Brownian motion with drift conditioned to have restricted $L^2$-norm

We study the long-time behaviour of Brownian motion with drift under a rare-event conditioning of the quadratic additive functional $Z_T={\int }_0^T{W}_s^{2}ds$, where $(W_t{)}_{t\ge 0}$ is a Brownian motion with drift $\mu$. More precisely, we consider the conditional law of $W$ given the event $Z_T\le \theta T$, as $T\to \mathrm{\infty }$, allowing the drift $\mu$ to depend on $T$. We derive sharp, uniform small-deviation asymptotics for $Z_T$, including the exact prefactor, and use them to show that the conditioned process converges weakly to an Ornstein–Uhlenbeck process. Remarkably, the limiting dynamics are independent of the original drift.