
$\mathrm{e}^{\beta e}$ on $e$. Therefore higher heat-bath energy levels, which
correspond to larger $e$ in (4.62), have a larger contribution in (4.62), and
this dependence gives a downward pressure on the energy of System A. This
downward pressure can be recognized by integrating $\mu_{\beta,e}$ over both
$\xi$ and $e$; we then obtain

$\frac{1}{{\mathsf{Z}}_{\beta,{\mathcal{E}}_{0}}}\int_{\mathbb{R}}\mathrm{e}^{\beta
e}\,\bm{\updelta}\big{[}{\mathcal{H}}_{zw}(z,w)-{\mathcal{E}}_{0}+e\big{]}(B)\;\\!\mathrm{d}e\\\
=\frac{1}{{\mathsf{Z}}_{\beta,{\mathcal{E}}_{0}}}\int_{B}\mathrm{e}^{-\beta{\mathcal{H}}_{zw}(z,w)+\beta{\mathcal{E}}_{0}}\;\\!\mathrm{d}z\;\\!\mathrm{d}w,\qquad\text{for
all }B\subset{\mathbf{Z}}{\times}{\mathbf{W}},$

or more concisely, as measures on ${\mathbf{Z}}{\times}{\mathbf{W}}$,

$\frac{1}{{\mathsf{Z}}_{\beta,{\mathcal{E}}_{0}}}\int_{\mathbb{R}}\mathrm{e}^{\beta
e}\,\bm{\updelta}\big{[}{\mathcal{H}}_{zw}(z,w)-{\mathcal{E}}_{0}+e\big{]}\;\\!\mathrm{d}e=\frac{1}{{\mathsf{Z}}_{\beta,{\mathcal{E}}_{0}}}\mathrm{e}^{-\beta{\mathcal{H}}_{zw}(z,w)+\beta{\mathcal{E}}_{0}}.$

(See Definition B.1 for this type of manipulation.) The expression on the
right-hand side coincides with (3.5), and shows how the increase in heat-bath
surface area with heat-bath energy penalizes energy increases in System A.

Summarizing, the formula ${\mathcal{S}}_{\mathrm{GEN}}(z,w,e)=\beta
e+\mathrm{constant}$ can be interpreted as a characterization of the
dependence of microcanonical heat-bath surface area on heat-bath energy, and
the discussion above shows how this dependence has a depressing effect on the
energy in System A.

##  5 Conclusion and discussion

While the coarse-graining of Hamiltonian systems is a classical topic, the
introduction of the compression hypothesis from dilation theory gives a new
type of insight into this question. Specifically, we have shown how the
compression hypothesis gives an abstract characterization of heat-bath
behaviour that automatically generates two important effects.

First, the existence of a compression means in particular that the Hamiltonian
evolution of the coupled system can be projected to the finite-dimensional
subspace ${\mathbf{Z}}{\times}{\mathbf{W}}$. Secondly, and just as
importantly, the compression property implies that the combination of random
heat-bath initial data, Hamiltonian propagation, and projection generates a
noise that is memoryless, leading to the Brownian noise terms in the equations
(3.3) and (4.16). This is best expressed by Lemma 3.2, and the proof of that
lemma clearly shows how the projection and Hamiltonian propagation combine to
give the covariance function of an Ornstein-Uhlenbeck process.

In addition, the same compression hypothesis has a clear connection to the
GENERIC structure. The operator ${\mathbb{D}}$ that appears in the compression
hypothesis generates terms in both the Poisson operator
${\mathbb{J}}_{\mathrm{GEN}}$ and the Onsager operator
${\mathbb{K}}_{\mathrm{GEN}}$: the skew-symmetic part of ${\mathbb{D}}$
contributes to ${\mathbb{J}}_{\mathrm{GEN}}$, and the symmetric part to
${\mathbb{K}}_{\mathrm{GEN}}$. In this way the compression hypothesis
generates new structure in the (formerly) Hamiltonian system, which turns out
to be GENERIC structure.

Finally, the fluctuations generated by the noise are linked to the Onsager
operator ${\mathbb{K}}_{\mathrm{GEN}}$ by the fluctuation-dissipation relation
(4.21). This puts the GENERIC framework on a firmer footing, and gives
modelling insight into the various components of GENERIC.

###  5.1 Relation with models and results in the literature

Coupled Hamiltonian systems consisting of a fixed system and a ‘heat bath’
have been studied by many authors. The earliest example that we could find is
by Zwanzig [Zwa73a], and similar systems return in [Zwa05, Zwa01b, KS∗02b]. In
all of these works the heat bath is finite-dimensional; infinite-dimensional
heat baths were used in the related work of Jaksic and Pillet [JaP98b] and
Rey-Bellet [Rey06].

The fact that a particular tuning of the properties of the heat bath and its
projector can lead to Markovianity of the projected subsystem was observed by
Ford, Kac, and Mazur [FKM65], and later used by many others [Zwa73b, Zwa05].
Stuart and co-workers [StW99, KS∗02b] appear to be the first to give a
rigorous convergence statement in the limit of infinite dimensions of the heat
bath (and the Compression Property 1.3 is effectively equivalent to the choice
of the frequencies of the $n$ oscillators in these two papers).

Very similar ideas were also used to explain dissipation in quantum systems
(so-called open quantum systems), in particular in the famous Caldeira-Leggett
model [CaL83b, CaL83a], see also [ToH20] where the heat bath is interpreted in
terms of a phonon-number representation. The straightforward usage of the
classical dilation theory leads to so-called quantum-state diffusion, see e.g.
[GiP92], which does not fully account for the true nature of quantum
probability. A genuine dilation theory for open quantum system needs a quantum
probability theory, see e.g. [AFG84, Mey95].

######  Remark 5.1.

The setup of Jaksic and Pillet [JaP98a] is similar to ours, but their
condition (H3) effectively excludes the case of this paper. This is due to the
already-remarked fact that
${\mathbf{W}}\cap\mathrm{dom}({\mathbb{J}}_{\mathcal{B}})=\\{0\\}$, which is
proved in Remark LABEL:re:Dom.JB.bfW. Jaksic and Pillet find from their (H3)
that their equivalent of the process $Y_{t}$ has $C^{1}$ dependence on the
initial datum $(z_{0},\eta_{0})$ (their Theorem 2.1), while because of the
lack of continuity we only obtain measurability.  ∎

The coarse-graining of this paper is closely related to the ‘Mori-Zwanzig’
reduction method [Mor65, Zwa61, Zwa01a, CHK00, Sch22]. The first step in this
method is the expression of the ‘hidden’ variables in terms of the ‘visible’
variables. In in this paper equation (2.10) plays the same role. The integral
term characterizes the changes in $z$ that are generated by $z$ itself but
through the heat bath:

$\int_{0}^{t}\underbrace{{\mathsf{C}}^{*}\mathrm{e}^{(t-s){\mathbb{J}}_{\mathcal{B}}}{\mathsf{C}}}_{=:\,K(t-s)}\dot{z}_{s}\;\\!\mathrm{d}s.$

The Compression Hypothesis (1.5) amounts to assuming that there exists a
finite-dimensional subspace ${\mathbf{W}}$ with corresponding orthogonal
projection ${\mathbb{P}}\colon{\mathbf{H}}\to{\mathbf{W}}$ and a dissipative
operator ${\mathbb{D}}$ on ${\mathbf{W}}$ such that

$\text{(a)}\qquad{\mathsf{C}}^{*}{\mathbb{P}}={\mathsf{C}}^{*}\qquad\text{and\qquad(b)}\qquad
K(t)={\mathsf{C}}^{*}\mathrm{e}^{-t{\mathbb{D}}}{\mathsf{C}}\qquad\text{for
}t\geq 0.$ (5.1)

Condition (a) means that ${\mathbf{W}}$ contains all information necessary to
characterize the effect of the heat bath on $z$. Condition (b), on the other
hand, amounts to an assumption that $K$ can be written as a finite sum of
exponentials (one for each of the coordinates); this is sometimes used as an
approximation in numerical algorithms [Sch22] (see also [StS19]).

The compression property bypasses the usual solving of the ‘hidden-variables’
equation, which in the case of this paper would involve the orthogonal
component $\xi_{t}=\widehat{\mathbb{Q}}\eta_{t}$. The formal generator of this
‘orthogonal dynamics’ should be either
${\mathbb{Q}}{\mathbb{J}}_{\mathcal{B}}$ or
${\mathbb{Q}}{\mathbb{J}}_{\mathcal{B}}{\mathbb{Q}}$, which both are not well
defined and tend to be difficult to study, even formally. In the context of
the compression property this dynamics is bypassed, and the effect of the
hidden variables on the visible ones is instead fully characterized by (5.1).

Mori-Zwanzig reduction schemes come in two forms, depending on whether the
split is performed in the state space itself (as in this paper) or in the
space of probability measures or ‘ensembles’ on the state space. The latter
approach has the advantage of lifting a nonlinear evolution to a linear one,
thereby allowing the application of linear theory such as the Mori-Zwanzig
reformulation. In the case of this paper, we can apply the simpler, first
type, since the nonlinearity is confined to System A.

###  5.2 Relation between positive and zero temperature

In the zero-temperature limit, $\beta\to\infty$, the positive-temperature
equations (3.3) and (4.16) converge to the corresponding deterministic ODEs
(3.2) and (4.15). This can be recognized in the characterization
$\Sigma\Sigma^{*}=\beta^{-1}({\mathbb{D}}+{\mathbb{D}}^{*})$ of the noise
intensity, which implies that $\Sigma\to 0$ as $\beta\to\infty$.

It is a natural question whether the GENERIC structure itself converges. The
answer is that it does not, at least not in this form. The entropy
${\mathcal{S}}_{\mathrm{GEN}}=\beta e+\text{constant}$ becomes singular as
$\beta\to\infty$, and the Onsager operator ${\mathbb{K}}_{\mathrm{GEN}}$
vanishes. The product
${\mathbb{K}}_{\mathrm{GEN}}{\mathrm{D}}{\mathcal{S}}_{\mathrm{GEN}}$,
however, is independent of $\beta$, and this suggests that in order to study
this limit it would make sense to rescale by

${\mathcal{S}}_{{\mathrm{GEN}}}^{\text{rescaled}}:=\frac{1}{\beta}{\mathcal{S}}_{\mathrm{GEN}},\qquad\text{and}\qquad{\mathbb{K}}_{\mathrm{GEN}}^{\text{rescaled}}:=\beta{\mathbb{K}}_{\mathrm{GEN}}.$

The rescaled objects ${\mathcal{S}}_{{\mathrm{GEN}}}^{\text{rescaled}}$ and
${\mathbb{K}}_{\mathrm{GEN}}^{\text{rescaled}}$ are independent of $\beta$,
and therefore trivially converge.

As observed in Section 1.4, the deterministic GENERIC evolution, however, is
independent of $\beta$.

###  5.3 Well-preparedness of the heat bath at time zero

The central construction of the microscopic evolution in this paper is done in
part 3 of Theorem 2.5. There the initial state of the ‘visible’ part of the
heat bath $w_{0}=\widehat{\mathbb{P}}\eta_{0}$ is assumed to be given, while
the ‘invisible’ part $\xi_{0}=\widehat{\mathbb{Q}}\eta_{0}$ is drawn randomly
from the corresponding conditioned equilibrium measure
$\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$ of the heat bath. This randomness
then generates the randomness in the SDE (3.3) of the projected system
$(z,w)$.

