######  Lemma A.1.

Given smooth vector fields $X,Y_{1},\ldots,Y_{N}$ on $\mathscr{M}$, consider
the Stratonovich SDE

(40)
$\mathrm{d}Z_{t}=X(Z_{t})\,\mathrm{d}t+Y_{i}(Z_{t})\circ\mathrm{d}W^{i}_{t}\,.$

Then its generator is given by $\mathcal{L}f=Xf+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over 2}}{1/2}}Y_{i}Y_{i}f$.
The Fokker–Planck operator, viewed as the formal adjoint of $\mathcal{L}$ in
$L^{2}(\mu_{\mathscr{M}})$ (if the boundary $\partial\mathscr{M}$ is non-empty
we restrict to functions that vanish on the boundary) is given by

(41)
$\mathcal{L}^{*}f=\text{div}_{\mu_{\mathscr{M}}}\big{(}-fX+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\text{div}_{\mu_{\mathscr{M}}}(fY_{i})Y_{i}\big{)}=\text{div}_{\mu_{\mathscr{M}}}\left(-fX+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}Y_{i}(f)Y_{i}+{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over
2}}{1/2}}f\text{div}_{\mu_{\mathscr{M}}}(Y_{i})Y_{i}\right)\,.$

###### Proof.

Vector fields satisfy Leibniz rule

$\displaystyle\int gX(f)\mu_{\mathscr{M}}=\int X(gf)\mu_{\mathscr{M}}-\int
fX(g)\mu_{\mathscr{M}},$

and note that
$\int_{\mathscr{M}}\text{div}(gX)\mu_{\mathscr{M}}=\int_{\mathscr{M}}\mathrm{d}i_{gX}\mu_{\mathscr{M}}=\int_{\partial\mathscr{M}}i_{gX}\mu_{\mathscr{M}}=\int_{\partial\mathscr{M}}gi_{X}\mu_{\mathscr{M}}$
which vanishes if $g|_{\partial M}=0$ (here
$\text{div}\equiv\text{div}_{\mu_{\mathscr{M}}}$). Hence

$\displaystyle\int gX(f)\mu_{\mathscr{M}}$ $\displaystyle=-\int
gf\text{div}(X)\mu_{\mathscr{M}}-\int fX(g)\mu_{\mathscr{M}}$
$\displaystyle=\int f\big{(}-g\text{div}(X)-X(g)\big{)}\mu_{\mathscr{M}}$
$\displaystyle=\int f\big{(}-\text{div}(gX)\big{)}\mu_{\mathscr{M}}=\int
f\mathcal{L}^{*}g\mu_{\mathscr{M}}.$

Now let us compute the adjoint of the diffusion component. First note that

$\displaystyle Y_{k}(gY_{k}f)$
$\displaystyle=\text{div}\big{(}gY_{k}fY_{k}\big{)}-gY_{k}f\text{div}(Y_{k})=$
$\displaystyle=\text{div}\big{(}gY_{k}fY_{k}\big{)}-\text{div}\big{(}fg\text{div}(Y_{k})Y_{k}\big{)}+f\text{div}\big{(}g\text{div}(Y_{k})Y_{k}\big{)}$

and

$\displaystyle
Y_{i}gY_{i}f=\text{div}(fY_{i}gY_{i})-f\text{div}(Y_{i}gY_{i}).$

It follows that if $f,g$ vanish on the boundary

$\displaystyle{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}\int gY_{i}Y_{i}f\mu_{\mathscr{M}}$
$\displaystyle={\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over
2}}{1/2}}\int\Big{(}Y_{i}\big{(}gY_{i}f\big{)}-Y_{i}gY_{i}f\Big{)}\mu_{\mathscr{M}}$
$\displaystyle={\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over
2}}{1/2}}\int\Big{(}f\text{div}\big{(}g\text{div}(Y_{i})Y_{i}\big{)}+f\text{div}(Y_{i}gY_{i})\Big{)}\mu_{\mathscr{M}}$
$\displaystyle=\int f{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over
2}}{1/2}}\text{div}\big{(}\text{div}(gY_{i})Y_{i}\big{)}\mu_{\mathscr{M}}=\int
f\mathcal{L}^{*}g\mu_{\mathscr{M}}.$

∎

###  A.2. Proof of Theorem 2.4

######  Theorem A.2.

The Gibbs measure (4) is preserved under the bracket diffusion (7), in the
sense that $\mathcal{L}^{*}p_{\infty}=0$, if and only if the vector field $Y$
satisfies

(42) $\text{div}_{\mu_{\mathscr{M}}}(X_{p_{\infty}}{B}-\beta
p_{\infty}Y)=0\,.$

###### Proof.

Let $p(x)=\frac{1}{Z}e^{-\beta x}$ for $x\in\mathbb{R}$, so $p_{\infty}=p\circ
H:\mathscr{M}\to\mathbb{R}$, and $X_{p_{\infty}}=-p_{\infty}X_{H}$. Then using
the Fokker-Planck operator defined in (6), we have

$\displaystyle\mathcal{L}^{*}p_{\infty}$
$\displaystyle=\text{div}\Bigg{(}-p_{\infty}\Big{(}X_{H}-\frac{\beta}{2}Y_{i}(H)Y_{i}+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\text{div}(Y_{i})Y_{i}+Y\Big{)}+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\text{div}\big{(}p_{\infty}Y_{i}\big{)}Y_{i}\Bigg{)}$
$\displaystyle=\text{div}\Bigg{(}\frac{1}{\beta}X_{p_{\infty}}-p_{\infty}Y-\frac{1}{2}Y_{i}(p_{\infty})Y_{i}-{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}p_{\infty}\text{div}(Y_{i})Y_{i}+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\text{div}\big{(}p_{\infty}Y_{i}\big{)}Y_{i}\Bigg{)}$
$\displaystyle=\text{div}\Bigg{(}\frac{1}{\beta}X_{p_{\infty}}-p_{\infty}Y-\frac{1}{2}Y_{i}(p_{\infty})Y_{i}-{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}p_{\infty}\text{div}(Y_{i})Y_{i}+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\text{div}\big{(}p_{\infty}Y_{i}\big{)}Y_{i}\Bigg{)}$
$\displaystyle=\text{div}\Bigg{(}\frac{1}{\beta}X_{p_{\infty}}-p_{\infty}Y-{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\text{div}\big{(}p_{\infty}Y_{i}\big{)}Y_{i}+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\text{div}\big{(}p_{\infty}Y_{i}\big{)}Y_{i}\Bigg{)}$
$\displaystyle=\text{div}\Big{(}\frac{1}{\beta}X_{p_{\infty}}-p_{\infty}Y\Big{)},$

thus $\mathcal{L}^{*}p_{\infty}=0$ iff $Y$ satisfies
$\text{div}\big{(}\frac{1}{\beta}X_{p_{\infty}}-p_{\infty}Y\big{)}=0$. ∎

###  A.3. Recovering The Euclidean Complete Recipe

######  Corollary A.3.

Let $\mathscr{M}=\mathbb{R}^{n}$ with flat metric,
$\mu_{\mathscr{M}}=\mathrm{d}x$, $\sigma_{ij}\equiv Y^{i}_{j}$,
$D\equiv{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}\sigma\sigma^{T}$ and
$Q_{ij}\equiv\mathzapfc{A}^{ij}$. Then (7) reduces to the Itô diffusion
considered in the complete recipe for SGMCMC [121]

(43) $\mathrm{d}Z_{t}=-\big{(}Q+D\big{)}\nabla
H\mathrm{d}t+\nabla\cdot\big{(}Q+D\big{)}\mathrm{d}t+\sqrt{2D}\mathrm{d}W_{t}\,,$

where we use the convention $(\nabla\cdot Q)_{i}\equiv\partial_{j}Q_{ij}$ as
in [121].

###### Proof.

In that case, the term
$Y_{i}(H)Y_{i}=Y^{j}_{i}Y^{k}_{i}\partial_{j}H\partial_{k}$ represents the
“symmetric semi-definite part" of the dynamics since in flat space this is
$(\sigma\sigma^{T})_{jk}\partial_{j}H\partial_{k}$ where
$\sigma_{ij}=\sigma^{i}_{j}=Y^{i}_{j}$, while the term
$\text{div}(Y_{i})Y_{i}=\sigma_{ki}\partial_{j}\sigma_{ji}\partial_{k}$
together with the Itô-to-Statonovich correction yields the divergence of the
diffusion matrix $D\equiv{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\sigma\sigma^{T}$. Using the convention $(\nabla\cdot
Q)_{i}\equiv\partial_{j}Q_{ij}$ in [121]

$\mathrm{d}Z_{t}=\left(\mathzapfc{A}^{ji}\partial_{j}H\partial_{i}-\frac{\beta}{2}(\sigma\sigma^{T})_{jk}\partial_{j}H\partial_{k}+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\sigma_{ki}\partial_{j}\sigma_{ji}\partial_{k}-\partial_{j}\mathzapfc{A}^{ji}\partial_{i}\right)\mathrm{d}t\\\
+\sigma_{ij}\partial_{i}\circ\mathrm{d}W^{j}_{t}\\\
=\left(\mathzapfc{A}^{ji}\partial_{j}H\partial_{i}-\frac{\beta}{2}(\sigma\sigma^{T})_{jk}\partial_{j}H\partial_{k}+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\sigma_{ki}\partial_{j}\sigma_{ji}\partial_{k}+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\sigma_{ki}\partial_{k}\sigma_{ji}\partial_{j}+\partial_{j}\mathzapfc{A}^{ij}\partial_{i}\right)\mathrm{d}t\\\
+\sigma_{ij}\partial_{i}\mathrm{d}W^{j}_{t}\\\ =\left(-Q\nabla H-\beta D\nabla
H+\nabla\cdot D+\nabla\cdot Q\right)\mathrm{d}t+\sqrt{2D}\mathrm{d}W_{t}.$

∎

###  A.4. Derivation that Curl is Modular Field

This result was proved in the thesis of one of the authors [14], but the proof
relies on introducing Schouten–Nijenhuis brackets. Thus we here include a new
direct and more constructive proof:

###### Proof.

We will prove the equivalence by demonstrating that the action of the modular
vector field and the curl vector field are equal for all smooth functions $f$,

$\text{curl}_{P}(\mathzapfc{A})(f)\cdot P=X^{P}_{\mathzapfc{A}}(f)\cdot P.$

The curl vector field is defined implicitly by the action

$i_{\text{curl}_{P}(\mathzapfc{A})}P=\mathbf{d}\left(P^{\flat}(\mathzapfc{A})\right)$

which implies that for any $f$

(44) $\mathbf{d}f\wedge
i_{\text{curl}_{P}(\mathzapfc{A})}P=\mathbf{d}f\wedge\mathbf{d}\left(P^{\flat}(\mathzapfc{A})\right).$

