ad}}_{(\xi,b)}(\tilde{\xi},\tilde{b})}\,,\,{(\mu,a)}\right\rangle}={\left\langle{\left({{\rm
ad}}_{\xi}\tilde{\xi},\tilde{b}\xi-b\tilde{\xi}\right)}\,,\,{(\mu,a)}\right\rangle}={\left\langle{({{\rm
ad}}^{*}_{\xi}\mu+b\diamond
a,-a\xi)}\,,\,{\left(\tilde{\xi},\tilde{b}\right)}\right\rangle}={\left\langle{({{\rm
ad}}^{*}_{(\xi,b)}(\mu,a))}\,,\,{\left(\tilde{\xi},\tilde{b}\right)}\right\rangle},$

where in the first equality we have used the left Lie algebra action in (A.2)
to obtain

$\displaystyle{{\rm ad}}^{*}_{(\xi,b)}(\mu,a)=({{\rm
ad}}^{*}_{\xi}\mu+b\diamond a,-a\xi).$ (A.6)

When $\mathfrak{g}=\mathfrak{X}$, it is formally the right Lie algebra of
${\rm Diff}(\mathcal{D})$, that is, its standard left Lie algebra bracket is
minus the usual Lie bracket for vector fields. To distinguish between these
brackets, we denote with $[u,v]$ the standard Jacobi-Lie bracket of the vector
fields where $u,v\in\mathfrak{X}$ such that ${{\rm ad}}_{u}v=-[u,v]$. Here
${{\rm ad}}$ operation is the adjoint action of the left Lie algebra to
itself. Then the adjoint action of $\mathfrak{s}=\mathfrak{X}\circledS V$ on
itself is then

$\displaystyle{{\rm
ad}}_{(\xi_{1},v_{1})}(\xi_{2},v_{2})=(-[\xi_{1},\xi_{2}],v_{2}\xi_{1}-v_{1}\xi_{2}),$
(A.7)

and the coadjoint action is taken to be the formal dual of (A.7). Identifying
$-a\xi=\mathcal{L}_{\xi}a=-\mathcal{L}^{T}_{\xi}a$ yields the expression for
the Lie-Poisson bracket for the left action of a semidirect-product Lie
algebra by simply computing the dual of the semidirect product Lie algebra
$\mathfrak{s}=\mathfrak{g}\circledS V$ action given in equation (A.2). Namely,
one computes

$\displaystyle\begin{split}\\{f,h\\}(\mu,a)&={\left\langle{(\mu,a)}\,,\,{{{\rm
ad}}_{(\frac{\delta h}{\delta\mu},\frac{\delta h}{\delta
a})}\left(\frac{\delta f}{\delta\mu},\frac{\delta f}{\delta
a}\right)}\right\rangle}\\\ &={\left\langle{\bigg{(}{\rm ad}^{*}_{\frac{\delta
h}{\delta\mu}}\mu-a\diamond\frac{\delta h}{\delta
a}\,,\,-\,\mathcal{L}_{\frac{\delta
h}{\delta\mu}}^{T}a\bigg{)}}\,,\,{\bigg{(}\frac{\delta
f}{\delta\mu}\,,\,\frac{\delta f}{\delta a}\bigg{)}}\right\rangle}\\\
&={\left\langle{{{\rm ad}}^{*}_{(\frac{\delta h}{\delta\mu},\frac{\delta
h}{\delta a})}(\mu,a)}\,,\,{\left(\frac{\delta f}{\delta\mu},\frac{\delta
f}{\delta a}\right)}\right\rangle}\,,\end{split}$ (A.8)

which of course agrees with (A.6).

##  Appendix B Stochastic advection by Lie transport (SALT)

###  B.1 Deterministic semidirect-product coadjoint motion for ideal fluids

######  Proposition B.1 (Euler-Poincaré theorem [41]).

The Euler-Poincaré (EP) equations of a reduced Lagrangian $\ell(u,a)$,
$\ell:\mathfrak{X}\times V^{*}\rightarrow\mathbb{R}$ defined over the space of
smooth vector fields with elements $u\in\mathfrak{X}$ acting by Lie derivative
on elements $a\in V^{*}$ of a vector space $V^{*}$are written as [41]

$\displaystyle\frac{d}{dt}\frac{\delta\ell}{\delta u}+{{\rm
ad}}^{*}_{u}\frac{\delta\ell}{\delta u}-\frac{\delta\ell}{\delta a}\diamond
a=0,\quad\frac{d}{dt}a+\mathsterling_{u}a=0.$ (B.1)

In the EP equation in (B.1), the variational derivative is defined as usual
by,

$\displaystyle\delta\ell(u,a)=\frac{d}{d{\epsilon}}\bigg{|}_{{\epsilon}=0}\ell(u_{\epsilon},a_{\epsilon})=\left\langle\,\frac{\delta\ell}{\delta
u}\,,\,\delta
u\,\right\rangle_{\mathfrak{X}}+\left\langle\,\frac{\delta\ell}{\delta
a}\,,\,\delta a\,\right\rangle_{V}\,.$ (B.2)