In the philosophical debate about irreversibility and coarse-graining (see
e.g. [Rob20]) such a setup has been criticized: it amounts to an assumption
that the microscopic state at time $t=0$ is rather special. This can be
recognized in the fact that while the initial hidden variables
$\xi=\widehat{\mathbb{Q}}\eta$ have distribution
$\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$ at time 0, they _will not_ have
that same distribution at later or earlier times. (The distribution
$\gamma_{\beta}$ is invariant under the heat-bath evolution, but
$\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$ is not.)

In fact, it turns out that the altered distribution of $\xi$ at times
different from zero _does not_ change the effect of the coarse-graining. To
illustrate this, consider starting the heat bath in part 3 of Theorem 2.5 at
some time $t_{0}\in{\mathbb{R}}$ with distribution
$\xi_{t_{0}}\sim\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$. At time $t=0$ the
corresponding state equals $\widehat{S}_{-t_{0}}\xi_{t_{0}}$. Note that this
propagated state is distributed according to
$(\widehat{S}_{-t_{0}}\widehat{\mathbb{Q}})_{\\#}\gamma_{\beta}$, and is not
completely hidden
($\widehat{\mathbb{P}}\widehat{S}_{-t_{0}}\xi_{t_{0}}\not=0$). In equation
(2.10) describing the evolution of $z$, only the first term changes:

$\dot{z}_{t}={\mathbb{J}}_{\mathcal{A}}{\mathsf{C}}^{*}Y_{t-t_{0}}(\xi_{t_{0}})+{\mathbb{J}}_{\mathcal{A}}\left({\mathrm{D}}{\mathcal{H}}_{\mathcal{A}}(z)+{\mathsf{C}}^{*}(\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}{\mathsf{C}}z_{0}-{\mathsf{C}}z_{t})\right)+{\mathbb{J}}_{\mathcal{A}}\int_{0}^{t}{\mathsf{C}}^{*}\mathrm{e}^{(t-s){\mathbb{J}}_{\mathcal{B}}}{\mathsf{C}}\dot{z}_{s}\;\\!\mathrm{d}s.$
(5.2)

However, under the compression property by Lemma 2.4 the process $t\mapsto
Y_{t-t_{0}}(\xi_{t_{0}})$ is again a non-stationary Ornstein-Uhlenbeck
process; in other words the _noise_ in this process is stationary, even though
the state at time $t=0$ is not in equilibrium. It follows that the noise
generated by the heat bath has the same distribution, also if one starts with
a hidden variable of the form $\widehat{S}_{-t_{0}}\xi_{t_{0}}$, where
$\xi_{t_{0}}\sim\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$.

###  5.4 Other derivations of GENERIC by coarse-graining

##### Öttinger’s coarse-graining of Hamiltonian systems.

The closest in spirit to this paper is the coarse-graining approach by
Öttinger [Ö05, Ch. 6]. The starting point is again a Hamiltonian system, and a
central role is played by a ‘coarse-graining map’ similar to $\pi_{z,w,e}$ in
(1.7). The two approaches differ in the definition of the coarse-grained
state: while in this paper we consider $\pi_{z,w,e}(z,\eta)$ itself to be the
coarse-grained state (up to renormalization of the infinite energy), in [Ö05,
Ch. 6] the coarse-grained state is in effect a _distribution_ $\rho_{z,w,e}$
on ${\mathbf{Z}}{\times}{\mathbf{X}}$, parametrized by $(z,w,e)$.

It is tempting to consider Öttinger’s distribution $\rho_{z,w,e}$ as an
approximation of the conditional distribution of $(z,\eta)$ given the value of
$\pi_{z,w,e}(z,\eta)$. Unfortunately this approach seems to fail: in simple
examples one can verify that the _actual_ conditional distribution of
$(z,\eta)$ is different from the $\rho_{z,w,e}$ postulated in Öttinger’s
approach, sufficiently different to make it unclear why $\rho_{z,w,e}$ should
be a good approximation for the purpose of coarse-graining. While there are
clear parallels between the two approaches, at this stage it is not clear to
us whether they should be considered effectively the same or not.

##### Obtaining GENERIC from large deviations.

GENERIC has also been derived from large-deviation principles of stochastic
processes. In [DPZ13], Duong and two of us showed how the GENERIC structure of
the kinetic Fokker-Planck equation can be recognized in the large-deviations
rate functional of a sequence of ‘more microscopic’ stochastic interacting
particle systems. Since a large-deviations principle can also be interpreted
as a form of coarse-graining, this gives an independent derivation of the
GENERIC structure, but from a process with stochastic forcing and
deterministic initial data, the opposite of the setting of this paper. In
[KL∗18, KL∗20] the arguments of [DPZ13] were generalized to a larger class of
systems.

##  Appendix A Properties of Gaussian measures

In this appendix we recall a number of properties of Gaussian measures that we
use in this paper. We will mostly use the notation and definitions of the book
on Gaussian Measures by Bogachev [Bog98].

###  A.1 Basic setup

In ${\mathbb{R}}$, a _centered Gaussian random variable_ $\eta$ with variance
$\sigma^{2}$, written as $\eta\sim{\mathcal{N}}(0,\sigma^{2})$, has the
probability density

$\mu({\mathrm{d}}\eta)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{\eta^{2}}{2\sigma^{2}}\right)\;\\!\mathrm{d}\eta.$

Note that such a scalar random variable has the equivalent formulation

$\text{for all $t\in{\mathbb{R}}$,
}\quad{\mathbb{E}}\left[\mathrm{e}^{it\eta}\right]=\mathrm{e}^{-\frac{\sigma^{2}t^{2}}{2}}.$

Centered Gaussian random variables on ${\mathbb{R}}^{n}$ have the probability
density

$\mu({\mathrm{d}}\eta)=\frac{1}{\sqrt{(2\pi)^{n}}\det(\Sigma)}\exp\left(-\frac{1}{2}\eta^{T}\Sigma^{-2}\eta\right)\;\\!\mathrm{d}\eta,$
(A.1)

where $\Sigma$ is a symmetric positive definite $n\times n$ matrix. We now
define the $n$-dimensional Hilbert space ${\mathbf{H}}$ to be
${\mathbb{R}}^{n}$ with the norm
$\|\eta\|_{\mathbf{H}}^{2}:=\eta^{T}\Sigma^{-2}\eta$, which implies that the
dual norm is $\|f\|_{{\mathbf{H}}^{*}}=f^{T}\Sigma^{2}f$. Then the expression
above takes the form

$\mu({\mathrm{d}}\eta)=\frac{1}{{\mathsf{Z}}}\exp\left(-\frac{1}{2}\|\eta\|_{\mathbf{H}}^{2}\right)\;\\!\mathrm{d}\eta$

This leads to the following equivalent characterizations of centered Gaussian
measures on any finite-dimensional Hilbert space ${\mathbf{H}}$.

######  Lemma A.1.

Let ${\mathbf{H}}$ be a finite-dimensional real Hilbert space, let
$\mu\in{\mathcal{P}}({\mathbf{H}})$, and fix $\beta>0$. The following are
equivalent:

$\displaystyle\mu({\mathrm{d}}\eta)=\frac{1}{{\mathsf{Z}}}\exp\left(-\frac{\beta}{2}\|\eta\|_{\mathbf{H}}^{2}\right)\;\\!\mathrm{d}\eta;$
(A.2) For all $f\in{\mathbf{H}}^{*}$,
$f(\eta)\sim{\mathcal{N}}_{\mathbb{R}}\big{(}0,\beta^{-1}\|f\|_{{\mathbf{H}}^{*}}^{2}\big{)}$;
(A.3) For all $h\in{\mathbf{H}}$,
$\left(h\middle|\eta\right)_{\mathbf{H}}\sim{\mathcal{N}}_{\mathbb{R}}\big{(}0,\beta^{-1}\|h\|_{\mathbf{H}}^{2}\big{)}$.
(A.4)

In this case, if $\eta\sim\mu$, then for any bounded linear operator
${\mathbb{O}}:{\mathbf{H}}\to{\mathbf{H}}$, we have

${\mathbb{E}}\left({\mathbb{O}}\eta\middle|\eta\right)_{{\mathbf{H}}}=\operatorname{tr}_{\mathbf{H}}{\mathbb{O}}.$
(A.5)

Formulations (A.3) and (A.4) above can be interpreted as stating that the
covariance structure of the measure $\mu$ is given by the norms of
${\mathbf{H}}$ and ${\mathbf{H}}^{*}$, as in (A.1). In this paper we need
$\eta$ to be infinite-dimensional (see for instance Remark
LABEL:re:Dom.JB.bfW), while still having covariance structure determined by
${\mathbf{H}}$ and ${\mathbf{H}}^{*}$. The following lemma allows us to
construct such Gaussian measures.

The critical step is that we need to consider larger spaces ${\mathbf{X}}$,
such that ${\mathbf{H}}$ is Hilbert-Schmidt-embedded into ${\mathbf{X}}$; the
measure is then defined on ${\mathbf{X}}$, and the random variable $\eta$
takes values in ${\mathbf{X}}$.

######  Lemma A.2.

Let ${\mathbf{H}}$ and ${\mathbf{X}}$ be finite- or infinite-dimensional real
Hilbert spaces, and let $\mu\in{\mathcal{P}}({\mathbf{X}})$. If the embedding
${\mathbf{H}}\hookrightarrow{\mathbf{X}}$ is Hilbert-Schmidt, then there
exists a measure $\mu\in{\mathcal{P}}({\mathbf{X}})$ with the properties

For all $f\in{\mathbf{X}}^{*}$,
$f(\eta)\sim{\mathcal{N}}_{\mathbb{R}}\big{(}0,\beta^{-1}\|f\|_{{\mathbf{H}}^{*}}^{2}\big{)}$.
(A.6) For all $x\in{\mathbf{X}}$,
$\left(x\middle|\eta\right)_{\mathbf{X}}\sim{\mathcal{N}}_{\mathbb{R}}\big{(}0,\left(\operatorname{cov}_{\mu}x\middle|x\right)_{\mathbf{X}}\big{)}$.
(A.7)

Here $\operatorname{cov}_{\mu}$ is a self-adjoint non-negative trace-class
operator on ${\mathbf{X}}$. If
$f(\eta)=\left(x\middle|\eta\right)_{\mathbf{X}}$ for some $x\in{\mathbf{X}}$,
then
$\left(\operatorname{cov}_{\mu}x\middle|x\right)_{\mathbf{X}}=\beta^{-1}\|f\|_{{\mathbf{H}}^{*}}^{2}$.

This generalises the finite-dimensional case, where one can take
${\mathbf{H}}={\mathbf{X}}$ and
$\operatorname{cov}_{\mu}=\beta^{-1}\mathrm{id}$, and the properties above
reduce to those of Lemma A.1.

