#  Stochastic geometric mechanics with diffeomorphisms

Darryl D. Holm and Erwin Luesink

Department of Mathematics, Imperial College London SW7 2AZ, UK
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Noether’s celebrated theorem associating symmetry and conservation laws in
classical field theory is adapted to allow for broken symmetry in geometric
mechanics and is shown to play a central role in deriving and understanding
the generation of fluid circulation via the Kelvin-Noether theorem for ideal
fluids with stochastic advection by Lie transport (SALT).

##  1 Noether’s theorem in geometric mechanics

###  1.1 Euler-Poincaré reduction

Geometric mechanics deals with group-invariant variational principles. In this
setting, Noether’s theorem Noether (1918, 1971) plays a key role. Given the
tangent lift action $G\times TM\to TM$ of a Lie group $G$ on the tangent
bundle $TM$ of a manifold $M$111$M$ is called the _configuration manifold_ in
classical mechanics. on which $G$ acts transitively, Noether’s theorem states
that each Lie symmetry of a Lagrangian $L:TM\to\mathbb{R}$ defined in the
action integral $S=\int L(q,v)dt$ for Hamilton’s variational principle $\delta
S=0$ with $(q,v)\in TM$ implies a conserved quantity for the corresponding
Euler-Lagrange equations defined on the cotangent bundle $T^{*}M$. The
conserved quantities arising from Noether’s theorem in the case where the
configuration manifold $M$ is a Lie group $G$ were studied by Smale, in Smale
(1970a, b), where it was shown that the reduction procedure $TG\to TG\setminus
G\simeq\mathfrak{g}$ leads to dynamics which take place on the dual
$\mathfrak{g}^{*}$ of the Lie algebra $\mathfrak{g}$. The dynamical variable
$m\in\mathfrak{g}^{*}$ in the dual Lie algebra is now called the momentum map
(Smale called it angular momentum). In general, the configuration manifold $M$
is not a Lie group. However, when a Lie group $G$ acts transitively on a
configuration manifold $M$ the proof of Noether’s theorem induces a cotangent-
lift momentum map $J:T^{*}M\to\mathfrak{g}^{*}$. The momentum map induced this
way is an infinitesimally equivariant Poisson map taking functions on the
cotangent bundle $T^{*}M$ of $M$ to the dual Lie algebra $\mathfrak{g}^{*}$ of
the Lie group $G$. The momentum map $J:T^{*}M\to\mathfrak{g}^{*}$ is
equivariant and Poisson, even if $G$ is not a Lie symmetry of the Lagrangian
in Hamilton’s principle. Momentum maps naturally lead from the Lagrangian side
to the Hamiltonian side. The Hamiltonian dynamics on $T^{*}M$ involves
symplectic transformations. However, as we shall discuss below, for the class
of Hamiltonians which can be defined as $H\circ
J:\mathfrak{g}^{*}\to\mathbb{R}$, the momentum map induces Euler-Poincaré
motion on the Lagrangian side and Lie-Poisson motion on the Hamiltonian side.
To illustrate these remarks, we return to the situation in which the
configuration manifold, $M$, is a Lie group, $G$.

For hyperregular Lagrangians, the Legendre transform to the Hamiltonian side
is invertible and one may reconstruct the solution on $G$ from its
representation on $T^{*}G\setminus G\simeq\mathfrak{g}^{*}$. In that case,
solving the equations describing the evolution of the momentum map on the dual
Lie algebra $\mathfrak{g}^{*}$ is equivalent to solving the equations on the
cotangent bundle $T^{*}G$ when the configuration manifold is $G$. When the Lie
group $G$ acts transitively, freely and properly on the configuration manifold
$M$, then one may reconstruct the solution on $M$ from its representation on
$T^{*}G\setminus G\simeq\mathfrak{g}^{*}$. The last statement is proven for
finite-dimensional Lie groups $G$ in, e.g., Abraham and Marsden (1978).

The Lie-group reduced equations defined on the dual Lie algebra
$\mathfrak{g}^{*}$ via Smale’s procedure of reduction by symmetry
$T^{*}G\setminus G\simeq\mathfrak{g}^{*}$ are called Euler-Poincaré equations
after Poincaré (1901). Provided the Lagrangian is hyperregular, the Euler-
Poincaré reduction procedure can be expressed in terms of the cube of linked
commutative diagrams shown in figure 1.

${L:TG\to\mathbb{R}}$${H:T^{*}G\to\mathbb{R}}$Euler-Lagrange eqnsHamilton’s
eqns${\ell:\mathfrak{g}\to\mathbb{R}}$${\hslash:\mathfrak{g}^{*}\to\mathbb{R}}$Euler-
Poincaré eqnsLie-Poisson eqnsLegendre transformLegendre transform Figure 1:
The cube of commutative diagrams for geometric mechanics on Lie groups. Euler-
Poincaré reduction (on the left side) and Lie-Poisson reduction (on the right
side) are both indicated by the arrows pointing down. The diagrams are all
commutative, provided the Legendre transformation and reduced Legendre
transformation are both invertible.

To summarise the notation in figure 1, $G$ denotes the configuration manifold
which is assumed to be isomorphic to a Lie group, $TG$ is the tangent bundle,
$T^{*}G$ is the cotangent bundle, $TG\setminus G\simeq\mathfrak{g}$ is the Lie
algebra and $T^{*}G\setminus G\simeq\mathfrak{g}^{*}$ is the dual of the Lie
algebra. The Lagrangian is a functional $L:TG\to\mathbb{R}$ and the
Hamiltonian is a functional $H:T^{*}G\to\mathbb{R}$. Euler-Poincaré reduction
takes advantage of Lie group symmetries to transform the Lagrangian and
Hamiltonian into group-invariant variables, which leads to a reduced
Lagrangian $\ell:\mathfrak{g}\to\mathbb{R}$ and a reduced Hamiltonian
$\hslash:\mathfrak{g}^{*}\to\mathbb{R}$. The diagram comprising the face of
the cube involving these functionals in figure 1 commutes if the Legendre
transform is a diffeomorphism. This is guaranteed if the Lagrangian or
Hamiltonian is hyperregular. The Euler-Lagrange equations and Hamilton’s
equations are related via a change of variables, which also holds for the
Euler-Poincaré equations and the Lie-Poisson equations. Many finite
dimensional mechanical systems may be described naturally in this framework.
The classic example is the rotating rigid body, discussed from the viewpoint
of symmetry reduction by Poincaré in Poincaré (1901). In his 1901 paper,
Poincaré also raised the issue of _symmetry breaking_ , by introducing the
vertical acceleration of gravity, which breaks the $SO(3)$ symmetry for free
rotation and restricts it to $SO(2)$ for rotations about the vertical axis.

