Reconstructing the drift of a diffusion from partially observed transition probabilities

The problem of reconstructing the drift of a diffusion in Rd, d⩾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 [S. Albeverio et al. J. Statist. Phys. 57(1-2) (1989) 347-356] 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.