11institutetext: Department of Computer Science, University of Copenhagen
11email<EMAIL_ADDRESS>22institutetext: Department of
Biomedical Engineering, University of Utah
22email<EMAIL_ADDRESS>
# Bridge Simulation on Lie Groups and Homogeneous Spaces with Application to
Parameter Estimation

Mathias Højgaard Jensen 11 Lennard Hilgendorf 11 Sarang Joshi 22 Stefan Sommer
11

###### Abstract

We present three simulation schemes for simulating Brownian bridges on
complete and connected Lie groups and homogeneous spaces and use numerical
results of the guided processes in the Lie group $\mathrm{SO}(3)$ and on the
homogeneous spaces $\operatorname{SPD}(3)=\mathrm{GL}_{+}(3)/\mathrm{SO}(3)$
and $\mathbb{S}^{2}=\mathrm{SO}(3)/\mathrm{SO}(2)$ to evaluate our sampling
scheme. Brownian motions on Lie groups can be defined via the Laplace-Beltrami
of a left- (or right-)invariant Riemannian metric. Given i.i.d. Lie group-
valued samples on $\mathrm{SO}(3)$ drawn from a Brownian motion with unknown
Riemannian metric structure, the underlying Riemannian metric on
$\mathrm{SO}(3)$ is estimated using an iterative maximum likelihood (MLE)
method. Furthermore, the re-sampling technique is applied to yield estimates
of the heat kernel on the two-sphere considered as a homogeneous space.
Comparing this estimate to the truncated version of the closed-form expression
for the heat kernel on $\mathbb{S}^{2}$ serves as a proof of concept for the
validity of the sampling scheme on homogeneous spaces.

###### Keywords:

Brownian Motion, Brownian Bridge Simulation, Lie groups, Homogeneous Spaces,
Metric estimation

##  1 Introduction

In this paper, we consider discrete time observations on Lie groups and
homogeneous spaces considered as incomplete observations of a continuous time
sample path of a left- (or right-)invariant Brownian motion. In order to
handle the discrete time observations and infer properties of the unknown
underlying distribution, we derive bridge simulation schemes on Lie groups and
homogeneous spaces. To the best of our knowledge, this paper is the first to
describe a simulation technique for diffusion bridges in the context of Lie
groups.

Simulation of conditioned diffusion processes is a highly non-trivial problem.
A common issue is that the transition densities are non-tractable and thus
simulating directly from the true distribution is non-feasible. Several papers
have described diffusion bridge simulation methods; see, e.g., [3, 2, 4, 14,
15, 17] which describes various methods for simulation of multi-dimensional
conditioned diffusion processes and their usage for parameter estimation and
likelihood inference for stochastic differential equations. These papers
introduced a variety of different guiding drift terms which replaced the drift
term depending on the intractable transition density. To the best of our
knowledge, this paper is the first to describe a simulation technique for
diffusion bridges in the context of Lie groups.

The idea of the present paper is based on the method presented in the seminal
paper by Delyon and Hu [4]. In here, the authors exchanged the intractable
guiding drift term $\nabla_{x=X_{t}}\log p_{T-t}(x,v)$ in the stochastic
differential equation (SDE) for the conditioned diffusion with the guiding
drift term in the SDE for a Brownian bridge $(BB_{t})_{t\geq 0}$ conditioned
at $BB_{T}=v$.

Geometric statistics and probability on non-linear spaces is still a widely
unexplored domain. It can be mentioned as an example that a closed form
expression for the transition density of a Brownian motion is only known on a
limited amount of geometries. Examples of these geometries include: Euclidean
spaces (zero curvature), hyper-spheres (constant positive curvature), and
hyperbolic spaces (constant negative curvature). Thompson [20] showed how
guiding to the nearest point yielded an approximation of the transition
density (heat kernel) of a Brownian motion on Riemannian manifolds. More
generally, Thompson obtained an expression of the integrated heat kernel over
a submanifold by conditioning a Brownian motion to end up in the submanifold
at a fixed positive time.

In this paper, we present three simulation schemes for simulating diffusion
bridges on homogeneous spaces. The first scheme builds on the idea of Thompson
[20] by conditioning on a submanifold in the Lie group $G$ obtained as a fiber
over the point $v\in M=G/K$, for some closed subgroup $K\subseteq G$. The
second scheme assumes the homogeneous space has a discrete fiber $\Gamma$ and
therefore the fiber over $v\in M$, $\pi^{-1}(v)$, is discrete. Using the
$k$-nearest-points from the fiber $\pi^{-1}(v)$ to the initial point $x_{0}$,
we obtain a truncated guiding drift term convergence to a subset of
$\pi^{-1}(v)$. The last scheme assumes that the fiber is connected. Sampling
$k$-points in the fiber over $v$ a similar conditioning is obtained.

The paper is organized as follows. In Section 2, we describe some background
theory of Lie groups, Brownian motions, and Brownian bridges in Riemannian
manifolds. Section 3 presents the theory and results of bridge sampling in Lie
groups, while Section 4 introduce bridge sampling on various homogeneous
spaces. Numerical experiments on certain Lie groups and homogeneous spaces are
presented in 5. Section 5.1 shows in practice the simulation scheme in the Lie
group $\mathrm{SO}(3)$. A Brownian motion on any nice smooth manifold is
intimately connected with its endowed metric through its covariance structure
(see Section 2.4). Sampling data points from a Brownian motion with unknown
covariance, an estimate of the underlying unknown metric is obtained by means
of importance sampling. In Section 5.3, estimates for the heat kernel are
visualized on the two-sphere $\mathbb{S}^{2}$ considered as the homogeneous
space $\mathrm{SO}(3)/\mathrm{SO}(2)$. The estimate on $\mathbb{S}^{2}$ is
compared to the exact heat kernel on $\mathbb{S}^{2}$ in case of a bi-
invariant metric.

##  2 Notation and Background

We briefly describe the basics of simulating conditioned diffusion in
$\mathbb{R}^{n}$ as was developed in [4], before reviewing some theory on
conditioned diffusion on Riemannian manifolds.

###  2.1 Conditioned Diffusions in $\mathbb{R}^{n}$

Suppose a strong solution to an SDE of the form

$\displaystyle dx_{t}=b(t,x_{t})dt+\sigma(t,x_{t})dw_{t},$

where $b$ and $\sigma$ satisfies certain regularity conditions and where $w$
denote a $\mathbb{R}^{n}$-valued Brownian motion. In this case $x$ is Markov
and its transition density exist. Suppose we define the function

$\displaystyle h(t,x)=\frac{p_{T-t}(x_{t},v)}{p_{T}(x_{0},v)},$

for some $x_{0},v\in\mathbb{R}^{n}$, then it is easily derived that $h$ is a
martingale on $[0,T)$ with $h(0,x_{0})=1$ and Doob’s $h$-transform implies
that the SDE of the conditioned diffusion $x|x_{T}=v$ is given by

$\displaystyle dy_{t}=\tilde{b}(t,y_{t})dt+\sigma(t,y_{t})dw_{t}$

where $\tilde{b}(t,y)=b(t,y)+(\sigma\sigma^{T})(t,y)\nabla_{y}\log
p_{T-t}(y,v)$. In case that the transition density is intractable, simulation
from the exact distribution is in-feasible. Delyon and Hu [4] suggested
substituting the latter term in $\tilde{b}$ with a drift term of the form
$-(y_{t}-v)/(T-t)$, which equals the drift term in a Brownian bridge. The
guided process obtained by making the above substitution yields a conditioning
and one obtain

$\mathbb{E}[f(x)|x_{T}=v]=C\mathbb{E}[f(y)\varphi_{T}],$ (1)

where $\varphi_{T}$ is a likelihood function that is tractable and easy to
compute, and $y$ is the guided process. Below we generalize this method to
manifolds and homogeneous spaces.

###  2.2 General Lie Group Notation

Throughout, we let $G$ denote a connected Lie Group of dimension $d$, i.e., a
smooth manifold with a group structure such that the group operations $G\times
G\ni(x,y)\overset{\mu}{\mapsto}xy\in G$ and $G\ni
x\overset{\iota}{\mapsto}x^{-1}\in G$ are smooth maps. If $x\in G$, the left-
multiplication map, $L_{x}y$, defined by $y\mapsto\mu(x,y)$, is a
diffeomorphism from $G$ to itself. Similarly, the right-multiplication map
$R_{x}y$ defines a diffeomorphism from $G$ to itself by $y\mapsto\mu(y,x)$.
Let $dL_{x}\colon TG\rightarrow TG$ denote the pushforward map given by
$(dL_{x})_{y}\colon T_{y}G\rightarrow T_{xy}G$. A vector field $V$ on $G$ is
said to be left-invariant if $(dL_{x})_{y}V(y)=V(xy)$. The space of left-
invariant vector fields is linearly isomorphic to $T_{e}G$, the tangent space
at the identity element $e\in G$. By equipping the tangent space $T_{e}G$ with
the Lie bracket we can identify the Lie algebra $\mathfrak{g}$ with $T_{e}G$.
The group structure of $G$ makes it possible to define an action of $G$ on its
Lie algebra $\mathfrak{g}$. The conjugation map $C_{x}:=L_{x}\circ
R_{x}^{-1}\colon y\mapsto xyx^{-1}$, for $x\in G$, fixes the identity $e$. Its
pushforward map at $e$, $(dC_{x})_{e}$, is then a linear automorphism of
$\mathfrak{g}$. Define $\operatorname{\mathrm{Ad}}(x):=(dC_{x})_{e}$, then
$\operatorname{\mathrm{Ad}}\colon x\mapsto\operatorname{\mathrm{Ad}}(x)$ is
the adjoint representation of $G$ in $\mathfrak{g}$. The map
$G\times\mathfrak{g}\ni(x,v)\mapsto\operatorname{\mathrm{Ad}}(x)v\in\mathfrak{g}$
is the adjoint action of $G$ on $\mathfrak{g}$. We denote by
$\langle\cdot,\cdot\rangle$ a Riemannian metric on $G$. The metric is said to
be left-invariant if $\langle
u,v\rangle_{y}=\left\langle(dL_{x})_{y}u,(dL_{x})_{y}v\right\rangle_{L_{x}(y)}$,
for every $u,v\in T_{y}G$, i.e., the left-multiplication maps are isometries,
for every $x\in G$. In particular, we say that the metric is
$\operatorname{\mathrm{Ad}}(G)$-invariant if $\langle
u,v\rangle_{e}=\left\langle\operatorname{\mathrm{Ad}}(x)u,\operatorname{\mathrm{Ad}}(x)v\right\rangle_{e}$,
for every $u,v\in\mathfrak{g}$. Note that an
$\operatorname{\mathrm{Ad}}(G)$-invariant metric on $G$ is equivalent to a bi-
invariant (left- and right-invariant) inner product on $\mathfrak{g}$. The
differential of the $\operatorname{\mathrm{Ad}}$ map at the identity yields a
linear map
$\operatorname{\mathrm{ad}}(x)=d/dt\operatorname{\mathrm{Ad}}(\exp(tx))|_{0}$.
This linear map defines the Lie bracket
$[v,w]=\operatorname{\mathrm{ad}}(v)w$, $v,w\in\mathfrak{g}$.

A one-parameter subgroup of $G$ is a continuous (Lie) group homomorphism
$\gamma\colon(\mathbb{R},+)\rightarrow G$. The Lie group exponential map
$\exp\colon\mathfrak{g}\rightarrow G$ is defined as $\exp(v)=\gamma_{v}(1)$,
for $v\in\mathfrak{g}$, where $\gamma_{v}$ is the unique one-parameter
subgroup of $G$ whose tangent vector at $e$ is $v$. For matrix Lie groups the
exponential map has the particular form:
$\exp(A)=\sum_{k=0}^{\infty}A^{k}/k!$, for a square matrix $A$. The resulting
matrix $\exp(A)$ is an invertible matrix. Given an invertible matrix $B$, if
there exist a square matrix $A$ such that $B=\exp(A)$, then $A$ is said to be
the logarithm of $B$. In general, the logarithm might not exist and if it does
it may fail to be unique. However, the matrix exponential and logarithms can
be computed numerically efficient (see [16, Chapter 5] and references
therein). Note that we can always find a domain in the Lie algebra
$\mathfrak{g}$ where the Lie group exponential map is a diffeomorphism.
Therefore, in a neighborhood sufficiently close to the identity the Lie group
logarithm exist and is unique. By means of left-translation (or right-
translation), the Lie group exponential map can be extended to a map
$\exp_{g}\colon T_{g}G\rightarrow G$, for all $g\in G$, defined by
$\exp_{g}(v)=g\exp(dL_{g^{-1}}v)$. Similarly, the Lie group logarithm at $g$
becomes $\log_{g}(v)=dL_{g}\log(g^{-1}v)$.