To simplify the left hand side note that

$\displaystyle 0$
$\displaystyle=i_{\text{curl}_{P}(\mathzapfc{A})}\left(\mathbf{d}f\wedge
P\right)$ $\displaystyle=i_{\text{curl}_{P}(\mathzapfc{A})}(\mathbf{d}f)\cdot
P-\mathbf{d}f\wedge i_{\text{curl}_{P}(\mathzapfc{A})}P$
$\displaystyle=\text{curl}_{P}(\mathzapfc{A})(f)\cdot P-\mathbf{d}f\wedge
i_{\text{curl}_{P}(\mathzapfc{A})}P,$

or

$\mathbf{d}f\wedge
i_{\text{curl}_{P}(\mathzapfc{A})}P=\text{curl}_{P}(\mathzapfc{A})(f)\cdot P.$

For the right hand side we take

$\displaystyle\mathbf{d}\left(\mathbf{d}f\wedge(P^{\flat}(\mathzapfc{A}))\right)$
$\displaystyle=\mathbf{d}^{2}f\wedge
P^{\flat}(\mathzapfc{A})-\mathbf{d}f\wedge\mathbf{d}\left(P^{\flat}(\mathzapfc{A})\right)$
$\displaystyle=-\mathbf{d}f\wedge\mathbf{d}\left(P^{\flat}(\mathzapfc{A})\right).$

Substituting both results in Equation (44) then gives

(45) $\text{curl}_{P}(\mathzapfc{A})(f)\cdot
P=-\mathbf{d}\left(\mathbf{d}f\wedge P^{\flat}(\mathzapfc{A})\right).$

Now we use the fact that for any function $P^{\sharp}\left(\mathbf{d}f\wedge
P^{\flat}(\mathzapfc{A})\right)=i_{\mathbf{d}f}\mathzapfc{A}$ [52], so that
equation (45) becomes

$\displaystyle\text{curl}_{P}(\mathzapfc{A})(f)\cdot
P=-\mathbf{d}P^{\flat}\left(i_{\mathbf{d}f}\mathzapfc{A}\right)=\mathbf{d}P^{\flat}\left(X_{f}{A}\right)=\text{div}_{P}(X^{\mathzapfc{A}}_{f})\cdot
P$

as desired. ∎

###  A.5. Derivation of Adjoint of Integration Pairing

######  Lemma A.4.

The Fokker–Planck operator of the Stratonovich SDE (5), viewed as the formal
adjoint of $\mathcal{L}$ with respect to the pairing $(f,P)\mapsto\int
f\mathbf{d}P$ between smooth, compactly supported functions $f$ and smooth
measures $P$, is given by

(46) $\mathcal{L}^{*}P=-\mathcal{L}_{X}P+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\mathcal{L}_{Y_{i}}\mathcal{L}_{Y_{i}}P.$

###### Proof.

Recall the generator is $\mathcal{L}=X+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over 2}}{1/2}}Y_{i}\circ
Y_{i}$. Thus for the deterministic drift we find

$\displaystyle\int X(f)P$ $\displaystyle=\int\mathbf{d}f(X)P=\int
i_{X}(\mathbf{d}f)P=\int\mathbf{d}f\wedge i_{X}P$
$\displaystyle=\int\mathbf{d}(f\wedge i_{X}P)-\int f\mathbf{d}i_{X}P=-\int
f\mathcal{L}_{X}P.$

Similarly, for the diffusion coefficient,

$\displaystyle\int Y_{i}Y_{i}(f)P$ $\displaystyle=-\int
Y_{i}(f)\mathcal{L}_{Y_{i}}P=\int f\mathcal{L}_{Y_{i}}\mathcal{L}_{Y_{i}}P.$

∎

###  A.6. Derivation of $\mathzapfc{A}$-diffusion expressed in terms of the
reference measure $\mu_{\mathscr{M}}$

Here, we include additional details to the derivation of the
$\mathzapfc{A}$-diffusion (11) in terms of reference measure
$\mu_{\mathscr{M}}$ from the abstract $\mathzapfc{A}$-diffusion (16) expressed
in terms of the target $P=e^{-\beta H}\mu_{\mathscr{M}}$. First note that if
$f$ is a non-vanishing function, then
$(f\mu_{\mathscr{M}})^{\flat}=f\mu_{\mathscr{M}}^{\flat}$ and so
$(f\mu_{\mathscr{M}})^{\sharp}=\mu_{\mathscr{M}}^{\sharp}\circ\frac{1}{f}$.
Hence

$\displaystyle\text{curl}_{f\mu_{\mathscr{M}}}(X)$
$\displaystyle=(f\mu_{\mathscr{M}})^{\sharp}\circ\mathbf{d}\circ(f\mu_{\mathscr{M}})^{\flat}(X)=(\mu_{\mathscr{M}})^{\sharp}\left(\frac{1}{f}\,\mathbf{d}\left(f\,(\mu_{\mathscr{M}})^{\flat}(X)\right)\right)$
$\displaystyle=(\mu_{\mathscr{M}})^{\sharp}\circ\mathbf{d}\circ(\mu_{\mathscr{M}})^{\flat}(X)+(\mu_{\mathscr{M}})^{\sharp}\left(\left(\frac{1}{f}\,\mathbf{d}f\right)\wedge(\mu_{\mathscr{M}})^{\flat}(X)\right)$
$\displaystyle=(\mu_{\mathscr{M}})^{\sharp}\circ\mathbf{d}_{f}\circ(\mu_{\mathscr{M}})^{\flat},$

where $\mathbf{d}_{f}\equiv\mathbf{d}+\mathbf{d}\log|f|\wedge$ is the
distorted de Rham derivative.

######  Proposition A.5.

Given $P=e^{-\beta H}\mu_{\mathscr{M}}$, we have

(47) $\text{div}_{P}(Y)=\text{div}_{\mu_{\mathscr{M}}}(Y)-\beta Y(H),$

for any vector field $Y\in\mathfrak{X}(\mathscr{M})$ and

(48) $\text{curl}_{P}(\mathzapfc{A})=X{A}^{\mu_{\mathscr{M}}}+\beta X_{H}{A},$

for any bi-vector field $\mathzapfc{A}\in\mathfrak{X}^{2}(\mathscr{M})$.

###### Proof.

Taking $f=e^{-\beta H}$ in the above, we have

$\displaystyle\text{div}_{P}(Y)=\text{curl}_{P}(Y)$
$\displaystyle=(\mu_{\mathscr{M}})^{\sharp}\circ\mathbf{d}\circ(\mu_{\mathscr{M}})^{\flat}(Y)-\beta(\mu_{\mathscr{M}})^{\sharp}\left(\mathbf{d}H\wedge(\mu_{\mathscr{M}})^{\flat}(Y)\right)$
$\displaystyle=\text{div}_{\mu_{\mathscr{M}}}(Y)-\beta(\mu_{\mathscr{M}})^{\sharp}\left(\mathbf{d}H\wedge
i_{Y}\mu_{\mathscr{M}}\right).$

Now, using the identity
$i_{Y}(\mathbf{d}H\wedge\mu_{\mathscr{M}})=i_{Y}\mathbf{d}H\wedge\mu_{\mathscr{M}}-\mathbf{d}H\wedge
i_{Y}\mu_{\mathscr{M}}$ and noting that $\mathbf{d}H\wedge\mu_{\mathscr{M}}=0$
(since $\mu_{\mathscr{M}}$ is a top degree twisted form on $\mathscr{M}$), we
have $\mathbf{d}H\wedge
i_{Y}\mu_{\mathscr{M}}=i_{Y}\mathbf{d}H\wedge\mu_{\mathscr{M}}=(\mu_{\mathscr{M}})^{\flat}(Y(H))$.
The first identity (47) then follows immediately.

Similarly, we have

$\displaystyle\text{curl}_{P}(\mathzapfc{A})$
$\displaystyle=\text{curl}_{\mu_{\mathscr{M}}}(\mathzapfc{A})-\beta(\mu_{\mathscr{M}})^{\sharp}\left(\mathbf{d}H\wedge(\mu_{\mathscr{M}})^{\flat}(\mathzapfc{A})\right).$

Noting that
$(\mu_{\mathscr{M}})^{\sharp}\left(\mathbf{d}H\wedge(\mu_{\mathscr{M}})^{\flat}(\mathzapfc{A})\right)=i_{\mathbf{d}H}\mathzapfc{A}\equiv-
X_{H}{A}$ (see [52]) and
$\text{curl}_{\mu_{\mathscr{M}}}(\mathzapfc{A})=X{A}^{\mu_{\mathscr{M}}}$ (see
§A.4), the second identity (48) follows. ∎

###  A.7. Derivation of Measure-Preserving Diffusion on Riemannian Manifolds

######  Theorem A.6.

Let $\mathscr{M}$ be a compact orientable Riemannian manifold and let
$\mu_{\mathscr{M}}\equiv\text{vol}$ and $\nabla\cdot$ denote respectively the
Riemannian measure and divergence. Then, any $e^{-H}\text{vol}$-preserving
diffusion has the form

(49) $\mathrm{d}Z_{t}\equiv\left(X_{H}+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\left(\nabla\cdot
Y_{i}-Y_{i}(H)\right)Y_{i}-\nabla\cdot\mathzapfc{A}+e^{H}\sharp\star^{-1}\zeta\right)\mathrm{d}t+Y_{i}\circ\mathrm{d}W^{i}_{t},$

where $\star$ is the Hodge star operator, $\sharp$ is the Riemannian musical
isomorphism, $\mathzapfc{A}$ is an antisymmetric bracket and $\zeta$ is a
harmonic $(n-1)$-form (i.e., it satisfies “Maxwell’s equations”
$\mathrm{d}\zeta=0$, $\mathrm{d}\star\zeta=0$).

###### Proof.

Let $\delta\propto\star\mathbf{d}\star$ be the co-differential. Since
$\text{div}_{\text{vol}}=\delta\circ\flat$, the Fokker-Planck operator (6) may
be written as

(50) $\mathcal{L}^{*}g=\delta\left(-gX+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\nabla\cdot(gY_{i})Y_{i}\right)^{\flat}\,.$

It then follows by the Hodge decomposition that $\mathcal{L}^{*}p_{\infty}=0$
iff $\delta\left(-p_{\infty}X+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\nabla\cdot(p_{\infty}Y_{i})Y_{i}\right)^{\flat}=0$ iff
$\mathrm{d}\star\left(-p_{\infty}X+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\nabla\cdot(p_{\infty}Y_{i})Y_{i}\right)^{\flat}=0$ iff

$\star\left(-p_{\infty}X+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\nabla\cdot(p_{\infty}Y_{i})Y_{i}\right)^{\flat}=\mathrm{d}\alpha-\gamma$

where $\gamma$ is Harmonic, that is co-exact and closed, and vanishes whenever
the $\dim(\mathscr{M})-1$ de Rham cohomology is trivial. Thus
$\left(-p_{\infty}X+{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over
2}}{1/2}}\nabla\cdot(p_{\infty}Y_{i})Y_{i}\right)^{\flat}=\delta\varepsilon-\star^{-1}\gamma$
where $\varepsilon\equiv\star^{-1}\alpha$ is a 2-form. Hence,

