First Passage Time Densities through H\"older curves

TL;DR

This paper establishes the existence of first-passage-time densities for Brownian motion through H"older continuous curves with exponent greater than 1/2, linking stochastic processes with PDE theory.

Contribution

It provides a sufficient condition for the existence of first-passage-time densities through H"older curves and relates it to the heat equation's Green function derivative.

Findings

Existence of density for H"older curves with exponent > 1/2

Density proportional to the space derivative of the heat equation Green function

Utilizes properties of local time and PDE theory

Abstract

We prove that for a standard Brownian motion, there exists a first-passage-time density function through a locally H\"older continuous curve with exponent greater than 1/2. By using a property of local time of a standard Brownian motion and the theories of partial differential equations in Cannon [2], we find a sufficient condition for existence of the density function. We also show that this density function is proportional to the space derivative of the Green function of the heat equation with Dirichlet boundary condition at the moving boundary.