######  Example 1.

Some common examples of Lie groups include; the $n$-dimensional Euclidean
space $(\mathbb{R}^{n},+)$ with the additive group structure,
$(\mathbb{R}_{+},\cdot)$ the positive real line with a multiplicative group
structure, the space of invertible real matrices $\mathrm{GL}(n)$ equipped
with a multiplication of matrices forms a Lie group, and the rotation group
$\mathrm{SO}(n)$, consisting of real orthogonal matrices with determinant one,
equipped with matrix multiplication is a Lie group.

###  2.3 Homogeneous Spaces

A homogeneous space is a special type of quotient manifold arising as a smooth
manifold endowed with a transitive smooth action by a Lie group $G$. The
homogeneous space is called a $G$-homogeneous space to indicate the Lie group
action on it. In fact, all $G$-homogeneous spaces arise as a quotient manifold
$G/H$, for some closed subgroup $H\subseteq G$. The fact that $H$ is a closed
subgroup of the Lie group $G$ makes $H$ into a Lie group. Any homogeneous
space is diffeomorphic to the quotient space $G/G_{x}$, where $G_{x}$ is the
stabilizer for the point $x$. The dimension of the $G$-homogeneous space is
equal to $\dim G-\dim H$ the quotient map $\pi\colon G\rightarrow G/H$ is a
smooth submersion, i.e., the differential of $\pi$ is surjective at every
point. We assume throughout that $G$ acts on itself by left-multiplication.

######  Example 2.

The rotation group $\mathrm{SO}(n)$ acts transitively on $\mathbb{S}^{n-1}$,
thus $\mathbb{S}^{n-1}$ is a $\mathrm{SO}(n)$-homogeneous space. Consider for
a moment the north-pole. Since any rotations that fix the north-pole are
rotations in the $xy$-plane, we see that the stabilizer or isotropy group is
the rotation group $\mathrm{SO}(n-1)$, thus
$\mathbb{S}^{n-1}=\mathrm{SO}(n)/\mathrm{SO}(n-1)$.

The set of three-by-three invertible matrices with positive determinant
$\mathrm{GL}_{+}(3)$ acts on the left on the set of covariance matrices, i.e.,
the symmetric positive definite matrices $\operatorname{SPD}(3)$. The isotropy
group is the rotation group $\mathrm{SO}(3)$ and thus
$\operatorname{SPD}(3)=\mathrm{GL}_{+}(3)/\mathrm{SO}(3)$.

A special type of homogeneous space arise when the subgroup is a discrete
subgroup of $G$. Such homogeneous spaces are typically denoted $G/\Gamma$. For
example, the space $\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$ defines the
$n$-torus as a homogeneous space.

###  2.4 Brownian Motion

The Brownian motion is a classical diffusion process defined in
$\mathbb{R}^{n}$ via the Laplace operator. The Brownian motion on Riemannian
manifolds and Lie groups can similarly be defined via the generalized
Laplacian or Laplace-Beltrami operator. This is described in the sections
below.

####  2.4.1 On Riemannian Manifolds

Endowing a smooth manifold $M$ with a Riemannian metric, $g$, allows us to
define the Laplace-Beltrami operator. This operator is the generalization of
the Euclidean Laplacian operator to manifolds. The Laplace-Beltrami operator
is defined as the gradient’s divergence, $\Delta_{M}f=$ div grad $f$. In terms
of local coordinates $(x_{1},\dots,x_{d})$ the expression for the Laplace-
Beltrami operator becomes

$\displaystyle\Delta_{M}f=\det(g)^{-1/2}\left(\frac{\partial}{\partial
x_{j}}g^{ji}\det(g)^{1/2}\frac{\partial}{\partial x_{i}}\right)f,$ (2)

where $\det(g)$ denotes the determinant of the Riemannian metric $g$ and
$g^{ij}$ are the coefficients of the inverse of $g$. An application of the
product rule implies that (2) can be rewritten as

$\displaystyle\Delta_{M}f$ $\displaystyle=a^{ij}\frac{\partial}{\partial
x_{i}}\frac{\partial}{\partial x_{j}}f+b^{j}\frac{\partial}{\partial x_{j}}f,$
(3)

where $a^{ij}=g^{ij}$, $b^{k}=-g^{ij}\Gamma^{k}_{ij}$, and $\Gamma$ denote the
Christoffel symbols related to the Riemannian metric. This diffusion operator
defines a Brownian motion on the $G$, valid up to its first exit time of the
local coordinate chart, i.e.,

$N^{f}(X_{t})=f(X_{t})-f(X_{0})-\frac{1}{2}\int_{0}^{t}\Delta_{M}f(X_{s})ds,$
(4)

is a local martingale, for all smooth functions $f$ on $G$. Equivalently, the
stochastic differential equation (SDE) for $X$ in terms of local coordinates
is

$dX_{t}^{k}=-\frac{1}{2}g^{ij}(X_{t})\Gamma_{ij}^{k}(X_{t})dt+\sigma^{k}_{j}(X_{t})dB_{t}^{j},$
(5)

for $t<\tau$, where $\tau$ is the explosion time of $X$ and
$\sigma=\sqrt{g^{-1}}$ the matrix square root of $g^{-1}$.

####  2.4.2 On Lie Groups

In the case of the Lie group $G$, the identification of the space of left-
invariant vector fields with the Lie algebra $\mathfrak{g}$ allows for a
global description of $\Delta_{G}$. Indeed, let $\\{v_{1},\dots v_{d}\\}$ be
an orthonormal basis of $T_{e}G$. Then $V_{i}(x)=(dL_{x})_{e}v_{i}$ defines
left-invariant vector fields on $G$ and the Laplace-Beltrami operator can be
written as (cf. [12, Proposition 2.5])

$\Delta_{G}f(e)=\sum_{i=1}^{d}V_{i}^{2}f(e)-V_{0}f(e),$

where $V_{0}=\sum_{i,j=1}^{d}C_{ij}^{j}V_{j}$ and $C^{k}_{ij}$ denote the
structure coefficients given by

$[V_{i},V_{j}]=C^{k}_{ij}V_{k}.$ (6)

By the left-invariance, the formula for the Laplace-Beltrami operator holds
globally, i.e., $\Delta_{G}f(a)=\Delta_{G}f\circ
L_{a}(e)=\left(dL_{a}\right)_{e}\Delta_{G}f(e)$. The corresponding stochastic
differential equation (SDE) for the Brownian motion on $G$, in terms of left-
invariant vector fields, then becomes

$dX_{t}=-\frac{1}{2}V_{0}(X_{t})dt+V_{i}(X_{t})\circ dB^{i}_{t},\qquad
X_{0}=e,$ (7)

where $\circ$ denotes integration in the Stratonovich sense. By [12,
Proposition 2.6], if the inner product is $\operatorname{\mathrm{Ad}}(G)$
invariant, then $V_{0}=0$. The solution of (7) is conservative or non-
explosive and is called the left-Brownian motion on $G$ (see [18] and
references therein).

###  2.5 Riemannian Bridges

In this section, we briefly review some classical facts on Brownian bridges on
Riemannian manifolds. As Lie groups themselves can be equipped with a
Riemannian manifold, the theory carries over mutatis mutandis. However, Lie
groups’ group structure allows the notion of left-invariant (resp. right-
invariant) vector fields. The identification of the Lie algebra with the
vector space of left-invariant vector fields makes Lie groups parallelizable.
The existence of smooth non-vanishing vector fields allows for constructing
semimartingales directly on the Lie groups, since the stochastic parallel
displacement is ensured by the left-invariant (resp. right-invariant) vector
fields.

####  2.5.1 Brownian Bridges

Let $\mathbb{P}^{t}_{x}=\mathbb{P}_{x}|_{\mathcal{F}_{t}}$ be the measure of a
Riemannian Brownian motion, $X_{t}$, at some time $t$ started at point $x$.
Suppose $p$ denotes the transition density of the Riemannian Brownian motion.
In that case, $d\mathbb{P}^{t}_{x}=p_{t}(x,y)d\operatorname{Vol}(y)$ describes
the measure of the Riemannian Brownian motion, where $d\operatorname{Vol}(y)$
is the Riemannian volume measure. Conditioning the Riemannian Brownian motion
to hit some point $v$ at time $T>0$ results in a Riemannian Brownian bridge.
Here, $\mathbb{P}_{x,v}^{T}$ denotes the corresponding probability measure.
The two measures are absolutely continuous (equivalent) over the time interval
$[0,T)$, however mutually singular at time $t=T$. This is an obvious
consequence of the fact that $\mathbb{P}_{x}(X_{T}=v)=0$, whereas
$\mathbb{P}_{x,v}^{T}(X_{T}=v)=1$. The corresponding Radon-Nikodym derivative
is given by

$\frac{d\mathbb{P}_{x,v}^{T}}{d\mathbb{P}_{x}}\big{|}_{\mathcal{F}_{s}}=\frac{p_{T-s}(X_{s},v)}{p_{T}(x,v)}\qquad\text{for
}0\leq s<T,$ (8)

which is a martingale for $s<T$. The Radon-Nikodym derivative defines the
density for the change of measure and provides the basis for the description
of Brownian bridges. In particular, it provides the conditional expectation
defined by

$\mathbb{E}[F(X_{t})|X_{T}=v]=\frac{\mathbb{E}[p_{T-t}(X_{t},v)F(X_{t})]}{p_{T}(x,v)},$
(9)

for any bounded and $\mathcal{F}_{s}$-measurable random variable $F(X_{s})$.
(In the case of an empty cut-locus or rather when the exponential map is a
covering map, and the underlying manifold is connected, then every point $p$
is a pole. In this case, Truman and Elworthy’s semi-classical Brownian bridges
apply.

As described in Hsu [8], the Brownian bridge on a Riemannian manifold, $M$, is
a nonhomogeneous diffusion on $M$ with an infinitesimal generator

$\displaystyle\mathcal{L}_{s}f(z)=\frac{t}{2}\Delta_{M}f(z)+t\nabla_{z}\log
p_{t(1-s)}(z,v)\cdot\nabla f(z),$

where $\Delta_{M}$ denotes the Laplacian on $M$. This infinitesimal generator
yields an SDE in the frame bundle, $\mathcal{FM}$, of the lifted $M$-valued
Brownian bridge, $X_{t}=\pi(U_{t})$, in terms the horizontal vector fields
$(H_{i})$ given by

$dU_{t}=H_{i}(U_{t})\circ\left(dB^{i}_{t}+\left(U_{t}^{-1}\left(\pi_{*}\left(\nabla_{u|u=U_{t}}\log\tilde{p}_{T-t}(u,v)\right)\right)\right)^{i}dt\right),\qquad
U_{0}=u_{0},$ (10)

where $\tilde{p}_{t}(u,v)=p_{t}(\pi(u),v)$ denotes the lift of the transition
density, $B$ an $\mathbb{R}^{d}$-valued Brownian motion, and $\pi_{*}\colon
T\mathcal{FM}\rightarrow TM$ is the pushforward of the canonical projection
$\pi\colon\mathcal{FM}\rightarrow M$. Essentially, the stochastic development
approach in the Lie group setting is redundant. The left invariant vector-
fields permits a notion of stochastic parallel transport independently of the
frame bundle construction.

####  2.5.2 Bridges to Submanifolds

A generalization of Riemannian Brownian bridges can be found in Thompson [20].
There, the author introduces Brownian bridges to submanifolds by considering
the transition density on a Riemannian manifold $M$ defined by

$p_{t}(x,N):=\int_{N}p_{t}(x,y)d\operatorname{Vol}_{N}(y),$ (11)

where $N\subset M$ is a submanifold of $M$ and $\operatorname{Vol}_{N}$
denotes the volume measure on $N$. The author terms these processes as Fermi
bridges, having infinitesimal generator given by

$\tfrac{1}{2}\Delta-\tfrac{r_{N}}{T-t}\tfrac{\partial}{\partial r_{N}},$ (12)

where $r_{N}(\cdot):=d(\cdot,N)=\inf_{y\in N}d(\cdot,y)$ and
$\tfrac{\partial}{\partial r_{N}}=\nabla d(\cdot,N)$. The resulting
conditional expectation becomes

$\mathbb{E}[F(X_{t})|X_{T}\in
N]=\frac{\mathbb{E}[p_{T-t}(X_{t},N)F(X_{t})]}{p_{T}(x,N)},$ (13)

which holds for all bounded $\mathcal{F}_{t}$-measurable random variables
$F(X_{t})$.

