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 $g\_T = 1$?



Brownian bridge on a Lie group as a stochastic differential equation
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>


