Let $(\mathcal{M}, d)$ be a connected compact Riemannian submanifold without boundary of $\mathbb{R}^m$, let $\mu(\mathrm{d} x)= p(x) \dd x$ be a probability measure on $\mathcal{M}$, where $\mathrm{d} x$ is the volume measure and $d $ denotes the geodesic distance. Consider the diffusion process $(X_t)_{t \ge0}$ generated by the operator $\cL:= \Delta + \dfrac{\nabla p}{p} \cdot \nabla$, which can be regarded as the limit of geometric random walks over i.i.d. points on the manifold \cite{paper2}.

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In this talk, we consider the problem of recovering information on $\mathcal{M}$, and in particular on the measure $\mu$ on $\mathcal{M}$, from its exploration by the diffusion process $(X_t)_{t\ge0}$. A natural estimator of $\mu$ is given by the occupation measure of $(X_t)_{t\ge0}$ whose speed of convergence has been established in \cite{Wang2023}. Based on ideas developed in the i.i.d. setting by \cite{Divol2022}, we consider the smoothed kernel estimator of the invariant density, $p_{T,h}$, of which we expect faster convergence rate:

$$p_{T,h}(y) := \frac{1}{T}\int_0^T K_h(X_t,y) \dd t,$$
with $K_h(x,y) := \rho_h(x)^{-1}K\PAR{\frac{\|x-y\|}{h}}$ and $\rho_h(x) = \int_\cM K\PAR{\frac{\|x,y\|}{h}} \dd y$, where $K: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ is a kernel function and where $\|\cdot \|$ denotes the Eulidean norm of $\mathbb{R}^m$.

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The convergence speed of $p_{T,h}$ is studied in Wasserstein distance and its dependence on the order of $K$ and the regularity of $p$ and $\cM$ are discussed.

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This is a joint work with Hélène Guérin, Viet-Chi Tran, and Vincent Divol.