  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.
