Large deviations at level 2.5 and for trajectories observables of diffusion processes : the missing parts with respect to their random-walks counterparts

Abstract

Behind the nice unification provided by the notion of the level 2.5 in the field of large deviations for time-averages over a long Markov trajectory, there are nevertheless very important qualitative differences between the meaning of the level 2.5 for diffusion processes on one hand, and the meaning of the level 2.5 for Markov chains either in discrete-time or in continuous-time on the other hand. In order to analyze these differences in detail, it is thus useful to consider two types of random walks converging towards a given diffusion process in dimension $d$ involving arbitrary space-dependent forces and diffusion coefficients, namely (i) continuous-time random walks on the regular lattice of spacing $b$ ; (ii) discrete-time random walks in continuous space with a small time-step $\tau$. One can then analyze how the large deviations at level 2.5 for these two types of random walks behave in the limits $b\to 0$ and $\tau \to 0$ respectively, in order to describe how the fluctuations of some empirical observables of the random walks are suppressed in the limit of diffusion processes. One can then also study the limits $b\to 0$ and $\tau \to 0$ for any trajectory observable of the random walks that can be decomposed on its empirical density and its empirical flows in order to see how it is projected on the appropriate trajectory observable of the diffusion process involving its empirical density and its empirical current.

arXiv commentv2=revised version (43 pages)

References in corpus (75)

• The large deviation approach to statistical mechanics
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Macroscopic fluctuation theory
• Dynamic first-order phase transition in kinetically constrained models of glasses
• Fluctuation theorems for stochastic dynamics
• First-order dynamical phase transition in models of glasses: an approach based on ensembles of histories
• Thermodynamic formalism for systems with Markov dynamics
• Direct evaluation of large-deviation functions
• Nonequilibrium Markov processes conditioned on large deviations
• Fluctuation Theorems
• Large deviations and ensembles of trajectories in stochastic models
• Nonequilibrium microcanonical and canonical ensembles and their equivalence
• Path-integral analysis of fluctuation theorems for general Langevin processes
• Simulating rare events in dynamical processes
• Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states
• Introduction to dynamical large deviations of Markov processes
• Unravelling the large deviation statistics of Markovian open quantum systems
• Aspects of non-equilibrium in classical and quantum systems: slow relaxation and glasses, dynamical large deviations, quantum non-ergodicity, and open quantum dynamics
• Variational and optimal control representations of conditioned and driven processes
• Ergodicity and large deviations in physical systems with stochastic dynamics
• Steady state statistics of driven diffusions
• Effective interactions and large deviations in stochastic processes
• A formal view on 2.5 large deviations and fluctuation relations
• Chaotic properties of systems with Markov dynamics
• Construction of a Coordinate Bethe Ansatz for the asymmetric simple exclusion process with open boundaries
• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
• Current large deviations for driven periodic diffusions
• Making rare events typical in Markovian open quantum systems
• A minimal model of dynamical phase transition
• Transition probabilities and dynamic structure factor in the ASEP conditioned on strong flux
• Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager-Machlup approach
• Large deviations and dynamical phase transitions in stochastic chemical networks
• Dynamical phase transition in drifted Brownian motion
• Large deviations conditioned on large deviations I: Markov chain and Langevin equation
• Density profiles, dynamics, and condensation in the ZRP conditioned on an atypical current
• Microscopic structure of shocks and antishocks in the ASEP conditioned on low current
• Generic Dynamical Phase Transition in One-Dimensional Bulk-Driven Lattice Gases with Exclusion
• Level 2 large deviation functionals for systems with and without detailed balance
• Second-order dynamic transition in a p=2 spin-glass model
• Effective driven dynamics for one-dimensional conditioned Langevin processes in the weak-noise limit
• Diffusions conditioned on occupation measures
• Large deviations conditioned on large deviations II: Fluctuating hydrodynamics
• Large-deviation theory for a Brownian particle on a ring: a WKB approach
• Current fluctuations in periodically driven systems
• Spectral theory of fluctuations in time-average statistical mechanics of reversible and driven systems
• Large deviations at level 2.5 for Markovian open quantum systems: quantum jumps and quantum state diffusion
• Dynamical large deviations of reflected diffusions
• Bethe Ansatz for the Weakly Asymmetric Simple Exclusion Process and phase transition in the current distribution
• Thermodynamic formalism and large deviation functions in continuous time Markov dynamics
• Exact Distributions of Currents and Frenesy for Markov Bridges
• Dynamical large deviations of linear diffusions
• Microscopic Fluctuation Theory (mFT) for interacting Poisson processes
• Large deviations for Markov processes with stochastic resetting : analysis via the empirical density and flows or via excursions between resets
• Large Deviations for the density and the current in Non-Equilibrium-Steady-States on disordered rings
• Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
• Path integrals and stochastic calculus
• Revisiting classical and quantum disordered systems from the unifying perspective of large deviations
• Large deviations of currents in diffusions with reflective boundaries
• Periodically driven jump processes conditioned on large deviations
• Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator
• Large Deviations for dynamical fluctuations of Open Markov processes, with application to random cascades on trees
• Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models
• Statistical physics of long dynamical trajectories for a system in contact with several thermal reservoirs
• Inference of Markov models from trajectories via Large Deviations at Level 2.5 with applications to random walks in disordered media
• Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates
• Effective Hamiltonians and Lagrangians for conditioned Markov processes at large volume
• Large deviations for the Skew-Detailed-Balance Lifted-Markov processes to sample the equilibrium distribution of the Curie-Weiss model
• Jump-Drift and Jump-Diffusion Processes : Large Deviations for the density, the current and the jump-flow and for the excursions between jumps
• Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment
• Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically-constrained East Model
• Dynamical large deviations of diffusions
• BEST statistics of Markovian fluxes: a tale of Eulerian tours and Fermionic ghosts
• Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
• Time-Slicing Path-integral in Curved Space
• Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and explicit large deviations at various levels

Cited by in corpus (3)

• Large deviations in statistics of the local time and occupation time for a run and tumble particle
• Markov generators as non-hermitian supersymmetric quantum Hamiltonians: spectral properties via bi-orthogonal basis and Singular Value Decompositions
• A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains