Dynamical large deviations of diffusions

Abstract

We solve two problems related to the fluctuations of time-integrated functionals of Markov diffusions, used in physics to model nonequilibrium systems. In the first we derive and illustrate the appropriate boundary conditions on the spectral problem used to obtain the large deviations of current-type observables for reflected diffusions. For the second problem we study linear diffusions and obtain exact results for the generating function associated with linear additive, quadratic additive and linear current-type observables by using the Feynman-Kac formula. We investigate the long-time behavior of the generating function for each of these observables to determine both the so-called rate function and the form of the effective process responsible for manifesting the fluctuations of the associated observable. It is found that for each of these observables, the effective process is again a linear diffusion. We apply our general results for a variety of linear diffusions in ${\mathbb{R}}^2$, with particular emphasis on investigating the manner in which the density and current of the original process are modified in order to create fluctuations.

v1: PhD thesis, 119 pages, 18 figures. v2: PhD thesis, 120 pages. Minor corrections, notation changes. v3: Final version, including all acknowledgements

Cited by in corpus (12)

• Dynamical large deviations of linear diffusions
• Large deviations of the stochastic area for linear diffusions
• Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
• Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and explicit large deviations at various levels
• Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics
• On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival
• Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts
• Large deviations for trajectory observables of diffusion processes in dimension $d>1$ in the double limit of large time and small diffusion coefficient
• A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains
• Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions
• Markov spin models for image generation : explicit large deviations with respect to the number of pixels
• Inverse problem in the conditioning of Markov processes on trajectory observables : what canonical conditionings can connect two given Markov generators ?