##  3 Simulation of Bridges on Lie Groups

In this section, we consider the task of simulating (7) conditioned to hit
$v\in G$, at time $T>0$. The potentially intractable transition density for
the solution of (7) inhibits simulation directly from (10). Instead, we
propose to add a guiding term mimicking that of Delyon and Hu [4], i.e., the
guiding term becomes the gradient of the distance to $v$ divided by the time
to arrival. The SDE for the guided diffusion becomes

$dY_{t}=-\frac{1}{2}V_{0}(Y_{t})dt+V_{i}(Y_{t})\circ\left(dB^{i}_{t}-\frac{\left(\nabla_{y|_{y=Y_{t}}}d(y,v)^{2}\right)^{i}}{2(T-t)}dt\right),\qquad
Y_{0}=e,$ (14)

where $d(\cdot,v)$ denotes the Riemannian distance to $v$. Note that we can
always, for convenience, take the initial value to be the identity $e$.
Equation (14) can equivalently be written as

$dY_{t}=-\frac{1}{2}V_{0}(Y_{t})dt+V_{i}(Y_{t})\circ\left(dB^{i}_{t}-\frac{\operatorname{Log}_{Y_{t}}(v)^{i}}{T-t}dt\right),\qquad
Y_{0}=e,$

where $\operatorname{Log}_{p}$ is the inverse of the Riemannian exponential
map $\operatorname{Exp}_{p}$. Numerical computations of the Lie group
exponential map is often faster than computing the Riemannian exponential map
(see [16] and references therein). Therefore, by a change of measures argument
the equation above can be expressed in terms of the inverse of the Lie group
exponential (this process is denoted $Y$ as well)

$dY_{t}=-\frac{1}{2}V_{0}(Y_{t})dt+V_{i}(Y_{t})\circ\left(d\bar{B}^{i}_{t}-\tau(Y_{t})\frac{\log_{Y_{t}}(v)^{i}}{T-t}dt-\left(1-\tau(Y_{t})\right)\frac{\operatorname{Log}_{Y_{t}}(v)^{i}}{T-t}dt\right)$
(15)

$Y_{0}=e$, where $\bar{B}$ is a Brownian motion under a new measure, say
$\bar{\mathbb{P}}$, and $\tau(y)$ is a smooth bump function defined in a
neighborhood of $v$ where the Lie group logarithm exists and is unique. The
measure $\bar{\mathbb{P}}$ can explicitly be expressed as

$\displaystyle\frac{d\bar{\mathbb{P}}}{d\mathbb{P}}|_{\mathcal{F}_{t}}=\exp\bigg{[}-$
$\displaystyle\int_{0}^{t}\bigg{\langle}\frac{\tau(Y_{s})\left(\log_{Y_{s}}(v)-\operatorname{Log}_{Y_{s}}(v)\right)}{T-s},V(Y_{s})dB_{s}\bigg{\rangle}$
$\displaystyle-\frac{1}{2}$
$\displaystyle\int_{0}^{t}\frac{\lVert\tau(Y_{s})\rVert^{2}\lVert\log_{Y_{s}}(v)-\operatorname{Log}_{Y_{s}}(v)\rVert^{2}}{(T-s)^{2}}ds\bigg{]},$
(16)

where $\mathbb{P}$ denotes the law of the SDE in (14). A bi-invariant metric
on the Lie group implies that the Riemannian exponential map and the Lie group
exponential map coincide, and thus the Radon-Nikodym above is identically one.
This justifies keeping the $Y$ notation.

Situations where a bi-invariant metric exists, however, are rare. A sufficient
condition for the existence of a bi-invariant metric is that the Lie group is
compact. In such cases, we can always choose the bi-invariant metric and work
with the Lie group exponential and logarithmic map. In the non-compact
situation, we restrict ourselves to neighborhoods of $v\in G$ and use the
group logarithmic map inside this neighborhood while using the Riemannian
logarithm outside.

The result below shows the validity of working with the group logarithm in a
small neighborhood. For $v\in\mathfrak{g}$ we define $\tilde{v}=(dL)_{e}v$ as
the corresponding left-invariant vector field. Let
$\alpha\colon\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ be a bi-
linear operator which uniquely defines the connection at the identity:
$\alpha(v,w)=\nabla_{\tilde{v}}\tilde{w}|_{e}$. Similar to [16, Chapter 5], we
define $\operatorname{\mathrm{ad}}^{*}$ as the metric adjoint of the adjoint
operator $\operatorname{\mathrm{ad}}$:
$\langle\operatorname{\mathrm{ad}}^{*}(\tilde{v},\tilde{w}),\tilde{z}\rangle=\langle[\tilde{w},\tilde{z}],\tilde{v}\rangle$.

######  Proposition 1.

The measures in (3) are equivalent, i.e., the Radon-Nikodym derivative is a
martingale.

###### Proof.

If $G$ is compact, there exists a bi-invariant metric such that the Riemannian
logarithmic map and the Lie group logarithmic map coincide (see [16, Chapter
5]).

If $G$ is not compact, endowing the Lie group $G$ with a left-invariant
metric, the bi-linear form $\alpha$ characterizing the Levi-Civita connection
of the left-invariant metric in the Lie algebra is given by

$\alpha(v,w)=\frac{1}{2}\left([v,w]-\operatorname{\mathrm{ad}}^{*}(v,w)-\operatorname{\mathrm{ad}}^{*}(w,v)\right),$

where
$\operatorname{\mathrm{ad}}^{*}(v,w):=\operatorname{\mathrm{ad}}^{*}(\tilde{v},\tilde{w})|_{e}$.
Whenever $\operatorname{\mathrm{ad}}^{*}(v,v)=0$, the one-parameter subgroups
$\gamma_{v}(t)$, passing through the identity with tangent vector $v$, are in
fact geodesics wrt. the left-invariant Riemannian metric. In case the Lie
group log equals such $v$’s the measures in (3) are equivalent. In such case
the Lie group log belongs to the center of $\mathfrak{g}$. This for example
holds if the group is abelian.

If $G$ is neither compact nor abelian, we chose a compact neighborhood of
$v\in G$ such that the group log is well defined on this compact neighborhood.
As the neighborhood is compact there exists constants $c$ and $C$ such that
$c\lVert\operatorname{Log}_{y}(v)\rVert\leq\lVert\log_{y}(v)\rVert\leq
C\lVert\operatorname{Log}_{y}(v)\rVert$. The triangle inequality then imply

$\lVert\tau(Y_{s})\rVert^{2}\lVert\log_{Y_{s}}(v)-\operatorname{Log}_{Y_{s}}(v)\rVert^{2}\leq\left(1+3C\right)\lVert\operatorname{Log}_{y}(v)\rVert^{2},$

which in conjunction with Novikov’s condition yields the result. ∎

####  3.0.1 Radial Process

We denote by $r_{v}(\cdot):=d(\cdot,v)$ the radial process. Due to the
singularities of the radial process on $\operatorname{Cut}(v)\cup\\{v\\}$, the
usual Itô’s formula only applies on subsets away from the cut-locus. The
extension beyond the cut-locus of a Brownian motion’s radial process was due
to Kendall [10]. Barden and Le [1, 11] generalized the result to
$M$-semimartingales. The radial process of the Brownian motion (7) is given by

$r_{v}(X_{t})=r_{v}(X_{0})^{2}+\int_{0}^{t}\left\langle\nabla
r_{v}(X_{s}),V(X_{s})dB_{s}\right\rangle+\frac{1}{2}\int_{0}^{t}\Delta_{G}r_{v}(X_{s})ds-L^{v}_{s}(X),$
(17)

where $L^{v}$ is the geometric local time of the cut-locus
$\operatorname{Cut}(v)$, which is non-decreasing continuous random functional
increasing only when $X$ is in $\operatorname{Cut}(v)$ (see [1, 10, 11]). Let
$W_{t}:=\int_{0}^{t}\left\langle\frac{\partial}{\partial{r}},V_{i}(X_{s})\right\rangle
dB^{i}_{s}$, which is the local-martingale part in the above equation. The
quadratic variation of $W_{t}$ satisfies $d[W,W]_{t}=dt,$ by the
orthonormality of $\\{V_{1},\dots,V_{d}\\}$, thus $W_{t}$ is a Brownian motion
by Levy’s characterization theorem. From the stochastic integration by parts
formula and (17), the squared radial process of $X$ satisfies

$r_{v}(X_{t})^{2}=r_{v}(X_{0})^{2}+2\int_{0}^{t}r_{v}(X_{s})dW_{s}+\int_{0}^{t}r_{v}(X_{s})\Delta_{G}r_{v}(X_{s})ds-2\int_{0}^{t}r(X_{s})dL^{v}_{s},$
(18)

where $dL^{v}_{s}$ is the random measure associated to $L^{v}_{s}(X)$.

Similarly, we obtain an expression for the squared radial process of $Y$.
Using the shorthand notation $r_{t}:=r_{v}(Y_{t})$ the radial process then
becomes

$r_{t}^{2}=r_{0}^{2}+2\int_{0}^{t}r_{s}dW_{s}+\int_{0}^{t}\frac{1}{2}\Delta_{G}r_{s}^{2}ds-\int_{0}^{t}\frac{r_{s}^{2}}{T-s}ds-2\int_{0}^{t}r_{s}dL^{v}_{s}.$
(19)

Imposing a growth condition on the radial process yields an $L^{2}$-bound on
the radial process of the guided diffusion, [20]. So assume there exist
constants $\nu\geq 1$ and $\lambda\in\mathbb{R}$ such that
$\tfrac{1}{2}\Delta_{G}r_{v}^{2}\leq\nu+\lambda r_{v}^{2}$ on
$D\backslash\operatorname{Cut}(v)$, for every regular domain $D\subseteq G$.
Then (19) satisfies

$\mathbb{E}[1_{t<\tau_{D}}r_{v}(Y_{t})^{2}]\leq\left(r^{2}_{v}(e)+\nu
t\left(\frac{t}{T-t}\right)\right)\left(\frac{T-t}{t}\right)^{2}e^{\lambda
t},$ (20)

where $\tau_{D}$ is the first exit time of $Y$ from the domain $D$.

####  3.0.2 Girsanov Change of Measure

Let $B$ be the Brownian motion in $\mathbb{R}^{d}$ defined on the filtered
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{s}),\mathbb{P})$ and $X$
the solution of (7). The process $\tfrac{\nabla r_{v}(X_{t})^{2}}{2(T-t)}$ is
an adapted process. As $X$ is non-explosive, we see that

$\int_{0}^{t}\left\lVert\frac{\nabla
r_{v}(X_{s})^{2}}{2(T-s)}\right\rVert^{2}ds=\int_{0}^{t}\frac{r_{v}(X_{s})^{2}}{(T-s)^{2}}ds\leq
C,$ (21)

for every $0\leq t<T$, almost surely, and for some fixed constant $C>0$.
Define a new measure $\mathbb{Q}$ by

$Z_{t}:=\frac{d\mathbb{Q}}{d\mathbb{P}}\bigg{|}_{\mathcal{F}_{t}}(X)=\exp\left[-\int_{0}^{t}\left\langle\frac{\nabla
r_{v}(X_{s})^{2}}{2(T-s)},V(X_{t})dB_{s}\right\rangle-\frac{1}{2}\int_{0}^{t}\frac{r_{v}(X_{s})^{2}}{(T-s)^{2}}ds\right].$
(22)

From (21), the process $Z_{t}$ is a martingale, for $t\in[0,T)$, and
$\mathbb{Q}_{t}$ defines a probability measure on each $\mathcal{F}_{t}$
absolutely continuous with respect to $\mathbb{P}$. By Girsanov’s theorem (see
e.g. [7, Theorem 8.1.2]), we get a new process $b_{s}$ which is a Brownian
motion under the probability measure $\mathbb{Q}$. Moreover, under the
probability $\mathbb{Q}$, equation (7) becomes

$dY_{t}=-\frac{1}{2}V_{0}(Y_{t})dt+V_{i}(Y_{t})\circ\left(db^{i}_{t}-\frac{r_{v}(Y_{t})}{T-t}\left(\frac{\partial}{\partial{r}}\right)_{v}^{i}dt\right),$
(23)

where $\left(\frac{\partial}{\partial{r}}\right)^{i}$ is the $i$’th component
of the unit radial vector field in the direction of $v$. The squared radial
vector field is smooth away from $\operatorname{Cut}(v)$ and thus we set it to
zero on $\operatorname{Cut}(v)$. Away from $\operatorname{Cut}(v)$, the
squared radial vector field is $2\operatorname{Log}_{v}$, which is the inverse
exponential at $v$. The added drift term acts as a guiding term, which pulls
the process towards $v$ at time $T>0$.

