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

TL;DR

This paper investigates the large deviations of trajectory observables for diffusion processes in dimensions greater than one, analyzing the interplay of large time and small diffusion coefficient limits through quantum Hamiltonian analogies.

Contribution

It introduces a comprehensive framework comparing different regimes of large deviations in diffusion processes, connecting quantum spectral analysis with classical trajectory approximations.

Findings

Large deviations can be characterized via quantum Hamiltonian ground states.

Classical trajectories dominate in the small diffusion limit.

The double limit simplifies the analysis of trajectory observables.

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, 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.