Short-time blowup statistics of a Brownian particle in repulsive potentials

We study the dynamics of an overdamped Brownian particle in a repulsive scale-invariant potential $V(x) \sim -x^{n+1}$. For $n > 1$, a particle starting at position $x$ reaches infinity in a finite, randomly distributed time. We focus on the short-time tail $T \to 0$ of the probability distribution $P(T, x, n)$ of the blowup time $T$ for integer $n > 1$. Krapivsky and Meerson [Phys. Rev. E \textbf{112}, 024128 (2025)] recently evaluated the leading-order asymptotics of this tail, which exhibits an $n$-dependent essential singularity at $T = 0$. Here we provide a more accurate description of the $T \to 0$ tail by calculating, for all $n = 2, 3, \dots$, the previously unknown large pre-exponential factor of the blowup-time probability distribution. To this end, we apply a WKB approximation -- at both leading and subleading orders -- to the Laplace-transformed backward Fokker--Planck equation governing $P(T, x, n)$. For even $n$, the WKB solution alone suffices. For odd $n$, however, the WKB solution breaks down in a narrow boundary layer around $x = 0$. In this case, it must be supplemented by an ``internal'' solution and a matching procedure between the two solutions in their common region of validity.