Concentration Inequalities for Fluctuations in Classical and Quantum Markov Processes - Bakewell-Smith, George (2025)
Markov processes describe a plethora of physical systems in nature, including the mechanics of molecular motors, chemical reactions, and even financial markets. In the framework of continuous-time Markov processes, there are many important quantities of interest. One example are dynamical observables: time-integrated functionals of stochastic trajectories. Another are first passage times (FPTs), which are the time taken for an observable to reach a fixed threshold. The fluctuations of these two quantities are bounded from below by the thermodynamic uncertainty relations (TURs), a fundamental result which captures the tradeoff between their precision, and physical parameters such as entropy production or activity. Whilst these lower bounds are well-known, less attention has been given to finding upper bounds on fluctuations.