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Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation

$$dg_t = dB_t \circ g_t$$

where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes differential in the sense of Stratonovich. We take $g_0 = 1$. (We assume the Lie group and its Lie algebra are embedded into a group of matrices and we suppressed the matrix multiplication indices in the above.)

What's the analogous equation for a Brownian bridge, i.e. condition at time $T$ so that $g_T = 1$?

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  • $\begingroup$ I am not a specialist here, but I guess that the correct definition is the pinned version: take the conditional distribution given that $g$ finishes in a neighborhood of $1$ and let the neighborhood vanish. $\endgroup$
    – zhoraster
    Commented Sep 5, 2015 at 6:55

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There is a general formula for the infinitesimal generator of the Brownian bridge on an arbitrary Riemannian manifold, and I do not see how it would become simpler in your particular case. For instance, see formula (2.2) on p.105 of

MR1027823 (90m:58227) Hsu, Pei(1-ILCC) Brownian bridges on Riemannian manifolds. Probab. Theory Related Fields 84 (1990), no. 1, 103–118

available at

http://www.math.northwestern.edu/~ehsu/Brownian%20Bridges%20on%20Riemannian%20Manifolds.pdf

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  • $\begingroup$ Thank you! I ultimately want an SDE that relates the Lie group variables to the Lie algebra variables, but that general formula will be enough for me to get going. $\endgroup$
    – Tim Nguyen
    Commented Sep 6, 2015 at 0:11

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