Bayesian Monitoring of a Diffusive Particle in One Dimension

Abstract

We study the Bayesian monitoring of a single particle diffusing along a line, while an observer tracks it continuously using noisy measurements at each spatial point. The average Shannon entropy $\overline{S(t)}$ associated with the posterior distribution of the particle's position quantifies the uncertainty that remains after conditioning on the sequence of measurements. We map the problem to the calculation of the moments of the partition functions of directed polymers in $1 + 1$ dimensions. For a constant monitoring rate, the entropy remains bounded for any finite measurement intensity and can be quantified using exact results from the Kardar-Parisi-Zhang equation, providing both the saturation value and the asymptotic behavior for long times. For a monitoring intensity that decreases according to a power law $\sim t^{-α}$, two distinct asymptotic regimes emerge: $\overline{S(t)} \simeq \tfrac12 \ln t$ for $α> 1/2$ and $\overline{S(t)} \simeq α\ln t$ for $0 < α< 1/2$. Both results are obtained using the replica method, which maps the problem onto an attractive Lieb-Liniger Hamiltonian with a time-dependent coupling: the two regimes can be understood from a perturbative expansion around free diffusion and around the attractive Lieb-Liniger ground state, respectively. We discuss the limiting case $α= 1/2$ and compare the predictions with simulations of discrete Gaussian and log-gamma polymer models.