# Mean Estimation on the Diagonal of Product Manifolds
## 3 Diffusion Mean Estimation
### Simulation in Coordinates  
We here develop a more efficient simulation scheme focusing on manifolds that can be covered by a single chart. The scheme follows the partial observation scheme developed [19]. Representing the product process in coordinates and using a transformation \(L\), whose kernel is the diagonal \(\operatorname{diag}(M^{n})\), gives a guided bridge process converging to the diagonal. An explicit expression for the likelihood is given.  
In the following, we assume that \(M\) can be covered by a chart in which the square root of the cometric tensor, denoted by \(\sigma\), is \(C^{2}\). Furthermore, \(\sigma\) and its derivatives are bounded; \(\sigma\) is invertible with bounded inverse. The drift \(b\) is locally Lipschitz and locally bounded.  
Let \(x_{1},\dots,x_{n}\in M\) be observations and let \(X_{1,t},\dots,X_{n,t}\) be independent Brownian motions with \(X_{1,0}=x_{1},\dots,X_{n,0}=x_{n}\). Using the coordinate SDE (15) for each \(X_{i,t}\), we can write the entire system on \(M^{n}\) as \[d\begin{pmatrix}X_{1,t}^{1}\\ \vdots\\ X_{1,t}^{d}\\ \vdots\\ X_{n,t}^{1}\\ \vdots\\ X_{n,t}^{d}\end{pmatrix}=\begin{pmatrix}b^{1}(X_{1,t})\\ \vdots\\ b^{d}(X_{1,t})\\ \vdots\\ b^{1}(X_{n,t})\\ \vdots\\ b^{d}(X_{n,t})\end{pmatrix}dt+\left[\begin{array}{c}\Sigma^{1}(X_{1,t}, \ldots,X_{n,t})\\ \vdots\\ \Sigma^{d}(X_{1,t},\ldots,X_{n,t})\end{array}\right]dW_{t}. \tag{18}\]  
In the product chart, \(\Sigma\) and \(b\) satisfy the same assumptions as the metric and cometric tensor and drift listed above.  
The conditioning \(X_{T}\in N\) is equivalent to the requiring \(X_{T}\in\operatorname{diag}((\mathbb{R}^{d})^{n})\) in coordinates. \(\operatorname{diag}((\mathbb{R}^{d})^{n})\) is a linear subspace of \((\mathbb{R}^{d})^{n}\), we let \(L\in M^{d\times nd}\) be a matrix with orthonormal rows and \(\ker L=\operatorname{diag}((\mathbb{R}^{d})^{n})\) so that the desired conditioning reads \(LX_{T}=0\). Define the following oblique projection, similar to [19],  
\[P_{t}(x)=a(x)L^{T}A(x)L \tag{19}\]  
where  
\[a(x)=\Sigma(x)\Sigma(x)^{T}\quad\text{and}\quad A_{t}(x)=(La(x)L^{T})^{-1}.\]  
Set \(\beta(x)=\Sigma(x)^{T}L^{T}A(x)\). The guiding scheme (13) then becomes  
\[dY_{t}=b(Y_{t})dt+\Sigma(Y_{t})dW_{t}-\Sigma(Y_{t})\beta(Y_{t})\frac{LY_{t}}{ T-t}1_{(T-e,T)}(t)dt,\quad Y_{0}=u. \tag{20}\]  
We have the following result.  
Lemma 1: _Equation (20) admits a unique solution on \([0,T)\). Moreover, \(\|LY_{t}\|^{2}\leq C(\omega)(T-t)\log\log[(T-t)^{-1}+e]\) a.s., where \(C\) is a positive random variable._  
Proof: Since \(LP=L\), the proof is similar to the proof of [19] Lemma 6.  
With the same assumptions, we also obtain the following result similar to [19] Theorem 3.  
Theorem 3: _Let \(Y_{t}\) be a solution of (20), and assume the drift \(b\) is bounded. For any bounded function \(f\),_  
\[\mathbb{E}[f(X)|X_{T}\in N]=\mathbb{CE}[f(y)\varphi(Y)], \tag{21}\]  
_where \(C\) is a positive constant and_  
\[\varphi(Y_{t})=\sqrt{\det(A(Y_{T}))}\exp\bigg{\{}-\frac{\|\beta _{T-e}(Y_{T-e})LY_{T-e}\|^{2}}{2\varepsilon}\] \[\qquad-\int_{T-e}^{T}\frac{2(LY_{s})^{T}Lb(Y_{s})ds-(LY_{s})^{T} d(A(Y_{s}))LY_{s}+d[A(Y_{s})^{ij},(LY_{s})_{i}(LY_{s})_{j}]}{2(T-s)}\bigg{\}}\]  
Proof: A direct consequence of [19] Theorem 3, for \(k=1\), and Lemma 1.  
The theorem can also be applied for unbounded drift by replacing \(b\) with a bounded approximation and performing a Girsanov change of measure.