Relations between L¹-completeness and continuity at infinity of stochastic flows and some applications

Abstract

We show that for a complete stochastic flow ξ(s) on a finite-dimensional manifold M there exists a proper function ψ on M such that for every orbit ξ _t,m (s) the inequality Eψ(ξ _t,m (s)) < ∞ for every s > t holds. The L ¹-completeness of a flow means that such ψ satisfies some additional conditions that make the situation closer to the property of orbits of a flow in Euclidean space to belong to the functional space L ¹ at each s and to be smooth in s in this space. We show that if a backward flow is L ¹-complete, the forward flow is continuous at infinity and if a flow with strictly elliptic generator is continuous at infinity and complete, it is L ¹-complete. Then we present two rather general examples where the forward and the backward infinitesimal generators of a flow coincide, and obtain some results of L ¹-completeness and continuity at infinity for such flows.

Keywords:

• stochastic flow
• completeness
• L ¹-completeness
• continuity at infinity
• forward and backward infinitesimal generators
• proper function

AMS Subject Classifications:

• 58J65
• 60H10

Acknowledgement

This research is supported in part by RFBR Grants No. 07-01-00137 and 08-01-00155.