The quantities $a\in V^{*}$ and $\delta\ell/\delta a\in V$ in (B.2) are dual
to each other under the $L^{2}$ pairing
$\langle\,\cdot\,,\,\cdot\,\rangle_{V}:V\times V^{*}\to\mathbb{R}$. Likewise,
the velocity vector field $u\in\mathfrak{X}$ and momentum 1-form density
$m:=\delta\ell/\delta u\in\mathfrak{X}^{*}$ are dual to each other under the
$L^{2}$ pairing
$\langle\,\cdot\,,\,\cdot\,\rangle_{\mathfrak{X}}:\mathfrak{X}\times\mathfrak{X}^{*}\to\mathbb{R}$.
In terms of these pairings and the Lie derivative operator $\mathcal{L}_{u}$
with respect to the vector field $u\in\mathfrak{X}$, the coadjoint operator
${{\rm ad}}_{u}^{*}$ and the diamond operator $(\diamond)$ in (B.1) are
defined by

$\displaystyle{\Big{\langle}{{{\rm ad}}^{*}_{u}\frac{\delta\ell}{\delta
u}}\,,\,{v}\Big{\rangle}}_{\mathfrak{X}}:={\Big{\langle}{\frac{\delta\ell}{\delta
u}}\,,\,{-\mathcal{L}_{u}v}\Big{\rangle}}_{\mathfrak{X}}=:{\Big{\langle}{\frac{\delta\ell}{\delta
u}}\,,\,{{{\rm ad}}_{u}v}\Big{\rangle}}_{\mathfrak{X}},$ (B.3)

where $v\in\mathfrak{X}$, and ${{\rm
ad}}:\mathfrak{X}\times\mathfrak{X}\to\mathfrak{X}$ is defined as ${{\rm
ad}}_{u}v:=-[u,v]:=-(u^{j}v_{,j}^{i}-v^{j}u_{,j}^{i}){\partial}_{i}$, and

$\displaystyle{\Big{\langle}{b\diamond
a}\,,\,{v}\Big{\rangle}}_{\mathfrak{X}}:={\Big{\langle}{b}\,,\,{-\mathcal{L}_{v}a}\Big{\rangle}}_{V}.$
(B.4)

As we shall see later, the coadjoint operator ${{\rm ad}}_{u}^{*}$ and the
diamond operator $(\diamond)$ enter together in (B.1) as a form of coadjoint
motion for semidirect product action of the Lie algebra

######  Proposition B.2 (The EP equation (B.1) also follows from the HPVP).

The EP equation in (B.1) can be derived by direct calculation from the
following Hamilton-Pontryagin variational principle (HPVP)

$\displaystyle 0=\delta
S=\delta\int^{b}_{a}\Big{(}\ell(u,a)+{\left\langle{m}\,,\,{\dot{g}g^{-1}-u}\right\rangle}+{\left\langle{b}\,,\,{a_{0}g^{-1}-a}\right\rangle}\Big{)}\,dt,$
(B.5)

where $m\in\mathfrak{X}^{*}$, $b\in V$, $g\in G$ and the variations are taken
to be arbitrary.

###  B.2 Stochastic semidirect-product coadjoint fluid motion with SALT noise

To add noise in the SALT form to the deterministic HPVP in (B.5), we introduce
the following stochastic variational principle, cf. [38],444Note: we can
choose separate (uncorrelated) Brownian motions in the $m$ and $a$ equations
in (B.6) by choosing $h_{i}(m,a)=h^{m}_{i}(m)+h^{a}_{i}(a)$ for the
Stratonovich noise, $\circ dW^{i}_{t}$. The choice of Stratonovich noise
enables the standard calculus chain rule and product rule to be used for the
operations of differentiation and integration by parts, in which variational
principles are defined.

$\displaystyle 0=\delta
S=\delta\int^{b}_{a}\Big{(}\ell(u,a)\,dt+{\left\langle{m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}-u\,dt}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}g^{-1}-a}\right\rangle}-\sum_{i}h_{i}(m,a)\circ
dW^{i}_{t}\Big{)}\,.$ (B.6)

For brevity of notation, we will suppress the subscript labels in the pairings
whenever the meaning is clear. The symbol
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm d}}$
in (B.6) abbreviates stochastic time integrations. The action integral $S$ in
(B.6) is defined in the framework of variational principles with
semimartingale constraints which was established in [66]. As we shall see
below, the semimartingale nature of a Lagrange multiplier which imposes one of
these semimartingale constraints emerges in the context of the full system of
equations, which is obtained after the variations have been taken. The
Hamiltonian functions $h_{i}(m,a)$ on $\mathfrak{X}^{*}\times V^{*}$ in (B.6)
will be prescribed here without discussing how they would be obtained in
practice, e.g., via data assimilation. The data assimilation procedure for
SALT is discussed, e.g., in [12, 13].