######  Remark A.3 (Definition of infinite-dimensional Gaussian measures).

The usual definition of a Gaussian measure on an infinite-dimensional space
${\mathbf{X}}$ is that for each bounded linear form $f$ on ${\mathbf{X}}$ the
projected measure $f_{\\#}\mu$ is a one-dimensional Gaussian measure (e.g.
[Bog98, Def. 2.2.1]); since ${\mathbf{X}}$ is Hilbert, each such $f$ is an
inner product with an $x\in{\mathbf{X}}$, and this coincides with (A.7).  ∎

######  Remark A.4 (Necessity of Hilbert-Schmidt embedding).

The formulation of Lemma A.2 suggests that the embedding
${\mathbf{H}}\hookrightarrow{\mathbf{X}}$ needs to be Hilbert-Schmidt for the
measure $\mu$ to exist, and indeed this is necessary. In particular there is
no Gaussian measure on an infinite-dimensional space with the identity
operator as covariance.  ∎

The construction above concerns centered Gaussian measures. Non-centered
measures are constructed by translating by $a\in{\mathbf{X}}$, resulting in a
random variable with expectation $a$. We write the corresponding measure as
${\mathcal{N}}_{\mathbf{X}}(a,\mathrm{cov})$. Formally, such a measure
$\gamma$ has the form

$\frac{1}{{\mathsf{Z}}}\mathrm{e}^{-\frac{1}{2}\left(\mathrm{cov}^{-1}(\eta-a)\middle|(\eta-a)\right)_{{\mathbf{X}}}}\;\\!\mathrm{d}\eta.$
(A.8)

This is the sense in which we construct the ‘canonical’ Gaussian measure with
formal structure (2.6).

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

Writing $\mathrm{id}$ for the embedding
${\mathbf{H}}\hookrightarrow{\mathbf{X}}$, the operator
${\mathbb{C}}:=\mathrm{id}\,\mathrm{id}^{*}$ is a self-adjoint non-negative
trace-class operator; by [Bog98, Th. 2.3.1] there exists a Gaussian measure
$\mu$ with mean zero and covariance operator
$\operatorname{cov}_{\mu}=\beta^{-1}{\mathbb{C}}$, which therefore satisfies
(A.7) by the definition of a Gaussian measure.

From the definition of $\mathrm{id}:{\mathbf{H}}\hookrightarrow{\mathbf{X}}$
we have
$\left(x\middle|\eta\right)_{\mathbf{X}}=\left(\mathrm{id}^{*}x\middle|\eta\right)_{\mathbf{H}}$.
For $f\in{\mathbf{X}}^{*}$ of the form
$f(\eta)=\left(x\middle|\eta\right)_{\mathbf{X}}=\left(\mathrm{id}^{*}x\middle|\eta\right)_{\mathbf{H}}$
for some $x\in{\mathbf{X}}$, we then calculate

$\displaystyle\|f\|_{{\mathbf{H}}^{*}}$
$\displaystyle=\sup_{\eta\in{\mathbf{H}}}\frac{f(\eta)}{\|\eta\|_{\mathbf{H}}}=\sup_{\eta\in{\mathbf{H}}}\frac{\left(\mathrm{id}^{*}x\middle|\eta\right)_{\mathbf{H}}}{\|\eta\|_{\mathbf{H}}}=\left(\mathrm{id}^{*}x\middle|\mathrm{id}^{*}x\right)_{\mathbf{H}}=\left({\mathbb{C}}x\middle|x\right)_{\mathbf{X}},$

by which also (A.6) follows from (A.7). ∎

###  A.2 Construction of ${\mathbf{X}}$ by Gelfand triple

In Section 2.2 we aim to construct a Gaussian measure of the formal structure

$\mathrm{e}^{-\frac{\beta}{2}\|\eta\|^{2}_{\mathbf{H}}}\;\\!\mathrm{d}\eta.$
(A.9)

This expression can be made rigorous as a Gaussian measure on a larger Hilbert
space ${\mathbf{X}}$ with the property that ${\mathbf{H}}$ admits a Hilbert-
Schmidt embedding into ${\mathbf{X}}$. While essentially ‘any’ such embedding
is possible (see [Bog98, Lemma 3.2.2]), we choose a particular one based on
the well-known Gelfand triple. We now describe this setting, which is
visualized Fig. A.1.

${{\mathbf{V}}}$${{{\mathbf{H}}\simeq{\mathbf{H}}^{*}}}$${{{\mathbf{V}}^{*}}}$${{{\mathbf{X}}^{*}}}$${{\mathbf{X}}}$${{\mathbb{A}}}$${{\mathbb{A}}}$
definition Figure A.1: The Gelfand-triple setup. The top row is the classical
Gelfand triple
${\mathbf{V}}\hookrightarrow{\mathbf{H}}\simeq{\mathbf{H}}^{*}\hookrightarrow{\mathbf{V}}^{*}$,
and the bottom row connects this triple to the notation of this paper. The
space ${\mathbf{X}}$ is defined as ${\mathbf{V}}^{*}$, and ${\mathbf{X}}^{*}$
is defined as the space of bounded linear maps on ${\mathbf{X}}$. The two
dotted lines indicate the two Riesz identifications that would be possible in
this diagram; because of the risk of confusion we make these identifications
explicitly, not implicitly (see (A.13) below).

The space ${\mathbf{X}}$ is implicitly characterized by the choice of an
unbounded, densely defined, self-adjoint linear operator
${\mathbb{A}}\colon\mathrm{Dom}({\mathbb{A}})\subset{\mathbf{H}}\to{\mathbf{H}}$
with the property that
${\mathbb{A}}^{-1}\colon{\mathbf{H}}\to\mathrm{Dom}({\mathbb{A}})$ is well-
defined and Hilbert-Schmidt. Given such an operator we construct
${\mathbf{X}}$ in the following steps.

  1. 1.

We define ${\mathbf{V}}:=\mathrm{Dom}({\mathbb{A}})$, which is a Hilbert space
with norm $\|v\|_{\mathbb{V}}:=\|{\mathbb{A}}v\|_{\mathbf{H}}$. This implies
that ${\mathbf{V}}={\mathbb{A}}^{-1}{\mathbf{H}}$.

  2. 2.

We identify ${\mathbf{H}}^{*}$ with ${\mathbf{H}}$ and define the norm
$\|\cdot\|_{{\mathbf{V}}^{*}}$ on ${\mathbf{H}}$ by

$\frac{1}{2}\|h\|_{{\mathbf{V}}^{*}}^{2}:=\sup_{v\in{\mathbf{V}}}\left(h\middle|v\right)_{\mathbf{H}}-\frac{1}{2}\|v\|_{\mathbf{V}}^{2}\qquad\text{for
}h\in{\mathbf{H}}.$

We then define ${\mathbf{V}}^{*}$ as the closure of ${\mathbf{H}}$ in the norm
$\|\cdot\|_{{\mathbf{V}}^{*}}$. By construction the elements of
${\mathbf{V}}^{*}$ are in duality with ${\mathbf{V}}$, and we write
${}_{{\mathbf{V}}^{*}}\\!\left\langle\xi,v\right\rangle_{\mathbf{V}}$ for the
dual product. We then also have ${\mathbf{H}}\hookrightarrow{\mathbf{V}}^{*}$,
in the sense that any $h\in{\mathbf{H}}$ is a bounded linear form on
${\mathbf{V}}$ by

${}_{{\mathbf{V}}^{*}}\\!\left\langle
h,v\right\rangle_{\mathbf{V}}:=\left(h\middle|v\right)_{\mathbf{H}}\qquad\text{for
all }v\in{\mathbf{V}}.$

  3. 3.

We extend ${\mathbb{A}}^{-1}$ as an operator ${\mathbf{V}}^{*}\to{\mathbf{H}}$
by setting for any $\xi\in{\mathbf{V}}^{*}$

$\left({\mathbb{A}}^{-1}\xi\middle|h\right)_{\mathbf{H}}:={}_{{\mathbf{V}}^{*}}\\!\left\langle\xi,{\mathbb{A}}^{-1}h\right\rangle_{\mathbf{V}}\qquad\text{for
all }h\in{\mathbf{H}}.$

We can then write

$\|\xi\|_{{\mathbf{V}}^{*}}=\|{\mathbb{A}}^{-1}\xi\|_{\mathbf{H}}\qquad\text{for
all }\xi\in{\mathbf{V}}^{*}.$ (A.12)

The extension is also a symmetric operator in ${\mathbf{V}}^{*}$, since for
$h_{1},h_{2}\in{\mathbf{H}}$ we have

$\left({\mathbb{A}}^{-1}h_{1}\middle|h_{2}\right)_{{\mathbf{V}}^{*}}=\left(h_{1}\middle|{\mathbb{A}}^{-1}h_{2}\right)_{\mathbf{H}}=\left(h_{1}\middle|{\mathbb{A}}^{-1}h_{2}\right)_{{\mathbf{V}}^{*}}.$

We similarly extend ${\mathbb{A}}$ as an operator
${\mathbf{H}}\to{\mathbf{V}}^{*}$.

  4. 4.

We define ${\mathbf{X}}:={\mathbf{V}}^{*}$. The space ${\mathbf{X}}$ is a
Hilbert space with norm
$\|\xi\|_{\mathbf{X}}=\|{\mathbb{A}}^{-1}\xi\|_{\mathbf{H}}$ for all
$\xi\in{\mathbf{X}}$. We have ${\mathbf{H}}\hookrightarrow{\mathbf{X}}$ as
mentioned above, with
$\|h\|_{{\mathbf{X}}}=\|{\mathbb{A}}^{-1}h\|_{\mathbf{H}}$; this shows that
the embedding ${\mathbf{H}}\hookrightarrow{\mathbf{X}}$ is Hilbert-Schmidt.

  5. 5.

We characterize elements of the dual space

${\mathbf{X}}^{*}:=\left\\{f\colon{\mathbf{X}}\to{\mathbb{R}}\text{ linear and
bounded }\right\\}$

in two different ways, by applying Riesz representation on either
${\mathbf{X}}$ or ${\mathbf{H}}$, leading to an element $x_{f}\in{\mathbf{X}}$
or equivalently an element $v_{f}\in{\mathbf{V}}$:

$\displaystyle f(\xi)$
$\displaystyle=\left(\xi\middle|x_{f}\right)_{\mathbf{X}}$
$\displaystyle\qquad\text{for all }\xi\in{\mathbf{X}},$ (A.13a)
$\displaystyle={}_{{\mathbf{V}}^{*}}\\!\left\langle\xi,v_{f}\right\rangle_{\mathbf{V}}$
$\displaystyle\qquad\text{for all }\xi\in{\mathbf{X}},$ (A.13b)
$\displaystyle=\left(\xi\middle|v_{f}\right)_{\mathbf{H}}$
$\displaystyle\qquad\text{whenever }\xi\in{\mathbf{H}}.$ (A.13c)

We have $x_{f}={\mathbb{A}}^{2}v_{f}$ and
$\|f\|_{{\mathbf{X}}^{*}}=\|x_{f}\|_{\mathbf{X}}=\|v_{f}\|_{\mathbf{V}}$.

The setup above defines the space ${\mathbf{X}}$. In order to define the
measure (A.9) we define the trace-class operator

${\mathbb{C}}:={\mathbb{A}}^{-2}\colon{\mathbf{X}}\to{\mathbf{X}}\qquad\text{or
equivalently}\qquad\left({\mathbb{C}}\xi\middle|\xi\right)_{\mathbf{X}}:=\|{\mathbb{A}}^{-1}\xi\|_{\mathbf{X}}^{2}\text{
for }\xi\in{\mathbf{X}}.$

We then set, for $\beta>0$, $\gamma_{\beta}$ to be the Gaussian measure on
${\mathbf{X}}$

$\gamma_{\beta}={\mathcal{N}}_{{\mathbf{X}}}(0,\beta^{-1}{\mathbb{C}}).$

This can be considered a rigorous version of (A.9), because from
$\|\eta\|_{\mathbf{X}}=\|{\mathbb{A}}^{-1}\eta\|_{\mathbf{H}}$ we deduce that

$\mathrm{e}^{-\frac{\beta}{2}\|\eta\|_{\mathbf{H}}^{2}}=\mathrm{e}^{-\frac{\beta}{2}\|{\mathbb{A}}\eta\|_{\mathbf{X}}^{2}}=\mathrm{e}^{-\frac{\beta}{2}\left({\mathbb{A}}^{2}\eta\middle|\eta\right)_{\mathbf{X}}}=\mathrm{e}^{-\frac{\beta}{2}\left({\mathbb{C}}^{-1}\eta\middle|\eta\right)_{\mathbf{X}}},$

which matches with (A.8).

