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The First Passage Time Density of Brownian Motion and the Heat Equation with Dirichlet Boundary Condition in Time Dependent Domains
Theory of Probability and Its Applications ( IF 0.6 ) Pub Date : 2021-05-06 , DOI: 10.1137/s0040585x97t990307
J. M. Lee

Theory of Probability &Its Applications, Volume 66, Issue 1, Page 142-159, January 2021.
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.


中文翻译:

布朗运动的第一通过时间密度和时变域中带狄利克雷边界条件的热方程

概率论及其应用,第 66 卷,第 1 期,第 142-159 页,2021 年 1 月。
在 [J. 李,ALEA 纬度。是。J. 概率。数学。Stat., 15 (2018), pp. 837--849] 证明了当边界是指数大于 1/ 的 Hölder 连续时,我们可以有一个连续的一维标准布朗运动的首次通过时间密度函数2. 为了扩展[J. 李,ALEA 纬度。是。J. 概率。数学。Stat., 15 (2018), pp. 837--849] 到多维域,我们表明在 $\mathbb{ 的移动边界中存在标准 $d$ 维布朗运动的连续第一通道时间密度函数R}^{d}$, $d\geq 2$,在 $C^{3}$-微分同胚下。与 [J. 李,ALEA 纬度。是。J. 概率。数学。Stat., 15 (2018), pp. 837--849],通过使用标准$d$维布朗运动的本地时间特性和具有狄利克雷边界条件的热方程,
更新日期:2021-07-15
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