SDE on a manifold

A stochastic differential equation

On this page, we showed how to use the tools of differential geometry to show that a distribution \(\rho_\infty\) is stationary under the ODE:

\[ \frac{d}{dt}\begin{bmatrix} x \\ u \end{bmatrix} = \begin{bmatrix} u \\ -P(u)(\nabla S(x)/(d − 1)) \end{bmatrix} \]

This isn't ergodic, so we add a stochastic term, upgrading our equation to a stochastic differential equation (SDE).

\[ \frac{d}{dt}\begin{bmatrix} x \\ u \end{bmatrix} = \begin{bmatrix} u \\ -P(u)(\nabla S(x)/(d − 1)) + \eta P(u)dW \end{bmatrix} \]

Note

In the paper introducing MCHMC stochastic momentum updates were added to the discrete process induced by a (slightly different version of the) ODE. But no SDE was directly discussed which when discretized would give rise to those updates. This is the topic of the follow up paper.

Stationarity

🚧 Under construction 🚧

Ergodicity

🚧 Under construction 🚧