Estimates for the volume variation of compact submanifolds driven by a stochastic flow

Abstract

Consider a compact submanifold N without the boundary of a Riemannian manifold M, and a stochastic flow ${\phi }_t$ associated with a stochastic differential equation. Let $N_t={\phi }_t(N)$ be the random compact submanifold obtained by the action of the stochastic flow. In this work, we present an Itô formula for the volume of the random variable $N_t$ and, as a main result, we obtain estimates for its average growth assuming that Ricci curvature is bounded. We first analyse the particular case where the submanifolds are closed curves, thus obtaining estimates for the arc length, and then we study the volume variation of compact submanifolds of dimensions greater than or equal to 2. In addition, we apply our results to the special case where the vector fields of stochastic differential equation are conformal Killing.

Keywords:

• Volume growth estimate
• stochastic flow
• compact submanifold
• Fréchet manifold

MSC2010 subject classifications:

• 60H10
• 60J60
• 58J65

Acknowledgements

We thank the anonymous reviewers for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

D. S. L. has been partially supported by São Paulo Research Foundation (FAPESP) [grant number 2012/18780-0], [grant number 2015/07278-0] and by CAPES (Finance Code 001), and F. B. S. has been supported by FAPESP [grant number 2018/16568-0].