$-p_{\infty}X+{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over
2}}{1/2}}\nabla\cdot(p_{\infty}Y_{i})Y_{i}=\sharp\delta\varepsilon+\sharp\star^{-1}\gamma=\nabla\cdot(\mathzapfc{A}p_{\infty})-\sharp\star^{-1}\gamma\,,$

for an appropriate antisymmetric bracket $\mathzapfc{A}$. Now,
$\nabla\cdot(\mathzapfc{A}p_{\infty})=p_{\infty}\nabla\cdot\mathzapfc{A}-p_{\infty}X_{H}$,
since $\nabla\cdot\mathzapfc{A}=\text{curl}_{\text{vol}}(\mathzapfc{A})$
(proved below), and for any $f$

$\displaystyle\text{curl}_{\mu}(p_{\infty}\mathzapfc{A})[f]$
$\displaystyle=-\text{div}_{\mu}(p_{\infty}X_{f})=-X_{f}(p_{\infty})-p_{\infty}\text{div}_{\mu}X_{f}=+p_{\infty}X_{f}(H)-p_{\infty}\text{div}_{\mu}X_{f}$
$\displaystyle=-p_{\infty}X_{H}(f)-p_{\infty}\text{div}_{\mu}X_{f},$

so

$\text{curl}_{\mu}(p_{\infty}\mathzapfc{A})=-p_{\infty}X_{H}+p_{\infty}\text{curl}_{\mu}\mathzapfc{A}.$

It follows that the drift vector field must take the form

$X=X_{H}+{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over
2}}{1/2}}\nabla\cdot(Y_{i})Y_{i}-{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}Y_{i}(H)Y_{i}-\nabla\cdot\mathzapfc{A}+e^{H}\sharp\star^{-1}\gamma.$

∎

Note that the Riemann divergence $\nabla\cdot\mathzapfc{A}$ in the previous
theorem is precisely the curl of the Riemann measure (up to a sign). Indeed,
if $(\mathscr{M},\mathzapfc{M})$ is a pseudo-Riemannian manifold, recall that
$\nabla\cdot\mathzapfc{A}=\text{Tr}\nabla\mathzapfc{A}$, where Tr is the trace
and $\nabla$ the covariant derivative. Then using the fact that
$S_{ab}\mathzapfc{A}^{ab}=0$ for any symmetric tensor $S_{ab}$ and anti-
symmetric tensor $\mathzapfc{A}^{ab}$, we see that for any smooth function $f$

$\displaystyle\nabla\cdot\big{(}f\nabla\cdot\mathzapfc{A}\big{)}$
$\displaystyle=\frac{1}{\sqrt{|\mathzapfc{M}|}}\partial_{b}\Big{(}f\partial_{a}(\sqrt{|\mathzapfc{M}|}\mathzapfc{A}^{ab})\Big{)}=\frac{1}{\sqrt{|\mathzapfc{M}|}}(\partial_{b}f)\Big{(}\partial_{a}(\sqrt{|\mathzapfc{M}|}\mathzapfc{A}^{ab})\Big{)}$
$\displaystyle=\frac{1}{\sqrt{|\mathzapfc{M}|}}\partial_{a}\Big{(}\sqrt{|G|}\mathzapfc{A}^{ab}(\partial_{b}f)\Big{)}=-\frac{1}{\sqrt{|\mathzapfc{M}|}}\partial_{a}\Big{(}\sqrt{|\mathzapfc{M}|}X_{f}^{a}\Big{)}=-\nabla\cdot
X_{f}{A}\,,$

so from (8) applied to $\mu_{\mathscr{M}}=\text{vol}$ and $f=p_{\infty}$, we
see that

$\nabla\cdot\mathzapfc{A}=-X{A}^{\text{vol}}=-\text{curl}_{\text{vol}}(\mathzapfc{A}).$

###  A.8. Derivations for the Reversibility Section

Recall that $X^{P}_{\mathzapfc{B}}(f)\equiv\text{div}_{P}(X{B}_{f})$ for any
bracket $\mathzapfc{B}$. Then

######  Corollary A.7.

The generator of a $P$-preserving expressed in the form of (2) can be written
as

$\mathcal{L}f=\underbrace{X^{P}{A}(f)+P^{\sharp}(\gamma)(f)}_{L^{2}(P)\text{-antisymmetric}}+\underbrace{{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}X^{P}_{\SS}(f)}_{L^{2}(P)\text{-symmetric}}.$

Moreover, ${\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}X^{P}_{\SS}$ is symmetric in
$L^{2}(P)$, while $X^{P}{A}$ and $P^{\sharp}(\gamma)$ are both antisymmetric
in $L^{2}(P)$,

$\left<X^{P}{A}f,h\right>_{P}=-\left<f,X^{P}{A}h\right>_{P},\qquad\left<X^{P}_{\SS}f,h\right>_{P}=\left<f,X^{P}_{\SS}h\right>_{P},$

where $\left<\cdot,\cdot\right>_{P}$ denotes the $L^{2}(P)$ pseudo-inner
product, $\left<f,h\right>_{P}\equiv\int fh\,\mathbf{d}P$. Hence,
$\mathcal{L}$ is symmetric if and only if $X{A}^{P}+P^{\sharp}(\gamma)=0$. In
general, the generator of (5) satisfies
$\mathcal{L}={\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}X^{P}_{\SS}$ if and only if the Fokker-
Planck current of $P$ vanishes, in which case, we say that $\mathcal{L}$
satisfies the detailed balance condition, and the diffusion is reversible.
Finally, ${\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}X^{P}_{\SS}$ is non-positive, i.e.,

$\left<{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}X^{P}_{\SS}(f),f\right>_{P}\leq 0$

for all $f\in C_{c}^{\infty}(\mathscr{M})$.

###### Proof.

The formula for the generator follows from (19) and
$\text{div}_{P}(X^{\SS}_{f})=\text{div}_{P}(Y_{i})Y_{i}(f)+Y_{i}Y_{i}f$, as we
prove using a local argument: given local coordinates $(z^{i})$, writing
$P=p_{\infty}\mathbf{d}z$ we have

$\displaystyle\text{div}_{P}(X^{\SS}_{f})=\frac{1}{p_{\infty}}\partial_{r}(p_{\infty}\SS^{jr}\partial_{j}f)=\frac{1}{p_{\infty}}\partial_{r}(p_{\infty}Y^{j}_{i}Y^{r}_{i}\partial_{j}f)=\text{div}_{P}(Y_{i})Y_{i}(f)+\partial_{r}(Y^{j}_{i}\partial_{j}f)Y^{r}_{i},$

which, combined with the fact that for any vector field $Y$,
$YY(f)=Y^{i}\partial_{i}(Y^{j}\partial_{j}f)=Y^{i}Y^{j}\partial_{i}\partial_{j}f+Y^{i}\partial_{i}Y^{j}\partial_{j}f$,
yields the result. From the proof of proposition §A.1, we know the (formal)
adjoint of $X_{P}{A}$ is $f\mapsto-\text{div}_{P}(fX{A}_{P})$, and
$-\text{div}_{P}(fX{A}_{P})=-X{A}_{P}(f)-f\text{div}_{P}(X{A}_{P})=-X{A}_{P}(f)$.
Indeed, this only requires the fact that the dynamics $X{A}_{P}$ preserves
$P$, and thus still holds when we include the topological obstruction
contribution, which has itself vanishing $\text{div}_{P}$. From (6) we also
know that the adjoint of $X^{\SS}_{P}$ is
$f\mapsto-\text{div}_{P}\left(f\text{div}_{P}(Y_{i})Y_{i}\right)+\text{div}_{P}\big{(}\text{div}_{P}(fY_{i})Y_{i}\big{)}=\text{div}_{P}\big{(}Y_{i}(f)Y_{i})=\text{div}_{P}\big{(}X{A}_{f})=X^{\SS}_{P}(f)$.
∎

We denote the pushforward with respect to a diffeomorphism $\mathcal{R}$ on
tensor fields by $\mathcal{R}_{*}$. Then

######  Corollary A.8.

Let $\mathcal{R}$ be a target-preserving diffeomorphism, which is an
$\mathzapfc{A}$-antimorphism and a $\SS$-morphism, that is

$\mathcal{R}_{*}\mathzapfc{A}=-\mathzapfc{A},\qquad\mathcal{R}_{*}\SS=\SS.$

Then, the generator of the $\mathzapfc{A}$-diffusion (21) is reversible up to
$\mathcal{R}$, that is we have

$\left<f,\mathcal{L}h\right>_{P}=\left<\mathcal{L}\mathcal{R}^{*}f,\mathcal{R}^{*}h\right>_{P},\quad\forall
f,h\in C_{c}^{\infty}(\mathscr{M}).$

###### Proof.

First note that for any $\mathzapfc{B}$-Hamiltonian vector field, since
$\mathcal{R}^{*}P=P$, then, using proposition 6.3.5 [2],
$\mathcal{R}^{*}\mathcal{L}_{X_{f}}P=\mathcal{L}_{(\mathcal{R}^{-1})_{*}X_{f}}\mathcal{R}^{*}P=\mathcal{L}_{(\mathcal{R}^{-1})_{*}X_{f}}P,$
so
$\text{div}_{P}(X_{f})\circ\mathcal{R}=\text{div}_{P}((\mathcal{R}^{-1})_{*}X_{f})$.
Now consider the antisymmetric part, $\mathcal{L}=X_{P}{A}$. Then
$\left<f,X_{P}{A}h\right>_{P}=-\left<X_{P}{A}f,h\right>_{P}=-\left<\mathcal{R}^{*}X_{P}{A}f,\mathcal{R}^{*}h\right>_{P}$,
where we have used $\mathcal{R}_{\sharp}P=P$ in the last equality. Moreover
$\mathcal{R}^{*}X_{P}{A}f=-X_{P}{A}\mathcal{R}^{*}f$ since $\mathcal{R}$ is an
$\mathzapfc{A}$-antimorphism,
$(\mathcal{R}^{-1})_{*}X_{f}{A}=-X_{f\circ\mathcal{R}}{A}$. The proof of the
symmetric part is analogous, except we have
$\mathcal{R}^{*}X_{P}^{\SS}f=X_{P}^{\SS}\mathcal{R}^{*}f$. Note that if
$\mathcal{R}_{*}Y_{i}=\pm Y_{i}$ then $\mathcal{R}$ is a $\SS$-morphism, since
$(\mathcal{R}^{-1})_{*}X_{f}^{\SS}=(\mathcal{R}^{-1})_{*}(Y_{i}(f)Y_{i})=Y_{i}(f)\circ\mathcal{R}(\mathcal{R}^{-1})_{*}Y_{i}=\left(\mathcal{R}_{*}(\mathcal{R}^{-1})_{*}Y_{i}\right)(f)\circ\mathcal{R}(\mathcal{R}^{-1})_{*}Y_{i}=\left((\mathcal{R}^{-1})_{*}Y_{i}\right)(\mathcal{R}^{*}f)(\mathcal{R}^{-1})_{*}Y_{i}=Y_{i}(\mathcal{R}^{*}f)Y_{i}=X^{\SS}_{\mathcal{R}^{*}f}$.
∎

###  A.9. Non-Degenerate and Overdamped Systems

######  Theorem A.9.

Any $P\propto p_{\infty}\text{vol}$ diffusion generated by (27) takes, at
least up to topological obstructions, the form (28) for some
$\mathzapfc{A}\in\mathfrak{X}^{2}(\mathscr{M})$ ($\nabla$ is the Riemannian
gradient)

