  For diffusion processes in dimension $d>1$, the statistics of trajectory
observables over the time-window $[0,T]$ can be studied via the Feynman-Kac
deformations of the Fokker-Planck generator, that can be interpreted as
euclidean non-hermitian electromagnetic quantum Hamiltonians. It is then
interesting to compare the four regimes corresponding to the time $T$ either
finite or large and to the diffusion coefficient $D$ either finite or small.
(1) For finite $T$ and finite $D$, one needs to consider the full
time-dependent quantum problem that involves the full spectrum of the
Hamiltonian. (2) For large time $T \to + \infty$ and finite $D$, one only needs
to consider the ground-state properties of the quantum Hamiltonian to obtain
the generating function of rescaled cumulants and to construct the
corresponding canonical conditioned processes. (3) For finite $T$ and $D \to
0$, one only needs to consider the dominant classical trajectory and its action
satisfying the Hamilton-Jacobi equation, as in the semi-classical WKB
approximation of quantum mechanics. (4) In the double limit $T \to + \infty$
and $D \to 0$, the simplifications in the large deviations in $\frac{T}{D}$ of
trajectory observables can be analyzed via the two orders of limits, i.e.
either from the limit $D \to 0$ of the ground-state properties of the quantum
Hamiltonians of (2), or from the limit of long classical trajectories $T \to
+\infty$ in the semi-classical WKB approximation of (3). This general framework
is illustrated in dimension $d=2$ with rotational invariance.
