A Poincaré Inequality and Exponential Decay for the Elephant Random Walk
AIPR assessment
Problem difficulty: a niche but active theory problem, with moderate competition in reinforced random walks and quasi-stationary analysis, but not a saturated benchmark domain. Compounding strengths: the operator decomposition, discrete Poincare9 inequality, and matching upper and lower exponential bounds reinforce one another and give the paper a coherent mathematical story. Compounding weaknesses: the time-dependent conditioning notation and some proof steps are not fully clean, so verificatio
Abstract
We study the long-time behaviour of a coninuous time one-dimensional elephant random walk with an absorbing boundary. By analyzing the associated evolution equation, we identify a proper limiting operator and establish a Poincaré inequality with spectral gap of order $N^{-2}$. As a consequence, we obtain matching exponential upper and lower bounds for the survival probability, showing that it decays at rate $e^{-ct/N^2}$. The proof relies on a decomposition of the generator into a limiting operator and a time-dependent perturbation, together with spectral estimates.