First Passage Time Densities through Holder curves

We prove that for a standard Brownian motion, there exists a firstpassage-time density function through a Holder curve with exponent greater than 1/2. With a property of local time of a standard Brownian motion and adopting the theories of partial differential equations in Cannon (1984) and the strategies in Fasano (2005) and Carinci et al. (2016), 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.