From (22), we see that $\mathbb{E}[f(Y_{t})]=\mathbb{E}[f(X_{t})Z_{t}]$. Using
(18) and the identity
$\Delta_{G}r_{v}=\frac{d-1}{r_{v}}+\frac{\partial}{\partial{r_{v}}}\log\Theta_{v}$
(see e.g. [19]), we equivalently write
$\mathbb{E}[f(Y_{t})\varphi_{t}]=\mathbb{E}[f(X_{t})\psi_{t}]$, with

$\psi_{t}:=\exp\left[\frac{-r(X_{t})^{2}}{2(T-t)}\right]\qquad\varphi_{t,v}:=\exp\left[\int_{0}^{t}\frac{r_{v}(Y_{s})^{2}}{T-s}\left(dA^{v}_{s}+dL^{v}_{s}\right)\right],$
(24)

where
$dA^{v}_{s}=\frac{\partial}{\partial{r_{v}}}\log\Theta_{v}^{-1/2}(Y_{s})ds$ is
a random measure supported on $G\backslash\operatorname{Cut}(v)$ and
$\Theta_{v}$ is the Jacobian determinant of $\operatorname{Exp}_{v}$.

####  3.0.3 Delyon and Hu in Lie Groups

This section generalizes the result of Delyon and Hu [4, Theorem 5] to the Lie
group setting. It is possible to modify the result to incorporate a
generalization of [4, Theorem 6]. The results in the remaining part of this
section are modified from the Riemmanian setting in[9]to the Lie group setting
presented here.

######  Theorem 3.1.

Let $X$ be the solution of (7). The SDE (14) yields a strong solution on
$[0,T)$ and satisfies $\lim_{t\uparrow T}Y_{t}=v$ almost surely. Moreover, the
conditional expectation of $X$ given $X_{T}=v$ is

$\mathbb{E}[f(X)|X_{T}=v]=C\mathbb{E}\left[f(Y)\varphi_{T,v}\right],$ (25)

for every $\mathcal{F}_{t}$-measurable non-negative function $f$ on $G$,
$t<T$, where $\varphi_{t}$ is given in (24).

###### Proof.

The result is a consequence of the change of measure together with Lemma 1,
Lemma 2, and Lemma 3. ∎

######  Lemma 1.

The solution of SDE (14) satisfies $\lim_{t\rightarrow T}Y_{t}=v$ almost
surely.

###### Proof.

Let $\\{D_{n}\\}_{n=1}^{\infty}$ be an exhaustion of $G$, that is, the
sequence consists of open, relatively compact subsets of $M$ such that
$\bar{D}_{n}\subseteq D_{n+1}$ and $G=\bigcup_{n=1}^{\infty}D_{n}$.
Furthermore, let $\tau_{D_{n}}$ denote the first exit time of $Y$ from
$D_{n}$, then from (20) we have that the sequence
$\bigl{(}\mathbb{E}[1_{\\{t<\tau_{D_{n}}\\}}r_{v}^{2}(Y_{t})]\bigr{)}_{n=1}^{\infty}$
is non-decreasing and bounded, hence from the monotone convergence theorem, it
has a limit which is bounded by the right-hand side of (20). Applying Jensen’s
inequality to the left-hand side of (20)

$\mathbb{E}[r_{v}(Y_{t})]\leq\left(r^{2}_{v}(e)+\nu
t\left(\frac{t}{T-t}\right)\right)^{\frac{1}{2}}\left(\frac{T-t}{t}\right)e^{\frac{\lambda
t}{2}}.$

Since obviously
$\mathbb{E}[r_{v}(Y_{T})]=r_{v}(Y_{T})\mathbb{Q}(r_{v}(Y_{T})\neq 0)$, by
Fatou’s lemma

$\mathbb{E}[r_{v}(Y_{T})]\leq\liminf_{t\rightarrow T}\mathbb{E}[r(Y_{t})]=0,$

we conclude that $r(Y_{t})\rightarrow 0$, $\mathbb{Q}$-almost surely. ∎

######  Lemma 2.

Let $0<t_{1}<t_{2}<\dots<t_{N}<T$ and $h$ be a continuous bounded function
function on $G^{N}$. With $\psi_{t}$ as in (24), then

$\lim_{t\rightarrow
T}\frac{\mathbb{E}\left[h\left(X_{t_{1}},X_{t_{2}},\dots,X_{t_{N}}\right)\psi_{t}\right]}{\mathbb{E}[\psi_{t}]}=\mathbb{E}\left[h\left(X_{t_{1}},X_{t_{2}},\dots,X_{t_{N}}\right)|X_{T}=v\right].$
(26)

###### Proof.

The proof is similar to that of [4, Lemma 7]. Let $(U,\phi)$ be a normal chart
centered at $v\in G$. First, since the cut locus of any complete connected
manifold has (volume) measure zero, we can integrate indifferently in any
normal chart. For any $t\in(t_{N},T)$ we have

$\mathbb{E}[h(x_{t_{1}},...,x_{t_{N}})\psi_{t}]=\int_{G}\Phi_{h}(t,z)e^{-\frac{r_{v}(z)^{2}}{2(T-t)}}d\operatorname{Vol}(z)$
(27)

where $d\operatorname{Vol}(z)=\sqrt{\det(A(z))}dz$ denotes the volume measure
on $G$, $dz$ the Lebesgue measure, and $A$ the metric tensor. Moreover,

$\Phi_{h}(t,z)=\int_{G^{N}}h(z_{1},...,z_{N})p_{t_{1}}(u,z_{1})\cdot\cdot\cdot
p_{t-{t_{N}}}(z_{N},z)d\operatorname{Vol}(z_{1})\cdot\cdot\cdot
d\operatorname{Vol}(z_{N}),$

and of course $\Phi_{1}(t,z)=p_{t}(e,z)$. Using the normal chart and applying
the change of variable $x=(T-t)^{1/2}y$ we get

$(T-t)^{-\frac{d}{2}}\mathbb{E}[h(x_{t_{1}},...,x_{t_{N}})\psi_{t}]\overset{t\rightarrow
T}{\rightarrow}\Phi_{h}(T,v)\det(A(v))^{\tfrac{d}{2}}\int_{\phi(G)}e^{-\frac{r_{v}(\phi^{-1}(y))^{2}}{2}}dy.$

The conclusion follows from Bayes’ formula. ∎

On simply connected spaces, we have the following $L_{1}$-convergence of
$\varphi$.

######  Lemma 3.

Suppose $G$ is simply connected. With $\varphi_{t,v}$ as defined above then
$\varphi_{t,v}\overset{L_{1}}{\rightarrow}\varphi_{T,v}$.

###### Proof.

Note that for each $t\in[0,T)$ we have
$\mathbb{E}^{\mathbb{Q}}[\varphi_{t}]<\infty$ as well as
$\varphi_{t}\overset{}{\rightarrow}\varphi_{T}$ almost surely by Lemma 1. The
result then follows from the uniform integrability of
$\left\\{\varphi_{t}\colon t\in[0,T)\right\\}$, which can be found in Appendix
C.2 in [19]. ∎

##  4 Simulation of Bridges in Homogeneous Spaces

Consider the homogeneous space $M=G/K$, where $K$ is a Lie subgroup of the Lie
group $G$ and let $\pi\colon G\rightarrow M$ denote the canonical projection.
Suppose that $G$ acts on $M$ on the left and that $g_{t}$ is a process in $G$.
As described in Liao [12], we obtain an induced process in $M$ induced by the
process $g_{t}$ in $G$. For any $x\in M$, the induced process $x_{t}=g_{t}x$
defines the one-point motion of $g_{t}$ in $M$, with initial value $x$. Using
one-point motions, we define conditioned processes in the homogeneous space
$M$. Throughout this section, let $\hat{X}=\pi(X)$ and $\hat{Y}=\pi(Y)$ denote
the one-point motions of respectively $X$ and $Y$ as defined above.

###  4.1 Guided Diffusions on Homogeneous Space as Guided Diffusions in Lie
Groups

To simulate bridges on homogeneous spaces, we develop simulation schemes on
the top space, that is, bridge simulation schemes on the Lie group $G$, which
we then project onto the homogeneous space $M$. We will be considering two
different schemes. Let $v\in M$.

  1. 1.

Whenever $\pi^{-1}(v)$ is closed, find closest point $\bar{v}$ in fiber above
$v$ and iterative update $\bar{v}$ at each time step.

  2. 2.

Sample $k$-points in fiber above $v\in M$ and consider the bridge $X$ in $G$
conditioned on $X_{T}\in\\{\bar{v}_{1},\dots,\bar{v}_{k}\\}$.

We make the following assumptions throughout on the Markov transition density
$p^{G}_{t}(\cdot,\cdot)$.

######  Assumption 1.

  1. 1.

The Markov transition density is symmetric, i.e.,

$p^{G}_{t}(x,y)=p_{t}^{G}(y,x).$

  2. 2.

The Chapman-Kolmogorov equation is satisfied, i.e.,

$p^{G}_{t+s}(x,y)=\int_{G}p^{G}_{t}(x,z)p^{G}_{s}(z,y)d\operatorname{Vol}_{G}(z).$

  3. 3.

For any bounded continuous function $f$ on $G$ and every fixed $y\in G$

$\lim_{t\downarrow
0}\int_{G}p_{t}^{G}(x,y)f(x)d\operatorname{Vol}_{G}(x)=f(y).$

###  4.2 Guiding to Nearest Point

This section addresses the approach outlined in the previous section when
guiding to a closed embedded manifold $N\subseteq G$. Considering the one-
point motion of the guided process $Y$ towards $N:=\pi^{-1}(v)$ yields a
conditioning in the homogeneous space $M$.

######  Lemma 4.

Let $v\in M$ and $\pi\colon G\rightarrow M$ be the canonical projection with
$N=\pi^{-1}(v)$ being the fiber over $v$ and let $e\in G$ denote the identity
element. Furthermore, define $Y$ as the process with infinitesimal generator
(12) and $\Delta_{G}$ being the Laplace-Beltrami operator on $G$. Then $N$ is
a submanifold of $G$ and $Y$ is the Fermi bridge from $Y_{0}=e$ to $N$ at time
$T$.

###### Proof.

The result follows from standard differential geometry and [20]. ∎

We reinvigorate the fact that the one-point motion, $X_{t}=g_{t}x$, of a
Brownian motion $g_{t}$ in $G$, started at $g_{0}=e$, is only a Brownian
motion in $M$ under certain regularity conditions (see [12, Proposition 2.7]).
In case of a bi-invariant metric, a Brownian motion on $G$ maps to a Brownian
motion in $M$ through its one-point motion. In the general case, one-point
processes might not even preserve the Markov property. A numerical example is
provided below which show how we can obtain anisotropic distribution on the
homogeneous space from a non-invariant metric on the top space.

The Riemannian volume measure $\operatorname{Vol}_{G}$ on $G$ decomposes into
a product measure consisting of the volume measure on fibers in $G$, e.g.
$\pi^{-1}(z)$, and the volume measure on its horizontal complement, i.e.,
$d\operatorname{Vol}_{G}=d\operatorname{Vol}_{\pi^{-1}(z)}d\operatorname{Vol}_{|H}(z)$,
where $d\operatorname{Vol}_{|H}$ is the horizontal restriction of the volume
measure in $G$.

Assume that we have an induced volume measure defined on the homogeneous space
$M$. This is, for example, the case when the metric on $G$ is bi-invariant.

######  Theorem 4.1.

Let $x\in M$. The Fermi bridge $Y_{t}$ converges almost surely to $N$ and the
one-point motion $\hat{Y}_{t}=Y_{t}x$ is a diffusion bridge starting at $x\in
M$ converging almost surely to $v$.

