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.
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