$\mathbf{d}Z_{t}={\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}\nabla\log
p_{\infty}(Z_{t})\mathbf{d}t-X_{\log
p_{\infty}}{A}(Z_{t})\mathbf{d}t-\nabla\cdot\mathzapfc{A}(Z_{t})\mathbf{d}t+\mathbf{d}B_{t}.$

###### Proof.

Since $\Delta$ is symmetric in $L^{2}(\mathscr{M},\text{vol})$,

$\mathcal{L}^{*}p_{\infty}=-\nabla\cdot(p_{\infty}X)+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over 2}}{1/2}}\Delta
p_{\infty}=\nabla\cdot(-p_{\infty}X+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over 2}}{1/2}}\nabla
p_{\infty}),$

Thus, following the proof in §A.7

$-p_{\infty}X+{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}\nabla
p_{\infty}=\nabla\cdot(\mathzapfc{A}p_{\infty})=p_{\infty}\nabla\cdot\mathzapfc{A}+p_{\infty}X_{\log
p_{\infty}}{A},$

at least locally (we refer to [90] for an analysis of non-degenrate measure-
preserving diffusions in the case of compact oriented manifolds), and thus the
drift has the form

$X={\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}\nabla\log
p_{\infty}-\nabla\cdot\mathzapfc{A}-X_{\log p_{\infty}}{A},$

has claimed. ∎

###  A.10. Rate of Change of Functionals along Measure-preserving Diffusions

######  Proposition A.10.

Let $F$ be a functional on the space of volume measures, and suppose
$\frac{\delta F}{\delta Q}\in C_{c}^{\infty}(\mathscr{M})$ (or that more
generally Stokes’ theorem holds). The rate of change of $F$ along the
$P$-preserving diffusion is given by

$\frac{\mathbf{d}}{\mathbf{d}t}F(\mu_{t})=\left\\{\log\frac{\mathbf{d}P}{\mathbf{d}\mu_{t}},\frac{\delta
F}{\delta\mu_{t}}\right\\}_{\int_{\mathcal{T}}}(\mu_{t})+\left<\frac{\mathbf{d}\mu_{t}}{\mathbf{d}P},\mu^{\sharp}_{t}(\gamma)\left[\frac{\delta
F}{\delta\mu_{t}}\right]\right>_{\mu_{t}}$

where $\mathcal{T}$ is the thermodynamic bracket $\SS/\sqrt{2}-\mathzapfc{A}$,
and $\gamma$ the topological obstruction.

###### Proof.

Differentiating we have

$\frac{\mathbf{d}}{\mathbf{d}t}F(\mu_{t})=\int\frac{\delta
F}{\delta\mu_{t}}\frac{\partial\mu_{t}}{\partial t}=\int\frac{\delta
F}{\delta\mu_{t}}\text{div}_{\mu_{t}}\left(X^{\mathzapfc{A}}_{\log\frac{\mathbf{d}P}{\mathbf{d}\mu_{t}}}-{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}X^{\SS}_{\log\frac{\mathbf{d}P}{\mathbf{d}\mu_{t}}}\right)\mu_{t}.$

If $\frac{\delta F}{\delta\mu_{t}}$ is compactly supported, or more generally
provided Stokes theorem holds, we have

$\frac{\mathbf{d}}{\mathbf{d}t}F(\mu_{t})=-\int\left(X^{\mathzapfc{A}}_{\log\frac{\mathbf{d}P}{\mathbf{d}\mu_{t}}}\left(\frac{\delta
F}{\delta\mu_{t}}\right)-{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}X^{\SS}_{\log\frac{\mathbf{d}P}{\mathbf{d}\mu_{t}}}\left(\frac{\delta
F}{\delta\mu_{t}}\right)\right)\mu_{t}.$

Hence

$\frac{\mathbf{d}}{\mathbf{d}t}F(\mu_{t})=-\left\\{\log\frac{\mathbf{d}P}{\mathbf{d}\mu_{t}},\frac{\delta
F}{\delta\mu_{t}}\right\\}_{\int_{\mathzapfc{A}}}(\mu_{t})+{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\left\\{\log\frac{\mathbf{d}P}{\mathbf{d}\mu_{t}},\frac{\delta
F}{\delta\mu_{t}}\right\\}_{\int_{\SS}}(\mu_{t}),$

which can be expressed as

$\frac{\mathbf{d}}{\mathbf{d}t}F(\mu_{t})=\left\\{\log\frac{\mathbf{d}P}{\mathbf{d}\mu_{t}},\frac{\delta
F}{\delta\mu_{t}}\right\\}_{\int_{\mathcal{T}}}(\mu_{t})$

via the thermodynamic bracket $\mathcal{T}=\SS/\sqrt{2}-\mathzapfc{A}$ of the
diffusion.

For the topological obstruction contribution, note that since
$\mu_{t}^{\sharp}(\gamma)$ is $\mu_{t}$ preserving,

$-\int\frac{\delta
F}{\delta\mu_{t}}\mu_{t}^{\sharp}(\gamma)\left(\frac{\mathbf{d}\mu_{t}}{\mathbf{d}P}\right)\mu_{t}=\int\mu_{t}^{\sharp}(\gamma)\left(\frac{\delta
F}{\delta\mu_{t}}\right)\frac{\mathbf{d}\mu_{t}}{\mathbf{d}P}\mu_{t}.$

Note that when $F\equiv\mathrm{KL}(\cdot\|P)$ we have

$\frac{\delta\mathrm{KL}(\cdot\|P)}{\delta\mu_{t}}=\log\frac{\mathbf{d}\mu_{t}}{\mathbf{d}P}=-\log\frac{\mathbf{d}P}{\mathbf{d}\mu_{t}},$

so we recover the formula for the rate of change of KL. ∎

###  A.11. Underdamped Langevin on Manifolds

######  Theorem A.11.

Suppose $(\mathzapfc{A},\mu_{T\mathscr{M}})$ is a Langevin pair. If we choose
the noise fields to be the vertical fields $Y_{i}\equiv\mathcal{V}\circ
X_{i}\circ\pi$ for $X_{i}\in\mathfrak{X}(\mathscr{M})$, then the
$\mathzapfc{A}$-diffusion generated by $e^{-H}\mu_{T\mathscr{M}}$, with
$H\equiv\pi^{*}V+{\mathchoice{\textstyle{1\over 2}}{\scriptstyle{1\over
2}}{\scriptscriptstyle{1\over 2}}{1/2}}\|\cdot\|^{2}$, is

$\mathrm{d}(q_{t},v_{t})=\underbrace{X_{H}{A}(q_{t},v_{t})\mathrm{d}t}_{\mathzapfc{A}-\text{Hamiltonian
dynamic}}-\underbrace{\frac{\beta}{2}\left<X_{i}(q_{t}),v_{t}\right>_{q_{t}}\mathcal{V}(X_{i}(q_{t}))\mathrm{d}t}_{\text{vertical
kinetic
Dissipation}}+\underbrace{\mathcal{V}(X_{i}(q_{t}))\circ\mathrm{d}W_{t}}_{\text{vertical
randomness}},$

or in tangent-lifted coordinates

$\mathrm{d}(q_{t},v_{t})=\Big{(}X_{H}{A}(q_{t},v_{t})-\frac{\beta}{2}M(q_{t})v_{t}\Big{)}\mathrm{d}t+\sigma(q_{t})\circ\mathrm{d}W_{t},$

where
$M(q)v=M(q)_{rj}v^{j}\partial_{v^{r}}\equiv(\sigma\sigma^{\top}\mathzapfc{M}(q))_{rj}v^{j}\partial_{v^{r}}$
and
$\sigma\equiv\sigma_{ji}\partial_{v^{j}}\equiv(X_{i})^{j}\partial_{v^{j}}$,
and
$\mathzapfc{M}_{ij}(q)=\left<\partial_{x^{i}},\partial_{x^{j}}\right>_{q}$. In
particular $(\Pi,\omega^{n}_{\flat})$ is a Langevin pair, where $\Pi$ is the
Poisson 2-vector field associated to $\omega_{\flat}$.

###### Proof.

Recall the definition of the vertical lift
$\mathcal{V}:v_{q}\mapsto\mathrm{ver}_{v_{q}}v_{q}\in T\mathscr{F}$, with
$\mathrm{ver}_{v_{q}}v_{q}:f\mapsto\frac{\mathrm{d}}{\mathrm{d}t}f(q,v_{q}+tv_{q})|_{t=0}$
for any $f\in C^{1}(\mathscr{F})$. For example if
$\mathscr{F}=\mathbb{R}^{n}\times\mathbb{R}^{\ell}$ is a vector bundle of
$\mathscr{M}=\mathbb{R}^{n}$, this is just the directional derivative of $f$
at $(q,v_{q})$ in the direction $(0,v_{q})$, $(0,v)\cdot\nabla
f(q,v)=v^{k}\partial_{v^{k}}f(q,v)$, or $\mathcal{V}=v^{k}\partial_{v^{k}}$,
where $(x^{s},v^{j})$ are coordinates on $\mathscr{F}$. If
$X\in\mathfrak{X}(\mathscr{M})$ is a vector field, with local expansion
$X=X^{k}(x)\partial_{x^{k}}$, then its composition with the canonical vector
field $\mathcal{V}\circ X\circ\pi\in\mathfrak{X}(T\mathscr{F})$ is
$\mathcal{V}\circ X\circ\pi(x,v)=X^{k}(x)\partial_{v^{k}}$. Then if
$\mathzapfc{M}$ is a vector bundle Riemannian metric on $\mathscr{F}$, and $T$
is the associated kinetic energy we find

$\displaystyle Y_{i}(H)(x,v)$
$\displaystyle=X^{k}_{i}(x)\partial_{v^{k}}\big{(}V(x)+T(x,v))=X^{k}_{i}(x)\partial_{v^{k}}\big{(}{\mathchoice{\textstyle{1\over
2}}{\scriptstyle{1\over 2}}{\scriptscriptstyle{1\over
2}}{1/2}}\mathzapfc{M}_{rj}(x)v^{r}v^{j})$
$\displaystyle=X^{k}_{i}(x)\mathzapfc{M}_{jk}(x)v^{j}=\left<Y_{i}(x),v\right>_{x}.$

Now consider $\mathscr{F}=T\mathscr{M}$. The above derivation shows we can
also write

$Y_{i}(H)(x,v)=\left<X_{i}(x),v\right>_{x},$

from which (31) follows. Moreover, setting $\sigma_{ji}\equiv X^{j}_{i}$, so
$Y_{i}(x,v)=\sigma_{ji}(x)\partial_{v^{j}}$ where $v^{j}$ are the tangent-
lifted coordinates [3]. We have

$\displaystyle Y_{i}(H)Y_{i}(x,v)$
$\displaystyle=\left<X_{i}(x),v\right>_{x}\mathrm{ver}_{X_{i}(x)}(X_{i}(x))=X^{k}_{i}(x)\mathzapfc{M}_{jk}(x)v^{j}X^{r}_{i}(x)\partial_{v^{r}}$
$\displaystyle=\sigma_{ki}(x)\mathzapfc{M}_{jk}(x)v^{j}\sigma_{ri}(x)\partial_{v^{r}}=\big{(}\sigma\sigma^{\top}(x)\big{)}_{rk}\mathzapfc{M}_{kj}(x)v^{j}\partial_{v^{r}},$

and the local expression (32) follows. Moreover the symplectic measure is
indeed horizontal (this fact may be traced back to the fact that the Liouville
1-form is horizontal). Indeed, in tangent-lifted coordinates
$\omega_{\flat}^{n}=|\mathzapfc{M}|\mathrm{d}x\mathrm{d}v$, so locally the
divergence of $Z\in\mathfrak{X}(T\mathscr{M})$ is
$\text{div}_{\omega_{\flat}}(Z)=\frac{1}{|\mathzapfc{M}|}\partial_{x^{i}}\big{(}|\mathzapfc{M}|Z^{i}\big{)}+\partial_{v^{j}}\overline{Z}^{j}$,
and from the previous local expressions we see
$\text{div}_{\omega_{\flat}}(Y_{i})=0$, and thus $\omega^{n}_{\flat}$ is
horizontal.

