Abrupt decorrelation for linear stochastic differential equations

Abstract: Understanding how a system loses memory of its initial state is a central problem in probability and statistics. In this manuscript, we introduce the notion of abrupt decorrelation, which explicitly characterises a sharp and sudden loss of correlation over time. We study this phenomenon within a class of linear stochastic differential equations (LSDEs), where explicit descriptions are available under various statistical distances. Our main focus is on the multivariate Ornstein-Uhlenbeck process, while in the one-dimensional case we extend the analysis to LSDEs with time-dependent drifts. The results highlight strong parallels with the cut-off phenomenon in Markov processes and contribute to a broader understanding of decorrelation in stochastic systems.