Stochastic Lagrangian flows on the group of volume-preserving homeomorphisms of the spheres

We consider stochastic differential equations on the group of volume-preserving homeomorphisms of the sphere . The diffusion part is given by the divergence-free eigenvector fields of the Laplacian acting on -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 when they are close to each other.