  We consider stochastic differential equations on the group of
volume-preserving homeomorphisms of the sphere $S^d\,(d\geq 2)$. The diffusion
part is given by the divergence free eigenvector fields of the Laplacian acting
on $L^2$-vector fields, while the drift is some other divergence free vector
field. We show that the equation generates a unique flow of measure-preserving
homeomorphisms when the drift has first order Sobolev regularity, and derive a
formula for the distance between two Lagrangian flows. We also compute the
rotation process of two particles on the sphere $S^2$ when they are close to
each other.

Probability