#### Euler-Poincaré (EP) Lagrangian formulation.

Taking arbitrary variations in the stochastic HPVP in equation (B.6) yields
the following determining relations among the variables,

$\displaystyle 0$
$\displaystyle=\int_{a}^{b}{\left\langle{\frac{\delta\ell}{\delta
u}}\,,\,{\delta u}\right\rangle}\,dt+{\left\langle{\frac{\delta\ell}{\delta
a}}\,,\,{\delta a}\right\rangle}\,dt+{\left\langle{\delta
m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}-u\,dt}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}\delta g^{-1}-\delta
a}\right\rangle}+{\left\langle{\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}g^{-1}-a}\right\rangle}$
$\displaystyle\quad-{\left\langle{\frac{\delta h_{i}}{\delta m}}\,,\,{\delta
m}\right\rangle}\circ dW_{t}^{i}-{\left\langle{\frac{\delta h_{i}}{\delta
a}}\,,\,{\delta a}\right\rangle}\circ
dW_{t}^{i}+{\left\langle{m}\,,\,{\delta({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1})-\delta u\,dt}\right\rangle}$
$\displaystyle=\int_{a}^{b}{\left\langle{\frac{\delta\ell}{\delta
u}}\,,\,{\delta u}\right\rangle}\,dt+{\left\langle{\frac{\delta\ell}{\delta
a}}\,,\,{\delta a}\right\rangle}\,dt+{\left\langle{\delta
m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}-u\,dt}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{-a\eta-\delta
a}\right\rangle}+{\left\langle{\delta{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}g^{-1}-a}\right\rangle}$
$\displaystyle\quad-{\left\langle{\frac{\delta h_{i}}{\delta m}}\,,\,{\delta
m}\right\rangle}\circ dW_{t}^{i}-{\left\langle{\frac{\delta h_{i}}{\delta
a}}\,,\,{\delta a}\right\rangle}\circ
dW_{t}^{i}+{\left\langle{m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\eta-{{\rm
ad}}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}\eta-\delta u\,dt}\right\rangle}\,.$

Here, $\eta=\delta g\,g^{-1}$ and natural boundary terms have been assumed.
Collecting terms among the variational relations gives the following set of
equations, which turn out to involve four semimartingales,

$\displaystyle\begin{split}&\frac{\delta\ell}{\delta
u}=m,\quad\frac{\delta\ell}{\delta
a}=b,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}=u\,dt+\sum_{i}\frac{\delta h_{i}}{\delta m}\circ
dW^{i}_{t},\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b=\frac{\delta\ell}{\delta a}\,dt-\sum_{i}\frac{\delta h_{i}}{\delta
a}\circ dW_{t}^{i},\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m=-{{\rm
ad}}^{*}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}m+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b\diamond
a,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a=-\mathsterling_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}a.\end{split}$ (B.7)

Thus the SALT EP equations are found to be

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\frac{\delta\ell}{\delta u}=-\,{{\rm
ad}}^{*}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}\frac{\delta\ell}{\delta u}+\left(\frac{\delta\ell}{\delta
a}\,dt-\sum_{i}\frac{\delta h_{i}}{\delta a}\circ dW_{t}^{i}\right)\diamond
a\,,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a=-\mathsterling_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}a\,,$ (B.8)

where the definition of
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}$ are taken from (B.7). For a similar, but more rigorous approach
to the derivation of these equations, see [66].

#### Lie-Poisson (LP) Hamiltonian formulation.

Using the Legendre transform,
$h(m,a)={\left\langle{m}\,,\,{u}\right\rangle}-\ell(u,a)$ and taking
variations yields

$\displaystyle\frac{\delta\ell}{\delta u}=m\,,\quad\frac{\delta h}{\delta
m}=u\,,\quad\frac{\delta h}{\delta a}=-\frac{\delta\ell}{\delta a}\,,$ (B.9)

and the corresponding SALT Lie-Poisson (LP) Hamiltonian equations obtained
after a Legendre transform are

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m=-{{\rm
ad}}^{*}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}}m-\left(\frac{\delta h}{\delta a}\,dt+\sum_{i}\frac{\delta
h_{i}}{\delta a}\circ dW_{t}^{i}\right)\diamond
a\,,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a=-\mathsterling_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}}a\,,\end{split}$ (B.10)

where the Lagrangian path
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}$ Legendre-transforms to the Hamiltonian side as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}=\frac{\delta h}{\delta m}\,dt+\sum_{i}\frac{\delta h_{i}}{\delta
m}\circ dW^{i}_{t}\,.$ (B.11)

By using the ${{\rm ad}}^{*}$ operator for semidirect product Lie algebras
defined in appendix A, the LP equations (B.10) can be written equivalently in
the following compact form,

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}(m,a)=-{{\rm
ad}}^{*}_{({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t},{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b)}(m,a),\quad\text{where
}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b=-\frac{\delta h}{\delta a}\,dt-\sum_{i}\frac{\delta h_{i}}{\delta a}\circ
dW_{t}^{i}.$

######  Remark B.3 (Stochastic reduced Hamiltonian phase space variational
principle for SALT).

The SALT equations (B.10) can be derived from a stochastic reduced Hamilton
phase-space variational principle, namely,

$\displaystyle 0=\delta
S=\delta\int^{b}_{a}{\left\langle{m}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}}\right\rangle}+{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}b}\,,\,{a_{0}g^{-1}-a}\right\rangle}-h(m,a)\,dt-\sum_{i}h_{i}(m,a)\circ
dW^{i}_{t},$

where the variations $\delta m,\delta g,\delta a$ and $\delta(b)$ are taken to
be arbitrary.