###### Proof.

Since the Fermi bridge converges almost surely to $N$ (cf. [20]), it converges
in the horizontal direction. Hence the one-point motion converges almost
surely in $M$ to $\pi(N)=v\in M$. ∎

Suppose endowing both $G$ and $M:=G/K$ with sigma algebras making them into
two measurable spaces. Suppose further that the sigma algebras $\sigma(G)$ and
$\sigma(M)$ on $G$ and $M$, respectively, are the Borel-sigma algebras of all
open sets. This assumption is, of course, dependent on the metric. If the
metric space is non-separable, then Borel algebra is too big for anything
sensible to be said. Let $\pi\colon G\rightarrow M$ be the smooth submersion
onto the homogeneous space $M$. Then $\pi$ is a measurable map and if $\mu$ is
a measure on $G$ the pushforward of $\mu$ by $\pi$, defined by
$\pi_{*}\mu(B)=\mu\left(\pi^{-1}(B)\right)$, for all $B\in\sigma(M)$, is a
measure on $M$. Then we have the following result.

Given an initial value $x_{0}\in G$ for the process $X_{t}$, we can, by left-
translation, assume that the process starts at the identity in $G$, i.e.,
$x_{0}=e$. This alleviates the dependency on the distribution of the initial
value in the quotient space.

######  Theorem 4.2.

With the assumptions above, let $p_{T}^{G}(x_{0},)$ denote the transition
density of a diffusion process $X_{t}$ on $G$ initiated at $x_{0}\in G$. The
density $p_{T}^{G}(x_{0},)$ pushes forward to a transition density
$p_{T}^{M}(\hat{x}_{0},)$ of the diffusion process $\hat{X}_{t}=\pi(X_{t})$ on
$M$, initiated from $\hat{x}_{0}=\pi(x_{0})$.

###### Proof.

Let $N_{z}:=\pi^{-1}(z)$ be the fiber over $z$, for any $z\in M$, then
$N_{v}\subseteq G$ is a submanifold of $G$ by Lemma 4. Also, let $\mathbb{P}$
be the corresponding probability measure defined by
$\mathbb{P}(B)=\int_{B}p_{T}(u,x)d\operatorname{Vol}_{G}(x)$, then we see that
for any Borel measurable set $B\subseteq M$

$\displaystyle\mathbb{P}\left(X_{T}\in\pi^{-1}(B)\right)=$
$\displaystyle\int_{\pi^{-1}(B)}p_{T}^{G}(x_{0},y)d\operatorname{Vol}_{G}(y)$
$\displaystyle=$
$\displaystyle\int_{B}\int_{N_{z}}p_{T}^{G}(x_{0},y)d\operatorname{Vol}_{N_{z}}(y)d\operatorname{Vol}_{|H}(z)$
$\displaystyle=$
$\displaystyle\int_{B}\int_{N_{z}}p_{T}^{G}(x_{0},y)d\operatorname{Vol}_{N_{z}}(y)d\operatorname{Vol}_{M}(z)$
$\displaystyle=$
$\displaystyle\int_{B}p_{T}^{G}\left(x_{0},N_{z}\right)d\operatorname{Vol}_{M}(z),$

where the third equality follows by assumption and the last equality by (11).
Since $\hat{X}_{T}\in B$ if and only of $X_{T}\in\pi^{-1}(B)$, we see that
$p_{T}^{G}\left(x_{0},\pi^{-1}(\cdot)\right)$ is the transition density for
$\hat{X}_{t}$ which is exactly the pushforward
$\pi_{*}p_{T}^{G}(x_{0},\cdot):=p^{M}_{T}(\hat{x}_{0},\cdot)$. ∎

Note that the above is equivalent to $\pi P(f)=P(f\circ\pi)$, for any
measurable function $f$ on $M$.

######  Lemma 5.

Let $X$ be a Markov process on $G$, started at $x_{0}\in G$, with density
$p^{G}_{t}(x_{0},\cdot)$ satisfying the conditions in Assumption 1. The
conditional expectation on $M$ satisfies

$\mathbb{E}[f(\hat{X})|\hat{X}_{T}=v]=\mathbb{E}\left[f(\hat{X})\frac{p^{M}_{T-t}(\hat{X}_{t},v)}{p_{T}^{M}(\hat{x}_{0},v)}\right],$
(28)

for all bounded, continuous, and non-negative measurable $f$ on $M$.
Furthermore,

$\mathbb{E}[\tilde{f}(X)|X_{T}\in N]=\mathbb{E}[f(\hat{X})|\hat{X}_{T}=v],$

where $\tilde{f}=f\circ\pi$.

###### Proof.

For the first part note that
$\frac{p^{M}_{T-t}(\hat{X}_{t},v)}{p_{T}^{M}(\hat{x}_{0},v)}$ imply a
conditioning. For the second part, let $f$ be a bounded, continuous, and non-
negative measurable function on $M$, and let $\tilde{f}=f\circ\pi$. Then it
follows directly from [19] and Theorem 4.2

$\displaystyle\mathbb{E}[\tilde{f}(X_{t})|X_{T}\in N]=$
$\displaystyle\mathbb{E}\left[f(\pi(X_{t}))\frac{p^{G}_{T-t}(X_{t},N)}{p^{G}_{T}(x_{0},N)}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[f(\pi(X_{t}))\frac{p^{G}_{T-t}(X_{t},\pi^{-1}(v))}{p^{G}_{T}(x_{0},\pi^{-1}(v))}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[f(\pi(X_{t}))\frac{\pi_{*}p^{G}_{T-t}(X_{t},v)}{\pi_{*}p^{G}_{T}(x_{0},v)}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[f(\hat{X}_{t})\frac{p^{M}_{T-t}(\hat{X}_{t},v)}{p^{M}_{T}(\hat{x}_{0},v)}\right].$

∎

We summarize the main theorem of this section, namely, how to simulate guided
bridges on homogeneous spaces. Let $\hat{X}_{t}$ be a process in $M$ arising
as the one-point process of a Markov process $X_{t}$ in $G$.

######  Theorem 4.3.

The projection $\hat{X}_{t}$ of the $G$-valued Markov $X_{t}$ onto $M$ started
at $x\in M$ conditioned on $\hat{X}_{T}=v$, for any $v\in M$, is identical in
law to the process $\hat{Y}_{t}:=\pi(Y_{t})$, where
$g\overset{\pi}{\mapsto}g\cdot x$ and $Y_{t}$ is the Fermi bridge on $G$
conditioned at $Y_{T}\in N$. Moreover, the conditional expectation of
$\hat{X}$ given $\hat{X}_{T}=v$ is

$\mathbb{E}[f(\hat{X})|\hat{X}_{T}=v]=C\mathbb{E}\left[f(\hat{Y})\varphi_{T}\right],$
(29)

for every $\mathcal{F}_{t}$-measurable non-negative function $f$ on $G$,
$t<T$, where $\varphi_{t}$ is given in (24).

###### Proof.

Note that for any bounded, continuous, and non-negative measurable function
$\tilde{f}$ on $G$ we have [9]

$\mathbb{E}\left[\tilde{f}(X)|X_{T}\in\pi^{-1}(v)\right]=C\mathbb{E}\left[\tilde{f}(Y)\varphi_{T,N}\right].$

Let $f$ be any bounded, continuous, and non-negative measurable function on
$M$. For any such $f$ on $M$, let $\tilde{f}=f\circ\pi$. Then by Lemma 5,

$\mathbb{E}\left[f(\hat{X})|\hat{X}_{T}=v\right]=\mathbb{E}\left[f\circ\pi(X)|X_{T}\in\pi^{-1}(v)\right]$

the conclusion follows. ∎

###  4.3 Guiding to $k$-Points in Fiber

For certain homogeneous spaces, the fiber $N=\pi^{-1}(v)$ is a discrete
subgroup in $G$. In this case, the volume measure $\operatorname{Vol}_{N}$ in
(11) is the counting measure and we can write the density as

$p_{t}^{G}(x,N)=\sum_{v\in N}p^{G}_{t}(x,v).$

From a numerical perspective, when the discrete subgroup is large restricting
to a smaller finite subgroup $N_{k}\subseteq N$ of $k$-nearest-points of the
initial starting point may speed up computation-time.

Recall the Radon-Nikodym derivative given in (13), which in the case of a
discrete subgroup $N$ takes the form

$\frac{d\mathbb{P}_{x,N}^{T}}{d\mathbb{P}_{x}^{T}}\big{|}_{\mathcal{F}_{s}}=\frac{p^{G}_{T-s}(X_{s},N)}{p^{G}_{T}(x,N)}=\sum_{v\in
N}\frac{p^{G}_{T-s}(X_{s},v)}{\sum_{v\in N}p^{G}_{T}(x,v)},$ (30)

for $0\leq s<T$. Suppose that $N_{k}:=\\{v_{1},\dots,v_{k}\\}\subseteq N$. By
Theorem 5.2.1 in Thompson [19], the heat kernel to $v_{i}$, $p_{t}(x,v_{i})$,
has the following expression

$p^{G}_{T}(x,v_{i})=q_{T}(x,v_{i})\lim_{t\uparrow
T}\mathbb{E}\left[\varphi_{t}\right],$ (31)

where $\varphi_{t}$ is defined in (24) and $q_{t}(\cdot,v_{i})$ is the
Euclidean normal density

$q_{t}(x,v_{i})=(2\pi
t)^{-\frac{(d-n)}{2}}\exp\left[-\frac{r_{v_{i}}^{2}(x)}{2t}\right].$ (32)

From (30) and (31), we can see that the Radon-Nikodym derivative is equal to

$\sum_{i=1}^{k}\frac{p^{G}_{T-s}(x,v_{i})}{\sum_{j=1}^{k}p^{G}_{T}(x_{0},v_{j})}=\sum_{i=1}^{k}\frac{c_{j}q_{T-s}(x,v_{i})}{\sum_{j=1}^{k}c_{j}q_{T}(x_{0},v_{j})},$
(33)

where $c_{i}$ are the constants $\lim_{t\uparrow
T}\mathbb{E}[\varphi_{t,v_{i}}]$, with subscript $v_{i}$ to indicate its
dependence on the conditioning point. From the arguments above, the
conditional expectation is equal to

$\displaystyle\mathbb{E}_{\mathbb{P}}[F(X_{t})|X_{T}\in N_{k}]=$
$\displaystyle\frac{\mathbb{E}_{\mathbb{P}}[p^{G}_{T-t}(X_{t},N_{k})F(X_{t})]}{p^{G}_{T}(x_{0},N_{k})}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}\frac{\mathbb{E}_{\mathbb{P}}[p^{G}_{T-t}(X_{t},v_{i})F(X_{t})]}{\sum_{j=1}^{k}p^{G}_{T}(x_{0},v_{k})}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}c_{i}\frac{\mathbb{E}_{\mathbb{P}}\left[\exp\left[-\frac{r^{2}_{v_{i}}(X_{t})}{2(T-t)}\right]F(X_{t})\right]}{\sum_{j=1}^{k}c_{j}\exp\left[-\frac{r^{2}_{v_{j}}(x_{0})}{2(T-t)}\right]}=\mathbb{E}_{\mathbb{Q}}\left[F(Y_{t})\right].$

Let $h(s,X_{s})$ be the function defined by the right hand side of (33). It is
a classical argument following Chapman-Kolmogorov that $h(s,X_{s})$ is a
martingale on $[0,T)$, namely, for $s<t$

$\displaystyle\mathbb{E}_{\mathbb{P}}\left[h(t,X_{t})|\mathcal{F}_{s}\right]=$
$\displaystyle\mathbb{E}_{\mathbb{P}}\left[h(t,X_{t})|X_{s}\right]$
$\displaystyle=$ $\displaystyle\int
h(t,z)p^{G}_{t-s}(X_{s},z)d\operatorname{Vol}_{G}(z)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}\frac{\int
p^{G}_{T-t}(z,v_{i})p^{G}_{t-s}(X_{s},z)d\operatorname{Vol}_{G}(z)}{\sum_{j=1}^{k}p^{G}_{T}(x_{0},v_{j})}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}\frac{p_{T-s}(X_{s},v_{i})}{\sum_{j=1}^{k}p_{T}(x_{0},v_{j})}=h(s,X_{s}),$

where we have assumed that $X$ is a Markov process and clearly $h(0,x)=1$.
Then it is easily seen that

$\nabla_{x_{|x=X_{t}}}\log
h(t,x)=-\sum_{i=1}^{k}c_{i}\frac{q_{T-t}(X_{t},v_{i})}{\sum_{j=1}^{k}c_{j}q_{T}(x_{0},v_{j})}\frac{\nabla_{x_{|x=X_{t}}}r_{v_{i}}^{2}(x)}{2(T-t)}$
(34)

which from Doob’s $h$-transform yields an SDE of the form

$dY_{t}=-\frac{1}{2}V_{0}(Y_{t})dt+V_{i}(Y_{t})\circ\left(dB^{i}_{t}-\sum_{i=1}^{k}c_{i}\frac{q_{T-t}(Y_{t},v_{i})}{\sum_{j=1}^{k}c_{j}q_{T}(y_{0},v_{j})}\frac{\nabla_{y_{|y=Y_{t}}}r_{v_{i}}^{2}(y)}{2(T-t)}dt\right),$
(35)

where $Y_{0}=y_{0}=e$. Combining the above arguments, we have the following
result.

