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For diffusion processes in dimension $d>1$, the statistics of trajectoryobservables over the time-window $[0,T]$ can be studied via the Feynman-Kacdeformations of the Fokker-Planck generator, that can be interpreted aseuclidean non-hermitian electromagnetic quantum Hamiltonians. It is theninteresting to compare the four regimes corresponding to the time $T$ eitherfinite or large and to the diffusion coefficient $D$ either finite or small.(1) For finite $T$ and finite $D$, one needs to consider the fulltime-dependent quantum problem that involves the full spectrum of theHamiltonian. (2) For large time $T \to + \infty$ and finite $D$, one only needsto consider the ground-state properties of the quantum Hamiltonian to obtainthe generating function of rescaled cumulants and to construct thecorresponding canonical conditioned processes. (3) For finite $T$ and $D \to0$, one only needs to consider the dominant classical trajectory and its actionsatisfying the Hamilton-Jacobi equation, as in the semi-classical WKBapproximation of quantum mechanics. (4) In the double limit $T \to + \infty$and $D \to 0$, the simplifications in the large deviations in $\frac{T}{D}$ oftrajectory 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 quantumHamiltonians of (2), or from the limit of long classical trajectories $T \to+\infty$ in the semi-classical WKB approximation of (3). This general frameworkis illustrated in dimension $d=2$ with rotational invariance.

cornell university

Large deviations for trajectory observables of diffusion processes in  dimension $d>1$ in the double limit of large time and small diffusion  coefficient