The SALT Hamiltonian equations in (B.10) can be arranged into the Lie-Poisson
(LP) operator form

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\begin{bmatrix}m\\\ a\end{bmatrix}=-\begin{bmatrix}{{\rm
ad}}^{*}_{\framebox{}}m&\framebox{}\diamond a\\\
\mathsterling_{\framebox{}}a&0\end{bmatrix}\begin{bmatrix}\frac{\delta
h}{\delta m}\,dt+\sum_{i}\frac{\delta h_{i}}{\delta m}\circ dW^{i}_{t}\\\
\frac{\delta h}{\delta a}\,dt+\sum_{i}\frac{\delta h_{i}}{\delta a}\circ
dW^{i}_{t}\end{bmatrix}\,,$ (B.12)

The Lie-Poisson operator in (B.12) preserves the Casimirs of its deterministic
counterpart, since the Poisson structures remains the same. However, the
Hamiltonian is now a semimartingale, so energy depends explicitly on time and,
hence, is no longer preserved.

#### Kelvin-Noether theorem.

The SALT Lie-Poisson (LP) Hamiltonian equations in (B.10) also possess a
Kelvin-Noether theorem. To understand this statement, consider the following
$G$-equivariant map $\mathcal{K}:C\times V^{*}\rightarrow\mathfrak{X}^{**}$ as
explained in [41],

${\left\langle{\mathcal{K}(gc,ag^{-1})}\,,\,{{{\rm
Ad}}^{*}_{g^{-1}}v}\right\rangle}={\left\langle{\mathcal{K}(c,a)}\,,\,{v}\right\rangle}\,,\quad\hbox{for
all}\quad g\in G\,,$

for a manifold $C$ on which $G$ acts from the left. For fluid dynamics, $C$ is
the space of loops the fluid domain $\mathcal{D}$ and the map $\mathcal{K}$ is
the circulation around the loop. More specifically, for all
$\alpha\in\Lambda^{1}$,

$\displaystyle{\left\langle{\mathcal{K}(c,a)}\,,\,{\alpha}\right\rangle}:=\oint_{c}\alpha,$
(B.13)

######  Theorem B.4 (SALT Kelvin-Noether theorem).

Given solutions $m(t),a(t)$ satisfying the SALT LP equations (B.10) and fixed
$c_{0}\in C$, the associated Kelvin-Noether quantity
${\left\langle{\mathcal{K}(g(t)c_{0},a(t))}\,,\,{m(t)}\right\rangle}$
satisfies the following stochastic Kelvin-Noether relation.

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{\mathcal{K}(g(t)c_{0},a(t))}\,,\,{m(t)}\right\rangle}={\left\langle{\mathcal{K}(g(t)c_{0},a(t))}\,,\,{\left(-\frac{\delta
h}{\delta a}\,dt-\sum_{i}\frac{\delta h_{i}}{\delta a}\circ
dW_{t}^{i}\right)\diamond a}\right\rangle}$ (B.14)

where we identify
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}=:{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}$.

######  Remark B.5.

The proof of theorem B.4 is in appendix B.3. The fluid mechanics counterpart
of the Kelvin-Noether theorem is expressed as

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\oint_{c_{t}}\frac{m}{D}=\oint_{c_{t}}\frac{1}{D}\left(-\frac{\delta
h}{\delta a}\,dt-\sum_{i}\frac{\delta h_{i}}{\delta a}\circ
dW_{t}^{i}\right)\diamond a\,,$

where the material loop $c_{t}=g(t)c_{0}$ moves with stochastic velocity
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}$ in equation (B.11) and the quantity $D$ is the mass density of the
fluid which is also advected as $D(t)=g_{t\,*}D_{0}$. That is, the mass
density $D$ satisfies the stochastic continuity equation
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}D+\mathcal{L}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}}D=0$.

######  Remark B.6 (SALT Hamiltonians).

In [38], the SEP equations are derived from a stochastic Clebsch variational
principle where the advection of phase space Lagrangian variables are through
a stochastic vector field
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}=u\,dt+\sum_{i}\xi_{i}(x)\circ dW_{t}^{i}$. Compared to the stochastic
vector field defined via
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}g\,g^{-1}$ in (B.7), they coincide with those in [38] when the noise
Hamiltonians $h_{i}$ are linear in $m$, i.e.
$h_{i}(m,a)={\left\langle{\xi_{i}(x)}\,,\,{m}\right\rangle}$. The SEP
equations then becomes

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\frac{\delta\ell}{\delta u}=-\,{{\rm
ad}}^{*}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}}\frac{\delta\ell}{\delta u}+\frac{\delta\ell}{\delta a}\diamond
a\,dt\,,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a=-\mathsterling_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}}a\,,\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}=u\,dt+\sum_{i}\xi_{i}(x)\circ dW_{t}^{i}\,.$

For data assimilation purposes, the choice $h_{i}(m,a)=\sum_{i}\langle
m,\xi_{i}(x)\rangle$ was made in [13, 12].