Stochasticity may also be included in the framework of Euler-Poincaré
reduction by symmetry. The first attempt to include noise consistently in
finite-dimensional symplectic Hamiltonian mechanics was by Bismut (1982) and
reduction by symmetry of stochastic systems was studied by Lázaro-Camı and
Ortega (2008).

#### Plan of the paper.

In the present work, we will review Euler-Poincaré reduction of stochastic
infinite dimensional variational systems with symmetry breaking. The infinite
dimensional case is interesting because it is the natural setting for fluid
dynamics, quantum mechanics and elasticity. The foundations of the finite
dimensional stochastic geometric mechanics are established in Cruzeiro et al.
(2018). We will explore the infinite dimensional case in context of fluid
dynamics, where symmetry under the smooth invertible maps of the flow domain
is broken by the spatial dependence of the initial mass density.

###  1.2 Sobolev class diffeomorphisms

Consider an $n$-dimensional compact and oriented smooth manifold $M$, equipped
with a Riemannian metric $\langle\,\cdot\,,\,\cdot\,\rangle$. This will be the
spatial domain of flow and $X\in M$ will denote the initial position of any
given fluid particle. The manifold $M$ is acted upon by a group of Sobolev
class diffeomorphisms. In Ebin and Marsden (1970) it is shown that the space
of $C^{\infty}$ diffeomorphisms, defined by $\mathfrak{D}=\\{g\in
C^{\infty}(M,M)|\,g\text{ is bijective and }g^{-1}\in C^{\infty}(M,M)\\}$, is
not the convenient setting to study fluid dynamics, but that one should use
$\mathfrak{D}^{s}=\\{g\in H^{s}(M,M)|\,g\text{ is bijective and }g^{-1}\in
H^{s}(M,M)\\}$, the space of Sobolev class diffeomorphisms with $s$ weak
derivatives. The reason for this choice is that the smooth diffeomorphisms
constitute a Fréchet manifold for which there is no inverse or implicit
function theorem and no general solution theorem for ordinary differential
equations. Each of these latter features would prohibit the study of
geodesics.

The space of Sobolev class diffeomorphisms is both a Hilbert manifold and a
topological group if $s>n/2+1$, as was shown by Ebin (1967). The Hilbert
manifold structure implies the existence of function inverses and the implicit
function theorem, as well as the existence of a general solution theorem for
ordinary differential equations. This additional structure also implies that
one can construct the tangent space of $\mathfrak{D}^{s}$ in the usual way and
study geodesics. The space $\mathfrak{D}^{s}$ is the configuration space for
continuum mechanics and each $g\in\mathfrak{D}^{s}$ is called a configuration.
A fluid trajectory starting from $X\in M$ at time $t=0$ is given by
$x(t)=g_{t}(X)=g(X,t)$, with $\mathfrak{D}^{s}\ni g:M\times\mathbb{R}^{+}\to
M$ being a continuous one-parameter subgroup of $\mathfrak{D}^{s}$. In the
deterministic case, computing the time derivative of this one-parameter
subgroup gives rise to the _reconstruction equation_ , given by

$\frac{\partial}{\partial t}g_{t}(X)=u(g_{t}(X),t),$ (1)

where $u_{t}(\,\cdot\,)=u(\,\cdot\,,t)\in\mathfrak{X}^{s}$ is a time dependent
vector field with flow $g_{t}(\,\cdot\,)=g(\,\cdot\,,t)$. The initial data is
given by $g(X,0)=X$. Here $\mathfrak{X}^{s}=H^{s}(TM)$ denotes the space of
Sobolev class vector fields on $M$, which is also the Lie algebra associated
to the Sobolev class diffeomorphisms.

###  1.3 Stochastic advection by Lie transport (SALT)

In the setting of stochastic advection by Lie transport (SALT), which was
introduced by Holm (2015), the deterministic reconstruction equation in (1) is
replaced by the semimartingale

${\sf d}g(X,t)=u(g_{t}(X),t)dt+\sum_{i=1}^{M}\xi_{i}(g_{t}(X))\circ
dW_{t}^{i},$ (2)

where the symbol $\circ$ means that the stochastic integral is taken in the
Stratonovich sense. The initial data is given by $g(X,0)=X$. The $W_{t}^{i}$
are independent, identically distributed Brownian motions, defined with
respect to the standard stochastic basis
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$. Such a noise
was shown to arise from a multi-time homogenisation argument in Cotter et al.
(2017). The $\xi_{i}(\,\cdot\,)\in\mathfrak{X}^{s}$ are called data vector
fields and are prescribed. These data vector fields represent the effects of
unresolved degrees of freedom on the resolved scales of fluid motion and
account for unrepresented processes. They are determined by applying empirical
orthogonal function analysis to appropriate numerical and/or observational
data. For instance, for an application to the two dimensional Euler equations
for an ideal fluid, see Cotter et al. (2019a) and for an application to a two-
layer quasi-geostrophic model, see Cotter et al. (2018). Stochastic models
enable the use of a variety of methods in data assimilation, which are
discussed in Cotter et al. (2019b). It is not difficult to make sense of (1),
but understanding (2) is more complicated. In Bethencourt de Leon et al.
(2020), a _stochastic chain rule_ is shown to exist. This stochastic chain
rule is called the _Kunita-Itô-Wentzell (KIW) formula_ and helps interpret the
semimartingale in (2). The KIW formula will also be used later to prove the
stochastic Kelvin circulation theorem. First, however, the space
$\mathfrak{D}^{s}$ needs to be given more structure.

