The first passage time density of Brownian motion and the heat equation
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 1, pp. 175-195
Voir la notice de l'article provenant de la source Math-Net.Ru
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.
Keywords:
first passage time, Brownian motion, heat equation, Dirichlet boundary condition.
@article{TVP_2021_66_1_a8,
author = {J. M. Lee},
title = {The first passage time density of {Brownian} motion and the heat equation},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {175--195},
publisher = {mathdoc},
volume = {66},
number = {1},
year = {2021},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a8/}
}
J. M. Lee. The first passage time density of Brownian motion and the heat equation. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 1, pp. 175-195. http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a8/