###  B.3 SALT Kelvin theorem via the Kunita-Itô-Wentzell theorem

A pair of results in Kunita [44, 45] provided the key to working with
stochastic advection by Lie transport (SALT) in ideal fluid dynamics. In
particular, if we choose the diffeomorphism $\phi_{t}$ as the stochastic
process obtained by homogenisation in [14]

${{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\phi_{t}(x):=u(\phi_{t}(x),t)dt+\sum_{i}\xi_{i}(\phi_{t}(x))\circ
dW_{t}^{i}\,,$

then the Kunita Itô-Wentzell change of variables formula discussed below leads
to the following differential form leads to the stochastic advection law,

${{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\big{(}\phi_{t}^{*}K(t,x)\big{)}=\phi_{t}^{*}\Big{(}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}K(t,x)+\mathcal{L}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{t}(x)}K(t,x)\Big{)}=0\,,\quad\hbox{a.s.}$

where
$\mathcal{L}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{t}(x)}$ is the _Lie derivative_ by the vector field
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{t}(x)$ whose time integral
$\int_{0}^{t}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{s}(x)=\phi_{t}(x)-\phi_{0}(x)$ generates the semimartingale flow
$\phi_{t}$ acting on semimartingale $k$-form,
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}K(t,x)=G(t,x)dt+H(t,x)\circ dW_{t}$. One may recall that the Lie derivative
$\mathcal{L}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{t}}K$ has both a dynamic and a geometric definition,

$\mathcal{L}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{t}}K=\lim_{\Delta s\to 0}\frac{1}{\Delta s}(\phi_{\Delta
s}^{*}K-K)={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{t}\makebox[11.99998pt][c]{\mbox{\rule{6.00006pt}{0.72229pt}\rule{0.72229pt}{6.00006pt}}}dK+d({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{t}\makebox[11.99998pt][c]{\mbox{\rule{6.00006pt}{0.72229pt}\rule{0.72229pt}{6.00006pt}}}K)$

in which the latter formula is attributed to Cartan.

Here is a simplified statement of the theorem for applying the Kunita Itô-
Wentzell change of variables formula to stochastic advection of differential
$k$-forms proved in [19], based on [44, 45]. See also [49].

######  Theorem B.7 (Kunita-Itô-Wentzell (KIW) formula for $k$-forms).

Consider a sufficiently smooth $k$-form $K(t,x)$ in space which is a
semimartingale in time

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}K(t,x)=G(t,x)dt+\sum_{i=1}^{M}H_{i}(t,x)\circ dW_{t}^{i},$ (B.15)

where $W_{t}^{i}$ are i.i.d. Brownian motions. Let $\phi_{t}$ be a
sufficiently smooth flow satisfying the SDE

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{t}(x)=b(t,\phi_{t}(x))dt+\sum_{i=1}^{N}\xi_{i}(t,\phi_{t}(x))\circ
dB_{t}^{i}\,,$

in which $B_{t}^{i}$ are i.i.d. Brownian motions. Then the pull-back
$\phi_{t}^{*}K$ satisfies the formula

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\,(\phi_{t}^{*}K)(t,x)=&\phi_{t}^{*}G(t,x)dt+\sum_{i=1}^{M}\phi_{t}^{*}H_{i}(t,x)\circ
dW_{t}^{i}\\\
&+\phi_{t}^{*}\mathcal{L}_{b}K(t,x)dt+\sum_{i=1}^{N}\phi_{t}^{*}\mathcal{L}_{\xi_{i}}K(t,x)\circ
dB_{t}^{i}.\end{split}$ (B.16)

Formulas (B.15) and (B.16) are compact forms of equations in [19] which are
written in integral notation to make the stochastic processes more explicit.

To understand the distinction between integral and differential notation for
SPEs, one may begin by writing the stochastic ‘fundamental theorem of
calculus’ as

$\phi_{t}^{*}K(t,x)-\phi_{0}^{*}K(0,x):=K(t,\phi_{t}(x))-K(0,x)=\int^{t}_{0}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\,(\phi_{s}^{*}K_{s})\,.$

In the integral notation, the Kunita-Itô-Wenzell (KIW) formula is written as

$\int^{t}_{0}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\,(\phi_{s}^{*}K_{s})=\int^{t}_{0}\phi_{s}^{*}\big{(}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}K(s,x)+\mathcal{L}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{s}(x)}K(s,x)\big{)}{}\,.$

So, in the differential notation the KIW formula ‘transfers’ to the equivalent
differential form