The space $\mathfrak{D}^{s}$ inherits a _weak Riemannian structure_ from the
underlying manifold $M$ in a natural way. For $g\in\mathfrak{D}^{s}$ and
$V,W\in T_{g}\mathfrak{D}^{s}$, one can define the following bilinear form

$(V,W)=\int_{M}\langle(V(X),W(X)\rangle_{g(X)}\mu(dX),$ (3)

where $\mu$ is the volume form on $M$ induced by the metric. The Riemannian
structure induced by (3) is weak because the topology is of type $L^{2}$,
which is strictly weaker than the $H^{s}$ topology. This bilinear form is a
linear functional on the Hilbert space $T_{g}\mathfrak{D}^{s}$ and can be used
to define the dual space $T_{g}^{*}\mathfrak{D}^{s}$. The pairing between
$V\in T_{g}\mathfrak{D}^{s}$ and $\alpha\in T_{g}^{*}\mathfrak{D}^{s}$ is
given by

$\langle\alpha,V\rangle=\int_{M}\alpha(X)\cdot V(X).$ (4)

Hence the metric on $M$ and the volume form $\mu(dX)$ can be used to construct
the isomorphism between $T\mathfrak{D}^{s}$ and $T^{*}\mathfrak{D}^{s}$ as
$V(X)\mapsto\alpha(X)=V^{\flat}(X)\mu(dX)$, where $\flat:TM\to T^{*}M$ is one
of the musical isomorphisms that are induced by the metric on $M$. The group
$\mathfrak{D}^{s}$ is not a Lie group; since right multiplication is smooth,
but left multiplication is only continuous. Hence $\mathfrak{D}^{s}$ is a
topological group with a weak Riemannian structure. In general, these
properties are not sufficient to guarantee the existence of an exponential
map. However, Ebin and Marsden (1970) showed that an exponential map can exist
in many important cases. In particular, they showed that the geodesic spray
associated to (3) (with and without forcing) is smooth.222The _geodesic spray_
is the vector field whose integral curves are the geodesics. The smoothness of
the geodesic spray persists even though $H^{s}$ diffeomorphisms are considered
rather than smooth diffeomorphisms. Combined with the existence of an
exponential map, the smoothness property implies a regular interpretation of
the Euler-Poincaré equations on $\mathfrak{D}^{s}$, provided that one uses
right translations and right representations of the group on itself and its
Lie algebra, as shown in Holm et al. (1998). However, due to the presence of
the volume form $\mu(dm)$, the bilinear form (3) is not right-invariant under
the action of the entire $H^{s}$ diffeomorphism group, although there is
right-invariance under the action of the isotropy subgroup
$\mathfrak{D}^{s}_{\mu}=\\{g\in\mathfrak{D}^{s}|\,g_{*}\mu=\mu\\}$. Since this
subgroup is a proper subgroup, as it is smaller than $\mathfrak{D}^{s}$
itself. Thus, one speaks of _symmetry breaking_.

In deriving the equations of ideal deterministic fluid dynamics, one needs to
keep track of the volume form as well. The appropriate mathematical setting
for this is an _outer semidirect product group_. This means that one
constructs a new group from two given groups with a particular type of group
operation. For continuum mechanics, the ingredients are $\mathfrak{D}^{s}$ and
$V^{*}$, where $V^{*}$ is a vector space of tensor fields. This vector space
is the space of _advected quantities_ and it will always contain at least the
volume form $\mu$.

######  Definition 1.1 (Advected quantity).

A fluid variable is said to be _advected_ , if it keeps its value along
Lagrangian particle trajectories. Advected quantities are sometimes called
_tracers_ , because the evolution histories of scalar advected quantities with
different initial values (labels) trace out the Lagrangian particle
trajectories of each label, or initial value, via the _push-forward_ of the
full diffeomorphism group, i.e., $a_{t}=g_{t\,*}a_{0}=a_{0}g_{t}^{-1}$, where
$g_{t}$ is a time-dependent curve on the manifold of diffeomorphisms that
represents the fluid flow.

######  Remark 1 (Advected quantities as order parameters).

When several advected quantities are involved, the space $V^{*}$ is the direct
sum of several vector spaces, where each summand space hosts a different
advected quantity. In general, each additional advected quantity decreases the
dimension of the isotropy subgroup. For example, consider an ideal
deterministic fluid with a buoyancy variable $b$, then the Lagrangian
corresponding to the model will depend on $\mu$ and $b$ in a parametric
manner. This Lagrangian will be right invariant under the action of the
isotropy subgroup
$\mathfrak{D}^{s}_{\mu,b}=\\{g\in\mathfrak{D}^{s}|\,g_{*}\mu=\mu\text{ and
}g_{*}b=b\\}$. Hence, advected quantities are _order parameters_ and each
additional order parameter breaks more symmetry. For the sake of notation, one
usually writes $\mathfrak{D}^{s}_{a_{0}}$ for the isotropy subgroup, no matter
how many advected quantities there are. One then uses $a$ to represent all
advected quantities and $a_{0}$ to denote the initial value of the advected
quantities.

###  1.4 Semidirect product group adjoint & coadjoint actions

The semidirect product group action is constructed in the following way. The
representation of $\mathfrak{D}^{s}$ on a vector space $V$ is by push-forward,
which is a left representation, as shown by Marsden et al. (1984). The
representation of the group on itself and on its Lie algebra is a right
representation. In terms of analysis, this means that all representations are
smooth and no derivatives need to be counted. The group action of the
semidirect product group is given by

$\displaystyle\bullet:(\mathfrak{D}^{s}\times V)\times(\mathfrak{D}^{s}\times
V)\to(\mathfrak{D}^{s}\times V)$ (5)
$\displaystyle\quad(g_{1},v_{1})\bullet(g_{2},v_{2}):=(g_{1}\circ
g_{2},v_{2}+(g_{2})_{*}v_{1})$

with $g_{1},g_{2}\in\mathfrak{D}^{s}$ and $v_{1},v_{2}\in V$. The semidirect
product group is often denoted as $\mathfrak{D}^{s}\circledS
V=(\mathfrak{D}^{s}\times V,\bullet)$. In the group action above,
$(g_{2})_{*}v_{1}$ denotes the _push-forward_ of $v_{1}$ by $g_{2}$ and
$\circ$ denotes composition. Note that the group affects both slots in (5),
but the vector space only appears in the second slot. The identity element of
the semidirect product group is $(e,0)$ where $e\in\mathfrak{D}^{s}$ is the
identity diffeomorphism and $0\in V$ is the zero vector. An inverse element is
given by

$(g,v)^{-1}=(g^{-1},-(g^{-1})_{*}v)=(g^{-1},-g^{*}v),$ (6)

where $g^{*}v$ denotes the pull-back of $v$ by $g$. To understand how
reduction works for semidirect products, it is helpful to know how the group
acts on its Lie algebra and on the dual of its Lie algebra. Duality will be
defined with respect to the sum of the pairing (4) and the dual linear
transformation $[\,\cdot\,]^{*}$ on $V$. This pairing induces another pairing
in a natural way on $\mathfrak{X}^{s}\times V$. Consider two at least $C^{1}$
one parameter subgroups
$(g_{t},v_{t}),(\widetilde{g}_{\epsilon},\widetilde{v}_{\epsilon})\in\mathfrak{D}^{s}\times
V$. Using these one parameter subgroups, one can compute the inner
automorphism, or adjoint action of the group on itself. This adjoint action is
defined by conjugation