Note that ${\mathbb{C}}$ also is trace-class as a map
${\mathbf{H}}\to{\mathbf{H}}$, since in terms of an orthonormal basis $e_{k}$
of ${\mathbf{H}}$ we can write

$\operatorname{tr}_{\mathbf{H}}{\mathbb{C}}=\sum_{k=1}^{\infty}\left({\mathbb{C}}e_{k}\middle|e_{k}\right)_{\mathbf{H}}=\sum_{k=1}^{\infty}\|{\mathbb{A}}^{-1}e_{k}\|_{\mathbf{H}}^{2},$

and ${\mathbb{A}}^{-1}$ is Hilbert-Schmidt on ${\mathbf{H}}$.

######  Remark A.5 (Positive temperature implies infinite energy).

A well-known property of infinite-dimensional Gaussian measures such as
$\gamma_{\beta}$ is that for any $\beta>0$, $\gamma_{\beta}({\mathbf{H}})=0$.
This shows that if $\eta$ is distributed according to $\gamma_{\beta}$, then
the natural energy $\frac{1}{2}\|\eta\|_{\mathbf{H}}^{2}$ is almost surely
infinite. See e.g. [Rey06, Sec. 2.1] or [Bog98, Rem. 2.2.3] for a discussion
of this property.  ∎

To facilitate applying the theorems of [Bog98] we make explicit the
relationship between the objects of this paper and the notation of [Bog98].
The covariance bilinear form $R_{\gamma_{\beta}}$ is given by

$R_{\gamma_{\beta}}(f)(f)=\frac{1}{\beta}\left({\mathbb{C}}x_{f}\middle|x_{f}\right)_{\mathbf{X}}\qquad\text{for
$f\in{\mathbf{X}}^{*}$ and for $x_{f}$ given
by~{}\eqref{eq:identification-X*-xf}},$ (A.14)

and the Cameron-Martin Hilbert space $H(\gamma_{\beta})$ coincides with
${\mathbf{H}}$, up to a scaling of the norm:

$\|h\|_{H(\gamma_{\beta})}^{2}=\beta\|h\|_{\mathbf{H}}^{2}\qquad\text{for all
}h\in{\mathbf{H}}.$ (A.15)

###  A.3 Measurable extensions of linear maps

In Section 2 we constructed several extensions of linear operators on
${\mathbf{H}}$ to ${\mathbf{X}}$. In this section we give the details of these
extensions.

######  Definition A.6 (Measurable linear extensions of [Bog98, Def.
2.10.1+3.7.1]).

Let ${\mathbf{Y}}$ be a normed linear space, and let
$f\colon{\mathbf{H}}\to{\mathbf{Y}}$ be given. If there exists a subspace
${\mathbf{L}}$ with ${\mathbf{H}}\subset{\mathbf{L}}\subset{\mathbf{X}}$ and
$\gamma_{\beta}({\mathbf{L}})=1$ and a linear map
$\widehat{f}\colon{\mathbf{L}}\to{\mathbf{Y}}$, measurable with respect to
$({\mathcal{B}}({\mathbf{X}})_{\gamma_{\beta}},{\mathcal{B}}({\mathbf{Y}}))$,
such that $\widehat{f}$ coincides with $f$ on ${\mathbf{L}}$, then we call
$\widehat{f}$ a _measurable linear extension_ of $f$.

Such an extension is characterized ${\gamma_{\beta}}$-a.e. by its values on
${\mathbf{H}}$ [Bog98, Cor. 2.10.8], and therefore can be considered
$\gamma_{\beta}$-unique.

Note that if $\eta\sim\gamma_{\beta}$ and $g$ is a measurable map from
${\mathbf{X}}$ to ${\mathbf{Y}}$, then $g(\eta)$ has distribution
$g_{\\#}\gamma_{\beta}$, the push-forward measure of $\gamma_{\beta}$ under
$g$, defined by

$\int_{\mathbf{Y}}\varphi(\xi)(g_{\\#}\gamma_{\beta})({\mathrm{d}}\xi)=\int_{\mathbf{X}}\varphi(g(\eta))\gamma_{\beta}({\mathrm{d}}\eta)\qquad\text{for
all }\varphi\in C_{b}({\mathbf{Y}}).$

######  Lemma A.7 (Extensions of bounded linear operators on ${\mathbf{H}}$).

Let $\eta$ have distribution $\gamma_{\beta}$.

  1. 1.

If $h\in{\mathbf{H}}$, then the bounded linear map
$f_{h}\colon{\mathbf{H}}\ni\eta\mapsto\left(h\middle|\eta\right)_{\mathbf{H}}\in{\mathbb{R}}$
can be extended to a measurable linear map $\widehat{f}_{h}$, and we have for
all $h_{1},h_{2}\in{\mathbf{H}}$

${\mathbb{E}}\left[\widehat{f}_{h_{1}}(\eta)\widehat{f}_{h_{2}}(\eta)\right]=\mathchoice{{\raise
2.05556pt\hbox{``}\hbox{$\displaystyle\;{\mathbb{E}}\big{[}\left(h_{1}\middle|\eta\right)_{\mathbf{H}}\left(h_{2}\middle|\eta\right)_{\mathbf{H}}\big{]}$}\raise
2.05556pt\hbox{''}}}{{\raise
2.05556pt\hbox{``}\hbox{$\textstyle\;{\mathbb{E}}\big{[}\left(h_{1}\middle|\eta\right)_{\mathbf{H}}\left(h_{2}\middle|\eta\right)_{\mathbf{H}}\big{]}$}\raise
2.05556pt\hbox{''}}}{{\raise
2.05556pt\hbox{``}\hbox{$\scriptstyle\;{\mathbb{E}}\big{[}\left(h_{1}\middle|\eta\right)_{\mathbf{H}}\left(h_{2}\middle|\eta\right)_{\mathbf{H}}\big{]}$}\raise
2.05556pt\hbox{''}}}{{\raise
2.05556pt\hbox{``}\hbox{$\scriptscriptstyle\;{\mathbb{E}}\big{[}\left(h_{1}\middle|\eta\right)_{\mathbf{H}}\left(h_{2}\middle|\eta\right)_{\mathbf{H}}\big{]}$}\raise
2.05556pt\hbox{''}}}=\frac{1}{\beta}\left(h_{1}\middle|h_{2}\right)_{\mathbf{H}}.\qquad$
(A.16)

  2. 2.

If ${\mathbb{O}}\colon{\mathbf{H}}\to{\mathbf{H}}$ is a bounded linear
operator on ${\mathbf{H}}$, then there exists a measurable extension
$\widehat{\mathbb{O}}$. Then $\widehat{\mathbb{O}}\eta$ is an
${\mathbf{X}}$-valued Gaussian random variable with covariance
$\beta^{-1}{\mathbb{C}}_{\mathbb{O}}$ defined by

$\frac{1}{\beta}\left({\mathbb{C}}_{\mathbb{O}}x\middle|x\right)_{\mathbf{X}}:=\frac{1}{\beta}\|{\mathbb{O}}^{*}{\mathbb{C}}x\|_{\mathbf{H}}^{2},\qquad
x\in{\mathbf{X}},$ (A.17)

where ${\mathbb{O}}^{*}$ is the adjoint of ${\mathbb{O}}$ in ${\mathbf{H}}$.

If in addition ${\mathbb{O}}$ has finite rank, then $\widehat{\mathbb{O}}\eta$
is an ${\mathbf{H}}$-valued Gaussian random variable with covariance
$\beta^{-1}{\mathbb{O}}{\mathbb{O}}^{*}$ on ${\mathbf{H}}$. In particular,
$\widehat{\mathbb{O}}\eta$ has finite ${\mathbf{H}}$-variance given by

${\mathbb{E}}\big{\lVert}\widehat{\mathbb{O}}\eta\big{\rVert}_{\mathbf{H}}^{2}=\frac{1}{\beta}\operatorname{tr}_{\mathbf{H}}({\mathbb{O}}{\mathbb{O}}^{*}).$
(A.18)

  3. 3.

Let ${\mathbb{P}}$ be an orthogonal projection on ${\mathbf{H}}$ and
${\mathbb{Q}}:=I-{\mathbb{P}}$ the complementary projection. Let
$X\sim\widehat{\mathbb{P}}_{\\#}\gamma_{\beta}$ and
$Y\sim\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$ be independent random
variables. Then $X+Y$ is distributed according to $\gamma_{\beta}$.

In addition, in order to recognize a probability measure
$\zeta\in{\mathcal{P}}({\mathbf{X}})$ as
$\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$ it is sufficient that there exists
a subspace $\widetilde{\mathbf{V}}\subset{\mathbf{V}}$, dense in
${\mathbf{H}}$ in the ${\mathbf{H}}$-norm, such that for all
$h\in\widetilde{\mathbf{V}}$

$\int_{{\mathbf{X}}}\mathrm{e}^{i\left(h\middle|\xi\right)_{\mathbf{H}}}\zeta({\mathrm{d}}\xi)=\exp\left(-\frac{1}{2\beta}\|{\mathbb{Q}}h\|_{\mathbf{H}}^{2}\right).$
(A.19)

######  Remark A.8.

Part 2 above implies in particular that if ${\mathbb{O}}$ is a unitary
operator on ${\mathbf{H}}$, then $\widehat{\mathbb{O}}\eta$ has the same
covariance as $\eta$.  ∎

###### Proof.

Part 1 is [Bog98, Th. 2.10.11] in combination with (A.15). In part 2, the
general case is [Bog98, Th. 3.7.6]. To calculate the covariance, first note
that since measurable linear extensions are characterized by their values on
${\mathbf{H}}$, the two linear extensions $\widehat{f}_{1}$ and
$\widehat{f}_{2}$ of

$f_{1}(\xi):=\left(h\middle|\widehat{\mathbb{O}}\xi\right)_{\mathbf{H}}\qquad\text{and}\qquad
f_{2}(\xi):=\left({\mathbb{O}}^{*}h\middle|\xi\right)_{\mathbf{H}}$

are $\gamma_{\beta}$-a.e. equal. It follows that

${\mathbb{E}}|f_{1}(\eta)|^{2}={\mathbb{E}}|f_{2}(\eta)|^{2}\stackrel{{\scriptstyle\eqref{eq:formula-
for-covariance}}}{{=}}\frac{1}{\beta}\|{\mathbb{O}}^{*}h\|_{\mathbf{H}}^{2}.$

The characterization (A.17) follows by remarking that for $x\in{\mathbf{X}}$,
$\left(x\middle|\xi\right)_{\mathbf{X}}=\left({\mathbb{C}}x\middle|\xi\right)_{\mathbf{H}}$,
so that $f_{1}$ equals $\left(x\middle|\xi\right)_{\mathbf{X}}$ whenever
$h={\mathbb{C}}x$.