${{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\big{(}\phi_{t}^{*}K(t,x)\big{)}=\phi_{t}^{*}\Big{(}{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}K(t,x)+\mathcal{L}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\phi_{t}(x)}K(t,x)\Big{)}\,,\quad\hbox{a.s.}$

######  Remark B.8.

In applications, one sometimes expresses equation (B.16) using the
differential notation

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\,(\phi_{t}^{*}K)(t,x)=\phi_{t}^{*}\left({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}K+\mathcal{L}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}}K\right)(t,x),$

where ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}$ is the stochastic vector field
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}x_{t}(x)=b(t,x){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}t+\sum_{i=1}^{N}\xi_{i}(t,x)\circ{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}B_{t}^{i}$. Importantly for fluid dynamics, this formula is also valid when
$K$ is a vector field rather than a $k$-form.

#### Stochastic Kelvin circulation theorem for the SALT theory.

Having understood the differential notation for stochastic integrals, now we
may assemble the stochastic fluid equations via the stochastic Kelvin
circulation theorem for the SALT theory [38]. For this purpose, we shall make
the argument that the stochastic Kelvin circulation theorem is fundamentally a
stochastic form of Newton’s law of motion,

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bf
d}\oint_{c({{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\phi_{t})}\\!\\!\\!\mathbf{v}\cdot
d\mathbf{x}=\oint_{c({{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\phi_{t})}\underbrace{\
({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bf
d}+\mathcal{L}_{{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\phi_{t}})(\mathbf{v}\cdot d\mathbf{x})\ }_{\sf KIW\
formula}=\oint_{c({{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\phi_{t})}\underbrace{\ \mathbf{f}\cdot d\mathbf{x}\ }_{\sf
Newton^{\prime}s\ Law}.$

This formula corresponds to the _motion equation_ derived from Hamilton’s
principle

$\Big{(}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bf
d}+\mathcal{L}_{{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\phi_{t}}\Big{)}\Big{(}\frac{1}{D}\frac{\delta\ell}{\delta\mathbf{u}}\cdot
d\mathbf{x}\Big{)}=\mathbf{f}\cdot d\mathbf{x}\,,$

along with the law of _advection of mass_ expressed in KIW form

$\Big{(}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bf
d}+\mathcal{L}_{{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\phi_{t}}\Big{)}\big{(}Dd^{3}x\big{)}=0\,,$

where the _flow velocity_ is given by the stochastic vector field

${{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}}\phi_{t}(x):=u(\phi_{t}(x),t)dt+{\sum_{i}\xi_{i}(\phi_{t}(x))\circ
dW_{t}^{i}}\,.$

Thus, the stochastic Kelvin’s circulation theorem for SALT simply describes
the rate of change of momentum of a _stochastically moving_ material loop.

##  Appendix C Itô form of the SFLP equation

######  Proposition C.1.

The Itô form of the SFLP equation (2.7)

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt+\sum_{i}{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}f_{i}\circ dW^{i}_{t}=0\,,$

is given by

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt+\frac{1}{2}\sum_{i}{{\rm
ad}}^{*}_{\sigma_{i}}f_{i}\,dt+\sum_{i}{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f_{i}dW_{t}^{i}=0,\quad\sigma_{i}={\left({-{{\rm ad}}^{*}_{\frac{\delta
h}{\delta m}}f_{i}}\,,\,{\frac{\delta^{2}h}{\delta m^{2}}}\right)},$ (C.1)

where the brackets ${\left({\cdot}\,,\,{\cdot}\right)}$ in the definition of
$\sigma_{i}$ denotes contraction, not $L^{2}$ pairing.

###### Proof.

This can be shown via direct computation. We ignore the drift term and
suppress the indices on constants $f_{i}$ for ease of notation, choosing an
arbitrary $\phi=\phi(x)\in\mathfrak{X}$, we have the Stratonovich stochastic
equation

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\big{\langle}{m}\,,\,{\phi}\big{\rangle}}={\Big{\langle}{-{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}f}\,,\,{\phi}\Big{\rangle}}\circ
dW_{t}={\left\langle{{{\rm ad}}^{*}_{\phi}f}\,,\,{\frac{\delta h}{\delta
m}}\right\rangle}\circ dW_{t}$ (C.2)

