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