######  Theorem 4.4.

Let $Y$ be the solution of the SDE (35), then $Y$ converges almost surely to
$N_{k}$ as $t\uparrow T$ and the conditional expectation is given as

$\mathbb{E}_{\mathbb{Q}}\left[F(Y)\right]=\sum_{i=1}^{k}c_{i}\frac{\mathbb{E}_{\mathbb{P}}\left[\exp\left[-\frac{r^{2}_{v_{i}}(Y_{t})}{2(T-t)}\right]F(Y)\right]}{\sum_{j=1}^{k}c_{j}\exp\left[-\frac{r^{2}_{v_{j}}(x_{0})}{2(T-t)}\right]}=\mathbb{E}_{\mathbb{P}}\left[F(X)|X_{T}\in
N_{k}\right].$

######  Remark 1.

Choosing which $k$-points to condition on introduce a certain bias. An
argument could be to choose the $k$-points closest to the initial value
$y_{0}\in G$ and somehow obtain a truncated version of the true bridge
process. However, if the fiber is connected and continuous such a choice is
not feasible.

Consider now the case where $N$ is a continuous, connected, and compact fiber
(Compactness assumption so we get the existence of a uniform distribution on
$N$). Suppose we sample $k$-points in $N$ uniformly at random.

######  Theorem 4.5.

Let $X$ be a Markov process defined on $(\Omega,\mathcal{F},\mathbb{P})$ with
values in $G$, $X_{0}=x_{0}$, and let $p_{t}^{G}(\cdot,\cdot)$ be its
transition density defined by $\mathbb{P}(X_{T}\in
du|X_{t}=x)=p^{G}_{T-t}(x,u)d\operatorname{Vol}_{G}(u)$. Assume $0\leq t<T$
and $X_{T}\sim f\cdot\operatorname{Vol}_{N}$ (e.g., uniform distribution on
fiber over $v$) under the probability measure $\mathbb{P}_{x_{0}}$ started at
$x_{0}$. The conditional law of $X_{t}$ given $X_{T}=v$,
$\mathbb{P}^{T}_{x_{0},v}$, has density wrt. the reference measure
$d\operatorname{Vol}_{G}$ given by

$\frac{p^{G}_{T-t}(y,v)p^{G}_{t}(y,x_{0})}{p^{G}_{T}(x_{0},v)},$ (36)

and the simultaneous distribution of $(X_{t},X_{T})$ has density given by

$\frac{\mathbb{P}(X_{t}\in dx,X_{T}\in
du)}{d\operatorname{Vol}_{N}(u)d\operatorname{Vol}_{G}(x)}=f(u)\frac{p^{G}_{T-t}(y,u)p^{G}_{t}(y,x)}{p^{G}_{T}(x_{0},u)}.$
(37)

Furthermore, if we define the $h$-function as

$h(t,X_{t})=\frac{\int
f(u)\frac{p^{G}_{T-t}(X_{t},u)}{p_{T}^{G}(x_{0},u)}d\operatorname{Vol}_{N}(u)}{\int_{N}f(y)d\operatorname{Vol}_{N}(y)},$
(38)

then for any non-negative measurable functional $F$ we have

$\displaystyle\mathbb{E}_{\mathbb{Q}}[F(X)]=$
$\displaystyle\mathbb{E}_{\mathbb{P}}[h(t,X_{t})F(X)]$ $\displaystyle=$
$\displaystyle\int\mathbb{E}_{\mathbb{P}}[F(X)|X_{T}=u]\frac{f(u)}{\int_{N}f(y)d\operatorname{Vol}_{N}(y)}d\operatorname{Vol}_{N}(u).$

The conditional distribution of $X_{t}$ given $X_{T}$ has density

$k_{t}(x,u)=\frac{f(u)}{\int_{N}f(y)d\operatorname{Vol}_{N}(y)}\frac{p^{G}_{T-t}(y,u)p^{G}_{t}(y,x)}{p^{G}_{T}(x_{0},u)},$
(39)

with respect to the volume measure $\operatorname{Vol}_{N}$. If the
distribution $X_{T}(\mathbb{P})$ has full mass in the fiber $N$, i.e.,
$X_{T}(\mathbb{P})(N)=\int_{N}fdVol=1$ the term above simplifies.

###### Proof.

The fact that (36) is the conditional density wrt. $d\operatorname{Vol}_{G}$
the volume measure on $G$ follows from (8), since
$\mathbb{P}^{t}_{x}=p_{t}(x,y)d\operatorname{Vol}_{G}(y)$ and therefore

$d\mathbb{P}^{T}_{x_{0},v}(X_{t})=\frac{p^{G}_{T-t}(y,v)p^{G}_{t}(x_{0},y)}{p^{G}_{T}(x_{0},v)}d\operatorname{Vol}_{G}(y).$

Hence (37) follows.

For the second part take $h$ as defined in (38). Without loss of generality
assume that $X_{T}(\mathbb{P})(N)=1$. Note that $h$ is a martingale with
$h(0,X_{0})=1$, since $X$ is a Markov process and

$\displaystyle\mathbb{E}[h(t,X_{t})|X_{s}]=$ $\displaystyle\int
p^{G}_{t-s}(X_{s},x)h(t,x)d\operatorname{Vol}_{G}(x)$ $\displaystyle=$
$\displaystyle\int p^{G}_{t-s}(X_{s},x)\int
f(u)\frac{p^{G}_{T-t}(x,u)}{p_{T}^{G}(x_{0},u)}d\operatorname{Vol}_{N}(u)d\operatorname{Vol}_{G}(x)$
$\displaystyle=$ $\displaystyle\int
f(u)\frac{p^{G}_{T-s}(X_{s},u)}{p_{T}^{G}(x_{0},u)}d\operatorname{Vol}_{N}(u)=h(s,X_{s})$

together with

$\mathbb{E}[h(t,X_{t})]=\int
f(u)\frac{p^{G}_{T}(x_{0},u)}{p_{T}^{G}(x_{0},u)}d\operatorname{Vol}_{N}(u)=1.$

Since $\lim_{t\downarrow 0}\int
p^{G}_{t}(x,y)f(y)d\operatorname{Vol}_{G}(y)=f(x)$, for any bounded continuous
function $f$, Fatou’s lemma ensures that $\mathbb{E}[h(T,X_{T})]=1$

$\displaystyle 1=$ $\displaystyle\limsup_{t\uparrow T}\mathbb{E}[h(t,X_{t})]$
$\displaystyle\leq$ $\displaystyle\mathbb{E}\left[\limsup_{t\uparrow
T}h(t,X_{t})\right]$ $\displaystyle=$ $\displaystyle\int_{G}\limsup_{t\uparrow
T}\int
f(u)\frac{p^{G}_{T-t}(x,u)}{p_{T}^{G}(x_{0},u)}d\operatorname{Vol}_{N}(u)d\operatorname{Vol}_{G}(x)$
$\displaystyle=$
$\displaystyle\int_{G}\frac{f(x)}{p_{T}^{G}(x_{0},x)}d\operatorname{Vol}_{G}(x)$
$\displaystyle=$ $\displaystyle\int_{G}\liminf_{t\uparrow T}\int
f(u)\frac{p^{G}_{T-t}(x,u)}{p_{T}^{G}(x_{0},u)}d\operatorname{Vol}_{N}(u)d\operatorname{Vol}_{G}(x)$
$\displaystyle\leq$ $\displaystyle\liminf_{t\uparrow
T}\mathbb{E}[h(t,X_{t})]=1.$

Hence $h$ is a true martingale on $[0,T]$ and thus defines a new probability
measure $\mathbb{Q}$ on $\mathcal{F}$ by
$\frac{d\mathbb{Q}}{d\mathbb{P}}|_{\mathcal{F}_{t}}(X)=h(t,X_{t})$.

For the second part of the proof, assume, temporarily, that $F$ is a
measurable function such that

$\displaystyle\mathbb{E}_{\mathbb{P}}[h(t,X_{t})F(X_{t})]=$ $\displaystyle\int
p^{G}_{t}(x_{0},x)h(t,x)F(x)d\operatorname{Vol}_{G}(x)$ $\displaystyle=$
$\displaystyle\int\int\frac{p^{G}_{T-t}(x,u)p^{G}_{t}(x_{0},x)}{p_{T}^{G}(x_{0},u)}F(x)d\operatorname{Vol}_{G}(x)f(u)d\operatorname{Vol}_{N}(u)$
$\displaystyle=$ $\displaystyle\int\int
F(x)d\mathbb{P}_{x_{0},u}^{T}(x)f(u)d\operatorname{Vol}_{N}(u)$
$\displaystyle=$
$\displaystyle\int\mathbb{E}_{\mathbb{P}}[F(X_{t})|X_{T}=u]f(u)d\operatorname{Vol}_{N}(u).$

In order to conclude, we need to show that for any finite distribution
$(X_{t_{1}},\dots,X_{t_{n}})$

$\mathbb{E}_{\mathbb{P}}[h(t,X_{t})F(X_{t_{1}},\dots,X_{t_{n}})]=\int\mathbb{E}_{\mathbb{P}}[F(X_{t_{1}},\dots,X_{t_{n}})|X_{T}=u]f(u)d\operatorname{Vol}_{N}(u).$

Therefore, let $0<t_{1}<\dots<t_{n}<T$ and $t\in(t_{n},T)$. Define $\Phi_{F}$
similar to how it was defined in Lemma 2. Then

$\displaystyle\mathbb{E}_{\mathbb{P}}[h(t,X_{t})F(X_{t_{1}},\dots,X_{t_{n}})]=$
$\displaystyle\int_{G}h(t,x)\Phi_{F}(t,x)d\operatorname{Vol}_{G}(x)$
$\displaystyle=$ $\displaystyle\int f(u)\int
F(z)P(z,x,u)d\operatorname{Vol}(z)d\operatorname{Vol}(x)d\operatorname{Vol}(u)$
$\displaystyle=$
$\displaystyle\int\mathbb{E}_{\mathbb{P}}[F(X_{t_{1}},\dots,X_{t_{n}})|X_{T}=u]f(u)d\operatorname{Vol}_{N}(u),$

where $z=(z_{1},\dots,z_{n})$ and
$d\operatorname{Vol}(z)=d\operatorname{Vol}(z_{1})\dots
d\operatorname{Vol}(z_{n})$ and where

$P(z,x,u)=\frac{p^{G}_{t_{1}}(x_{0},z_{1})\dots
p^{G}_{t-t_{n}}(z_{n},x)p^{G}_{T-t}(x,u)}{p_{T}^{G}(x_{0},u)}.$

∎

The next result uses the type of conditioning found in van der Meulen and
Schauer [13]. By imposing noise on the conditioning point they showed that the
endpoint is tilted to have a given density. We adapt here this type of
conditioning by imposing noise in the on the conditioning point.