$\displaystyle{\rm AD}:(\mathfrak{D}^{s}\times V)\times(\mathfrak{D}^{s}$
$\displaystyle\times V)\to(\mathfrak{D}^{s}\times V),$ (7) $\displaystyle{\rm
AD}_{(g_{t},v_{t})}(\widetilde{g}_{\epsilon},\widetilde{v}_{\epsilon})$
$\displaystyle:=(g_{t},v_{t})\bullet(\widetilde{g}_{\epsilon},\widetilde{v}_{\epsilon})\bullet(g_{t},v_{t})^{-1}$
$\displaystyle=\big{(}g_{t}\circ\widetilde{g}_{\epsilon}\circ
g_{t}^{-1},g_{t}^{*}(\widetilde{v}_{\epsilon}-v_{t}+\widetilde{g}_{\epsilon*}v_{t})\big{)}.$

To see how the group acts on its Lie algebra, one can compute the derivative
with respect to $\epsilon$ and evaluate at $\epsilon=0$ in the adjoint action
of the group on itself. Let
$\mathfrak{X}^{s}\ni\widetilde{u}=\frac{d}{d\epsilon}|_{\epsilon=0}\widetilde{g}_{\epsilon}$
and
$V\ni\widetilde{b}=\frac{d}{d\epsilon}|_{\epsilon=0}\widetilde{v}_{\epsilon}$.
This choice for a vector field is guided by the deterministic reconstruction
equation in (1). For any tensor $S_{\epsilon}\in T_{s}^{r}(M)$ whose
dependence on $\epsilon$ is at least $C^{1}$ it holds that

$\frac{d}{d\epsilon}\widetilde{g}_{\epsilon*}S_{\epsilon}=\widetilde{g}_{\epsilon*}\left(\frac{d}{d\epsilon}S_{\epsilon}-\mathcal{L}_{\widetilde{u}}S_{\epsilon}\right).$
(8)

Important here is that the Lie derivative does not commute with pull-backs and
push-forwards that depend on parameters, see Abraham and Marsden (1978). The
adjoint action of the group on its Lie algebra can be computed as

$\displaystyle{\rm Ad}:(\mathfrak{D}^{s}\times V)\times(\mathfrak{X}^{s}$
$\displaystyle\times V)\to(\mathfrak{X}^{s}\times V),$ (9) $\displaystyle{\rm
Ad}_{(g_{t},v_{t})}(\widetilde{u},\widetilde{b})$
$\displaystyle:=\frac{d}{d\epsilon}\Big{|}_{\epsilon=0}{\rm
AD}_{(g_{t},v_{t})}(\widetilde{g}_{\epsilon},\widetilde{v}_{\epsilon})$
$\displaystyle=(g_{t*}\widetilde{u},g^{*}_{t}\widetilde{b}-g_{t}^{*}\mathcal{L}_{\widetilde{u}}v_{t}).$

By means of the pairing on $\mathfrak{X}^{s}\times V$, one can compute the
dual action to the adjoint action (9). This is called the coadjoint action of
the group on the dual of its Lie algebra. Let
$(\widetilde{m},\widetilde{a})\in(\mathfrak{X}^{s}\times V)^{*}$, then the
coadjoint action is given by

$\displaystyle{\rm Ad}^{*}:(\mathfrak{D}^{s}\times V)\times(\mathfrak{X}^{s}$
$\displaystyle\times V)^{*}\to(\mathfrak{X}^{s}\times V)^{*},$ (10)
$\displaystyle\langle{\rm
Ad}^{*}_{(g_{t}^{-1},-g_{t}^{-1}v_{t})}(\widetilde{m},\widetilde{a}),(\widetilde{u},\widetilde{b})\rangle$
$\displaystyle:=\langle(\widetilde{m},\widetilde{a}),{\rm
Ad}_{(g_{t},v_{t})}(\widetilde{u},\widetilde{b})\rangle,$ $\displaystyle{\rm
Ad}^{*}_{(g_{t}^{-1},-g_{t}^{-1}v_{t})}(\widetilde{m},\widetilde{a})$
$\displaystyle=(g_{t}^{*}\widetilde{m}+v_{t}\diamond
g_{t*}\widetilde{a},g_{t*}\widetilde{a}).$

######  Definition 1.2 (The diamond operator).

The coadjoint action (10) features the _diamond operator_ , which is defined
for $a\in V^{*}$, $u\in\mathfrak{X}^{s}$ and fixed $v\in V$ as

$\langle v\diamond
a,u\rangle_{\mathfrak{X}^{s*}\times\mathfrak{X}^{s}}:=-\langle
a,\mathcal{L}_{u}v\rangle_{V^{*}\times V}.$ (11)

Note that the diamond operator is the dual of the Lie derivative regarded as a
map $\mathcal{L}_{(\,\cdot\,)}v:\mathfrak{X}^{s}\to V$, hence
$v\diamond(\,\cdot\,):V^{*}\to\mathfrak{X}^{s*}$. The diamond operator shows
how an element from the dual of the vector space acts on the dual of the Lie
algebra.

When evaluated at $t=0$, the $t$-derivatives of ${\rm Ad}$ in (9) and ${\rm
Ad}^{*}$ in (10) define, respectively, the adjoint and coadjoint actions of
the Lie algebra on itself and on its dual. Denote by $\mathfrak{X}^{s}\ni
u=\frac{d}{dt}|_{t=0}g_{t}$ and $V\ni b=\frac{d}{dt}|_{t=0}v_{t}$. The adjoint
action of the Lie algebra on itself is

$\displaystyle{\rm ad}:(\mathfrak{X}^{s}\times V)\times(\mathfrak{X}^{s}$
$\displaystyle\times V)\to(\mathfrak{X}^{s}\times V),$ (12) $\displaystyle{\rm
ad}_{(u,b)}(\widetilde{u},\widetilde{b})$
$\displaystyle:=\frac{d}{dt}\Big{|}_{t=0}{\rm
Ad}_{(g_{t},v_{t})}(\widetilde{u},\widetilde{b}),$ $\displaystyle{\rm
ad}_{(u,b)}(\widetilde{u},\widetilde{b})$
$\displaystyle=(-\mathcal{L}_{u}\widetilde{u},\mathcal{L}_{u}\widetilde{b}-\mathcal{L}_{\widetilde{u}}b)$
$\displaystyle=(-[u,\widetilde{u}],\mathcal{L}_{u}\widetilde{b}-\mathcal{L}_{\widetilde{u}}b),$

where the bracket $[\,\cdot\,,\,\cdot\,]$ in (12) is the commutator of vector
fields. The minus sign is due to fact that group acts on itself from the
right. The coadjoint action of the Lie algebra on its dual can be obtained by
computing the dual to (12) or by taking the derivative with respect to $t$ and
evaluate at $t=0$ in (10). Either way, one arrives at