The corresponding Itô form is then

$\displaystyle\begin{split}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\big{\langle}{\phi}\,,\,{m}\big{\rangle}}&={\left\langle{{{\rm
ad}}^{*}_{\phi}f}\,,\,{\frac{\delta h}{\delta
m}}\right\rangle}\,dW_{t}+\frac{1}{2}{\left[{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{\frac{\delta h}{\delta m}}\,,\,{{{\rm
ad}}^{*}_{\phi}f}\right\rangle}}\,,\,{dW_{t}}\right]}\\\ &={\left\langle{{{\rm
ad}}^{*}_{\phi}f}\,,\,{\frac{\delta h}{\delta
m}}\right\rangle}\,dW_{t}+\frac{1}{2}{\left[{{\left\langle{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\frac{\delta h}{\delta m}}\,,\,{{{\rm
ad}}^{*}_{\phi}f}\right\rangle}}\,,\,{dW_{t}}\right]}\\\ &={\left\langle{{{\rm
ad}}^{*}_{\phi}f}\,,\,{\frac{\delta h}{\delta
m}}\right\rangle}\,dW_{t}+\frac{1}{2}{\left[{{\left\langle{{\left({-\frac{\delta^{2}h}{\delta
m^{2}}}\,,\,{{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f}\right)}\,dW_{t}}\,,\,{{{\rm
ad}}^{*}_{\phi}f}\right\rangle}}\,,\,{dW_{t}}\right]}\\\
&=-{\left\langle{{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f}\,,\,{\phi}\right\rangle}\,dW_{t}-\frac{1}{2}{\left\langle{{{\rm
ad}}^{*}_{\sigma}f}\,,\,{\phi}\right\rangle}{\left[{dW_{t}}\,,\,{dW_{t}}\right]}\\\
\end{split}$ (C.3)

Since $\phi$ is arbitrary, the Itô form of (2.7) is

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt+\sum_{i}{{\rm
ad}}^{*}_{\frac{\delta h}{\delta
m}}f_{i}\,dW_{t}^{i}+\frac{1}{2}\sum_{i,j}{{\rm
ad}}^{*}_{\sigma_{j}}f_{i}{\left[{dW_{t}^{i}}\,,\,{dW_{t}^{j}}\right]}=0.$
(C.4)

For Brownian motion, the quadratic variation term simplifies to
${\left[{dW_{t}^{i}}\,,\,{dW_{t}^{j}}\right]}=\delta^{ij}dt$ which recovers
Itô form as presented. ∎

######  Proposition C.2.

The Itô form of the SFLP equation with advected quantity (2.19)

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt+\frac{\delta h}{\delta
a}\diamond a\,dt+\sum_{i}{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}f^{i}\circ
dW^{i}_{t}+\frac{\delta h}{\delta a}\diamond g^{i}\circ dW^{i}_{t}=0\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a+\mathcal{L}_{\frac{\delta h}{\delta
m}}a\,dt+\sum_{i}\mathcal{L}_{\frac{\delta h}{\delta m}}g^{i}\circ
dW^{i}_{t}=0\end{split}$ (C.5)

is given by

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt+\frac{\delta h}{\delta
a}\diamond a\,dt+\sum_{i}{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f^{i}\,dW^{i}_{t}+\frac{\delta h}{\delta a}\diamond
g^{i}\,dW^{i}_{t}+\frac{1}{2}\sum_{i}({{\rm
ad}}^{*}_{\sigma_{i}}f^{i}+\theta_{i}\diamond g^{i})\,dt=0\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a+\mathcal{L}_{\frac{\delta h}{\delta
m}}a\,dt+\sum_{i}\mathcal{L}_{\frac{\delta h}{\delta
m}}g^{i}\,dW^{i}_{t}+\frac{1}{2}\sum_{i}\mathcal{L}_{\sigma_{i}}g^{i}\,dt=0\end{split}$
(C.6)

where $\sigma_{i}$ and $\theta_{i}$ are given as

$\displaystyle\sigma_{i}:={\left({\frac{\delta^{2}h}{\delta
m^{2}}}\,,\,{-{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}f^{m}_{i}-\frac{\delta
h}{\delta a}\diamond
f^{a}_{i}}\right)}\quad\hbox{and}\quad\theta_{i}:={\left({\frac{\delta^{2}h}{\delta
a^{2}}}\,,\,{-\mathcal{L}_{\frac{\delta h}{\delta m}}f^{a}_{i}}\right)}.$

###### Proof.

Consider similarly by ignoring drift terms and suppress the indices on the
$f^{i}\in\mathfrak{X}^{*}$, $g^{i}\in V^{*}$, $\sigma_{i}\in\mathfrak{X}$ and
$\theta_{i}\in V$. Consider constants $\phi=\phi(x)\in\mathfrak{X}$ and
$\psi=\psi(x)\in V$, the following Stratonovich stochastic equations hold

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{m}\,,\,{\phi}\right\rangle}=-{\left\langle{{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}f+\frac{\delta h}{\delta a}\diamond
g}\,,\,{\phi}\right\rangle}\circ
dW_{t},\quad{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{a}\,,\,{\psi}\right\rangle}=-{\left\langle{\mathcal{L}_{\frac{\delta
h}{\delta m}}g}\,,\,{\psi}\right\rangle}\circ dW_{t}=0$ (C.7)

