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 Holder 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 R-d, d >= 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.