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

Abstract

Abstract 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, which can be interpreted as Euclidean non-Hermitian electromagnetic quantum Hamiltonians. It is 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 → + ∞ 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 → 0, one only needs to consider the dominant classical trajectory and its action satisfying the Hamilton–Jacobi equation, as in the semi-classical Wentzel–Kramers–Brillouin (WKB) approximation of quantum mechanics. (4) In the double limit T → + ∞ and D → 0, the simplifications in the large deviations in T D of trajectory observables can be analyzed via the two orders of limits, i.e. either from the limit D → 0 of the ground-state properties of the quantum Hamiltonians of (2), or from the limit of long classical trajectories T → + ∞ in the semi-classical WKB approximation of (3). This general framework is illustrated in dimension d = 2 with rotational invariance.