Finally we mention that if the noise vector fields are chosen to be
$\Pi$-Hamiltonian vector fields associated to “noise” Hamiltonians
$U_{i}:\mathscr{M}\to\mathbb{R}$, then $Y_{i}\equiv
X_{U_{i}\circ\pi}^{\Pi}=-\mathcal{V}\circ\nabla U_{i}\circ\pi$, i.e., they are
the vertical lift of the Riemannian gradients, and
$Y_{i}(H)=\omega_{\flat}\big{(}X_{H}^{\Pi},X_{U_{i}\circ\pi}^{\Pi}\big{)}=-X_{H}^{\Pi}(U_{i}\circ\pi)=-\mathrm{d}U_{i}\circ\partial\pi(X_{H}^{\Pi})=-\mathrm{d}U_{i}\circ\partial\pi(X_{T}^{\Pi})$
since $X_{V}$ is vertical.

∎

##  Appendix B Unpublished Result

The following results were proven in the thesis of one of the authors [14],
and are being submitted as part of an article discussing the intrinsic
geometry of smooth measures and its relations to various fields of
mathematics. When this latter paper will be available online, this section
will be erased and the mentions of it in the main article will be replaced by
citations, but in the mean time, for completeness, we include the
characterisation §B.1 of measure-preserving dynamical systems and its proof
that we will use in the main article.

Denote by $\Omega_{\text{Or}}^{k}(\mathscr{M})$ the space of twisted
differential $k$-forms, that is differential $k$-forms taking value in the
orientation bundle. In particular the smooth positive measure $P$ can be
identified with a twisted form of top rank. For any
$X\in\mathfrak{X}^{k}(\mathscr{M})$, using
$(\mathfrak{X}^{\ell}(\mathscr{M}))^{*}\cong\Omega^{\ell}(\mathscr{M})$, we
define the right interior product by $i_{X}P(A)\equiv\left<P,X\wedge
A\right>_{*}$ for any $A\in\mathfrak{X}^{n-k}(\mathscr{M})$. This induces the
$C^{\infty}(\mathscr{M})$-linear musical isomorphism
$P^{\flat}:\mathfrak{X}^{k}(\mathscr{M})\rightarrow\Omega_{\text{Or}}^{n-k}(\mathscr{M})$
by $P^{\flat}(X)\equiv i_{X}P$, and we denoted its inverse by $P^{\sharp}$,
$P^{\sharp}\circ P^{\flat}=\mathrm{Id}$ [52, Sec. 2.5].

Note that since $P^{\flat}(X)$ is twisted, it is a form that takes value in
the orientation line bundle. Since the orientation bundle is flat, we can find
transition functions that are locally constant (in fact these are given by the
sign of the Jacobian of the transition functions of $\mathscr{M}$). Hence the
exterior derivative $\mathbf{d}$ on differential forms extend to an operator
on twisted forms, which we will use in the definition of $\text{curl}_{P}$
below. Moreover $\mathbf{d}$ generates a canonical twisted de Rham complex by
extending $\mathbf{d}$ using any trivialisation of the orientation line bundle
induced by a trivialisation of $\mathscr{M}$, as explained in section 7 [27]
(on orientable manifolds this reduces do the standard de Rham complex). Using
this (extended) exterior derivative, we define the $P$-rotationnel as

$\text{curl}_{P}=P^{\sharp}\circ\mathbf{d}\circ P^{\flat},$

which satisfies $\text{curl}_{P}\circ\text{curl}_{P}=0$ since
$\mathbf{d}\circ\mathbf{d}=0$. In particular when applied to vector fields the
$P$-rotationnel acts as the divergence operator
$\text{curl}_{P}=\text{div}_{P}:\mathfrak{X}(\mathscr{M})\to
C^{\infty}(\mathscr{M})$, which follows from

$\mathcal{L}_{X}P=\mathbf{d}i_{X}P=P^{\flat}\text{curl}_{P}(X)=\text{curl}_{P}(X)P,$

together with the definition of $\text{div}_{P}(X)$ as the function satisfying
$\text{div}_{P}(X)P=\mathcal{L}_{X}P$. If $f$ is a function, observe that
$(fP)^{\flat}=fP^{\flat}$, and so if $f$ is non-vanishing,
$(fP)^{\sharp}=P^{\sharp}\circ\frac{1}{f}$. Hence

(51) $\text{curl}_{fP}=P^{\sharp}\circ\frac{1}{f}\circ\mathbf{d}\circ f\circ
P^{\flat}\equiv P^{\sharp}\circ\mathbf{d}_{f}\circ
P^{\flat},\qquad\text{or}\qquad\text{curl}_{Q}=P^{\sharp}\circ\mathbf{d}_{\frac{\mathbf{d}Q}{\mathbf{d}P}}\circ
P^{\flat},$

where $\mathbf{d}_{f}\equiv\mathbf{d}+\mathbf{d}\log|f|\wedge$ is the
distorted de Rham derivative. In particular, the $P$-rotationnel does not
depend on the normalisation constant of $P$, an important requirement in many
statistical applications, where the target distribution or statistical model
is only known up to normalisation. Importantly, we have the following key
result showing the homology groups defined by the boundary operator
$\text{curl}_{P}$ (since $\text{curl}_{P}\circ\text{curl}_{P}=0$) are
isomorphic to the twisted de Rham cohomology groups. As usual we denote by
$[\cdot]$ the equivalence classes.

######  Theorem B.1.

The isomorphism $P^{\flat}$ descends to an isomorphism between the homology
groups $\mathcal{H}_{P}^{\ell}(\mathscr{M})$ of $\text{curl}_{P}$ and the
twisted de Rham cohomology groups $\mathcal{H}^{n-\ell}_{dR}(\mathscr{M})$.
Hence

$\text{Dyn}(P)\cong\text{curl}_{P}\left(\mathfrak{X}^{2}(\mathscr{M})\right)\oplus
P^{\sharp}\left(\mathcal{H}^{n-1}_{dR}\left(\mathscr{M}\right)\right),$

and any $P$-preserving dynamics will be globally the $P$-rotationnel of some
$\mathzapfc{A}\in\mathfrak{X}^{2}(\mathscr{M})$ iff the $(n-1)$ de twisted
Rham cohomology is trivial. Moreover, if $U\subset\mathscr{M}$ is an open
subset, then $\text{curl}_{P}|_{U}=\text{curl}_{P|_{U}}$. Hence the set of
$P$-preserving dynamics is precisely the set of locally curl vector fields

$\text{Dyn}(P)=\text{Curl}_{loc}(P)$

where $\text{Curl}_{loc}(P)=\\{X\in\mathfrak{X}(\mathscr{M}):\forall
q,\text{there is a neighbourhood }U\text{ and
}\mathzapfc{A}\in\mathfrak{X}^{2}(U){s.t.,}X=\text{curl}_{P|_{U}}(\mathzapfc{A})\\}$.

###### Proof.

The $P$-derivative $\text{curl}_{P}\equiv P^{\sharp}\circ\mathrm{d}\circ
P^{\flat}:\mathfrak{X}^{k}(\mathscr{M})\to\mathfrak{X}^{k-1}(\mathscr{M})$ is
a vector space homomorphism, satisfying
$\text{curl}_{P}\circ\text{curl}_{P}=0$, that it is a boundary operator on the
chain complex of $k$-multi-vector fields, and thus
$\text{Im}\left(\text{curl}_{P}:\mathfrak{X}^{\ell+1}(\mathscr{M})\to\mathfrak{X}^{\ell}(\mathscr{M})\right)$
is a linear subspace of
$\ker\left(\text{curl}_{P}:\mathfrak{X}^{\ell}(\mathscr{M})\to\mathfrak{X}^{\ell-1}(\mathscr{M})\right)$.
We can then define the $\ell^{th}$ Holomology group

$\mathcal{H}_{P}^{\ell}(\mathscr{M})\equiv\frac{\ker\left(\text{curl}_{P}:\mathfrak{X}^{\ell}(\mathscr{M})\to\mathfrak{X}^{\ell-1}(\mathscr{M})\right)}{\text{Im}\left(\text{curl}_{P}:\mathfrak{X}^{\ell+1}(\mathscr{M})\to\mathfrak{X}^{\ell}(\mathscr{M})\right)},$

and in particular the first one provides information on $P$-preserving vector
fields (using $\text{curl}_{P}(X)=\text{div}_{P}(X)$)

$\mathcal{H}_{P}^{1}(\mathscr{M})\equiv\frac{\ker\left(\text{div}_{P}:\mathfrak{X}(\mathscr{M})\to
C^{\infty}(\mathscr{M})\right)}{\text{Im}\left(\text{curl}_{P}:\mathfrak{X}^{2}(\mathscr{M})\to\mathfrak{X}(\mathscr{M})\right)}.$

Notice that the map $[\cdot]\circ
P^{\flat}:\mathfrak{X}^{\ell}(\mathscr{M})\to\mathcal{H}^{n-\ell}_{dR}(\mathscr{M})$
descends to a map
$\mathcal{H}_{P}^{\ell}(\mathscr{M})\to\mathcal{H}^{n-\ell}_{dR}(\mathscr{M})$,
since $[\cdot]\circ
P^{\flat}(X+\text{curl}_{P}(Y))=[P^{\flat}(X)+\mathbf{d}i_{Y}P]=[P^{\flat}(X)]$.
The map is surjective since $P^{\flat}$ is, and injective since
$[P^{\flat}(X)]=[P^{\flat}(Y)]$ $\implies$ $[P^{\flat}(X-Y)]=0\implies
P^{\flat}(X-Y)=\mathbf{d}\alpha\implies
X-Y=\text{curl}_{P}(Z)\implies[X]=[Y]$. In particular,
$\mathcal{H}^{n-1}_{dR}(\mathscr{M})$ is trivial iff
$\mathcal{H}_{P}^{1}(\mathscr{M})$ is, in which case every divergence-free
vector field is the $\text{curl}_{P}$ of a bi-vector field $\mathzapfc{A}$.