The case of finite rank follows from part 1, by choosing orthonormal bases
$\\{e_{k}\\}_{k=1}^{\infty}$ and $\\{f_{k}\\}_{k=1}^{\infty}$ of
${\mathbf{H}}$ such that that
$\mathrm{Range}({\mathbb{O}})\subset\operatorname{span}\left(\\{e_{k}\\}_{k=1}^{d}\right)$
and
$\mathrm{Ker}({\mathbb{O}})^{\perp}\subset\operatorname{span}\left(\\{f_{k}\\}_{k=1}^{d}\right)$,
and expanding the squared norm (see also [Bog98, Ex. 3.7.14]).

In part 3 the distribution of $X+Y$ is given by [Bog98, Lemma 3.11.25]. The
characterization $\zeta=\widehat{\mathbb{Q}}_{\\#}\gamma_{\beta}$ can be
obtained as follows. By [Bog98, Cor. A.3.13], the identity (A.19) for all
elements of $\widetilde{\mathbf{V}}$ uniquely identifies $\zeta$ as a centered
Gaussian measure on ${\mathbf{X}}$. Writing for $h={\mathbb{C}}x$ the exponent
as

$\left(h\middle|\xi\right)_{\mathbf{H}}=\left({\mathbb{C}}x\middle|\xi\right)_{\mathbf{H}}=\left(x\middle|\xi\right)_{\mathbf{X}},$

we observe that $\zeta$ has covariance operator

$\frac{1}{\beta}\left({\mathbb{C}}_{\zeta}x\middle|x\right)_{\mathbf{X}}:=\frac{1}{\beta}\|{\mathbb{Q}}h\|_{\mathbf{H}}^{2}=\frac{1}{\beta}\|{\mathbb{Q}}{\mathbb{C}}x\|_{\mathbf{H}}^{2}.$

Using part 2 we recognize this Gaussian measure as the law of
$\widehat{\mathbb{Q}}\eta$ where $\eta\sim\gamma_{\beta}$. ∎

An important consequence of the above extension results is that the extended
group $(\widehat{S}_{t})_{t\in{\mathbb{R}}}$ on ${\mathbf{X}}$ preserves the
measure $\gamma_{\beta}$.

######  Lemma A.9 (Invariance of the Gaussian measure).

The extension $\widehat{S}_{t}:{\mathbf{X}}\to{\mathbf{X}}$ of
$\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}:{\mathbf{H}}\to{\mathbf{H}}$ (see
Assumption 2.3) leaves the Gaussian measure $\gamma_{\beta}$ invariant.

###### Proof.

We apply Part 2 of Lemma A.7 with the unitary operator
${\mathbb{O}}=\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}$ and its extension
$\widehat{\mathbb{O}}=\widehat{S}_{t}$. Hence, starting from
$\eta\sim\gamma_{\beta}$ we see that the random variable
$\widehat{\mathbb{O}}\eta=\widehat{S}_{t}\eta$ is a centered Gaussian in
${\mathbf{X}}$ with covariance $\beta^{-1}{\mathbb{C}}_{t}$ characterized by

$\frac{1}{\beta}\left({\mathbb{C}}_{t}x\middle|x\right)_{\mathbf{X}}=\frac{1}{\beta}\|\mathrm{e}^{-t{\mathbb{J}}_{\mathcal{B}}}{\mathbb{C}}x\|_{\mathbf{H}}^{2}=\frac{1}{\beta}\|{\mathbb{C}}x\|_{\mathbf{H}}^{2}.$

In particular, this means $\beta^{-1}{\mathbb{C}}_{t}=\beta^{-1}{\mathbb{C}}$;
hence $\widehat{S}_{t}\eta$ has the same distribution $\gamma_{\beta}$. ∎

######  Remark A.10 (Invariance of $\gamma_{\beta}$ via inner products).

The invariance of $\gamma_{\beta}$ can also be shown by working in
${\mathbf{X}}$ rather than in ${\mathbf{X}}^{*}$ as in the above proof. We do
not have the commutation property
${\mathbb{C}}\widehat{S}_{t}=\widehat{S}_{t}{\mathbb{C}}$ but we have the
invariance property

$\widehat{S}_{t}^{*}{\mathbb{C}}\widehat{S}_{t}={\mathbb{C}},$ (A.20)

which implies
$\left({\mathbb{C}}\widehat{S}_{t}\eta\middle|\widehat{S}_{t}\eta\right)=\left(\widehat{S}_{t}^{*}{\mathbb{C}}\widehat{S}_{t}\eta\middle|\eta\right)=\left({\mathbb{C}}\eta\middle|\eta\right)$.

To see the validity of (A.20), we need to calculate the Hilbert-space adjoint
$\widehat{S}_{t}^{*}$ of $\widehat{S}_{t}$. For an operator
$A\colon{\mathbf{U}}\to{\mathbf{V}}$ denote the Banach-space adjoint by
$A^{\top}\colon{\mathbf{V}}^{*}\to{\mathbf{U}}^{*}$, which is defined via
$\langle A^{\top}\nu,u\rangle_{\mathbf{U}}=\langle\nu,Au\rangle_{\mathbf{V}}$.
Using the dual space ${\mathbf{X}}^{*}$ satisfying
${\mathbf{X}}^{*}\subset{\mathbf{H}}\subset{\mathbf{X}}$ and the duality
isomorphism ${\mathbb{C}}\colon{\mathbf{X}}\to{\mathbf{X}}^{*}$ we have the
relation
$\widehat{S}_{t}^{*}={\mathbb{C}}\,\widehat{S}_{t}^{\top}\,{\mathbb{C}}^{-1}$,
where $\widehat{S}_{t}^{\top}\colon{\mathbf{X}}^{*}\to{\mathbf{X}}^{*}$. As
$\widehat{S}_{t}$ is the extension of
$\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}$ from ${\mathbf{H}}$ to
${\mathbf{X}}$, the Banach-space adjoint is simply the restriction
$\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}|_{{\mathbf{X}}^{*}}$. Together with
$(\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}})^{\top}=\mathrm{e}^{-t{\mathbb{J}}_{\mathcal{B}}}$
in ${\mathbf{H}}$, we found the representation
$\widehat{S}_{t}^{*}={\mathbb{C}}\,\mathrm{e}^{-t{\mathbb{J}}_{\mathcal{B}}}|_{{\mathbf{X}}^{*}}\,{\mathbb{C}}^{-1}$,
and the desired identity (A.20) follows.  ∎

##  Appendix B Delayed proofs

###  B.1 Section 2: Microscopic setup

###### Proof of Lemma 2.4.

Part 1 follows from the assumption that
$\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}$ is a strongly continuous group on
${\mathbf{H}}$ and from the fact that on ${\mathbf{H}}$ the operator
$\widehat{\mathbb{P}}$ reduces to the bounded linear operator ${\mathbb{P}}$.

For part 2, first note that $Y(\eta_{0})$ is stationary since $\gamma_{\beta}$
is invariant under $\widehat{S}_{t}$. We next show that both processes are
measurable with respect to the product $\sigma$-algebra
${\mathcal{B}}({\mathbb{R}}){\times}{\mathcal{B}}({\mathbf{X}})_{\gamma_{\beta}}$.
The following decomposition shows that $Y_{t}(\eta_{0})$ is a measurable
function of $(t,\eta_{0})$, with each arrow a measurable map between the two
$\sigma$-algebras:

$\displaystyle(t,\eta_{0})\qquad$ $\displaystyle\quad\mapsto\quad$
$\displaystyle\widehat{S}_{t}\eta_{0}\quad$ $\displaystyle\quad\mapsto\quad$
$\displaystyle\widehat{\mathbb{P}}\widehat{S}_{t}\eta_{0}$
$\displaystyle{\mathcal{B}}({\mathbb{R}}){\times}{\mathcal{B}}({\mathbf{X}})_{\gamma_{\beta}}$
$\displaystyle{\mathcal{B}}({\mathbf{X}})_{\gamma_{\beta}}$ $\displaystyle\
{\mathcal{B}}({\mathbf{W}})$

The first arrow is measurable by the continuity given by Assumption 2.3, and
the second by the measurability of $\widehat{\mathbb{P}}$ as a map from
$({\mathbf{X}},{\mathcal{B}}({\mathbf{X}})_{\gamma_{\beta}})$ to
$({\mathbf{W}},{\mathcal{B}}({\mathbf{W}}))$ (Lemma A.7).

To show that $Y=Y(\widehat{\mathbb{Q}}\eta_{0})$ similarly is measurable we
decompose it as

$Y_{t}(\widehat{\mathbb{Q}}\eta_{0})=\widehat{\mathbb{P}}\widehat{S}_{t}\widehat{\mathbb{Q}}\eta_{0}=\widehat{\mathbb{P}}\widehat{S}_{t}\eta_{0}-\widehat{\mathbb{P}}\widehat{S}_{t}\widehat{\mathbb{P}}\eta_{0}.$

We just showed that the first term is measurable from
${\mathcal{B}}({\mathbb{R}}){\times}{\mathcal{B}}({\mathbf{X}})_{\gamma_{\beta}}$
to ${\mathcal{B}}({\mathbf{W}})$. The measurability of the second follows from
remarking that
$\widehat{\mathbb{P}}\widehat{S}_{t}\widehat{\mathbb{P}}\eta_{0}={\mathbb{P}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}\widehat{\mathbb{P}}\eta_{0}$,
since the range of $\widehat{\mathbb{P}}$ is included in
${\mathbf{W}}\subset{\mathbf{H}}$. With this we then have

$\displaystyle(t,\eta_{0})\qquad$ $\displaystyle\quad\mapsto\quad$
$\displaystyle(t,\widehat{\mathbb{P}}\eta_{0})\qquad$
$\displaystyle\quad\mapsto\quad$
$\displaystyle\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}\widehat{\mathbb{P}}\eta_{0}$
$\displaystyle\quad\mapsto\quad$
$\displaystyle{\mathbb{P}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}\widehat{\mathbb{P}}\eta_{0}.$
$\displaystyle{\mathcal{B}}({\mathbb{R}}){\times}{\mathcal{B}}({\mathbf{X}})_{\gamma_{\beta}}$
$\displaystyle{\mathcal{B}}({\mathbb{R}}){\times}{\mathcal{B}}({\mathbf{W}})$
$\displaystyle{\mathcal{B}}({\mathbf{H}})\ $ $\displaystyle\
{\mathcal{B}}({\mathbf{W}})\ $

Here the first arrow is measurable again by the construction of
$\widehat{\mathbb{P}}$ (Lemma A.7), the second by the strong continuity of
$\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}$ on ${\mathbf{H}}$, and the third by
the continuity of ${\mathbb{P}}$ on ${\mathbf{H}}$.

Finally, to show that $Y(\eta_{0})$ and $Y(\widehat{\mathbb{Q}}\eta_{0})$ have
sample paths in $L^{2}_{\mathrm{loc}}$, we calculate for $t\in{\mathbb{R}}$
that

${\mathbb{E}}_{\eta_{0}\sim\gamma_{\beta}}\|Y_{t}(\eta_{0})\|_{\mathbf{H}}^{2}={\mathbb{E}}_{\eta_{0}}\big{\lVert}\widehat{\mathbb{P}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}\eta_{0}\big{\rVert}_{\mathbf{H}}^{2}\stackrel{{\scriptstyle\eqref{eq:variance-
of-finite-rank-
operator}}}{{=}}\frac{1}{\beta}\operatorname{tr}_{\mathbf{H}}\big{(}{\mathbb{P}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}\left({\mathbb{P}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}\right)^{*}\big{)}=\frac{1}{\beta}\operatorname{tr}_{\mathbf{H}}{\mathbb{P}}=\frac{\dim{\mathbf{W}}}{\beta}.$

The measurability allows us to apply Fubini and calculate for
$-\infty<a<b<\infty$

${\mathbb{E}}_{\eta_{0}\sim\gamma_{\beta}}\int_{a}^{b}\|Y_{t}(\eta_{0})\|_{\mathbf{H}}^{2}\;\\!\mathrm{d}t=\int_{a}^{b}{\mathbb{E}}_{\eta_{0}}\|Y_{t}(\eta_{0})\|_{\mathbf{H}}^{2}=(b{-}a)\\!\;\frac{\dim{\mathbf{W}}}{\beta}.$

This shows that almost surely $Y(\eta_{0})$ is an element of
$L^{2}_{\mathrm{loc}}({\mathbb{R}};{\mathbf{W}})$.

