Asymptotic flux across hypersurfaces for diffusion processes

We suggest a rigorous definition of the pathwise flux across the boundary of a bounded open set for transient finite energy diffusion processes. The expectation of such a flux has the property of depending only on the current velocity $v$, the nonsymmetric (as regards time reversibility) part of the drift. In the case the diffusion has a limiting velocity we then define the asymptotic (as $R\uparrow \mathrm{\infty }$) flux across subsets of the sphere of radius $R$ and compute its expectation, again in terms of $v$.