In general, we can still use Poincaré lemma (or Volterra theorem, as it was
proved by Vito Volterra [157]) and the properties of $\text{curl}_{P}$ to show
that any $P$-preserving vector field is locally a curl vector field. Denoting
the inclusion by $\iota_{U}:U\hookrightarrow\mathscr{M}$, we have
$\iota_{U}^{*}(P^{\flat}(Y))=\iota_{U}^{*}(i_{Y}P)=i_{Y_{U}}\iota_{U}^{*}P=i_{Y_{U}}P_{U}=P_{U}^{\flat}(Y_{U})$,
where $P_{U},Y_{U}$ denote their restriction to $U$, and we have used
proposition 7.4.10 [2]; hence $\iota_{U}^{*}\circ
P^{\flat}=P_{U}^{\flat}\circ|_{U}$. Setting $Y=P^{\sharp}(\alpha)$ this yields
$P^{\sharp}(\alpha)|_{U}=P_{U}^{\sharp}(\iota^{*}_{U}\alpha)$ for any twisted
form $\alpha$. Hence

$|_{U}\circ\text{curl}_{P}\equiv|_{U}\circ P^{\sharp}\circ\mathbf{d}\circ
P^{\flat}=P^{\sharp}_{U}\circ\iota_{U}^{*}\circ\mathbf{d}\circ
P^{\flat}=P^{\sharp}_{U}\circ\mathbf{d}\circ\iota_{U}^{*}\circ
P^{\flat}=P^{\sharp}_{U}\circ\mathbf{d}\circ
P^{\flat}_{U}=\text{curl}_{P_{U}}\circ|_{U}.$

Thus, $\text{curl}_{P}(Y)=0$ iff $P^{\sharp}\circ\mathrm{d}\circ
P^{\flat}(Y)=0$ iff $\mathrm{d}\circ P^{\flat}(Y)=0$ (since $P^{\sharp}$ is a
linear isomorphism). By Poincaré Lemma, this holds iff around any point there
is an open neighbourhood $U$ over which $P^{\flat}(Y)|_{U}=\mathrm{d}\alpha$
for some twisted ($\dim(\mathscr{M})-2)$-form $\alpha$ on $U$. Then
$\mathrm{d}\alpha=\iota_{U}^{*}(P^{\flat}(Y))=P_{U}^{\flat}(Y_{U})$, and
$P_{U}^{\flat}(Y|_{U})=\mathrm{d}\alpha$ iff
$Y_{U}=P^{\sharp}_{U}\mathrm{d}\alpha=\text{curl}_{P_{U}}(\mathzapfc{A})$
where $\mathzapfc{A}\equiv P^{\sharp}_{U}(\alpha)$ is a 2-vector field on $U$.

Finally we also mention that when $P=\mathbf{d}x$ is the Lebesgue measure on
Euclidean space,
$\text{Dyn}(\mathbf{d}x)=\text{curl}_{\mathbf{d}x}\left(\mathfrak{X}^{2}(\mathscr{M})\right)$
was essentially already proved by Vito Volterra [157], that the statement
$\text{Dyn}(P)\cong\text{curl}_{P}\left(\mathfrak{X}^{2}(\mathscr{M})\right)\oplus
P^{\sharp}\left(\mathcal{H}^{n-1}_{dR}\left(\mathscr{M}\right)\right)$ appears
in implicit form (essentially written as
$P^{\flat}\left(\text{Dyn}(P)\right)=\mathbf{d}\Omega^{n-2}(\mathscr{M})\oplus
H^{n-1}(\mathscr{M})$) in [130, Thm. 6] under the assumption that
$\mathscr{M}$ is orientable, and that by Poincaré duality if $\mathscr{M}$ has
a finite good cover (in which case the cohomology groups are finite
dimensional) we may alternatively work with the first compactly supported de
Rham cohomology group (proposition 5.3.1 and theorem 7.8 [27]). ∎

Note that by Poincaré duality if $\mathscr{M}$ has a finite good cover we may
alternatively work with the first compactly supported de Rham cohomology
group. The de Rham cohomology groups may be very large, though they must be
finite dimensional when $\mathscr{M}$ is compact. Moreover, in that case, they
are isomorphic to the vector spaces of harmonic forms, and the following
results follows:

######  Corollary B.2.

If $\mathscr{M}$ is compact and orientable, then

$\text{Dyn}(P)\cong\text{curl}_{P}\mathfrak{X}^{2}(\mathscr{M})\oplus
P^{\sharp}\left(\mathcal{H}^{n-1}\left(\mathscr{M}\right)\right),$

where $\mathcal{H}^{n-1}$ is the space of harmonic $n-1$-forms associated to
an arbitrary Riemannian metric. In other words, any $P$-preserving vector
field on a compact orientable manifold has the form
$X=\text{curl}_{P}(\mathzapfc{A})+P^{\sharp}(\gamma)$.

