Geodesic
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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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},
     year = {2021},
     volume = {66},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a8/}
}
TY  - JOUR
AU  - J. M. Lee
TI  - The first passage time density of Brownian motion and the heat equation
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2021
SP  - 175
EP  - 195
VL  - 66
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a8/
LA  - ru
ID  - TVP_2021_66_1_a8
ER  - 
%0 Journal Article
%A J. M. Lee
%T The first passage time density of Brownian motion and the heat equation
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2021
%P 175-195
%V 66
%N 1
%U http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a8/
%G ru
%F 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/

[1] Jimyeong Lee, “First passage time densities through Hölder curves”, ALEA Lat. Am. J. Probab. Math. Stat., 15:2 (2018), 837–849 | DOI | MR | Zbl

[2] K. Burdzy, Zhen-Qing Chen, J. Sylvester, “The heat equation and reflected Brownian motion in time-dependent domains”, Ann. Probab., 32:1B (2004), 775–804 | DOI | MR | Zbl

[3] S. Kou, Haowen Zhong, “First-passage times of two-dimensional Brownian motion”, Adv. in Appl. Probab., 48:4 (2016), 1045–1060 | DOI | MR | Zbl

[4] M. E. Taylor, Partial differential equations, v. 1, Appl. Math. Sci., 115, Basic theory, 2nd ed., Springer-Verlag, Berlin, 2011, xxii+654 pp. | DOI | MR | Zbl

[5] S. Lang, Fundamentals of differential geometry, Grad. Texts in Math., 191, Springer-Verlag, New York, 1999, xviii+535 pp. | DOI | MR | Zbl

[6] A. Friedman, Partial differential equations of parabolic type, Dover Books on Math., Paperback ed., Dover Publications, Mineola, NY, 2008, 368 pp. | MR | Zbl | Zbl

[7] J. R. Cannon, The one-dimensional heat equation, Encyclopedia Math. Appl., 23, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984, xxv+483 pp. | DOI | MR | Zbl

[8] J. L. Lewis, M. A. M. Murray, The method of layer potentials for the heat equation in time-varying domains, Mem. Amer. Math. Soc., 114, no. 545, Amer. Math. Soc., Providence, RI, 1995, viii+157 pp. | DOI | MR | Zbl