We study the dynamics of an overdamped Brownian particle in a repulsive scale-invariant potential $V\left(x\right)\sim -x^{n+1}$. 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 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 $n=2,3,\cdots$, the previously unknown large preexponential factor of the blowup-time probability distribution. To this end, we apply a WKB (after Wentzel, Kramers and Brillouin) 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.

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• Fluctuations & noise
• Stochastic processes
• First passage problems
• Semiclassical methods