This paper is concerned with the integration (of 1-forms) against
the Markov stochastic process associated with a second-order elliptic
differential operator in divergence form. It focuses on the limiting behavior
of the integral as the process leaves a fixed point or goes to infinity. This
extends previous work in the area where advantage was usually taken of the fact
that the operator was self adjoint and started with the associated invariant
measure. Applications are given. For example, it is a trivial consequence that
the diffusion associated to a uniformly elliptic operator on a negatively
curved Cartan–Hadamard manifold has an asymptotic direction (recovering
and strengthening the previous arguments of Pratt, Sullivan and others). The
approach can also be used to construct a Lévy area for such processes,
to study the thinness of sets for the elliptic operator, and probably has wider
applications.
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