The first passage time density of Brownian motion and the heat equation with Dirichlet boundary condition in time dependent domains

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 ${\mathbb{R}}^d$, $d\ge 2$, 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.