On the notion of $L^1$-completeness of a stochastic flow on a manifold

Abstract

We introduce the notion of $L^1$-completeness for a stochastic flow on manifold and prove a necessary and sufficient condition for a flow to be $L^1$-complete. $L^1$-completeness means that the flow is complete (i.e., exists on the given time interval) and that it belongs to some sort of $L^1$-functional space, natural for manifolds where no Riemannian metric is specified.