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

Abstract

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.

Keywords::

• Navier–Stokes equation
• Brownian motion
• group of volume-preserving homeomorphisms
• rotation process

MSC (2010) Classification::

• 58J65
• 60H10

Acknowledgements

The author would like to thank Professor Shizan Fang for suggesting him to study the stochastic Lagrangian flows on the sphere, as an example of the general framework of M. Arnaudon and A.B. Cruzeiro [3,4]. He is also grateful to the anonymous referees for their valuable suggestions which helped to correct the mistakes in the earlier version.

Additional information

Funding

This study was supported in part by the Key Laboratory of RCSDS, CAS [grant number 2008DP173182]; the NSFC [grant number 11101407], [grant number 11371099] and the AMSS [grant number Y129161ZZ1].