Time Development of Explosion of Stochastic Process with Repulsive Drift of Polynomial Growth - Nakamura, Toru; Ezawa, Hiroshi; Watanabe, Keiji (AIP conference proceedings, 2008)
For a diffusion process X(t) representing the momentum of a particle subject to an external force f(x) and a random force dB(t)/dt in one-dimensional space, almost all sample paths explode to infinity in finite times if f(x) grows repulsively to infinity faster than linear as |x|'[, so that the survival probability P(t) of the particle by time t decreases to zero as time passes. It is shown that P(t) decays at least exponentially in time and the rate of the exponential decay is strictly positive and equal to the lowest eigenvalue of a Hamiltonian of an imaginary-time Schrodinger equation.