On the heat equation with a moving boundary and applications to hitting times for Brownian motion

Abstract

In this paper we provide conditions under which the hitting-time problem for Brownian motion is equivalent to solving a heat equation with moving boundary and distributional initial conditions. Motivated by the hitting-time problem, we study the heat equation with absorbing moving boundaries. Using Fourier analysis we develop a procedure to solve this problem for a family of curves that includes the square root, quadratic, and cubic boundaries. As an application of our results, and using Sturm-Liouvile theory, we compute the density of the hitting time of a Brownian motion to a family of quadratic boundaries.

Keywords:

• Heat equation
• Brownian motion
• hitting times
• Sturm-Liouville theory

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 35K05
• 35K08
• 60J65
• 60G40