The Itô form thus satisfies

$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{m}\,,\,{\phi}\right\rangle}$
$\displaystyle=-{\left\langle{{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f+\frac{\delta h}{\delta a}\diamond
g}\,,\,{\phi}\right\rangle}dW_{t}-\frac{1}{2}{\left[{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}f+\frac{\delta
h}{\delta a}\diamond g}\,,\,{\phi}\right\rangle}}\,,\,{dW_{t}}\right]}$
$\displaystyle=-{\left\langle{{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f+\frac{\delta h}{\delta a}\diamond
g}\,,\,{\phi}\right\rangle}dW_{t}+\frac{1}{2}{\left[{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{{{\rm ad}}^{*}_{\phi}f}\,,\,{\frac{\delta h}{\delta
m}}\right\rangle}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{\mathcal{L}_{\phi}g}\,,\,{\frac{\delta h}{\delta
a}}\right\rangle}}\,,\,{dW_{t}}\right]}$ $\displaystyle=-{\left\langle{{{\rm
ad}}^{*}_{\frac{\delta h}{\delta m}}f+\frac{\delta h}{\delta a}\diamond
g}\,,\,{\phi}\right\rangle}dW_{t}+\frac{1}{2}{\left[{{\left\langle{{{\rm
ad}}^{*}_{\phi}f}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\frac{\delta h}{\delta
m}}\right\rangle}+{\left\langle{\mathcal{L}_{\phi}g}\,,\,{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}\frac{\delta h}{\delta a}}\right\rangle}}\,,\,{dW_{t}}\right]}$
$\displaystyle=-{\left\langle{{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f+\frac{\delta h}{\delta a}\diamond
g}\,,\,{\phi}\right\rangle}dW_{t}+\frac{1}{2}{\left[{{\left\langle{{{\rm
ad}}^{*}_{\phi}f}\,,\,{\sigma_{i}\,dW_{t}}\right\rangle}+{\left\langle{\mathcal{L}_{\phi}g}\,,\,{\theta_{i}\,dW_{t}}\right\rangle}}\,,\,{dW_{t}}\right]}$
$\displaystyle=-{\left\langle{{{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f+\frac{\delta h}{\delta a}\diamond
g}\,,\,{\phi}\right\rangle}dW_{t}-\frac{1}{2}{\left\langle{{{\rm
ad}}^{*}_{\sigma}f+\mathcal{L}_{\theta}g}\,,\,{\phi}\right\rangle}\,{\left[{dW_{t}}\,,\,{dW_{t}}\right]}$
$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{a}\,,\,{\psi}\right\rangle}$
$\displaystyle=-{\left\langle{\mathcal{L}_{\frac{\delta h}{\delta
m}}g}\,,\,{\psi}\right\rangle}\,dW_{t}-\frac{1}{2}{\left[{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{\mathcal{L}_{\frac{\delta h}{\delta
m}}g}\,,\,{\psi}\right\rangle}}\,,\,{dW_{t}}\right]}$
$\displaystyle=-{\left\langle{\mathcal{L}_{\frac{\delta h}{\delta
m}}g}\,,\,{\psi}\right\rangle}\,dW_{t}+\frac{1}{2}{\left[{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}{\left\langle{\psi\diamond g}\,,\,{\frac{\delta h}{\delta
m}}\right\rangle}}\,,\,{dW_{t}}\right]}$
$\displaystyle=-{\left\langle{\mathcal{L}_{\frac{\delta h}{\delta
m}}g}\,,\,{\psi}\right\rangle}\,dW_{t}-\frac{1}{2}{\left[{{\left\langle{\psi\diamond
g}\,,\,{\sigma\,dW_{t}}\right\rangle}}\,,\,{dW_{t}}\right]}$
$\displaystyle=-{\left\langle{\mathcal{L}_{\frac{\delta h}{\delta
m}}g}\,,\,{\psi}\right\rangle}\,dW_{t}-\frac{1}{2}{\left\langle{\mathcal{L}_{\sigma}g}\,,\,{\psi}\right\rangle}\,{\left[{dW_{t}}\,,\,{dW_{t}}\right]}$

Since $\phi$ and $\psi$ are arbitrary, the Itô form of
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm d}}m$
and ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a$ with drift are

$\displaystyle\begin{split}&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}m+{{\rm ad}}^{*}_{\frac{\delta h}{\delta m}}m\,dt+\frac{\delta h}{\delta
a}\diamond a\,dt+\sum_{i}\left({{\rm ad}}^{*}_{\frac{\delta h}{\delta
m}}f^{i}+\frac{\delta h}{\delta a}\diamond
g^{i}\right)\,dW^{i}_{t}+\frac{1}{2}\sum_{i,j}({{\rm
ad}}^{*}_{\sigma_{j}}f^{i}+\theta_{j}\diamond
g^{i}){\left[{dW^{i}_{t}}\,,\,{dW^{j}_{t}}\right]}=0\\\
&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\rm
d}}a+\mathcal{L}_{\frac{\delta h}{\delta
m}}a\,dt+\sum_{i,j}\mathcal{L}_{\sigma_{j}}g^{i}\,{\left[{dW^{i}_{t}}\,,\,{dW^{j}_{t}}\right]}=0\end{split}$
(C.8)

For Brownian motion, the quadratic variation term simplifies to
${\left[{dW_{t}^{i}}\,,\,{dW_{t}^{j}}\right]}=\delta^{ij}dt$ which completes
the proof. ∎

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