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)∼−xⁿ⁺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→0 of the probability distribution P(T,x,n) of the blowup time T for integer n>1. Krapivsky and Meerson [Phys. Rev. E112, 024128 (2025) 2470-0045 10.1103/1hds-9ttg] 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→0 tail by calculating, for all n=2,3,⋯, 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.