$\displaystyle{\rm ad}^{*}:(\mathfrak{X}^{s}\times V)\times(\mathfrak{X}^{s}$
$\displaystyle\times V)^{*}\to(\mathfrak{X}^{s}\times V)^{*},$ (13)
$\displaystyle\langle{\rm
ad}^{*}_{(u,b)}(\widetilde{m},\widetilde{a}),(\widetilde{u},\widetilde{b})\rangle$
$\displaystyle:=\langle(\widetilde{m},\widetilde{a}),{\rm
ad}_{(u,b)}(\widetilde{u},\widetilde{b})\rangle,$ $\displaystyle{\rm
ad}^{*}_{(u,b)}(\widetilde{m},\widetilde{a})$
$\displaystyle=(\mathcal{L}_{u}\widetilde{m}+b\diamond\widetilde{a},-\mathcal{L}_{u}\widetilde{a}),$

in which (12) implies the last line in (13). Alternatively, one can also
obtain (13) by taking the derivative with respecto to $t$ in (10) and evaluate
at $t=0$.

######  Remark 2 (Coadjoint action and the diamond operator).

The coadjoint action is an important operator in geometric mechanics and
representation theory. It was shown by Kirillov (1962) and in further work by
Kostant (1970) and Souriau (1970) that the coadjoint orbits of a Lie group $G$
have the structure of symplectic manifolds and are connected with Hamiltonian
mechanics. See Kirillov (1999) for a review. The computations of the adjoint
and coadjoint actions for the semidirect product group is valuable for fluid
mechanics, as they introduce the two fundamental operators that appear in the
equations of motion. The Lie derivative is responsible for transport of
tensors along vector fields and its dual action given by the diamond operator
encodes the symmetry breaking. In particular, the diamond operator introduces
the effect of symmetry breaking into the Euler-Poincaré equations of motion.

##  2 Deterministic geometric fluid dynamics

With the adjoint and coadjoint actions defined, one can derive continuum
mechanics equations with advected quantities by using symmetry reduction.
Euler-Poincaré reduction for a semidirect product group
$\mathfrak{D}^{s}\times V$ as developed in Holm et al. (1998) is sketched
below in figure 2.

${L:T\mathfrak{D}^{s}\times
V^{*}\to\mathbb{R}}$${H:T^{*}(\mathfrak{D}^{s}\times V)\to\mathbb{R}}$Euler-
Lagrange eqnsHamilton’s eqns${\ell:\mathfrak{X}^{s}\times
V^{*}\to\mathbb{R}}$${\hslash:(\mathfrak{X}^{s}\times
V)^{*}\to\mathbb{R}}$Euler-Poincaré eqnsLie-Poisson eqnsLegendre
transformLegendre transform Figure 2: The cube of continuum mechanics in the
semidirect product group setting. Reduction is indicated by the arrows
pointing down.

As shown by comparison of figure 2 with figure 1, several new features arise
in semidirect product Lie group reduction which differ from Euler-Poincaré
reduction by symmetry when the configuration space itself is a Lie group.
These differences can be conveniently explained by introducing the physical
concept of an order parameter. As discussed earlier, the order parameters in
continuum mechanics are the elements of $V^{*}$ which are advected by the
action of the diffeomorphism group $\mathfrak{D}^{s}$. The advection is
defined simply as the semidirect product action on the elements of $V^{*}$.
The introduction of each additional advected state variable (or, order
parameter) into the physical problem further breaks the original symmetry
$\mathfrak{D}^{s}$. The remaining symmetry of the Lagrangian in Hamilton’s
principle is the isotropy subgroup $\mathfrak{D}^{s}_{a_{0}}$ of the initial
conditions, $a_{0}$, for the entire set of advected quantities, $a$. The
action of the diffeomorphism group $\mathfrak{D}^{s}$ on these initial
conditions then describes their advection as the action of $\mathfrak{D}^{s}$
on its coset space $\mathfrak{D}^{s}\setminus\mathfrak{D}^{s}_{a_{0}}=V^{*}$.
Once the inital values of the order parameters, $a_{0}$, have been set, one
must still define a Legendre transform from the Lagrangian formulation into
the Hamiltonian formulation and vice versa. The Legendre transform in the
setting of semidirect products is a partial Legendre transform, since it
transforms between $T\mathfrak{D}^{s}$ and $T^{*}\mathfrak{D}^{s}$ or
$T\mathfrak{D}^{s}\setminus\mathfrak{D}^{s}_{a_{0}}\simeq\mathfrak{X}^{s}$ and
$T^{*}\mathfrak{D}^{s}\setminus\mathfrak{D}^{s}_{a_{0}}\simeq\mathfrak{X}^{s*}$
only after having fixed the value $a_{0}$ of the order parameters, which live
in $V^{*}$. This coset reduction is what figure 2 shows. The remaining
invariance of a functional under the action of the isotropy subgroup is called
its _particle relabelling symmetry_.

Our exploration continues on the Lagrangian side in figure 2. Consider a fluid
Lagrangian $L:T\mathfrak{D}^{s}\times V^{*}\to\mathbb{R}$. By fixing the value
of $a_{0}\in V^{*}$, one can construct
$L_{a_{0}}:T\mathfrak{D}^{s}\to\mathbb{R}$. If this Lagrangian is right
invariant under the action of the isotropy subgroup
$\mathfrak{D}_{a_{0}}^{s}$, then one can construct

$\displaystyle L\left(\frac{d}{dt}g\circ g^{-1},e,a_{0}\right)$
$\displaystyle=L_{a_{0}}\left(\frac{d}{dt}g\circ g^{-1},e\right)$ (14)
$\displaystyle=\ell_{a_{0}}\left(\frac{d}{dt}g\circ
g^{-1}\right)=\ell\left(\frac{d}{dt}g\circ g^{-1},g_{*}a_{0}\right).$

Here $\circ$ means composition of functions. The same procedure applies to the
Hamiltonian. Since the coadjoint action is known, it is straightforward to
formulate the Lie-Poisson equations. The details of Hamiltonian semidirect
product reduction and also more information on the Lagrangian semidirect
product reduction can be found in Holm et al. (1998).