######  Theorem 4.6.

Let $X$ be a Markov process defined on $(\Omega,\mathcal{F},\mathbb{P})$ with
values in $G$, $X_{0}=x_{0}$, and let $p_{t}^{G}(\cdot,\cdot)$ be its
transition density defined by $\mathbb{P}(X_{T}\in
du|X_{t}=x)=p^{G}_{T-t}(x,u)d\operatorname{Vol}_{G}(u)$. Assume $0\leq t<T$
and $X_{T}\sim f\cdot\operatorname{Vol}_{N}$ (e.g., uniform distribution on
fiber over $v$). The simultaneous distribution of $(X_{t},X_{T})$ has density
given by

$\frac{\mathbb{P}(X_{t}\in dx,X_{T}\in
du)}{d\operatorname{Vol}_{N}(u)d\operatorname{Vol}_{G}(x)}=f(u)p^{G}_{T-t}(x,u)$
(40)

Furthermore, the conditional distribution of $X_{t}$ given $X_{T}$ has density

$k_{t}(x,u)=\frac{f(u)p^{G}_{T-t}(x,u)}{\int_{N}f(y)p^{G}_{T}(x_{0},y)d\operatorname{Vol}_{N}(y)}.$
(41)

(NB. $\sigma$-finite measures exists on any connected locally compact Lie
group)

###### Proof.

The first part is a direct consequence of standard measure theory. For the
second part define

$Z_{t}=h(t,X_{t})=\frac{\int
f(u)p^{G}_{T-t}(X_{t},u)d\operatorname{Vol}_{N}(u)}{\int
f(u)p^{G}_{T}(x_{0},u)d\operatorname{Vol}_{N}(u)}.$ (42)

Then, for $0\leq s<t$

$\displaystyle\mathbb{E}^{\mathbb{P}}[Z_{t}|\mathcal{F}_{s}]=\mathbb{E}^{\mathbb{P}}[h(t,X_{t})|X_{s}]=$
$\displaystyle\int\frac{\int
f(u)p^{G}_{T-t}(x,u)d\operatorname{Vol}_{N}(u)}{\int
f(u)p^{G}_{T}(x_{0},u)d\operatorname{Vol}_{N}(u)}p_{t-s}(X_{s},x)d\operatorname{Vol}_{G}(x)$
$\displaystyle=$ $\displaystyle\frac{\int f(u)\int
p^{G}_{T-t}(x,u)p_{t-s}(X_{s},x)d\operatorname{Vol}_{G}(x)d\operatorname{Vol}_{N}(u)}{\int
f(u)p^{G}_{T}(x_{0},u)d\operatorname{Vol}_{N}(u)}$ $\displaystyle=$
$\displaystyle\frac{\int
f(u)p^{G}_{T-s}(X_{s},u)d\operatorname{Vol}_{N}(u)}{\int
f(u)p^{G}_{T}(x_{0},u)d\operatorname{Vol}_{N}(u)}$ $\displaystyle=$
$\displaystyle Z_{s},$

and we see that $Z$ is a local martingale with $Z_{0}=1$. Defining
$\mathbb{Q}_{t}=Z_{t}\cdot\mathbb{P}$, we see from the abstract Bayes formula

$\displaystyle\mathbb{E}^{\mathbb{Q}}[F(X_{t})|X_{s}]=$
$\displaystyle\frac{\mathbb{E}^{\mathbb{P}}[Z_{t}F(X_{t})|X_{s}]}{Z_{s}}$
$\displaystyle=$ $\displaystyle\frac{\int F(x)p_{t-s}^{G}(X_{s},x)\frac{\int
f(u)p^{G}_{T-t}(x,u)d\operatorname{Vol}_{N}(u)}{\int
f(u)p^{G}_{T}(x_{0},u)d\operatorname{Vol}_{N}(u)}d\operatorname{Vol}_{G}(x)}{\frac{\int
f(u)p^{G}_{T-s}(X_{s},u)d\operatorname{Vol}_{N}(u)}{\int
f(u)p^{G}_{T}(x_{0},u)d\operatorname{Vol}_{N}(u)}}$ $\displaystyle=$
$\displaystyle\frac{\int F(x)p_{t-s}^{G}(X_{s},x)\int
f(u)p^{G}_{T-t}(x,u)d\operatorname{Vol}_{N}(u)d\operatorname{Vol}_{G}(x)}{\int
f(u)p^{G}_{T-s}(X_{s},u)d\operatorname{Vol}_{N}(u)}$ $\displaystyle=$
$\displaystyle\int\int\frac{F(x)p_{t-s}^{G}(X_{s},x)p^{G}_{T-t}(x,u)}{p_{T-s}^{G}(X_{s},u)}d\operatorname{Vol}_{G}(x)k_{u}(X_{s})d\operatorname{Vol}_{N}(u)$
$\displaystyle=$
$\displaystyle\int\mathbb{E}^{\mathbb{P}}\left[F(X_{t})|\mathcal{F}_{s},X_{T}=u\right]k_{u}(X_{s})d\operatorname{Vol}_{N}(u),$

where exactly

$k_{t}(X_{s},u)=\frac{f(u)p_{T-s}^{G}(X_{s},u)}{\int
f(u)p^{G}_{T-s}(X_{s},u)d\operatorname{Vol}_{N}(u)}.$

∎

We note that the density $k_{u}(x)$ is proportional to the expression

$k_{T}(x_{0},u)=\frac{f(u)p^{G}_{T}(x,u)}{\int_{N}f(y)p^{G}_{T}(x_{0},y)d\operatorname{Vol}_{N}(y)}\propto
f(u)q_{T}(x_{0},u)\lim_{t\uparrow T}\mathbb{E}[\varphi_{t,u}].$ (43)

Assume that the constants $c_{i}$ exist. We can obtain unbiased estimates of
the constants $c_{i}$ via simulation of sample paths of the guided process.
Let $y^{i}_{t}=Y_{t}(\omega_{i})$ be realizations of the guided bridge
process. We can then use the unbiased estimator

$\bar{\varphi}_{T,v_{i}}(y)=\frac{1}{m}\sum_{n=1}^{m}\exp\left[\int_{0}^{T}\frac{r_{v_{i}}(y^{n}_{s})}{T-s}\left(dA^{v_{i}}_{s}+dL_{s}^{v_{i}}\right)\right]$
(44)

to approximate the constants $c_{i}=\mathbb{E}[\varphi_{T,v_{i}}]$.

// Initialization

Choose initial point $e\in G$ and $v_{1}\in N$ closest to $e$. Simulate a
guided bridge process to $v_{1}$ and obtain an unbiased estimate of
$\mathbb{E}[\varphi_{T,v_{1}}]$.

// Main loop

while  _$k$ points not reached_ do

// Step 1:

Propose $u$ from the proposal density $f(v_{i},u)$ (e.g. uniform density in
$N$ centered at $v_{i}$ or normal density in $N$ centered at $v_{i}$) and
sample estimator for $\mathbb{E}[\varphi_{T,u}]$

// Step 2:

Calculate the acceptance ratio
$g(u,v_{i})=\min\left\\{1,\frac{f(u,v_{i})q_{T}(x_{0},u)\bar{\varphi}_{T,u}}{f(v_{i},u)q_{T}(x_{0},v_{i})\bar{\varphi}_{T,v_{i}}}\right\\}$
(Note that if $f$ is symmetric it cancels out in the acceptance probability)

// Step 3

Accept with probability g$(u,v_{i})$ and set $v_{i+1}=u$ as well as
$c_{i+1}=\bar{\varphi}_{T,v_{i+1}}$ otherwise do nothing.

end while

// Output:

$\\{(v_{1},c_{1}),\dots,(v_{k},c_{k})\\}$

Algorithm 1 Stochastic Metropolis-Hastings Algorithm

###### Proof.

By repeated use of the tower property and Markov property

$\displaystyle\mathbb{E}[g(X_{T})F(X)]=$
$\displaystyle\mathbb{E}\left[g(X_{T})\mathbb{E}[F(X)|X_{T}]\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[F(X)\mathbb{E}[g(X_{T})|X_{s}]\right],$

for any $\mathcal{F}_{s}$-measurable functional $F$ and
$\mathcal{F}_{T}$-measurable function $g$. From the Markov property it follows
that

$\displaystyle\mathbb{E}\left[F(X)\mathbb{E}[g(X_{T})|X_{s}]\right]=$
$\displaystyle\mathbb{E}\left[F(X)\int_{G}g(v)p_{T-s}^{G}(X_{s},v)d\operatorname{Vol}_{G}(v)\right]$
$\displaystyle=$
$\displaystyle\int_{G}\mathbb{E}[F(X)p_{T-s}^{G}(X_{s},v)]g(v)d\operatorname{Vol}_{G}(v).$

Using the fact that $X_{T}(\mathbb{P})(v)=f(v)d\operatorname{Vol}_{G}(v)$,
where $f(v)\equiv 0$ for any $v\in G\backslash N$,

$\displaystyle\mathbb{E}\left[g(X_{T})\mathbb{E}[F(X)|X_{T}]\right]=$
$\displaystyle\int_{G}g(v)\mathbb{E}[F(X)|X_{T}=v]dX_{T}(\mathbb{P})(v)$
$\displaystyle=$
$\displaystyle\int_{G}\mathbb{E}[F(X)|X_{T}=v]f(v)g(v)d\operatorname{Vol}_{G}(v).$

From this we see that for any non-negative measurable function $g$

$\int_{G}\mathbb{E}[F(X)p_{T-s}^{G}(X_{s},v)]g(v)d\operatorname{Vol}_{G}(v)=\int_{G}\mathbb{E}[F(X)|X_{T}=v]f(v)g(v)d\operatorname{Vol}_{G}(v),$

which implies that almost everywhere, for any $F\in\mathcal{F}_{s}$,

$\mathbb{E}[F(X)|X_{T}=v]=\frac{\mathbb{E}[F(X)p_{T-s}^{G}(X_{s},v)]}{f(v)}.$

∎

##  5 Numerical Experiments

In this section, we present numerical results of bridge sampling on specific
Lie groups and homogeneous spaces. The specific Lie groups in question are the
three-dimensional rotation group $\mathrm{SO}(3)$ and the general linear group
of invertible matrices with positive determinant $\mathrm{GL}_{+}(3)$.

The rotation group $\mathrm{SO}(3)$ is a Lie group with Lie algebra
$\mathfrak{so}(3)$, consisting of skew-symmetric matrices. The Lie group
$\mathrm{SO}(3)$ is well-studied group, where closed form expressions of the
Lie group exponential and logarithmic map are available. Furthermore, the
structure coefficients of $\mathrm{SO}(3)$ are particularly simple to work
with. Exploiting the bridge sampling scheme described above, we show below how
to estimate the underlying true metric on $\mathrm{SO}(3)$ via an iterative
MLE method.

The space of covariance matrices, i.e., the symmetric positive definite
matrices $\operatorname{SPD}(n)$, is an example of a non-linear space in which
geometric data appear in many applications. The space $\operatorname{SPD}(3)$
can be obtained as the homogeneous space $\mathrm{GL}_{+}(3)/\mathrm{SO}(3)$,
where $\mathrm{GL}_{+}$ is the space of invertible matrices with positive
determinant.

Lastly, considering the two-sphere $\mathbb{S}^{2}$ as the homogeneous space
$\mathrm{SO}(3)/\mathrm{SO}(2)$, we verify that the bridge sampling scheme on
this homogeneous space yields admissible heat kernel estimates on
$\mathbb{S}^{2}$.

####  5.0.1 Numerical Simulations

The Euler-Heun scheme leads to approximation of the Stratonovich integral.
With a time discretization $t_{1},\ldots,t_{k}$, $t_{k}-t_{k-1}=\Delta t$ and
corresponding noise $\Delta B_{t_{i}}\sim N(0,\Delta t)$, the numerical
approximation of the Brownian motion (7) takes the form

$x_{t_{k+1}}=x_{t_{k}}-\frac{1}{2}\sum_{j,i}C^{j}_{ij}V_{i}(x_{t_{k}})\Delta
t+\frac{v_{t_{k+1}}+V_{i}(v_{t_{k+1}}+x_{t_{k}})\Delta B_{t_{k}}^{i}}{2}$ (45)

where $v_{t_{k+1}}=V_{i}(x_{t_{k}})\Delta B_{t_{k}}^{i}$ is only used as an
intermediate value in integration. Adding the logarithmic term in (23) to (45)
we obtain a numerical approximation of a guided diffusion (14).

###  5.1 Importance Sampling and Metric Estimation on $\mathrm{SO}(3)$

This section takes $G$ to be the special orthogonal group of rotation
matrices, $\mathrm{SO}(3)$, a compact connected matrix Lie group. In the
context of matrix Lie groups, computing left-invariant vector fields is
straightforward.

The Lie algebra of the rotation group $\mathrm{SO}(3)$ is the space of three-
by-three skew symmetric matrices, $\mathfrak{so}(3)$. The exponential map
$\exp\colon\mathfrak{so}(3)\rightarrow\mathrm{SO}(3)$ is a well-defined
surjective map which coincide with usual matrix exponential $e^{A}$. With
$a\in\mathbb{R}^{3}$, we can express any element $A\in\mathfrak{so}(3)$ in
terms of the standard basis $\\{e_{1},e_{2},e_{3}\\}$ of $\mathbb{R}^{3}$ as

$A=\begin{pmatrix}0&-a_{3}&a_{2}\\\ a_{3}&0&-a_{1}\\\
-a_{2}&a_{1}&0\end{pmatrix}.$

Let $\theta=\lVert a\rVert_{2}$ and assume that $\theta\neq 0$. By Rodrigues’
formula the matrix Lie group exponential map
$\exp\colon\mathfrak{so}(3)\rightarrow\mathrm{SO}(3)$ is given by

$R:=e^{A}=I+\frac{\sin(\theta)}{\theta}A+\frac{(1-\cos(\theta))}{\theta^{2}}A^{2}$

and the corresponding inverse matrix Lie group exponential map
$\log\colon\mathrm{SO}(3)\rightarrow\mathfrak{so}(3)$

$\log(R)=\frac{\sin^{-1}(\theta)}{2\theta}(R-R^{T}).$

The rotation group $\mathrm{SO}(3)$ is semi-simple, hence there exist a bi-
invariant inner product. In this case, the Riemannian exponential map
$\operatorname{Exp}$ coincide with the Lie group exponential map $\exp$ and
thus the Riemannian distance function
$d(R,I)^{2}=\lVert\operatorname{Log}_{I}(R)\rVert^{2}$, from the rotation $R$
to the identity $I$, satisfies $\nabla_{R}d(R,I)^{2}=2\log(R)$.