## References

  * [1] Abdulle, A., Pavliotis, G.A., Vilmart, G.: Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations. Comptes Rendus Mathematique 357(4), 349–354 (2019)
  * [2] Abraham, R., Marsden, J.E., Ratiu, T.S.: Manifolds, tensor analysis, and applications (1988)
  * [3] Abraham, R., Marsden, J.E.: Foundations of Mechanics. American Mathematical Society, second edn. (May 2008), with the assistance of Tudor Ratiu and Richard Cushman
  * [4] Ahn, S., Korattikara, A., Welling, M.: Bayesian posterior sampling via stochastic gradient fisher scoring. arXiv preprint arXiv:1206.6380 (2012)
  * [5] Akhmatskaya, E., Bou-Rabee, N., Reich, S.: A comparison of generalized hybrid monte carlo methods with and without momentum flip. Journal of Computational Physics 228(6), 2256–2265 (2009)
  * [6] Armstrong, J., Brigo, D.: Coordinate-free stochastic differential equations as jets. arXiv preprint arXiv:1602.03931 (2016)
  * [7] Arnaudon, A., Barp, A., Takao, S.: Irreversible langevin mcmc on lie groups. arXiv preprint arXiv:1903.08939 (2019)
  * [8] Arnaudon, M., Thalmaier, A., et al.: The differentiation of hypoelliptic diffusion semigroups. Illinois Journal of Mathematics 54(4), 1285–1311 (2010)
  * [9] Baez, J., Muniain, J.P.: Gauge fields, knots and gravity, vol. 4. World Scientific Publishing Company (1994)
  * [10] Bakry, D., Gentil, I., Ledoux, M.: Analysis and geometry of Markov diffusion operators, vol. 348. Springer Science & Business Media (2013)
  * [11] Barbour, A., Chen, L.H.Y.: An introduction to Stein’s method. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore (2005)
  * [12] Barbour, A.D.: Stein’s method and poisson process convergence. Journal of Applied Probability 25(A), 175–184 (1988)
  * [13] Barp, A.: Hamiltonian monte carlo on lie groups and constrained mechanics on homogeneous manifolds. arXiv preprint arXiv:1903.04662 (2019)
  * [14] Barp, A.: The bracket geometry of statistics. Ph.D. thesis, Imperial College London (2020)
  * [15] Barp, A., Briol, F.X., Kennedy, A.D., Girolami, M.: Geometry and Dynamics for Markov Chain Monte Carlo. Annual Review of Statistics and its Application (2018). https://doi.org/https://doi.org/10.1146/annurev-statistics-031017-100141
  * [16] Barp, A., Briol, F.X., Duncan, A., Girolami, M., Mackey, L.: Minimum stein discrepancy estimators. In: Advances in Neural Information Processing Systems. pp. 12964–12976 (2019)
  * [17] Barp, A., Kennedy, A., Girolami, M.: Hamiltonian monte carlo on symmetric and homogeneous spaces via symplectic reduction. arXiv preprint arXiv:1903.02699 (2019)
  * [18] Barp, A., Oates, C., Porcu, E., Girolami, M., et al.: A riemannian-stein kernel method. arXiv preprint arXiv:1810.04946 (2018)
  * [19] Batrouni, G., Kawai, H., Rossi, P.: Coordinate-independent formulation of the langevin equation. Journal of mathematical physics 27(6), 1646–1648 (1986)
  * [20] Bell, D.R.: Stochastic differential equations and hypoelliptic operators. In: Real and stochastic analysis, pp. 9–42. Springer (2004)
  * [21] Bell, D.R., Mohammed, S.E.A., et al.: An extension of hormander’s theorem for infinitely degenerate second-order operators. Duke Mathematical Journal 78(3), 453–476 (1995)
  * [22] Beskos, A., Stuart, A.: MCMC methods for sampling function space. In: ICIAM 07?6th International Congress on Industrial and Applied Mathematics. pp. 337–364. European Mathematical Society (2009)
  * [23] Betancourt, M.: A conceptual introduction to hamiltonian monte carlo. arXiv preprint arXiv:1701.02434 (2017)
  * [24] Bismut, J.M.: Martingales, the malliavin calculus and hypoellipticity under general hörmander’s conditions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 56(4), 469–505 (1981)
  * [25] Bismut, J.M.: Mécanique aléatoire. In: Ecole d’Eté de Probabilités de Saint-Flour X-1980, pp. 1–100. Springer (1982)
  * [26] Bloch, A., Krishnaprasad, P., Marsden, J.E., Ratiu, T.S.: The euler-poincaré equations and double bracket dissipation. Communications in mathematical physics 175(1), 1–42 (1996)
  * [27] Bott, R., Tu, L.W.: Differential forms in algebraic topology, vol. 82. Springer Science & Business Media (2013)
  * [28] Bou-Rabee, N.: Time integrators for molecular dynamics. Entropy 16(1), 138–162 (2014)
  * [29] Bou-Rabee, N., Donev, A., Vanden-Eijnden, E.: Metropolis integration schemes for self-adjoint diffusions. Multiscale Modeling & Simulation 12(2), 781–831 (2014)
  * [30] Bou-Rabee, N., Sanz-Serna, J.M.: Geometric integrators and the hamiltonian monte carlo method. Acta Numerica 27, 113–206 (2018)
  * [31] Bouchet, F., Venaille, A.: Statistical mechanics of two-dimensional and geophysical flows. Physics reports 515(5), 227–295 (2012)
  * [32] Brockett, R.W.: Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems. Linear Algebra and its applications 146, 79–91 (1991)
  * [33] Brubaker, M., Salzmann, M., Urtasun, R.: A family of mcmc methods on implicitly defined manifolds. In: Artificial intelligence and statistics. pp. 161–172 (2012)
  * [34] Byrne, S., Girolami, M.: Geodesic Monte Carlo on Embedded Manifolds. Scandinavian Journal of Statistics, Theory and Applications (2013). https://doi.org/10.1111/sjos.12036
  * [35] Cances, E., Legoll, F., Stoltz, G.: Theoretical and numerical comparison of some sampling methods for molecular dynamics. ESAIM: Mathematical Modelling and Numerical Analysis 41(2), 351–389 (2007)
  * [36] Cattiaux, P., Mesnager, L.: Hypoelliptic non-homogeneous diffusions. Probability Theory and Related Fields 123(4), 453–483 (2002)
  * [37] Chen, T., Fox, E., Guestrin, C.: Stochastic gradient hamiltonian monte carlo. In: International conference on machine learning. pp. 1683–1691 (2014)
  * [38] Chen, W.Y., Barp, A., Briol, F.X., Gorham, J., Girolami, M., Mackey, L., Oates, C., et al.: Stein point markov chain monte carlo. arXiv preprint arXiv:1905.03673 (2019)
  * [39] Cheng, X., Chatterji, N.S., Abbasi-Yadkori, Y., Bartlett, P.L., Jordan, M.I.: Sharp convergence rates for langevin dynamics in the nonconvex setting. arXiv preprint arXiv:1805.01648 (2018)
  * [40] Cheng, X., Chatterji, N.S., Bartlett, P.L., Jordan, M.I.: Underdamped langevin mcmc: A non-asymptotic analysis. arXiv preprint arXiv:1707.03663 (2017)
  * [41] Chirikjian, G.S.: Information theory on lie groups and mobile robotics applications. In: 2010 IEEE International Conference on Robotics and Automation. pp. 2751–2757. IEEE (2010)
  * [42] Clark, M.A., Joó, B., Kennedy, A.D., Silva, P.J.: Improving dynamical lattice qcd simulations through integrator tuning using poisson brackets and a force-gradient integrator. Physical Review D 84(7), 071502 (2011)
  * [43] Cong, Y., Chen, B., Liu, H., Zhou, M.: Deep latent dirichlet allocation with topic-layer-adaptive stochastic gradient riemannian mcmc. In: Proceedings of the 34th International Conference on Machine Learning-Volume 70. pp. 864–873. JMLR. org (2017)
  * [44] Cruzeiro, A.B., Holm, D.D., Ratiu, T.S.: Momentum maps and stochastic clebsch action principles. Communications in Mathematical Physics 357(2), 873–912 (2018)
  * [45] De Rham, G.: Sur l’analysis situs des variétés à n dimensions, vol. 1305. Gauthier-Villars (1931)
  * [46] De Rham, G.: Variétés différentiables: formes, courants, formes harmoniques, vol. 3. Editions Hermann (1973)
  * [47] Diaconis, P., Holmes, S.P., Shahshahani, M.: Sampling from a manifold. Advances in Modern Statistical Theory and Applications: A Festschrift in honor of Morris L. Eaton (2013). https://doi.org/10.1214/12-IMSCOLL1006
  * [48] Ding, M., Tu, Z., Xing, X.: Covariant formulation of non-linear langevin theory with multiplicative guassian white noises. arXiv preprint arXiv:2007.16131 (2020)
  * [49] Dobson, P., Fursov, I., Lord, G., Ottobre, M.: Reversible and non-reversible markov chain monte carlo algorithms for reservoir simulation problems. arXiv preprint arXiv:1903.06960 (2019)
  * [50] Duane, S., Kennedy, A.D., Pendleton, B.J., Roweth, D.: Hybrid Monte Carlo. Physics Letters B 195(2), 216–222 (1987)
  * [51] Dufour, J.P., Haraki, A.: Rotationnnels et structures de poisson quadratiques. Comptes rendus de l’Académie des sciences. Série 1, Mathématique 312(1), 137–140 (1991)
  * [52] Dufour, J.P., Zung, N.T.: Poisson structures and their normal forms, vol. 242. Springer Science & Business Media (2006)
  * [53] Duncan, A., Pavliotis, G., Zygalakis, K.: Nonreversible langevin samplers: Splitting schemes, analysis and implementation. arXiv preprint arXiv:1701.04247 (2017)
  * [54] Durmus, A., Majewski, S., Miasojedow, B.: Analysis of langevin monte carlo via convex optimization. J. Mach. Learn. Res. 20, 73–1 (2019)
  * [55] Durmus, A., Moulines, E., Saksman, E.: On the convergence of Hamiltonian Monte Carlo. arXiv preprint arXiv:1705.00166 (2017)
  * [56] Eberle, A., Guillin, A., Zimmer, R., et al.: Couplings and quantitative contraction rates for langevin dynamics. The Annals of Probability 47(4), 1982–2010 (2019)
  * [57] Elworthy, D.: Geometric aspects of diffusions on manifolds. In: École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–87, pp. 277–425. Springer (1988)
  * [58] Émery, M.: Stochastic calculus in manifolds. Springer Science & Business Media (2012)
  * [59] Erdogdu, M.A., Mackey, L., Shamir, O.: Global non-convex optimization with discretized diffusions. In: Advances in Neural Information Processing Systems. pp. 9671–9680 (2018)
  * [60] Eyink, G.L., Lebowitz, J.L., Spohn, H.: Hydrodynamics and fluctuations outside of local equilibrium: driven diffusive systems. Journal of Statistical physics 83(3-4), 385–472 (1996)
  * [61] Fang, Y., Sanz-Serna, J.M., Skeel, R.D.: Compressible generalized hybrid monte carlo. The Journal of chemical physics 140(17), 174108 (2014)
  * [62] Fokker, A.D.: Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Annalen der Physik 348(5), 810–820 (1914)
  * [63] Frankel, T.: The geometry of physics: an introduction. Cambridge university press (2011)
  * [64] Franzke, C.L., O’Kane, T.J., Berner, J., Williams, P.D., Lucarini, V.: Stochastic climate theory and modeling. Wiley Interdisciplinary Reviews: Climate Change 6(1), 63–78 (2015)
  * [65] Gangbo, W., Kim, H.K., Pacini, T.: Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems. American Mathematical Soc. (2010)
  * [66] Gao, X., Gürbüzbalaban, M., Zhu, L.: Global convergence of stochastic gradient hamiltonian monte carlo for non-convex stochastic optimization: Non-asymptotic performance bounds and momentum-based acceleration. arXiv preprint arXiv:1809.04618 (2018)
  * [67] Gay-Balmaz, F., Holm, D.D.: Selective decay by casimir dissipation in inviscid fluids. Nonlinearity 26(2), 495 (2013)
  * [68] Girolami, M., Calderhead, B.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods. Journal of the Royal Statistical Society Series B: Statistical Methodology 73(2), 123–214 (2011)
  * [69] Glatt-Holtz, N.: Notes on statistically invariant states in stochastically driven fluid flows. arXiv preprint arXiv:1410.8622 (2014)
  * [70] Gorham, J., Duncan, A.B., Vollmer, S.J., Mackey, L.: Measuring sample quality with diffusions. arXiv preprint arXiv:1611.06972 (2016)
  * [71] Gorham, J., Mackey, L.: Measuring sample quality with stein’s method. In: Advances in Neural Information Processing Systems. pp. 226–234 (2015)
  * [72] Graham, M.M., Thiery, A.H., Beskos, A.: Manifold markov chain monte carlo methods for bayesian inference in a wide class of diffusion models. arXiv preprint arXiv:1912.02982 (2019)
  * [73] Graham, R.: Covariant formulation of non-equilibrium statistical thermodynamics. Zeitschrift für Physik B Condensed Matter 26(4), 397–405 (1977)
  * [74] Grmela, M.: Bracket formulation of diffusion-convection equations. Physica D: Nonlinear Phenomena 21(2-3), 179–212 (1986)
  * [75] Guedira, F., Lichnerowicz, A.: Géométrie des algébres de lie locales de kirillov. Journal de mathématiques pures et appliquées 63(4), 407–484 (1984)
  * [76] Guha, P.: Metriplectic structure, leibniz dynamics and dissipative systems. Journal of Mathematical Analysis and Applications 326(1), 121–136 (2007)
  * [77] Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, vol. 31. Springer Science & Business Media (2006)
  * [78] Heber, F., Trst?anová, Ž., Leimkuhler, B.: Posterior sampling strategies based on discretized stochastic differential equations for machine learning applications. Journal of Machine Learning Research 21(228), 1–33 (2020)
  * [79] Holbrook, A.: Note on the geodesic Monte Carlo. arXiv preprint arXiv:1805.05289 (2018)
  * [80] Holbrook, A., Lan, S., Vandenberg-Rodes, A., Shahbaba, B.: Geodesic Lagrangian Monte Carlo over the space of positive definite matrices: with application to Bayesian spectral density estimation. Journal of Statistical Computation and Simulation (2017). https://doi.org/10.1080/00949655.2017.1416470
