On singularity as a function of time of a conditional distribution of an exit time

TL;DR

This paper investigates the singularity of the conditional exit time distribution of two Brownian motions, revealing conditions under which the distribution's density with respect to Lebesgue measure becomes singular over time.

Contribution

It establishes the singularity of the conditional exit time distribution as a function of time and analyzes the behavior of solutions to the heat equation with Brownian boundary conditions.

Findings

Conditional exit time distribution is singular with respect to Lebesgue measure over time.

Normal derivative of the heat equation solution vanishes at the boundary with high diffusion.

Results connect stochastic exit times with PDE boundary behavior.

Abstract

We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability that the sum of two one-dimensional Brownian motions will exit from the unit interval before time $t$, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.