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

TL;DR

The paper develops a martingale-based framework to study first-passage times of time-additive observables in finite Markov jump processes, linking large-deviation properties of O_t to first-passage statistics through Perron martingales derived from the tilted generator. By exploiting Doob's optional stopping and a one-parameter family of martingales for each eigenpair of the tilted matrix, it derives both large-threshold results (splitting probabilities and cumulant generating functions) and finite-threshold solutions, illustrated with run-and-tumble and random-walker models. The work reveals three qualitative regimes for boundary-events suppression (exponential, super-exponential, or sub-exponential) and introduces a dual process governing negative-threshold statistics, establishing connections to effective affinity and thermodynamic bounds on cumulant generating functions. These results unify first-passage theory with large-deviation principles, extend fluctuation-relations to negative thresholds, and provide practical tools for exact calculations in nonequilibrium stochastic systems. The framework offers a versatile route to universal predictions for large thresholds and actionable insights for finite-threshold problems in active matter and driven systems.

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.

Figures (6)

• Figure 1: Left: Illustration of the standard version of Gambler's ruin problem. Right: Gambler's ruin problem for a time-additive observable $O_t$ in a Markov jump process.
• Figure 2: Scaled cumulant generating functions $\lambda_O(a)$ for three qualitatively different cases corresponding with a time-additive observable $O$ that is (i) a fluctuating current $J$ with $\overline{j}> 0$ (Left Panel); (ii) a fluctuation current $J$ with $\overline{j}=0$ (Middle Panel); and (iii) the time that the process the process $X$ spends in a certain state (Right Panel). In these cases, the scaled cumulant generating function has the following qualitative properties: (i) Equation (\ref{['eq:roots']}) admits two solutions, and the minimum value of $\lambda_O(a)$ is nonzero (Left Panel); (ii) Equation (\ref{['eq:roots']}) admits two solutions, and the minimum value of $\lambda_O(a)$ equals zero (Middle Panel); and (iii) Equation (\ref{['eq:roots']}) admits one solution (Right Panel). The specific $\lambda_O(a)$ functions plotted are the following: (i) plot of $\lambda_J$ given by Eq. (\ref{['eq:lambdaJ2D']}) for $\nu=5$, $\rho=2$, and $\Delta = 0.7$ (Left Panel); (ii) plot of $\mu_+$ given by Eq. (\ref{['eq:muPMRT']}) for parameters $k_{\rm b}=1$, $k_{\rm f}=2$, and $\alpha=0.01$ (Middle Panel); and (iii) plot of $\lambda_O$ given by (\ref{['eq:lambdaOTime']}) (Right Panel). The two invertible branches of $\lambda_O(a)$ that yield $m_+$ and $m_-$ are plotted with different colour and line style.
• Figure 3: Escape problem of an active particle that leaves a strip in the two-dimensional plane of finite width. Panel (a): illustration of the considered random walk process on a two-dimensional lattice (Figure is taken from Ref. neri2022estimating). Panel (b): Three representative trajectories of $X_t$. Parameters chosen are: the rates are given by (\ref{['eq:rate1']}) and (\ref{['eq:rate2']}) with $\nu=1/2$ and $\rho = 1$, the current is given by (\ref{['eq:Jexam']}) with $\Delta = 1/2$, and the threshold parameters are setto $\ell_-=5$ and $\ell_+=10$. The length $n$ of the lattice is taken to be large, $n\gg 1$.
• Figure 4: Top Left: The scaled cumulant generating functions $m_+(\mu)$ (solid line) and $m_-(\mu)$ (dashed-dotted line) for the first passage time $T$ of a current $J$ in the two-dimensional random walk model of Sec. \ref{['sec:RW2D']}, and comparison with the bounds Eqs. (\ref{['eq:bound1x']}) (dashed line) and (\ref{['eq:bound2x']}) (dotted line), respectively. The current $J$ is of the form Eq. (\ref{['eq:Jexam']}) with $\Delta=0.7$. The cumulant generating functions $m_+$ and $m_-$ are obtained from Eqs. (\ref{['eq:mPx']}) and (\ref{['eq:mMx']}), respectively, where $a^\ast$ is the nonzero solution of (\ref{['eq:aAst2D']}), and where $-a_-(\mu)$ and $a_+(\mu)$ are the two solutions of Eq. (\ref{['eq:tobeExpv2x']}); $\dot{s}$ is the rate of dissipation given by (\ref{['eq:sdot2D']}). Top Right: Comparison between the average currents $\overline{j}$ (Eq. (\ref{['eq:JMean2D']}), solid line) and $\overline{j}^\ast$ (Eq. (\ref{['eq:JMean2Dx']}), dashed line) in the forward process and the dual processes, respectively. The vertical dotted line shows $\Delta=1/3$, corresponding with $J_t=2 S_t/( \nu (\rho+1))$. Bottom: similar plot as for the Top Right Panel, but now for the second cumulants $\sigma_J$ and $\sigma^\ast_J$ obtained from the Eqs. (\ref{['eq:sigma2J2D']}) and Eq. (\ref{['eq:sigmaJAst']}), respectively. In all panels the model parameters are $\nu=5$ and $\rho=2$.
• Figure 5: Escape problem for a one-dimensional run-and-tumble particle. Top Panel: Illustration of the rates in the studied model for one-dimensional run-and-tumble motion. Bottom Panel: kymographs (y-axis is time and x-axis is space) of two trajectories corresponding with $k_{\rm f}=2$, $k_{\rm b}=1$, $\ell_-=\ell_+=1000$ and $\alpha=0.1$ (left) and $\alpha=0.01$ (right).