To show the analogous result for $Y(\widehat{\mathbb{Q}}\eta_{0})$, we
similarly obtain

${\mathbb{E}}_{\eta_{0}\sim\gamma_{\beta}}\int_{a}^{b}\big{\lVert}Y_{t}(\widehat{\mathbb{Q}}\eta_{0})\big{\rVert}_{\mathbf{H}}^{2}\;\\!\mathrm{d}t=\frac{1}{\beta}(b{-}a)\operatorname{tr}_{\mathbf{H}}({\mathbb{P}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}{\mathbb{Q}}^{2}\mathrm{e}^{-t{\mathbb{J}}_{\mathcal{B}}}{\mathbb{P}}).$

To estimate the trace we choose an orthonormal basis $(e_{k})_{k=1}^{d}$ of
${\mathbf{W}}$ of size $d=\dim{\mathbf{W}}$, and calculate

$\displaystyle\operatorname{tr}_{\mathbf{H}}({\mathbb{P}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}{\mathbb{Q}}^{2}\mathrm{e}^{-t{\mathbb{J}}_{\mathcal{B}}}{\mathbb{P}})$
$\displaystyle=\sum_{k=1}^{d}\left\lvert\left({\mathbb{P}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}{\mathbb{Q}}^{2}\mathrm{e}^{-t{\mathbb{J}}_{\mathcal{B}}}{\mathbb{P}}e_{k}\middle|e_{k}\right)_{\mathbf{H}}\right\rvert\leq\sum_{k=1}^{d}\left\lVert{\mathbb{P}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}{\mathbb{Q}}^{2}\mathrm{e}^{-t{\mathbb{J}}_{\mathcal{B}}}{\mathbb{P}}e_{k}\right\rVert_{\mathbf{H}}\left\lVert
e_{k}\right\rVert_{\mathbf{H}}$ $\displaystyle\leq\sum_{k=1}^{d}\left\lVert
e_{k}\right\rVert_{\mathbf{H}}^{2}=d.$

Again it follows that $Y(\widehat{\mathbb{Q}}\eta_{0})$ is almost surely in
$L^{2}_{\mathrm{loc}}({\mathbb{R}};{\mathbf{W}})$. ∎

###  B.2 Microcanonical measures and Section 3.2

In the physics literature it is common to define ‘microcanonical ensembles’,
which effectively are measures on the set of possible states. A useful tool
for this is the following delta-function-like object.

Let $f\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$ be continuous. For
$a\in{\mathbb{R}}$ the non-negative measure
$\mu\in{\mathcal{M}}({\mathbb{R}}^{n})$

$\mu({\mathrm{d}}x):=\bm{\updelta}\left[f(x)-a\right]({\mathrm{d}}x)$

is defined by the property

$\mu(A)=\lim_{h\downarrow 0}\frac{1}{h}\int_{A}\mathds{1}\big{\\{}a\leq
f(x)<a+h\big{\\}}\;\\!\mathrm{d}x\in[0,\infty]\qquad\text{for all
}A\subset{\mathbb{R}}^{n},$

whenever this definition makes sense.

One way of making this definition rigorous is by integration against test
functions:

######  Definition B.1 ([LRS10, (1.25)]).

The measure $\bm{\updelta}\left[f(x)-a\right]$ on ${\mathbb{R}}^{n}$ is
defined by the property that for all $F\in C_{c}({\mathbb{R}}^{n})$ and $G\in
C({\mathbb{R}})$,

$\int_{\mathbb{R}}G(a)\int_{{\mathbb{R}}^{n}}F(x)\bm{\updelta}\left[f(x)-a\right]({\mathrm{d}}x)\;\\!\mathrm{d}a=\int_{{\mathbb{R}}^{n}}G(f(x))F(x)\;\\!\mathrm{d}x.$
(B.1)

The right-hand side above is finite for all $F\in C_{c}({\mathbb{R}}^{n})$ and
$G\in C({\mathbb{R}})$, since $\operatorname{supp}F$ is compact and $G\circ f$
therefore is bounded on this support. This implies that
$\bm{\updelta}\left[f(x)-a\right]({\mathrm{d}}x)\;\\!\mathrm{d}a$ can be
considered a locally finite non-negative measure on
$(x,a)\in{\mathbb{R}}^{n}{\times}{\mathbb{R}}$. For almost all
$a\in{\mathbb{R}}$ the disintegration
$\bm{\updelta}\left[f(x)-a\right]({\mathrm{d}}x)$ is then a locally finite
non-negative measure on ${\mathbb{R}}^{n}$.

By the co-area formula (see, e.g., [LRS10, Sec. 3.2.1]) this measure can also
be expressed with help of the $(n{-}1)$-dimensional Hausdorff measure
${\mathcal{H}}^{n-1}$ on the set $S_{a}:=\\{x\in{\mathbb{R}}^{n}:f(x)=a\\}$ as

$\bm{\updelta}\left[f(x)-a\right]({\mathrm{d}}x)=\frac{1}{|\nabla
f(x)|}{\mathcal{H}}^{n-1}\big{|}_{S_{a}}({\mathrm{d}}x).$ (B.2)

In Section 4.6 we made use of the following property, which allows us to write
a microcanonical measure on a product space as the product of two such
measures on the individual spaces. The crucial element is the fact that the
‘energy’ function has the additive structure $(x,y)\mapsto f(x)+g(y)$.

######  Lemma B.2.

Let $X$ and $Y$ be finite-dimensional spaces, and let $f\colon
X\to{\mathbb{R}}$ and $g\colon Y\to{\mathbb{R}}$ be continuous. Set

$T\colon X{\times}Y\to
X{\times}Y{\times}{\mathbb{R}},\qquad(x,y)\mapsto(x,y,e:=g(y)).$

For Lebesgue almost all $a\in{\mathbb{R}}$ we then have

$\Big{(}T_{\\#}\bm{\updelta}\left[f(x)+g(y)-a\right]\Big{)}({\mathrm{d}}x\;\\!\mathrm{d}y\;\\!\mathrm{d}e)=\bm{\updelta}\left[f(x)-a+e\right]({\mathrm{d}}x)\bm{\updelta}\left[g(y)-e\right]({\mathrm{d}}y)\;\\!\mathrm{d}e.$

###### Proof.

By the definition of the push-forward $T_{\\#}\bm{\updelta}$ we need to show
that for all $\varphi\in C_{c}(X{\times}Y{\times}{\mathbb{R}})$ we have for
almost all $a\in{\mathbb{R}}$

$\int_{X{\times}Y}\varphi(x,y,g(y))\bm{\updelta}\left[f(x)+g(y)-a\right]({\mathrm{d}}x\;\\!\mathrm{d}y)\\\
=\int_{X{\times}Y{\times}{\mathbb{R}}}\varphi(x,y,e)\bm{\updelta}\left[f(x)-a+e\right]({\mathrm{d}}x)\bm{\updelta}\left[g(y)-e\right]({\mathrm{d}}y)\;\\!\mathrm{d}e.$

Since in the right-hand integral we can replace $\varphi(x,y,e)$ by
$\varphi(x,y,g(y))$, it is sufficient to show that for almost all
$a\in{\mathbb{R}}$ and for all measurable $A\subset X$, $B\subset Y$

$\bm{\updelta}\left[f(x)+g(y)-a\right](A{\times}B)=\int_{\mathbb{R}}\bm{\updelta}\left[f(x)+e-a\right](A)\bm{\updelta}\left[g(y)-e\right](B)\;\\!\mathrm{d}e.$
(B.5)

To prove this identity we take $G\in C({\mathbb{R}})$ and $F\in
C_{c}(X{\times}Y)$ and rewrite the left-hand side of (B.5) using (B.1),  In
particular, $\Phi\colon{\mathbf{W}}\to\overline{\mathbf{H}}$ is an isometry
and $\Phi\colon{\mathbf{W}}\to{\mathbf{G}}$ is unitary with inverse
$\Phi^{*}|_{\mathbf{G}}$. Moreover, ${\mathbb{P}}$ is the orthogonal projector
from $\overline{\mathbf{H}}$ onto ${\mathbf{G}}$.

* 2.

Defining the strongly continuous unitary shift group
$({\mathbb{S}}_{t})_{t\in{\mathbb{R}}}$ on
$\overline{\mathbf{H}}={\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})$ via
${\mathbb{S}}_{t}f=f(\cdot-t)$ the semigroup
$\big{(}{\mathbf{W}},(\mathrm{e}^{-t{\mathbb{D}}})_{t\geq 0}\big{)}$ has the
generalized dilation
$\big{(}\overline{\mathbf{H}},\Phi,({\mathbb{S}}_{t})_{t\in{\mathbb{R}}}\big{)}$.
Moreover, for all $t\in{\mathbb{R}}$ and $g\in{\mathbf{G}}=\Phi{\mathbf{W}}$
we have the relation

${\mathbb{P}}\,{\mathbb{S}}_{t}g={\mathbf{R}}(t)g,\quad\text{where
}{\mathbf{R}}(t)=\left\\{\begin{array}[]{cl}\Phi\,\mathrm{e}^{-t{\mathbb{D}}}\,\Phi^{*}&\text{for
}t\geq 0,\\\\[-1.99997pt]
\Phi\,\mathrm{e}^{\,t\,{\mathbb{D}}^{*}}\,\Phi^{*}&\text{for }t\leq
0.\end{array}\right.$ (C.5)

###### Proof.

_Part LABEL:pr:ExplicitDilation:i:projection_. The property of an isometry is
based on the standard relation

$\displaystyle\int_{-\infty}^{0}\\!\\!\big{(}\Sigma_{\mathbb{D}}\mathrm{e}^{y{\mathbb{D}}}\big{)}^{*}\,\Sigma_{\mathbb{D}}\mathrm{e}^{y{\mathbb{D}}}\;\\!\mathrm{d}y=\int_{-\infty}^{0}\\!\\!\mathrm{e}^{y{\mathbb{D}}^{*}}({\mathbb{D}}^{*}{+}{\mathbb{D}})\,\mathrm{e}^{y{\mathbb{D}}}\;\\!\mathrm{d}y=\int_{-\infty}^{0}\\!\frac{{\mathrm{d}}}{{\mathrm{d}}y}\big{(}\mathrm{e}^{y{\mathbb{D}}^{*}}\mathrm{e}^{y{\mathbb{D}}}\big{)}\;\\!\mathrm{d}y={\mathbb{I}}_{\mathbf{W}}.$
(C.6)

To establish $\Phi^{*}\Phi={\mathbb{I}}_{\mathbf{W}}$ we simply calculate, for
$w_{1},w_{2}\in{\mathbf{W}}$,

$\displaystyle(\Phi w_{1}|\Phi w_{2})_{{\mathbf{H}}}$
$\displaystyle=\int_{-\infty}^{0}\\!\\!\big{(}\Sigma_{\mathbb{D}}\mathrm{e}^{y{\mathbb{D}}}w_{1}\big{|}\Sigma_{\mathbb{D}}\mathrm{e}^{y{\mathbb{D}}}w_{2}\big{)}_{\mathbf{W}}\;\\!\mathrm{d}y=\int_{-\infty}^{0}\\!\\!\big{(}\mathrm{e}^{y{\mathbb{D}}^{*}}\Sigma_{\mathbb{D}}^{2}\mathrm{e}^{y{\mathbb{D}}}w_{1}\big{|}w_{2}\big{)}_{\mathbf{W}}\;\\!\mathrm{d}y\overset{\text{\eqref{eq:bbD.relation}}}{=}(w_{1}|w_{2})_{\mathbf{W}}.$

With this and ${\mathbb{P}}=\Phi\Phi^{*}$ the identities
${\mathbb{P}}^{2}={\mathbb{P}}$ and ${\mathbb{P}}={\mathbb{P}}^{*}$ follow
immediately, i.e., ${\mathbb{P}}$ is an orthogonal projector. Moreover,
${\mathbb{P}}f=\Phi(\Phi^{*}f)$ implies
$\mathrm{range}\,{\mathbb{P}}\subset\mathrm{range}\,\Phi={\mathbf{G}}$,
whereas ${\mathbb{P}}\Phi v=\Phi\Phi^{*}\Phi w=\Phi w$ shows
$\mathrm{range}\,{\mathbb{P}}\supset\mathrm{range}\,\Phi={\mathbf{G}}$.

