Conditional splitting probabilities for hidden-state inference in drift-diffusive processes
Abstract
For a one-dimensional process in a bounded interval, splitting probabilities quantify the likelihood of exiting said interval for the first time via either boundary. For two-dimensional Markov processes {X(t), Y(t)} t∈T , a joint analogue of the splitting probabilities can be defined, which captures the likelihood that the variable X(t), having been initialized at x 0 ∈ L, exits L for the first time via either of the interval boundaries and that the variable Y(t), initialized at y 0 , is y exit at the time of exit. Drawing on Bayes' theorem, we introduce the related notion of conditional splitting probabilities and formalize a simple scheme that leverages them to partially infer the assumedly hidden state Y(t) from detecting the particle exiting at either interval boundary. We demonstrate the viability of this boundary-based inference scheme for two classes of processes: processes where X(t) is Brownian motion and Y(t) is a decoupled internal state, and unidirectionally coupled processes where X(t) is drift-diffusive and depends on Y(t), while Y(t) evolves independently. For the first class we obtain generic expressions in terms of the eigensystem of the Fokker-Planck operator for the Y dynamics, while for the second we carry out explicit derivations for three paradigmatic cases (run-and-tumble motion, diffusion in an intermittent piecewise-linear potential and diffusion with stochastic resetting). Drawing on Bayes' theorem, we subsequently introduce the related notion of conditional splitting probabilities, defined as the posterior likelihoods of the internal state Y \emph{given} that the observable degree of freedom X has undergone a specific exit event. After computing these conditional splitting probabilities, we propose a simple scheme that leverages them to partially infer the assumedly hidden state Y(t) from point-wise detection events.
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BibTeX
@article{sezik2026conditional,
title = {{Conditional splitting probabilities for hidden-state inference in drift-diffusive processes}},
author = {Emir Sezik and Jacob Knight and Henry Alston and Connor Roberts and Thibault Bertrand and Gunnar Pruessner and Luca Cocconi},
journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
year = {2026},
month = jun,
doi = {10.1098/rspa.2025.0716},
eprint = {2508.07386},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05758232v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Conditional splitting probabilities for hidden-state inference in drift-diffusive processes
AU - Emir Sezik
AU - Jacob Knight
AU - Henry Alston
AU - Connor Roberts
AU - Thibault Bertrand
AU - Gunnar Pruessner
AU - Luca Cocconi
PY - 2026
DA - 2026/06/10
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
DO - 10.1098/rspa.2025.0716
UR - https://hal.science/hal-05758232v1
AB - For a one-dimensional process in a bounded interval, splitting probabilities quantify the likelihood of exiting said interval for the first time via either boundary. For two-dimensional Markov processes {X(t), Y(t)} t∈T , a joint analogue of the splitting probabilities can be defined, which captures the likelihood that the variable X(t), having been initialized at x 0 ∈ L, exits L for the first time via either of the interval boundaries and that the variable Y(t), initialized at y 0 , is y exit at the time of exit. Drawing on Bayes' theorem, we introduce the related notion of conditional splitting probabilities and formalize a simple scheme that leverages them to partially infer the assumedly hidden state Y(t) from detecting the particle exiting at either interval boundary. We demonstrate the viability of this boundary-based inference scheme for two classes of processes: processes where X(t) is Brownian motion and Y(t) is a decoupled internal state, and unidirectionally coupled processes where X(t) is drift-diffusive and depends on Y(t), while Y(t) evolves independently. For the first class we obtain generic expressions in terms of the eigensystem of the Fokker-Planck operator for the Y dynamics, while for the second we carry out explicit derivations for three paradigmatic cases (run-and-tumble motion, diffusion in an intermittent piecewise-linear potential and diffusion with stochastic resetting). Drawing on Bayes' theorem, we subsequently introduce the related notion of conditional splitting probabilities, defined as the posterior likelihoods of the internal state Y \emph{given} that the observable degree of freedom X has undergone a specific exit event. After computing these conditional splitting probabilities, we propose a simple scheme that leverages them to partially infer the assumedly hidden state Y(t) from point-wise detection events.
SN - 1364-5021
SN - 1471-2946
ER -