  * [81] Holbrook, A., Vandenberg-Rodes, A., Shahbaba, B.: Bayesian inference on matrix manifolds for linear dimensionality reduction. arXiv preprint arXiv:1606.04478 (2016)
  * [82] Holm, D.D., Marsden, J.E., Ratiu, T.S.: The euler–poincaré equations and semidirect products with applications to continuum theories. Advances in Mathematics 137(1), 1–81 (1998)
  * [83] Hörmander, L., et al.: Hypoelliptic second order differential equations. Acta Mathematica 119, 147–171 (1967)
  * [84] Horowitz, A.M.: A generalized guided monte carlo algorithm. Physics Letters B 268(2), 247–252 (1991)
  * [85] Hsu, E.P.: A brief introduction to brownian motion on a riemannian manifold. lecture notes (2008)
  * [86] Hsu, P.: Brownian motion and Riemannian geometry. Contemp. Math 73, 95–104 (1988)
  * [87] Hwang, C.R., Hwang-Ma, S.Y., Sheu, S.J.: Accelerating Gaussian diffusions. The Annals of Applied Probability pp. 897–913 (1993)
  * [88] Hwang, C.R., Hwang-Ma, S.Y., Sheu, S.J., et al.: Accelerating diffusions. The Annals of Applied Probability 15(2), 1433–1444 (2005)
  * [89] Ichihara, K., Kunita, H.: A classification of the second order degenerate elliptic operators and its probabilistic characterization. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 30(3), 235–254 (1974)
  * [90] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Elsevier (2014)
  * [91] Ivancevic, V.G., Ivancevic, T.T.: Dynamics and control of humanoid robots: A geometrical approach. Paladyn, Journal of Behavioral Robotics 1(4), 204–218 (2010)
  * [92] Izaguirre, J.A., Hampton, S.S.: Shadow hybrid monte carlo: an efficient propagator in phase space of macromolecules. Journal of Computational Physics 200(2), 581–604 (2004)
  * [93] Kaufman, A.N.: Dissipative hamiltonian systems: A unifying principle. Physics Letters A 100(8), 419–422 (1984)
  * [94] Kennedy, A.D., Silva, P.J., Clark, M.A.: Shadow Hamiltonians, Poisson Brackets, and Gauge Theories. Physical Review D87(3), 034511 (2013). https://doi.org/10.1103/PhysRevD.87.034511
  * [95] Khas’minskii, R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution to the cauchy problem for parabolic equations. Theory of Probability & Its Applications 5(2), 179–196 (1960)
  * [96] Khesin, B., Lee, P.: Poisson geometry and first integrals of geostrophic equations. Physica D: Nonlinear Phenomena 237(14-17), 2072–2077 (2008)
  * [97] Kolmogoroff, A.: Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Mathematische Annalen 104(1), 415–458 (1931)
  * [98] Koszul, J.L.: Crochet de schouten-nijenhuis et cohomologie. Astérisque 137, 257–271 (1985)
  * [99] Law, K., Stuart, A., Zygalakis, K.: Data assimilation. Cham, Switzerland: Springer (2015)
  * [100] Law, K.J.: Proposals which speed up function-space MCMC. Journal of Computational and Applied Mathematics 262, 127–138 (2014)
  * [101] Lázaro-Camí, J.A., Ortega, J.P.: Stochastic hamiltonian dynamical systems. arXiv preprint math/0702787 (2007)
  * [102] Le, H., Lewis, A., Bharath, K., Fallaize, C.: A diffusion approach to stein’s method on riemannian manifolds. arXiv preprint arXiv:2003.11497 (2020)
  * [103] Lee, J.M.: Smooth manifolds. In: Introduction to Smooth Manifolds, pp. 1–31. Springer (2013)
  * [104] Leimkuhler, B., Patrick, G.W.: A symplectic integrator for Riemannian Manifolds. Journal of Nonlinear Science (1996). https://doi.org/https://doi.org/10.1007/BF02433475
  * [105] Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics. Cambridge University Press, cambirdge edn. (2004)
  * [106] Leimkuhler, B., Matthews, C.: Efficient molecular dynamics using geodesic integration and solvent–solute splitting. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472(2189), 20160138 (2016)
  * [107] Leimkuhler, B., Matthews, C., Stoltz, G.: The computation of averages from equilibrium and nonequilibrium langevin molecular dynamics. IMA Journal of Numerical Analysis 36(1), 13–79 (2016)
  * [108] Leimkuhler, B., Pouchon, T., Vlaar, T., Storkey, A.: Constraint-based regularization of neural networks. arXiv preprint arXiv:2006.10114 (2020)
  * [109] Lelièvre, T., Nier, F., Pavliotis, G.A.: Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion. Journal of Statistical Physics 152(2), 237–274 (2013)
  * [110] Lelièvre, T., Rousset, M., Stoltz, G.: Hybrid monte carlo methods for sampling probability measures on submanifolds. arXiv preprint arXiv:1807.02356 (2018)
  * [111] Lelièvre, T., Rousset, M., Stoltz, G.: Hybrid monte carlo methods for sampling probability measures on submanifolds. Numerische Mathematik 143(2), 379–421 (2019)
  * [112] Lelievre, T., Stoltz, G.: Partial differential equations and stochastic methods in molecular dynamics. Acta Numerica 25, 681–880 (2016)
  * [113] Li, C., Chen, C., Fan, K., Carin, L.: High-order stochastic gradient thermostats for bayesian learning of deep models. In: Thirtieth AAAI Conference on Artificial Intelligence (2016)
  * [114] Liu, C., Zhu, J.: Riemannian stein variational gradient descent for bayesian inference. In: Thirty-second aaai conference on artificial intelligence (2018)
  * [115] Liu, C., Zhu, J., Song, Y.: Stochastic gradient geodesic mcmc methods. In: Advances in neural information processing systems. pp. 3009–3017 (2016)
  * [116] Liu, Q., Lee, J., Jordan, M.: A kernelized Stein discrepancy for goodness-of-fit tests. In: Proceedings of the International Conference on Machine Learning. pp. 276–284 (2016)
  * [117] Livingstone, S., Girolami, M.: Information-Geometric Markov Chain Monte Carlo Methods Using Diffusions. Entropy 16(6), 3074–3102 (2014)
  * [118] Livingstone, S., Betancourt, M., Byrne, S., Girolami, M.: On the geometric ergodicity of Hamiltonian Monte Carlo. arXiv preprint arXiv:1601.08057 (2016)
  * [119] Lott, J.: Some geometric calculations on wasserstein space. arXiv preprint math/0612562 (2006)
  * [120] Ma, Y.A., Chatterji, N., Cheng, X., Flammarion, N., Bartlett, P., Jordan, M.I.: Is there an analog of nesterov acceleration for mcmc? arXiv preprint arXiv:1902.00996 (2019)
  * [121] Ma, Y.A., Chen, T., Fox, E.: A complete recipe for stochastic gradient mcmc. In: Advances in Neural Information Processing Systems. pp. 2917–2925 (2015)
  * [122] Majda, A., Wang, X., et al.: Linear response theory for statistical ensembles in complex systems with time-periodic forcing. Communications in Mathematical Sciences 8(1), 145–172 (2010)
  * [123] Majda, A.J., Franzke, C., Khouider, B.: An applied mathematics perspective on stochastic modelling for climate. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366(1875), 2427–2453 (2008)
  * [124] Majda, A.J., Timofeyev, I., Vanden Eijnden, E.: A mathematical framework for stochastic climate models. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences 54(8), 891–974 (2001)
  * [125] Malliavin, P., Dehen, D., Michel, D.: Géométrie différentielle stochastique, vol. 64. Montreal, Presses de l’universite de Montreal (1978)
  * [126] Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley Series in Probability and Statistics (1999)
  * [127] Marle, C.M.: The schouten-nijenhuis bracket and interior products. Journal of Geometry and Physics 23(3), 350 – 359 (1997). https://doi.org/https://doi.org/10.1016/S0393-0440(97)80009-5, http://www.sciencedirect.com/science/article/pii/S0393044097800095
  * [128] Masoliver, J., Garrido, L., Llosa, J.: Geometrical derivation of the intrinsic fokker-planck equation and its stationary distribution. Journal of statistical physics 46(1-2), 233–248 (1987)
  * [129] Materassi, M.: Entropy as a metric generator of dissipation in complete metriplectic systems. Entropy 18(8), 304 (2016)
  * [130] McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numerica 11, 341–434 (2002)
  * [131] Modin, K., Perlmutter, M., Marsland, S., McLachlan, R.: Geodesics on lie groups: Euler equations and totally geodesic subgroup (2010)
  * [132] Morrison, P.J.: A paradigm for joined hamiltonian and dissipative systems. Physica D: Nonlinear Phenomena 18(1-3), 410–419 (1986)
  * [133] Neal, R.M., et al.: Mcmc using hamiltonian dynamics. Handbook of markov chain monte carlo 2(11), 2 (2011)
  * [134] Nelson, E., et al.: The adjoint markoff process. Duke Mathematical Journal 25(4), 671–690 (1958)
  * [135] Nielsen, F.: An elementary introduction to information geometry. Entropy 22(10), 1100 (2020)
  * [136] Norris, J.R.: A complete differential formalism for stochastic calculus in manifolds. Séminaire de probabilités de Strasbourg 26, 189–209 (1992)
  * [137] Osada, H., et al.: Diffusion processes with generators of generalized divergence form. Journal of Mathematics of Kyoto University 27(4), 597–619 (1987)
  * [138] Ottobre, M.: Markov chain monte carlo and irreversibility. Reports on Mathematical Physics 77(3), 267–292 (2016)
  * [139] Ottobre, M., Pillai, N.S., Pinski, F.J., Stuart, A.M., et al.: A function space hmc algorithm with second order langevin diffusion limit. Bernoulli 22(1), 60–106 (2016)
  * [140] Pavliotis, G.A.: Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations, vol. 60. Springer (2014)
  * [141] Planck, V.: Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie. Sitzungberichte der (1917)
  * [142] Radivojević, T., Akhmatskaya, E.: Modified hamiltonian monte carlo for bayesian inference. Statistics and Computing pp. 1–28 (2019)
  * [143] Raginsky, M., Rakhlin, A., Telgarsky, M.: Non-convex learning via stochastic gradient langevin dynamics: a nonasymptotic analysis. arXiv preprint arXiv:1702.03849 (2017)
  * [144] Rapaport, D.C., Rapaport, D.C.R.: The art of molecular dynamics simulation. Cambridge university press (2004)
  * [145] Rey-Bellet, L., Spiliopoulos, K.: Irreversible langevin samplers and variance reduction: a large deviations approach. Nonlinearity 28(7), 2081 (2015)
  * [146] Risken, H.: Fokker-Planck equation. In: The Fokker-Planck Equation, pp. 63–95. Springer (1996)
  * [147] Roberts, G.O., Tweedie, R.L., et al.: Exponential convergence of langevin distributions and their discrete approximations. Bernoulli 2(4), 341–363 (1996)
  * [148] Rohm, R., Witten, E.: The antisymmetric tensor field in superstring theory. Annals of Physics 170(2), 454–489 (1986)
  * [149] Schwartz, L.: Geometrie differentielle du 2 ème ordre, semi-martingales et equations differentielles stochastiques sur une variete differentielle. In: Séminaire de Probabilités XVI, 1980/81 Supplément: Géométrie Différentielle Stochastique, pp. 1–148. Springer (1982)
  * [150] Shahbaba, B., Lomeli, L.M., Chen, T., Lan, S.: Deep markov chain monte carlo. arXiv preprint arXiv:1910.05692 (2019)
  * [151] Shi, J., Chen, T., Yuan, R., Yuan, B., Ao, P.: Relation of a new interpretation of stochastic differential equations to ito process. Journal of Statistical physics 148(3), 579–590 (2012)
  * [152] Sohl-Dickstein, J., Mudigonda, M., DeWeese, M.R.: Hamiltonian monte carlo without detailed balance. arXiv preprint arXiv:1409.5191 (2014)
  * [153] Stoltz, G., Rousset, M., et al.: Free energy computations: A mathematical perspective. World Scientific (2010)
  * [154] Sweet, C.R., Hampton, S.S., Skeel, R.D., Izaguirre, J.A.: A separable shadow hamiltonian hybrid monte carlo method. The Journal of chemical physics 131(17), 174106 (2009)
  * [155] Takaishi, T., De Forcrand, P.: Testing and tuning symplectic integrators for the hybrid monte carlo algorithm in lattice qcd. Physical Review E 73(3), 036706 (2006)
  * [156] Tuckerman, M.: Statistical mechanics: theory and molecular simulation. Oxford university press (2010)
  * [157] Volterra, V., et al.: Sur une genéralisation de la théorie des fonctions d’une variable imaginaire: Ier Mémoire. Acta Mathematica 12, 233–286 (1889)
  * [158] Weinstein, A.: Hamiltonian structure for drift waves and geostrophic flow. The Physics of Fluids 26(2), 388–390 (1983)
  * [159] Weinstein, A.: The modular automorphism group of a poisson manifold. Journal of Geometry and Physics 23(3-4), 379–394 (1997)
  * [160] Welling, M., Teh, Y.W.: Bayesian learning via stochastic gradient langevin dynamics. In: Proceedings of the 28th international conference on machine learning (ICML-11). pp. 681–688 (2011)
  * [161] Xifara, T., Sherlock, C., Livingstone, S., Byrne, S., Girolami, M.: Langevin diffusions and the metropolis-adjusted langevin algorithm. Statistics & Probability Letters 91, 14–19 (2014)
  * [162] Xu, P., Chen, J., Zou, D., Gu, Q.: Global convergence of langevin dynamics based algorithms for nonconvex optimization. In: Advances in Neural Information Processing Systems. pp. 3122–3133 (2018)
  * [163] Yin, L., Ao, P.: Existence and construction of dynamical potential in nonequilibrium processes without detailed balance. Journal of Physics A: Mathematical and General 39(27), 8593 (2006)
  * [164] Zappa, E., Holmes-Cerfon, M., Goodman, J.: Monte carlo on manifolds: sampling densities and integrating functions. Communications on Pure and Applied Mathematics 71(12), 2609–2647 (2018)