The coadjoint action of the Lie algebra on its dual is also required for the
Lagrangian semidirect product reduction. One can use the deterministic
reconstruction equation to see that the argument of the Lagrangians in (14) is

$\frac{d}{dt}g\circ g^{-1}=u.$ (15)

Using this information, one can integrate the Lagrangian in time to construct
the action functional. By requiring the variational derivative of the action
functional to vanish, one can compute the equations of motion. However, due to
the removal of symmetries, the variations are no longer free.

##  3 Stochastic geometric fluid dynamics

In the situation where noise is present, that is, when the reconstruction
equation is (2), the Euler-Poincaré variations become stochastic. Consider
$g:\mathbb{R}^{2}\to\mathfrak{D}^{s}$ with $g_{t,\epsilon}=g(t,\epsilon)$ to
be a two parameter subgroup with smooth dependence on $\epsilon$, but only
continuous dependence on $t$. Let us denote

${\sf d}\chi_{t,\epsilon}(X)=({\sf d}g_{t,\epsilon}\circ
g_{t,\epsilon})(X)=u_{t,\epsilon}(X)dt+\sum_{i=1}^{N}\xi_{i}(X)\circ
dW_{t}^{i}$

and

$v_{t,\epsilon}(X)=(\frac{\partial}{\partial\epsilon}g_{t,\epsilon}\circ
g_{t,\epsilon})(X)\,.$

When a $\circ$ symbol is followed by $dW_{t}$ it means Stratonovich
integration and in every other context the $\circ$ symbol is used to denote
composition. Note that the data vector fields $\xi_{i}$ are prescribed and
hence will not have a dependence on $\epsilon$.

In order to compute with these stochastic subgroups and their associated
vector fields, one needs a stochastic Lie chain rule. The Kunita-Itô-Wentzell
(KIW) formula is the stochastic generalisation of the Lie chain rule (8). A
proof of the KIW formula is given in Bethencourt de Leon et al. (2020) for
differential $k$-forms and vector fields. That proof includes the technical
details on regularity that will be omitted here. In the KIW formula, the
$k$-form is allowed to be a semimartingale itself. Let $K$ be a continuous
adapted semimartingale that takes values in the $k$-forms and satisfies

$K_{t}=K_{0}+\int_{0}^{t}G_{s}ds+\sum_{i=1}^{N}\int_{0}^{t}H_{i\,s}\circ
dB_{s}^{i},$ (16)

where the $B_{t}^{i}$ are independent, identically distributed Brownian
motions. The drift of the semimartingale $K$ is determined by $G$ and the
diffusion by $H_{i}$, both of which are $k$-form valued continuous adapted
semimartingales with suitable regularity. Let $g_{t}$ satisfy (2), then
Bethencourt de Leon et al. (2020) shows that the following holds

${\sf d}(g_{t}^{*}K_{t})=g_{t}^{*}\big{(}{\sf
d}K_{t}+\mathcal{L}_{u_{t}}K_{t}\,dt+\mathcal{L}_{\xi_{i}}K_{t}\circ
dW_{t}^{i}\big{)}.$ (17)

Equation (16) helps to interpret the ${\sf d}K_{t}$ term in the KIW formula
(17). This formula will be particularly useful in computing the variations of
the variables in the Lagrangian. To compute these variations, one needs the
variational derivative.

#### The variational derivative.

The variational derivative of a functional $F:\mathcal{B}\to\mathbb{R}$, where
$\mathcal{B}$ is a Banach space, is denoted $\delta F/\delta\rho$ with
$\rho\in\mathcal{B}$. The variational derivative can be defined by the first
variation of the functional

$\delta
F[\rho]:=\frac{d}{d\epsilon}\Big{|}_{\epsilon=0}F[\rho+\epsilon\delta\rho]=\int\frac{\delta
F}{\delta\rho}(x)\delta\rho(x)\,dx=\left\langle\frac{\delta
F}{\delta\rho},\delta\rho\right\rangle.$ (18)

In the definition above, $\epsilon\in\mathbb{R}$ is a parameter,
$\delta\rho\in\mathcal{B}$ is an arbitrary function and the first variation
can be understood as a Fréchet derivative. A precise and rigorous definition
can be found in Gelfand et al. (2000). With the definition of the functional
derivative in place, the following lemma can be formulated.

######  Lemma 3.

With the notation as above, the variations of $u$ and any advected quantity
$a$ are given by

$\delta u(t)={\sf d}v(t)+[{\sf d}\chi_{t},v(t)],\quad\delta
a(t)=-\mathcal{L}_{v(t)}a(t),$ (19)

where $v(t)\in\mathfrak{X}^{s}$ is arbitrary.

###### Proof.

The proof of the variation of $a(t)$ is a direct application of the Kunita-
Itô-Wentzell formula to $a(t,\epsilon)=g_{t,\epsilon*}a_{0}$. Note that the
data vector fields $\xi_{i}$ are prescribed and do not depend on $\epsilon$.
Denote by $x_{t,\epsilon}=g_{t,\epsilon}(X)$. Then one has

${\sf d}g_{t,\epsilon}(X)={\sf
d}x_{t,\epsilon}=u_{t,\epsilon}(x_{t,\epsilon})\,dt+\sum_{i=1}^{N}\xi_{i}(x_{t,\epsilon})\circ
dW_{t}^{i}=:{\sf d}\chi_{t,\epsilon}(x_{t,\epsilon}).$ (20)

The vector field associated to the $\epsilon$-dependence of the two parameter
subgroup is given by

$\frac{\partial}{\partial\epsilon}g_{t,\epsilon}=\frac{\partial}{\partial\epsilon}x_{t,\epsilon}=v_{t,\epsilon}(x_{t,\epsilon}).$
(21)

Computing the derivative with respect to $\epsilon$ of (20) gives

$\displaystyle\frac{\partial}{\partial\epsilon}{\sf d}x_{t,\epsilon}$
$\displaystyle=\frac{\partial}{\partial\epsilon}\big{(}{\sf
d}\chi_{t,\epsilon}(x_{t,\epsilon})\big{)}$ (22)
$\displaystyle=\left(\frac{\partial}{\partial\epsilon}u_{t,\epsilon}+v_{t,\epsilon}\cdot\frac{\partial}{\partial
x_{t,\epsilon}}{\sf d}\chi_{t,\epsilon}\right)(x_{t,\epsilon}),$

where the independence of the data vector fields $\xi_{i}$ on $\epsilon$ was
used. Taking the differential with respect to time of (21) gives

