  The problem of reconstructing the drift of a diffusion in $\erre^d$, $d\geq
2$, from the transition probability density observed outside a domain is
considered. The solution of this problem also solves a new inverse problem for
a class of parabolic partial differential equations. This work considerably
extends \cite{jsp} in terms of generality, both concerning assumptions on the
drift coefficient, and allowing for non-constant diffusion coefficient.
Sufficient conditions for solvability of this type of inverse problem for $d=1$
are also given.
