## 2 Notation and Background
### Brownian Bridges  
In this section, we briefly review some facts on Brownian bridges on Riemannian manifolds, including Lie groups. On Lie groups, the existence of left-invariant (respectively right-invariant) vector fields allows identification of the Lie algebra with the vector space of left-invariant vector fields making the Lie group parallelizable. This allows constructing general semimartingales directly on the Lie groups.  
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 \(x\). Let \(p_{t}\) denote the transition density of \(X_{t}\) so that \(d\mathbb{P}^{t}_{x}=p_{t}(x,y)d\operatorname{Vol}(y)\) with \(d\operatorname{Vol}(y)\) the Riemannian volume measure. Conditioning the Riemannian Brownian motion to hit some point \(v\in M\) at time \(T>0\) defines a Riemannian Brownian bridge. We let \(\mathbb{P}^{T}_{x,v}\) denote 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 consequence is obvious because \(\mathbb{P}_{x}(X_{T}=v)=0\), whereas \(\mathbb{P}^{T}_{x,v}(X_{T}=v)=1\). The corresponding Radon-Nikodym derivative is  
\[\frac{d\mathbb{P}^{T}_{x,v}}{d\mathbb{P}_{x}}|_{\mathcal{F}_{s}}=\frac{p_{T-s }(X_{s},v)}{p_{T}(x,v)}\qquad\text{for $0\leq s<T$} \tag{7}\]  
which is a martingale for \(s<T\). The Radon-Nikodym derivative defines the density for the change of measure, and it provides the conditional expectation  
\[\mathbb{E}[F(X_{t})|X_{T}=v]=\frac{\mathbb{E}[p_{T-t}(X_{t},v)F(X_{t})]}{p_{T} (x,v)}, \tag{8}\]  
for any bounded and \(\mathcal{F}_{s}\)-measurable random variable \(F(X_{s})\). The Brownian bridge is a non-homogeneous diffusion on \(M\) with infinitesimal generator  
\[\mathcal{L}_{s}f(z)=\frac{t}{2}\Delta_{M}f(z)+t\
abla_{z}\log p_{t(1-s)}(z,v) \cdot\
abla f(z).\]  
The bridge can be described by an SDE in the frame bundle \(FM\) of \(M\). Let \(U_{t}\) be a lift of \(X_{t}=\pi_{FM}(U_{t})\), and using the horizontal vector fields \(H_{t},\ldots,H_{d}\in\mathfrak{X}(FM)\), we have  
\[dU_{t}=H_{i}(U_{t})\circ\left(dB^{i}_{t}+\left(U_{t}^{-1}\Big{(}\pi_{*}\Big{(} \
abla^{H}_{u|u=U_{t}}\log\bar{p}_{T-t}(u,v)\Big{)}\Big{)}\right)^{i}dt\right),\quad U_{0}=u_{0}, \tag{9}\]  
where \(\bar{p}_{t}(u,v)=p_{t}(\pi(u),v)\) denotes the lift of the transition density, \(B\) is an \(\mathbb{R}^{d}\)-valued Brownian motion, and \((\pi_{FM})_{*}\colon T\mathcal{FM}\to TM\) is the pushforward of the projection \(\pi_{FM}\colon\mathcal{FM}\to M\). Here, \(u_{0}\in FM\) is an orthonormal frame such that \(\pi_{FM}(u_{0})=x_{0}\).  
Further types of Riemannian bridges can be found in Thompson [15]. Brownian bridges to submanifolds are here introduced by considering the transition density evaluated at a submanifold \(N\subset M\) by  
\[p_{t}(x,N):=\int_{N}p_{t}(x,y)d\operatorname{Vol}_{N}(y), \tag{10}\]  
where \(\operatorname{Vol}_{N}\) denotes the volume measure on \(N\). Conditioning on \(X_{T}\in N\) gives  
\[\mathbb{E}[f(X_{t})|X_{T}\in N]=\frac{\mathbb{E}[p_{T-t}(X_{t},N)f(X_{t})]}{p _{T}(x,N)}, \tag{11}\]  
which holds for all bounded \(\mathcal{F}_{t}\)-measurable random variables \(f(X_{t})\). Fibers of homogeneous spaces are embedded submanifolds of a Lie group. We will later use this to derive a simulation scheme on homogeneous spaces by conditioning on the fibers.  
The notion of _Fermi bridges_ was also introduced in [15]. Fermi bridges have infinitesimal generator  
\[\tfrac{1}{2}\Delta-\tfrac{r_{N}}{T-t}\tfrac{\partial}{\partial t_{N}}, \tag{12}\]  
where \(r_{N}(\cdot):=d(\cdot,N)=\inf_{y\in N}d(\cdot,y)\) and \(\tfrac{\partial}{\partial t_{N}}=\
abla d(\cdot,N)\).