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Addendum to “Hitting time distributions for efficient simulations of drift‐diffusion processes”

In the paper “Hitting time distributions for efficient simulations of drift-diffusion processes”,1 we exhibited density functions and cumulative distributions for diffusion processes with and without drift. In particular, the previously well-known cumulative distribution for the hitting time T to a sphere of radius R of a particle released at the center of the sphere, and undergoing only diffusive transfer was shown in equation (4) loc. cit. It was there remarked that the distribution is convergent for T > 0 but not for T = 0 (as is immediately obvious from the series). Further, the series, as well as the series for the corresponding density of the distribution, were indicated in the paper as being derived from a previous equation (1) loc. cit., which gives the moment generating function. The derivation was said to follow when an inverse Laplace transform is computed by evaluation of the residues in a contour integral. However, during the peer-review process of a separate paper,2 we were alerted by an external reviewer, Dr. Enzhi Li, that the indicated derivation is also restricted to T > 0 only, a fact not explicitly mentioned in the paper. We would like to restate here that none of the results reported in Reference 1 are affected by this omission. Series like equation (4) of the paper, that are directly obtained from the moment generating functions are indeed not useful for nonzero times other than “large” times (ie, well away from time zero) where the probability of hitting the sphere is close to unity. Moreover, the point T = 0 is excluded in any algorithms or numerical methods for simulation, since the sphere is of radius unity (nonzero radius) and hence a zero hitting time has zero probability. In Reference 1, we emphasize the need for a Poisson resummation of such series for numerical evaluation for smaller, indeed for most times that occur in a simulation. These resummed series converge very fast near T = 0, and their T = 0 limit is also zero, though the series is not the expansion of any analytic function at time zero since all derivatives vanish. The fact that T = 0 is excluded arises from the following. The probability measure of time intervals close to zero is exponentially small in the inverse of the hitting times. When computing the hitting times near zero, we actually cut off the calculations at small enough times because the numerical programs have to evaluate large negative exponentials which result in underflow. The probabilities of hitting the sphere are set to zero at the smallest (nonzero) time for which the probabilities can be evaluated numerically. The hitting probabilities for the times at which we have to cut off the evaluations are less than e−1000, which is too small a probability to be distinguishable from zero for all practical purposes. We have introduced remarks similar to the above in the paper “Inverse functions for Monte-Carlo simulations with applications to hitting times distributions”2 following Dr. Li’s comment.