$\displaystyle{\sf
d}\left(\frac{\partial}{\partial\epsilon}x_{t,\epsilon}\right)$
$\displaystyle={\sf d}\big{(}v_{t,\epsilon}(x_{t,\epsilon})\big{)}$ (23)
$\displaystyle=\left({\sf d}v_{t,\epsilon}(x_{t,\epsilon})+{\sf
d}\chi_{t,\epsilon}\cdot\frac{\partial}{\partial
x_{t,\epsilon}}v_{t,\epsilon}\right)(x_{t,\epsilon}).$

One can then evaluate at $\epsilon=0$ and call upon equality of cross
derivative-differential to obtain the result by subtracting. Since
$g_{t,\epsilon}$ depends on $t$ in a $C^{0}$ manner, the integral
representation is required. The particle relabelling symmetry permits one to
stop writing the explicit dependence on space,

$\delta u(t)\,dt={\sf d}v(t)+[{\sf d}\chi_{t},v(t)].$ (24)

This completes the proof of formula (19) for the variation of $u(t)$. ∎

The notation in (20) needs careful explanation, because it comprises both a
stochastic differential equation and a definition. The symbol ${\sf
d}\chi_{t,\epsilon}$ is used to define a vector field, whereas ${\sf
d}x_{t,\epsilon}$ denotes a stochastic differential equation. This lemma makes
the presentation of the stochastic Euler-Poincaré theorem particularly simple.

######  Theorem 4 (Stochastic Euler-Poincaré).

With the notation as above, the following are equivalent.

  1. i)

The constrained variational principle

$\delta\int_{t_{1}}^{t_{2}}\ell(u,a)\,dt=0$ (25)

holds on $\mathfrak{X}^{s}\times V^{*}$, using variations $\delta u$ and
$\delta a$ of the form

$\delta u={\sf d}v+[{\sf d}\chi_{t},v],\qquad\delta a=-\mathcal{L}_{v}a,$ (26)

where $v(t)\in\mathfrak{X}^{s}$ is arbitrary and vanishes at the endpoints in
time for arbitrary times $t_{1},t_{2}$.

  2. ii)

The stochastic Euler-Poincaré equations hold on $\mathfrak{X}^{s}\times V^{*}$

${\sf d}\frac{\delta\ell}{\delta u}+\mathcal{L}_{{\sf
d}\chi_{t}}\frac{\delta\ell}{\delta u}=\frac{\delta\ell}{\delta a}\diamond
a\,dt,$ (27)

and the advection equation

${\sf d}a+\mathcal{L}_{{\sf d}\chi_{t}}a=0.$ (28)

###### Proof.

Using integration by parts and the endpoint conditions $v(t_{1})=0=v(t_{2})$,
the variation can be computed to be

$\displaystyle\delta\int_{t_{1}}^{t_{2}}\ell(u,a)\,dt$
$\displaystyle=\int_{t_{1}}^{t_{2}}\left\langle\frac{\delta\ell}{\delta
u},\delta u\right\rangle+\left\langle\frac{\delta\ell}{\delta a},\delta
a\right\rangle\,dt$ (29)
$\displaystyle=\int_{t_{1}}^{t_{2}}\left\langle\frac{\delta\ell}{\delta
u},{\sf d}v+[{\sf
d}\chi_{t},v]\right\rangle+\left\langle\frac{\delta\ell}{\delta
a}\,dt,-\mathcal{L}_{v}a\right\rangle$
$\displaystyle=\int_{t_{1}}^{t_{2}}\left\langle-{\sf
d}\frac{\delta\ell}{\delta u}-\mathcal{L}_{{\sf
d}\chi_{t}}\frac{\delta\ell}{\delta u}+\frac{\delta\ell}{\delta a}\diamond
a\,dt,v\right\rangle$ $\displaystyle=0\,.$

Since the vector field $v$ is arbitrary, one obtains the stochastic Euler-
Poincaré equations. Finally, the advection equation (28) follows by applying
the KIW formula to $a(t)=g_{t*}a_{0}$. ∎

######  Remark 5.

The stochastic Euler-Poincaré theorem is equivalent to the version presented
in Holm (2015), which uses stochastic Clebsch constraints. In Holm (2015) one
can also find an investigation the Itô formulation of the stochastic Euler-
Poincaré equation.

#### Stochastic Lie-Poisson formulation.

The stochastic Euler-Poincaré equations have an equivalent stochastic Lie-
Poisson formulation. To obtain the Lie-Poisson formulation, one must Legendre
transform the reduced Lagrangian. The Legendre transformation in the presence
of stochasticity becomes itself stochastic in the following way

$m:=\frac{\delta\ell}{\delta u},\qquad\hslash(m,a)\,dt+\sum_{i=1}^{N}\langle
m,\xi_{i}\rangle\circ dW_{t}^{i}=\langle m,{\sf
d}\chi_{t}\rangle-\ell(u,a)\,dt.$ (30)

The stochasticity enters the Legendre transformation because the momentum map
$m$ is coupled to the stochastic vector field ${\sf d}\chi_{t}$. The left hand
side of the transformation determines the Hamiltonian, which is a
semimartingale. The underlying semidirect product group structure has not
changed, it is still the $H^{s}$ diffeomorphisms with a vector space, but the
Hamiltonian has become a semimartingale. This implies that in the stochastic
case the energy is not conserved, because Hamiltonian depends on time
explicitly. Note that (30) emphasises that the Lagrangian does not feature
stochasticity in this framework. Instead, the Lagrangian represents the
physics in the problem, which does not change. The stochasticity is supposed
to account for the difference between observed data and deterministic
modelling. The stochastic Lie-Poisson equations are given by

${\sf d}(m,a)=-{\rm ad}^{*}_{(\frac{\delta\hslash}{\delta
m},\frac{\delta\hslash}{\delta a})}(m,a)\,dt-\sum_{i=1}^{N}{\rm
ad}^{*}_{(\xi_{i},0)}(m,a)\circ dW_{t}^{i},$ (31)

where ${\rm ad}^{*}$ is given in (13). Since both the drift and the diffusion
part use the same operator in (31), the stochastic Lie-Poisson equations
preserve the same family of Casimirs (or integral conserved quantities) as the
deterministic Lie-Poisson equations. The stochastic Euler-Poincaré theorem has
a stochastic Kelvin-Noether circulation theorem as a corollary.

