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 $Z_T=\int_0^T W_s^2ds$, where $(W_t)_{t\geq 0}$ is a Brownian motion with drift $μ$. More precisely, we consider the conditional law of $W$ given the event $Z_T\le θT$,