Martingale approach for first-passage problems of time-additive observables in Markov processes

Abstract

We develop a method based on martingales to study first-passage problems of time-additive observables exiting an interval of finite width in a Markov process. In the limit that the interval width is large, we derive generic expressions for the splitting probability and the cumulants of the first-passage time. These expressions relate first-passage quantities to the large deviation properties of the time-additive observable. We find that there are three qualitatively different regimes depending on the properties of the large deviation rate function of the time-additive observable. These regimes correspond to exponential, super-exponential, or sub-exponential suppression of events at the unlikely boundary of the interval. Furthermore, we show that the statistics of first-passage times at both interval boundaries are in general different, even for symmetric thresholds and in the limit of large interval widths. While the statistics of the times to reach the likely boundary are determined by the cumulants of the time-additive observables in the original process, those at the unlikely boundary are determined by a dual process. We obtain these results from a one-parameter family of positive martingales that we call Perron martingales, as these are related to the Perron root of a tilted version of the transition rate matrix defining the Markov process. Furthermore, we show that each eigenpair of the tilted matrix has a one-parameter family of martingales. To solve first-passage problems at finite thresholds, we generally require all one-parameter families of martingales, including the non-positive ones. We illustrate this by solving the first-passage problem for run-and-tumble particles exiting an interval of finite width.

45 pages, 6 figures. This version corrects a few typos, and corrects Eq. (131)

References in corpus (46)

• The large deviation approach to statistical mechanics
• A Gallavotti-Cohen Type Symmetry in the Large Deviation Functional for Stochastic Dynamics
• Dissipation bounds all steady-state current fluctuations
• Light fields in complex media: mesoscopic scattering meets wave control
• Tug-of-war as a cooperative mechanism for bidirectional cargo transport by molecular motors
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Bacterial hydrodynamics
• Diffusive transport without detailed balance in motile bacteria: Does microbiology need statistical physics?
• Nonequilibrium Markov processes conditioned on large deviations
• Universal bounds on current fluctuations
• Large deviations and ensembles of trajectories in stochastic models
• Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension
• Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states
• Introduction to dynamical large deviations of Markov processes
• Simple bounds on fluctuations and uncertainty relations for first-passage times of counting observables
• Fundamental Bounds on First Passage Time Fluctuations for Currents
• Steady state statistics of driven diffusions
• The dissipation-time uncertainty relation
• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
• Statistics of Infima and Stopping Times of Entropy Production and Applications to Active Molecular Processes
• Decision Making in the Arrow of Time
• Current large deviations for driven periodic diffusions
• Two refreshing views of Fluctuation Theorems through Kinematics Elements and Exponential Martingale
• Understanding Mechanochemical Coupling in Kinesins Using First-Passage Time Processes
• Integral Fluctuation Relations for Entropy Production at Stopping Times
• Large Deviation Function for a Driven Underdamped Particle in a Periodic Potential
• Waiting for rare entropic fluctuations
• Large-deviation theory for a Brownian particle on a ring: a WKB approach
• Current fluctuations in periodically driven systems
• Thermodynamic uncertainty relation for first-passage times on Markov chains
• Computation of current cumulants for small nonequilibrium systems
• Kinetic uncertainty relation on first passage time for accumulated current
• Splitting Probabilities of Jump Processes
• Martingales for physicists: A treatise on stochastic thermodynamics and beyond
• Estimating Entropy Production Rates with First-Passage Processes
• Universal tradeoff relation between speed, uncertainty, and dissipation in nonequilibrium stationary states
• Methods and Conversations in (Post)Modern Thermodynamics
• Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates
• Extreme value statistics of edge currents in Markov jump processes and their use for entropy production estimation
• Active Random Walks in One and Two Dimensions
• Run-and-tumble particle in one-dimensional potentials: mean first-passage time and applications
• Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle
• Large deviations of the stochastic area for linear diffusions
• Microcanonical conditioning of Markov processes on time-additive observables
• Work statistics at first-passage times
• On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities

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