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Recall that the Ornstein–Uhlenbeck (OU) process in $\mathbb{R}^d$ is defined by the following SDE, $$ d Z_t=\frac{-1}{2} Z_t d t+d W_t, \quad t \geqslant 0 $$ where $\left(W_t\right)_{t \geqslant 0}$ is a standard $d$ dimensional Brownian motion. As it is well known, the process $Z$ admits the following explicit representation $$ Z_t=Z_0 e^{-t / 2}+e^{-t / 2} \int_0^t e^{s / 2} d W_s, \quad t \geqslant 0. $$ Furthermore, the infinitesimal generator of the above process is the following operator $$L:=\Delta -x\cdot \nabla.$$

I was wondering what would be a generalization of this operator to Riemannian Manifolds? I know that Brownian motion can be constructed on manifolds and its generator is simply the Laplace Beltrami operator, however, the OU process has a drift and so I am not sure what is the appropriate way to generalize this to a sphere, for instance.

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    $\begingroup$ Just take the negative gradient field of the distance function with respect to a fixed reference point. $\endgroup$
    – R W
    Commented Jan 5, 2023 at 0:20
  • $\begingroup$ @RW thanks for your comment, but will that operator be the generator of the OU process on the manifold? $\endgroup$
    – Student
    Commented Jan 5, 2023 at 0:48
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    $\begingroup$ Are you familiar with Dirichlet forms? That's how I would extend this, fix a smooth measure $\mu$ on $(M,g)$, and consider the Dirichlet form on $L^2(M, \mu)$ defined formally by $\mathcal{E}(f) = \int_M |\nabla f|^2\,d\mu$. It induces a continuous Markov process $X_t$ symmetric with respect to $\mu$. On $\mathbb{R}^n$ with $\mu$ being Gaussian measure, the resulting $X_t$ is the Ornstein-Uhlenbeck process. $\endgroup$
    – Nate Eldredge
    Commented Jan 5, 2023 at 3:10
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    $\begingroup$ Also, if $\mu$ is the Riemannian volume measure, then the resulting $X_t$ is Brownian motion on $(M,g)$. So Ornstein-Uhlenbeck and its generalizations are, in some sense, the Brownian motion for some "weighting" of $(M,g)$. $\endgroup$
    – Nate Eldredge
    Commented Jan 5, 2023 at 3:14
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    $\begingroup$ @Student: I would say the natural analogue of the Gaussian is the heat kernel. Note that by Radon-Nikodym, saying $\mu = e^{-f} d\mathrm{vol}$ is equivalent to saying "smooth measure" and just adding the condition "strictly positive density". Any two such measures are mutually absolutely continuous. $\endgroup$
    – Nate Eldredge
    Commented Jan 5, 2023 at 5:20

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Indeed as mentioned in the comments, there is a natural analogue whose Dirichlet form is in terms of a Gaussian kernel eg. see sections 3,4 in "Ornstein-Uhlenbeck Type Processes on Wasserstein Space"

First they construct the generator and Dirichlet form $dG_{Q}$ (for covariance kernel $Q$) on the tangent space. And then we can use the exponential map sending $Exp: T_{x}\to M$ to precompose to create a Dirichlet form measure $dN_{Q}=dG_{Q}\circ Exp^{-1}$ on the manifold.

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