A Poincar\'e Inequality and Exponential Decay for the Elephant Random Walk

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 Poincar\'e 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.