  In [4], it is proved that we can have a continuous first-passage-time density
function of one dimensional standard Brownian motion when the boundary is
H\"older continuous with exponent greater than 1/2. For the purpose of
extending [4] into multidimensional domains, we show that there exists a
continuous first-passage-time density function of standard $d$-dimensional
Brownian motion in moving boundaries in $\mathbb{R}^{d}$, $d\geq 2$, under a
$C^{3}$-diffeomorphism. Similarly as in [4], by using a property of local time
of standard $d$-dimensional Brownian motion and the heat equation with
Dirichlet boundary condition, we find a sufficient condition for the existence
of the continuous density function.