Let the manifold $M$ be a submanifold of $\mathbb{R}^{n}$ with coordinates
$X$. Then the volume form can be expressed with respect to a density. That is,
$\mu(d^{n}X)=\rho_{0}(X)d^{n}X$. By pushing forward $\rho_{0}$ along the
stochastic flow $g_{t}$, one obtains $\rho$. Let $\mathfrak{C}^{s}$ be the
space of loops $\gamma:S^{1}\to\mathfrak{D}^{s}$, which is acted upon from the
left by $\mathfrak{D}^{s}$. Given an element $m\in\mathfrak{X}^{s}$, one can
obtain a 1-form by formally dividing $m$ by the density $\rho$.

The circulation map $\mathcal{K}:\mathfrak{C}^{s}\times
V^{*}\to\mathfrak{X}^{s**}$ is defined by

$\langle\mathcal{K}(\gamma,a),m\rangle=\oint_{\gamma}\frac{m}{\rho}\,.$ (32)

Given a Lagrangian $\ell:\mathfrak{X}^{s}\times V^{*}\to\mathbb{R}$, the
_Kelvin-Noether quantity_ is defined by

$I(\gamma,u,a):=\oint_{\gamma}\frac{1}{\rho}\frac{\delta\ell}{\delta u}\,.$
(33)

One can now formulate the following stochastic Kelvin-Noether circulation
theorem.

######  Theorem 6 (Stochastic Kelvin-Noether).

Let $u_{t}=u(t)$ satisfy the stochastic Euler-Poincaré equation (27) and
$a_{t}=a(t)$ the stochastic advection equation (28). Let $g_{t}$ be the flow
associated to the vector field ${\sf d}\chi_{t}$. That is, ${\sf
d}\chi_{t}={\sf d}g_{t}\circ g_{t}^{-1}=u_{t}\,dt+\sum_{i=1}^{N}\xi_{i}\circ
dW_{t}^{i}$. Let $\gamma_{0}\in\mathfrak{C}^{s}$ be a loop. Denote by
$\gamma_{t}=g_{t}\circ\gamma_{0}$ and define the Kelvin-Noether quantity
$I(t):=I(\gamma_{t},u_{t},a_{t})$. Then

${\sf d}I(t)=\oint_{\gamma_{t}}\frac{1}{\rho}\frac{\delta\ell}{\delta
a}\diamond a\,dt\,.$ (34)

###### Proof.

The statement of the stochastic Kelvin-Noether circulation theorem involves a
loop that is moving with the stochastic flow. One can transform to stationary
coordinates by pulling back the flow to the initial condition. This pull-back
yields

$I(t)=\oint_{\gamma_{t}}\frac{1}{\rho}\frac{\delta\ell}{\delta
u}=\oint_{\gamma_{0}}g_{t}^{*}\left(\frac{1}{\rho}\frac{\delta\ell}{\delta
u}\right)=\oint_{\gamma_{0}}\frac{1}{\rho_{0}}g_{t}^{*}\left(\frac{\delta\ell}{\delta
u}\right).$ (35)

An application of the Kunita-Itô-Wentzell formula (17) leads to

${\sf d}I(t)=\oint_{\gamma_{0}}\frac{1}{\rho_{0}}g_{t}^{*}\left({\sf
d}\frac{\delta\ell}{\delta u}+\mathcal{L}_{{\sf
d}\chi_{t}}\frac{\delta\ell}{\delta
u}\right)=\oint_{\gamma_{0}}\frac{1}{\rho_{0}}g_{t}^{*}\left(\frac{\delta\ell}{\delta
a}\diamond a\right)\,dt,$ (36)

since $u$ satisfies the stochastic Euler-Poincaré theorem. Transforming back
to the moving coordinates by pushing forward with $g_{t}$ yields the final
result. ∎

Thus, Theorem 6 explains how particle relabelling symmetry gives rise to the
Kelvin-Noether circulation theorem via Noether’s theorem. When the only
advected quantity present is the mass density, the loop integral of the
diamond terms vanishes. This means that circulation is conserved according to
Noether’s theorem for an incompressible fluid, or for a barotropically
compressible fluid. The presence of other advected quantities breaks the
symmetry further and introduces the _diamond terms_ which generate
circulation, as one can see in the Kelvin-Noether circulation theorem in
equation (34). Consequently, the symmetry breaking due to additional order
parameters can provide additional mechanisms for the generation of Kelvin-
Noether circulation in ideal fluid dynamics.

#### Outlook.

Stochastic geometric mechanics is an active field of mathematics which has
recently established its utility for a broad range of applications in science.
Basically, everything that can be done with Hamilton’s principle for
deterministic geometric mechanics can also be made stochastic in the sense of
Stratonovich. This is possible because the variational calculus in Hamilton’s
principle requires only the product rule and chain rule from ordinary
calculus. The happy emergence of the new science of stochastic geometric
mechanics was celebrated with the publication of the book Albeverio et al.
(2017). This book showcases some of the recent developments in stochastic
geometric mechanics. Another collection of recent developments can be found in
Castrillón López et al. (2019). An ongoing development is in the direction of
_rough geometric mechanics_ , initiated with a rough version of the Euler-
Poincaré theorem in Crisan et al. (2020). Remarkably, variational principles
which are driven by geometric rough paths again only require the product rule
and the chain rule. Other directions involve the inclusion of jump processes,
fractional derivatives and non-Markovian processes in geometric mechanics. For
example, recent work by Albeverio et al. (2020) shows that SDEs driven by
semimartingales with jumps have weak symmetries and a corresponding extension
of the reduction and reconstruction technique is discussed.

## Acknowledgments

We are enormously grateful for many encouraging discussions over the years
with T.S. Ratiu, F. Gay-Balmaz, C. Tronci, S. Albeverio, A.B. Cruzeiro, F.
Flandoli, and also with our friends in project STUOD (stochastic transport in
upper ocean dynamics) and in the geometric mechanics research group at
Imperial College London. The work of DDH was partially supported by European
Research Council (ERC) Synergy grant STUOD - DLV-856408. EL was supported by
EPSRC grant [grant number EP/L016613/1] and is grateful for the warm
hospitality at the Imperial College London EPSRC Centre for Doctoral Training
in the Mathematics of Planet Earth during the course of this work.

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