A White Noise Approach to Occupation Times of Brownian Motion

Herry Pribawanto Suryawan

Abstract


Occupation times of a stochastic process models the amount of time the process spends inside a spatial interval during a certain finite time horizon. It appears in the fiber lay-down process in nonwoven production industry. The occupation time can be interpreted as the mass of fiber material deposited inside some region. From application point of view, it is important to know the average mass per unit area of the final fleece. In this paper we use white noise theory to prove the existence of the occupation times of one-dimensional Brownian motion and provide an expression for the expected value of the occupation times.


Full Text:

PDF

References


W. Bock et al,"Parameter estimation from occupation times—a white noise approach," {em Communications on Stochastic Analysis}, textbf{26}(3), 29-40, 2014.

W. Bock, J. L. da Silva, and H. P. Suryawan, "Local times for multifractional Brownian motion in higher dimensions: A white noise approach," {em Infinite Dimensional Analysis, Quantum Probability and Related Topics}, textbf{19}(4), id. 1650026, 16 pp, 2016.

W. Bock, J. L. da Silva, and H. P. Suryawan, "Self-intersection local times for multifractional Brownian motion in higher dimensions: A white noise approach," {em Infinite Dimensional Analysis, Quantum Probability and Related Topics}, textbf{23}(1), id. 2050007, 18 pp, 2020.

G"otz et al, "A stochastic model and associated Fokker-Planck equation for the fiber lay-down process in nonwoven production processes," {em SIAM Journal of Applied Mathematics}, textbf{67}(6), 1704-1717, 2007.

M. Grothaus et al, "Application of a three-dimensional fiber lay-down model to non-woven production

processes," {em Journal of Mathematics in Industry}, textbf{4}(4), 1-19, 2014.

M. Grothaus, F. Riemann, and H. P. Suryawan, "A White Noise approach to the Feynman integrand for electrons in random media," {em Journal of Mathematical Physics}, textbf{55}(1), id. 013507, 16 pp, 2014.

M. Herty et al, "A smooth model for fiber lay-down processes and its diffusion approximations,"' {em Kinetic and Related Models}, textbf{2}(3), 489-502, 2009.

T. Hida et al, "White Noise. An Infinite Dimensional Calculus," textit{Kluwer Academic Publisher}, textit{Dordrecht}, 1993.

Z.Y. Huang and J. Yan, "Introduction to Infinite Dimensional Stochastic Analysis," textit{Kluwer Academic Publisher}, textit{Dordrecht}, 2000.

A. Klar, J. Maringer, and R. Wegener, "A 3D model for fiber lay-down nonwoven production processes,"' {em Mathematical Models and Methods in Applied Sciences}, textbf{22}(9), 1-18, 2012.

Y. Kondratiev et al, "Generalized functionals in Gaussian spaces: The characterization theorem revisited," {em Journal of Functional Analysis}, textbf{141} article number 0130, 301-318, 1996.

H.H. Kuo, "White Noise Distribution Theory," textit{CRC Press}, textit{Boca Raton}, 1996.

N. Obata, "White Noise Calculus and Fock Space," textit{Springer Verlag}, textit{Berlin}, 1994.

H.P. Suryawan, "A white noise approach to the self intersection local times of a Gaussian process,"' {em Journal of Indonesian Mathematical Society}, textbf{20}(2), 111-124, 2014.

H.P. Suryawan, "Weighted local times of a sub-fractional Brownian motion as Hida distributions," {em Jurnal Matematika Integratif}, textbf{15}(2), 81-87, 2019.

H.P. Suryawan, "Derivative of the Donsker delta functionals," {em Mathematica Bohemica}, textbf{144}(2), 161-176, 2019.

H.P. Suryawan, "Donsker’s delta functional of stochastic processes with memory," {em Journal of Mathematical and Fundamental Sciences}, textbf{51}(3), 265-277, 2019.

H.P. Suryawan, "Pendekatan analisis derau putih untuk arus stokastik dari gerak Brown subfraksional," {em Limits: Journal of Mathematics and Its Applications}, textbf{19}(1), 15-25, 2022.




DOI: https://doi.org/10.24071/ijasst.v4i2.5322

Article Metrics

Abstract view : 709 times
PDF view: 372 times

Refbacks

  • There are currently no refbacks.









Publisher : Faculty of Science and Technology

Society/Institution : Sanata Dharma University

 

 

 

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.