_Part 2_. Now we consider $t\geq 0$ and $v\in{\mathbf{W}}$ and obtain

$\displaystyle\Phi^{*}\big{(}{\mathbb{S}}_{t}\Phi w\big{)}$
$\displaystyle=\int_{-\infty}^{0}\\!\\!\mathrm{e}^{s{\mathbb{D}}^{*}}\,\Sigma_{\mathbb{D}}\big{(}\Phi
w\big{)}(s{-}t)\;\\!\mathrm{d}s=\int_{-\infty}^{0}\\!\\!\mathrm{e}^{s{\mathbb{D}}^{*}}\Sigma^{2}_{\mathbb{D}}\,\mathrm{e}^{(s-t){\mathbb{D}}}w\,\bm{1}_{]{-}\infty,0]}(s{-}t)\,\;\\!\mathrm{d}s$
$\displaystyle=\int_{-\infty}^{0}\\!\\!\mathrm{e}^{s{\mathbb{D}}^{*}}\Sigma_{\mathbb{D}}^{2}\,\mathrm{e}^{s{\mathbb{D}}}\;\\!\mathrm{d}s\
\mathrm{e}^{-t{\mathbb{D}}}w\overset{\text{\eqref{eq:bbD.relation}}}{=}\mathrm{e}^{-t{\mathbb{D}}}w.$

Hence,
$\big{(}\overline{\mathbf{H}},\Phi,({\mathbb{S}}_{t})_{t\in{\mathbb{R}}}\big{)}$
is a generalized dilation in the sense of Definition LABEL:de:GenerDilat.

For $t\leq 0$ we have $\bm{1}_{]{-}\infty,0]}(s{-}t)=0$ for $s>t$ and obtain

$\displaystyle\Phi^{*}\big{(}{\mathbb{S}}_{t}\Phi w\big{)}$
$\displaystyle=\int_{-\infty}^{t}\\!\\!\mathrm{e}^{s{\mathbb{D}}^{*}}\Sigma^{2}_{\mathbb{D}}\mathrm{e}^{(s-t){\mathbb{D}}}w\,\;\\!\mathrm{d}s=\int_{-\infty}^{0}\\!\\!\mathrm{e}^{t{\mathbb{D}}^{*}}\mathrm{e}^{r{\mathbb{D}}^{*}}\Sigma_{\mathbb{D}}^{2}\,\mathrm{e}^{r{\mathbb{D}}}w\,\;\\!\mathrm{d}s=\mathrm{e}^{t{\mathbb{D}}^{*}}w.$

Applying the operator $\Phi$ to the two relations for
$\Phi^{*}\big{(}{\mathbb{S}}_{t}\Phi w\big{)}$ (for $t\geq 0$ and $t\leq 0$,
respectively) and replacing $\Phi w$ by $g=\Phi w$ such that $w=\Phi^{*}g$, we
find the final relation (C.5). ∎

The major advantage of this explicit generalized dilation is that it
introduces, in a very natural way, the time shift ${\mathbb{S}}_{t}$ on the
Hilbert space
$\overline{\mathbf{H}}={\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})$ which will
take over the place of the linear evolution
$\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}$. The specification to the time
shift is advantageous as the time shift is also the natural variable in the
random processes $(Y_{t})_{t\in{\mathbb{R}}}$.

######  Remark C.5 (Minimality).

The above construction of a generalized dilation also allows us to show that
the dilation is minimal. For this we have to show that

$\mathfrak{G}:=\mathrm{span}\big{\\{}\,{\mathbb{S}}_{t}g\>\big{|}\>t\in{\mathbb{R}},\
g\in{\mathbf{G}}\,\big{\\}}=\mathrm{span}\big{\\{}\,{\mathbb{S}}_{t}(\Phi
w)\>\big{|}\>t\in{\mathbb{R}},\ w\in{\mathbf{W}}\,\big{\\}}$

is dense in
$\overline{\mathbf{H}}={\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})$. For
$h\in{]0,1[}$ and $w\in{\mathbf{W}}$, we define the functions

$\Psi_{h,w}:={\mathbb{S}}_{h}\big{(}\Phi w)-\Phi w:\
y\mapsto\begin{cases}0&\text{for }y>h,\\\
\Sigma_{\mathbb{D}}\mathrm{e}^{(y-h){\mathbb{D}}}w&\text{for }y\in{]0,h]},\\\
\Sigma_{\mathbb{D}}\mathrm{e}^{y{\mathbb{D}}}\big{(}\mathrm{e}^{-h{\mathbb{D}}}{-}{\mathbb{I}}\big{)}w&\text{for
}y\leq 0.\end{cases}$

Clearly, $\Psi_{h,w}\in\mathfrak{G}$. Using
$\mathrm{dim}({\mathbf{W}})<\infty$ we have
$\|\mathrm{e}^{-h{\mathbb{D}}}{-}{\mathbb{I}}\|\leq Ch$ for $h\in[0,1]$ and
obtain

$\big{\|}\Psi_{h,w}-w\bm{1}_{{[0,h[}}\big{\|}_{{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})}\leq
C_{*}h\|w\|_{\mathbf{W}},$

with a constant $C_{*}$ depending only on ${\mathbb{D}}$.

To show that $f\in{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})$ can be
approximated by linear combinations of ${\mathbb{S}}_{t}\Psi_{h,w}$, we fix
$\varepsilon>0$ and find a piecewise constant
$\overline{f}_{N}\in{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})$ such that

$\big{\|}f-\overline{f}_{N}\big{\|}_{{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})}<\varepsilon/2\quad\text{and}\quad\overline{f}_{N}=\sum_{j=-N^{2}}^{N^{2}}w^{N}_{j}\bm{1}_{{[j/N,(j+1)/N[}}\
\text{ with }N\in{\mathbb{N}}\text{ and }w^{N}_{j}\in{\mathbf{W}}.$

Here $N$ can be made as large as desired, by refining the partition. We now
approximate $\overline{f}$ by
$\widehat{f}_{N}=\sum_{-N^{2}}^{N^{2}}{\mathbb{S}}_{j/N}\Psi_{1/N,w^{N}_{j}}$,
which clearly lies in $\mathfrak{G}$, and obtain

$\displaystyle\textstyle\big{\|}\overline{f}_{N}-\widehat{f}_{N}\|_{{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})}^{2}=\sum_{j=-N^{2}}^{N^{2}}\big{\|}w_{j}^{N}\bm{1}_{{[j/N,(j+1)/N[}}-\overline{\mathbb{S}}_{j/n}\Psi_{1/N,w_{j}^{N}}\|_{{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})}^{2}$
$\displaystyle\textstyle\leq\
\sum_{j=-N^{2}}^{N^{2}}C_{*}^{2}(1/N)^{2}\|w^{N}_{j}\|_{\mathbf{W}}^{2}\ =\
\frac{C_{*}}{N}\,\big{\|}\overline{f}_{N}\big{\|}^{2}_{{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})}\
\leq\
\frac{C_{*}}{N}\,\big{(}\,\|f\|^{2}_{{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})}{+}\frac{\varepsilon}{2}\,\big{)}^{2}.$

Choosing $N\in{\mathbb{N}}$ sufficiently big, the last term is less than
$\varepsilon^{2}/4$, and thus we conclude
$\|f-\widehat{f}_{N}\|_{{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})}<\varepsilon$
as desired. Hence, $\mathfrak{G}$ is dense in
$\overline{\mathbf{H}}={\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})$.  ∎

###  C.2 Continuous extension of the group
$(\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}})_{t\in{\mathbb{R}}}$ from
${\mathbf{H}}$ to ${\mathbf{X}}$

In Assumption 2.3 we assumed that the unitary group
$\big{(}{\mathbf{H}},(\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}})_{t\in{\mathbb{R}}}\big{)}$
describing the heat bath can be extended to a strongly continuous group on the
larger space ${\mathbf{X}}$, on which we are able to realize the Gaussian
measure $\gamma_{\beta}$ for the finite-temperature heat bath. We will show in
Proposition C.6 that this assumption is already a consequence of the dilation
property from Assumption 1.3.

As a first step we construct the Gaussian measure $\overline{\gamma}_{\beta}$
on an extension $\overline{\mathbf{X}}$ of the special space
$\overline{\mathbf{H}}={\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})$. This
leads to a classical white noise process, but now taking values in the finite-
dimensional Hilbert space ${\mathbf{W}}$. While translations
${\mathbb{S}}_{t}$ can easily defined for the white noise process, we have to
be more careful and construct a Hilbert space $\overline{\mathbf{X}}$ on which
we can extend the unitary group
$\big{(}\overline{\mathbf{H}},({\mathbb{S}}_{t})_{t\in{\mathbb{R}}}\big{)}$ to
a strongly continuous group
$\big{(}\overline{\mathbf{X}},({\mathbb{S}}^{\mathbf{X}}_{t})_{t\in{\mathbb{R}}}\big{)}$,
which will be no longer unitary. If
${\mathbb{N}}\colon\overline{\mathbf{H}}\to\overline{\mathbf{X}}$ is the
embedding operator, then the extension property means

$\forall\>t\in{\mathbb{R}}\ \
\forall\>h\in\overline{\mathbf{H}}:\quad{\mathbb{S}}^{\mathbf{X}}_{t}{\mathbb{N}}h={\mathbb{N}}{\mathbb{S}}_{t}h.$
(C.7)

As explained in Section 2.2 we further need that
${\mathbb{C}}={\mathbb{N}}{\mathbb{N}}^{*}\colon\overline{\mathbf{X}}\to\overline{\mathbf{X}}$
is trace class.