The structure coefficients of $\mathfrak{so}(3)$ are particularly simple. Let
$A_{i}=A$ with $a_{j}=1$ if $i=j$ and zero otherwise. In this case,
$\\{A_{1},A_{2},A_{3}\\}$ defines a basis of $\mathfrak{so}(3)$. The structure
coefficients satisfy the relation
$[A_{i},A_{j}]=C^{k}_{ij}A_{k}=\epsilon^{ijk}A_{k}$, where $\epsilon^{ijk}$
denotes the Levi-Civita symbols. The Levi-Civita symbols are defined as $+1$,
for $(i,j,k)$ an even permutation of $(1,2,3)$, $-1$ for every odd
permutation, and zero otherwise.

####  5.1.1 Numerical Bridge Sampling Algorithm on $\mathrm{SO}(3)$

Utilizing the simple expressions for the structure coefficients and the Lie
group logarithmic map, we can explicitly write up the numerical approximation
of the guided bridge processes (Brownian bridge) on $\mathrm{SO}(3)$ as

$x_{t_{k+1}}=x_{t_{k}}-\frac{1}{2}\sum_{j,i}\epsilon^{ijj}V_{i}(x_{t_{k}})\Delta
t+\frac{v_{t_{k+1}}+V_{i}(v_{t_{k+1}}+x_{t_{k}})\left(\Delta
B_{t_{k}}^{i}-\frac{\log(x_{k})}{T-t_{k}}\Delta t\right)}{2},$ (46)

where in this case we have $v_{t_{k+1}}=V_{i}(x_{t_{k}})\left(\Delta
B_{t_{k}}^{i}-\frac{\log(x_{k})}{T-t_{k}}\Delta t\right)$. Figure 1
illustrates the numerical approximation by showcasing three different sample
paths from the guided diffusion conditioned to hit the rotation represented by
the black vectors.

(a)

(b)

(c)

Figure 1: Three sample paths $(a)-(c)$ of the guided diffusion process on
$\mathrm{SO}(3)$ visualized by its action on the basis vectors
$\\{e_{1},e_{2},e_{3}\\}$ (red, green, blue) arrows of $\mathbb{R}^{3}$
(rotated view). The sample paths are conditioned to hit the rotation
represented by the black vectors.

Another way of visualizing the guided bridge on the rotation group
$\mathrm{SO}(3)$ is through the angle-axis representation. Figure 2 represents
a guided process on $\mathrm{SO}(3)$ by presenting the axis representation on
$\mathbb{S}^{2}$ and its corresponding angle of rotation.

(a) The axis representation of the rotation matrices corresponding to the axis
around which the rotation happens.

(b) The angle-representation of the rotation around the ”fixed” rotation axis.

Figure 2: Angle-axis representation of a guided bridge process on the rotation
group $\mathrm{SO}(3)$

####  5.1.2 Metric Estimation on $\mathrm{SO}(3)$

In the $d$-dimensional Euclidean case, importance sampling yields the estimate
[15]

$p_{T}(u,v)=\left(\frac{\det\left(A(T,v)\right)}{2\pi
T}\right)^{\tfrac{d}{2}}e^{-\frac{\lVert
u-v\rVert^{2}_{A}}{2T}}\mathbb{E}[\varphi_{v,T}],$

where $\lVert x\rVert_{A}=x^{T}A(0,u)x$. Thus, from the output of the
importance sampling we get an estimate of the transition density. Similar to
the Euclidean case, we obtain an expression for the heat kernel $p_{T}(e,v)$
as $p_{T}(e,v)=q(T,e)\mathbb{E}\left[\varphi_{v,T}\right]$, where

$\displaystyle q(T,e)=$ $\displaystyle\left(\frac{\det A(v)}{2\pi
T}\right)^{\tfrac{3}{2}}\exp\left(-\frac{d(e,v)^{2}}{2T}\right)$
$\displaystyle=$ $\displaystyle\left(\frac{\det A(T,v)}{2\pi
T}\right)^{\tfrac{3}{2}}\exp\left(-\frac{\lVert\operatorname{Log}_{v}(e)\rVert^{2}_{A}}{2T}\right),$
(47)

where the equality holds almost everywhere and
$A\in\mathrm{Sym^{+}}(\mathfrak{g})$ denotes the metric $A(e):=A(0,e)$. The
$\operatorname{Log}_{v}$ map in (5.1.2) is the Riemannian inverse exponential
map.

Figure 3 illustrates how importance sampling on $\mathrm{SO}(3)$ leads to a
metric estimation of the underlying unknown metric, which generated the
Brownian motion. We sampled $128$ points as endpoints of a Brownian motion
from the metric $\text{diag}(0.2,0.2,0.8)$, and used $20$ time steps to sample
$4$ bridges per observation. An iterative MLE method using gradient descent
with a learning rate of $0.2$, and initial guess of the metric being
$\text{diag}(1,1,1)$ yielded a convergence to the true metric. Note that in
iteration the logarithmic map changes.

(a) Estimation of the unknown underlying metric using bridge sampling. Here
the true metric is the diagonal matrix $\text{diag}(0.2,0.2,0.8)$.

(b) The iterative log-likelihood.

Figure 3: The importance sampling technique applies to metric estimation on
the Lie group $\mathrm{SO}(3)$. Sampling a Brownian motion from an underlying
unknown metric, we obtain convergence to the true underlying metric using an
iterative MLE method. Here we sampled $4$ guided bridges per observation,
starting from the metric $\text{diag}(1,1,1)$, providing a relatively smooth
iterative likelihood in 3(b).

// Initialization

Given $n$ data points $\\{v_{1},\dots,v_{n}\\}$.

// Specify initial parameters $\theta_{0}=(g_{0},A)$ and a learning rate
$\eta$.

for  _$k=1$ to $K$_ do

for  _$j=1$ to $n$_ do

Sample $m$ bridges conditioned on $v_{j}$ to get estimate for
$\mathbb{E}[\varphi_{v_{j},T}]\approx\frac{1}{m}\sum_{i=1}^{m}\varphi_{v_{j},T}^{i}$

end for

$\ell_{\theta_{k-1}}(v_{1},\dots,v_{n})=\prod_{j=1}^{n}\left(\frac{\det
A(T,v_{j})}{2\pi
T}\right)^{3/2}e^{-\frac{\lVert\operatorname{Log}_{v_{j}}(e)\rVert^{2}_{A}}{2T}}\frac{1}{m}\sum_{i=1}^{m}\varphi_{v_{j},T}^{i}$

// Compute the gradient

$v_{k-1}=\nabla_{\theta_{k-1}}\log\ell_{\theta_{k-1}}(v_{1},\dots,v_{n})$

// Update the parameters

$\theta_{k}=\theta_{k-1}-\eta v_{k}$

end for

// Return final parameters $\theta_{K}$

Algorithm 2 Parameter Estimation: Iterative MLE.

###  5.2 Diffusion-Mean Estimation on $\operatorname{SPD}(3)$

The space of symmetric positive definite ($\operatorname{SPD}$) matrices is an
essential class of matrices arising as geometric data in many applications.
For example, in diffusion tensor imaging, $\operatorname{SPD}(3)$ matrices are
used to model the anisotropic diffusion of water molecules in each voxel. The
$\operatorname{SPD}$ matrices constitute a smooth incomplete manifold when
endowed with the Euclidean metric of matrices (Pennec et al. [16]). The space
of $\operatorname{SPD}(3)$ matrices can be regarded as the homogeneous space
$\mathrm{GL}_{+}(3)/\mathrm{SO}(3)$ of invertible matrices with positive
determinants being rotationally invariant to three-dimensional rotations.

In this section, the bridge sampling scheme derived above allow us to obtain
an estimate of the diffusion-mean [6, 5] on $\operatorname{SPD}(3)$, by
sampling guided bridge processes in the space of invertible matrices with
positive determinants $\mathrm{GL}_{+}(3)$. This sampling method provides an
estimate of the density on $\mathrm{GL}_{+}(3)$ which projects to a density in
$\operatorname{SPD}(3)$. Exploiting the resulting density in
$\operatorname{SPD}(3)$, an iterative MLE method then yield a convergence to
the diffusion mean.

(a) Convergence of the diffusion mean on $\operatorname{SPD}(3)$.

(b) Iterative log-likelihood for the diffusion mean.

Figure 4: Given 64 data points in $\operatorname{SPD}(3)$, simulating three
bridges per observation in $\mathrm{GL}_{+}(3)$, conditioned on the fibers, we
obtain convergence of the diffusion-mean using the iterative MLE with a
learning rate of $0.75$. The true mean is the identity matrix illustrated by
the red lines in (a). The yellow, blue, and purple lines visualize the
diagonal elements in the $\operatorname{SPD}(3)$ matrix, while the remaining
lines represent the off-diagonal.

Fig. 4 visualize the MLE approach to estimating the diffusion mean. As
$\operatorname{SPD}(3)$ matrices are six-dimensional, we only need six
parameters (three in the diagonal and three in the off-diagonal) to estimate
the diffusion mean.

###  5.3 Density Estimation on $\mathbb{S}^{2}:=\mathrm{SO}(3)/\mathrm{SO}(2)$

####  5.3.1 Bi-invariant Metric

A simulation scheme on specific homogeneous spaces was introduced in Section
4.2 by using guided bridges in the top space conditioned to arrive in the
fiber at time $T$. The two-sphere $\mathbb{S}^{2}$ can be considered as the
homogeneous space $\mathrm{SO}(3)/\mathrm{SO}(2)$ of three-dimensional
rotations, identifying the subgroup of two-dimensional rotations as a single
point. Conditioning on the fiber $\mathrm{SO}(2)$ in $\mathrm{SO}(3)$, we
obtain guided bridges on $\mathbb{S}^{2}$.

(a) The estimated heat kernel on $\mathbb{S}^{2}$ obtained from bridge
sampling in $\mathrm{SO}(3)$ for $T=0.5$.

(b) The estimated heat kernel, for $T=0.5$, on $\mathbb{S}^{2}$ along a
geodesic from the north pole to the south pole against the truncated exact
heat kernel, where $T=0.4$.

Figure 5: From $128$ sample points on $\mathbb{S}^{2}$, sampling guided
bridges in $\mathrm{SO}(3)$ conditioned on the fibers over the sample points,
an estimate of the heat kernel on $\mathbb{S}^{2}$ is obtained.

Comparing the estimated heat kernel on $\mathbb{S}^{2}$, obtained from bridge
samples in $\mathrm{SO}(3)$, with the truncated version of the exact heat
kernel, as described by Zhao and Song [21], the bridge estimated density
offers a good approximation of the (truncated) exact density. Fig. 5 shows the
relation between the estimated heat kernel and the truncated exact heat
kernel.

####  5.3.2 Non-invariant Metric

Changing the metric structure on $\mathrm{SO}(3)$ results in an anisotropic
distribution on $\mathbb{S}^{2}$, arising as the pushforward measure from
$\mathrm{SO}(3)$. Fig. 6 illustrate the anisotropic distributions on
$\mathbb{S}^{2}$ induced by a left-invariant metric on $\mathrm{SO}(3)$ for
$T=0.5,1.0,1.5$, and $2.0$, respectively.

Figure 6: Anisotropic distributions on $\mathbb{S}^{2}$ arising from a non-
invariant metric on $\mathrm{SO}(3)$, for $T=0.5,1.0,1.5$, and $2.0$,
respectively. The density estimates are based on $512$ sample points on
$\mathrm{SO}(3)$. For each sample point and end-time, three guided bridge
processes were sampled in $\mathrm{SO}(3)$ yielding estimates of the densities
in $\mathrm{SO}(3)$, which projected to anisotropic densities in
$\mathbb{S}^{2}$.

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