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

In [J. Lee, ALEA Lat. Am. J. Probab. Math. Stat., 15 (2018), pp. 837–849] it is proved that we can have a continuous first-passage-time density function of one-dimensional standard Brownian motion when the boundary is Hölder continuous with exponent greater than $1/2$. For the purpose of extending the results of [J. Lee, ALEA Lat. Am. J. Probab. Math. Stat., 15 (2018), pp. 837–849] to 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 [J. Lee, ALEA Lat. Am. J. Probab. Math. Stat., 15 (2018), pp. 837–849], 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.