We give an explicit construction following the ideas in [Rey06]. We define the
self-adjoint operator
${\mathbb{A}}\colon\mathrm{Dom}({\mathbb{A}})\subset\overline{\mathbf{H}}\to\overline{\mathbf{H}}$
via

$\displaystyle\mathrm{Dom}({\mathbb{A}})=\big{\\{}\,u\in{\mathrm{H}}^{2}({\mathbb{R}};{\mathbf{W}})\>\big{|}\>t\mapsto(1{+}t^{2})u(t)\in{\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})\,\big{\\}},$
$\displaystyle({\mathbb{A}}u)(t)=-\frac{1}{2}u^{\prime\prime}(t)+\frac{t^{2}}{2}\,u(t).$

Here ${\mathrm{H}}^{2}({\mathbb{R}};{\mathbf{W}})$ is the Sobolev space of
measurable ${\mathrm{L}}^{2}$-functions $u\colon{\mathbb{R}}\to{\mathbf{W}}$
with two weak derivatives in ${\mathrm{L}}^{2}$. This is a
${\mathbf{W}}$-valued version of the quantum harmonic oscillator, and hence is
self-adjoint with the discrete spectrum
$\big{\\{}\,n{-}1/2\>\big{|}\>n\in{\mathbb{N}}\,\big{\\}}\subset{[1/2,\infty)}$,
where all eigenvalues have multiplicity $\dim({\mathbf{W}})<\infty$.
Consequently ${\mathbb{A}}$ is an invertible operator, and its inverse
${\mathbb{A}}^{-1}$ is a Hilbert-Schmidt operator. This implies that
${\mathbb{A}}$ satisfies the assumptions for the construction in Section A.2,
i.e. we can define

$\overline{\mathbf{V}}=\mathrm{Dom}({\mathbb{A}})={\mathrm{H}}^{2}({\mathbb{R}};{\mathbf{W}})\cap{\mathrm{L}}^{2}_{(2)}({\mathbb{R}};{\mathbf{W}})\quad\text{and}\quad\overline{\mathbf{X}}=\overline{\mathbf{V}}^{*}={\mathrm{H}}^{-2}({\mathbb{R}};{\mathbf{W}})+{\mathrm{L}}^{2}_{(-2)}({\mathbb{R}};{\mathbf{W}}),$

where the subscripts “(m)” denote the ${\mathrm{L}}^{2}$ space with measure
$(1{+}t^{2})^{m}{\mathrm{d}}t$. In $\overline{\mathbf{V}}$ we choose the graph
norm
$\|u\|_{\overline{\mathbf{V}}}=\|{\mathbb{A}}u\|_{\overline{\mathbf{H}}}$.

Concerning the shift group
$(\overline{\mathbf{H}},{\mathbb{S}}_{t})_{t\in{\mathbb{R}}}$ we observe that
it can be restricted to $\overline{\mathbf{V}}$, since
$u\in\overline{\mathbf{V}}$ implies
${\mathbb{S}}_{t}u\in\overline{\mathbf{V}}$ for all $t$. Setting
${\mathbb{S}}^{\mathbf{V}}_{t}={\mathbb{S}}_{t}|_{\overline{\mathbf{V}}}:\overline{\mathbf{V}}\to\overline{\mathbf{V}}$
we have

$\displaystyle\|{\mathbb{S}}^{\mathbf{V}}_{t}u\|_{\overline{\mathbf{V}}}$
$\displaystyle\leq
C\big{(}\|({{\mathbb{S}}}_{t}u)^{\prime\prime}\|_{{\mathrm{L}}^{2}}+\|(1{+}s^{2})({{\mathbb{S}}}_{t}u)(s)\|_{{\mathrm{L}}^{2}}\big{)}$
$\displaystyle\leq
2C\big{(}\|u^{\prime\prime}\|_{{\mathrm{L}}^{2}}+(1{+}t^{2})\|(1{+}r^{2})u(r)\|_{{\mathrm{L}}^{2}}\big{)}\leq(1{+}t^{2})C\|u\|_{\overline{\mathbf{V}}}.$

Hence,
$\big{(}\overline{\mathbf{V}},({\mathbb{S}}^{\mathbf{V}}_{t})_{t\in{\mathbb{R}}}\big{)}$
forms a strongly continuous group of shift operators. By dualization (with
respect to the duality in $\overline{\mathbf{H}}$) we can define the extension

${\mathbb{S}}^{\mathbf{X}}_{t}\colon\overline{\mathbf{X}}\to\overline{\mathbf{X}};\
\
{\mathbb{S}}^{\mathbf{X}}_{t}=\big{(}{\mathbb{S}}^{\mathbf{V}}_{-t}\big{)}^{*}$

which provides the strongly continuous group
$\big{(}\overline{\mathbf{X}},({\mathbb{S}}^{\mathbf{X}}_{t})_{t\in{\mathbb{R}}}\big{)}$
satisfying $\|{\mathbb{S}}^{\mathbf{X}}_{t}\|_{\overline{\mathbf{X}}}\leq
C^{2}(1{+}t^{2})$.

######  Proposition C.6 (Assumption 2.3 holds).

If the contraction semigroup
$\big{(}{\mathbf{G}},(\mathrm{e}^{-t{\mathbb{D}}})_{t\geq 0}\big{)}$ satisfies
(LABEL:eq:Dilat.Assum) and
$\big{(}{\mathbf{H}},(\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}})_{t\in{\mathbb{R}}}\big{)}$
is an arbitrary minimal dilation, then there exists a larger space
${\mathbf{X}}$ with embedding ${\mathbb{N}}:{\mathbf{H}}\to{\mathbf{X}}$ and a
strongly continuous group $(\widehat{S}_{t})_{t\in{\mathbb{R}}}$ such

$\displaystyle{\mathbb{N}}{\mathbb{N}}^{*}\colon{\mathbf{X}}\to{\mathbf{X}}\quad\text{is
trace class}$ $\displaystyle\forall\,t\in{\mathbb{R}}\
\forall\,h\in{\mathbf{H}}:\quad\widehat{S}_{t}{\mathbb{N}}h={\mathbb{N}}\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}h.$

###### Proof.

We use the explicit construction of the generalized dilation
$\big{(}\overline{\mathbf{H}},({\mathbb{S}}_{t})_{t\in{\mathbb{R}}}\big{)}$
and of the explicit extension to
$\big{(}\overline{\mathbf{X}},({\mathbb{S}}^{\mathbf{X}}_{t})_{t\in{\mathbb{R}}}\big{)}$.

By Theorem LABEL:th:Dilation the two dilations are unitarily equivalent by a
unitary map $U\colon{\mathbf{H}}\to\overline{\mathbf{H}}$. If
$\overline{\mathbb{N}}$ is the embedding of $\overline{\mathbf{H}}$ into
$\overline{\mathbf{X}}$, then we can define

${\mathbf{X}}:=\overline{\mathbf{X}},\quad{\mathbb{N}}:=\overline{\mathbb{N}}\,U,\quad\widehat{S}_{t}={\mathbb{S}}^{\mathbf{X}}_{t}.$

Clearly,
${\mathbb{N}}{\mathbb{N}}^{*}=\overline{\mathbb{N}}U\,U^{*}\overline{\mathbb{N}}^{*}=\overline{\mathbb{N}}\,\overline{\mathbb{N}}^{*}$
is trace class by the explicit construction. Moreover,

$\widehat{S}_{t}{\mathbb{N}}h={\mathbb{S}}^{\mathbf{X}}_{t}\overline{\mathbb{N}}Uh\overset{\text{(1)}}{=}\overline{\mathbb{N}}{\mathbb{S}}_{t}Uh=\overline{\mathbb{N}}\,UU^{-1}{\mathbb{S}}_{t}Uh\overset{\text{(2)}}{=}{\mathbb{N}}\,\mathrm{e}^{t{\mathbb{J}}_{\mathcal{B}}}h$

holds for all $h\in{\mathbf{H}}$. In $\overset{\text{(1)}}{=}$ we used the
explicit extension of ${\mathbb{S}}^{\mathbf{X}}_{t}$ of ${\mathbb{S}}_{t}$
with embedding $\overline{\mathbb{N}}$, and in $\overset{\text{(2)}}{=}$ we
used the definition of ${\mathbb{N}}$ and that the two dilations are
equivalent via the unitary operator $U$. ∎

The advantage of using the explicit generalized dilation involving
$\overline{\mathbf{H}}={\mathrm{L}}^{2}({\mathbb{R}};{\mathbf{W}})$ becomes
even more apparent if we use the structure of the mapping
$\Phi\colon{\mathbf{W}}\to{\mathbf{G}}\subset\overline{\mathbf{H}}$ and
$\Phi^{*}\colon\overline{\mathbf{H}}\to{\mathbf{W}}$. Combining $\Phi^{*}$ and
the shift ${\mathbb{S}}_{t}$ we obtain the mapping

${\mathcal{U}}:\overline{\mathbf{H}}\to\overline{\mathbf{H}};\quad({\mathcal{U}}f)(t):=\Phi^{*}\big{(}{\mathbb{S}}_{t}f\big{)}.$

This corresponds to the mapping $t\mapsto
Y_{t}(\eta_{0}):=\widehat{\mathbb{P}}\widehat{S}_{t}\eta_{0}$ from Lemma 2.4.
We simply “dropped” the Hilbert-space isomorphism
$\Phi:{\mathbf{W}}\overset{1\text{-}1}{\longleftrightarrow}{\mathbf{G}}$.

We now show that ${\mathcal{U}}$ has a bounded linear extension
$\widehat{\mathcal{U}}\colon\overline{\mathbf{X}}\to\overline{\mathbf{X}}$
that is consistent with $Y_{t}$ in Lemma 2.4 and is defined everywhere, i.e.,
we do not need the measurable extension $\widehat{\mathbb{P}}$ defined
$\gamma_{\beta}$ a.e. For this we observe that

$\big{(}{\mathcal{U}}f\big{)}(t)=\int_{-\infty}^{0}\Sigma_{\mathbb{D}}\,\mathrm{e}^{s{\mathbb{D}}^{*}}f(s{-}t)\;\\!\mathrm{d}s.$

Thus, ${\mathcal{U}}$ is realized by convolution with the kernel
$s\mapsto\Sigma_{\mathbb{D}}\,\mathrm{e}^{s{\mathbb{D}}^{*}}\bm{1}_{]-\infty,0]}$.
In Fourier space the convolution is realized by the multiplication with the
Fourier multiplier

${\mathbb{R}}\ni\sigma\mapsto
M(\sigma)^{*}:=\Sigma_{\mathbb{D}}\big{(}{\mathbb{D}}^{*}-\mathrm{i}\sigma{\mathbb{I}}\big{)}^{-1}$

which is smooth and bounded. As the operator ${\mathbb{A}}$ stays invariant
(up to changing constants) under Fourier transform $\mathfrak{F}$, we also see
that $\mathfrak{F}$ maps ${{\mathbf{V}}}$ into itself, and hence by duality
also ${{\mathbf{X}}}$ into ${{\mathbf{X}}}$. Clearly, multiplying by $M$
defines a bounded linear operator in ${{\mathbf{V}}}$, because $M$ lies in
${\mathrm{C}}^{2}_{\mathrm{b}}\big{(}{\mathbb{R}};\mathrm{Lin}({\mathbf{W}})\big{)}$.
By duality the multiplication multiplying by $M^{*}$ defines a bounded linear
operator in ${{\mathbf{X}}}={{\mathbf{V}}}^{*}$. This shows that there is a
bounded linear extension
$\widehat{\mathcal{U}}\colon{{\mathbf{X}}}\to{{\mathbf{X}}}$ of
${\mathcal{U}}\colon{{\mathbf{H}}}\to{{\mathbf{H}}}$.

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