Adarsh Raghu

Martingales and First Passage Problems in Stochastic Thermodynamics

Mesoscopic systems out of equilibrium are often modelled as stochastic processes. The field of stochastic thermodynamics studies the thermodynamic structure of such systems and its connection to fluctuations of trajectory observables. In this talk I introduce an approach based on the analysis of first-passage problems for generic current-like observables using martingales. We construct an exponential family of martingales associated with fluctuating currents in Markov processes, which provides a natural framework for analysing stopping times and threshold-crossing events. Using this framework, we derive new results including thermodynamic bounds on the rate of dissipation in terms of the splitting probability and first-passage time of the current, and identify symmetry relations governing fluctuations of generic currents. Beyond these results, the martingale framework provides a unified perspective on several existing approaches to current fluctuations. In particular, we introduce an effective affinity associated with generic currents, extending the notion of thermodynamic affinities to arbitrary currents and connecting thermodynamics, large-deviation theory, and first-passage statistics. Together, these results illustrate how martingale methods provide a systematic and unifying approach to first-passage problems in stochastic thermodynamics.