A White Noise Approach to Occupation Times of Brownian Motion

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.


Introduction
Technical textiles have attracted great attention to diverse branches of industry over the last decades due to their comparatively cheap manufacturing. By overlapping thousands of individual slender fibers, random fiber webs emerge yielding nonwoven materials that find applications e.g. in textile, building and hygiene industry as integral components of baby diapers, closing textiles, filters and medical devices, to name but a few. They are (Received 02- ; Revised 17-11-2022; Accepted 18-11-2022) Corresponding Author: produced in melt-spinning operations: hundreds of individual endless fibers are obtained by the continuous extrusion of a molten polymer through narrow nozzles that are densely and equidistantly placed in a row at a spinning beam. The viscous or viscoelastic fibers are streched and spun until they solidify due to cooling air streams. Before the elastic fibers lay down on a moving conveyor belt to form a web, they become entangled and form loops due to the highly turbulent air flows. The homogeneity and load capacity of the fiber web are the most important textile properties for quality assesment of industrial nonwoven fabrics. The optimization and control of the fleece quality require modeling and simulation of fiber dynamics and lay-down. Available data to judge the quality, at least on the industrial scale, are usually the mass per unit area of the fleece.
Since the mathematical treatment of the whole process at a stroke is not possible due to its complexity, a hierarchy of models that adequately describe partial aspects of the process chain has been developed in research during the last years. A stochastic model for the fiber deposition in the nonwoven production was proposed and analyzed in [4,5,7,10]. The model is based on stochastic differential equations describing the resulting position of the fiber on the belt under the influence of turbulent air flows. In [1] parameter estimation of the Ornstein-Uhlenbeck process from available mass per unit area data, the occupation time in mathematical terms, was done. Motivated by the above mentioned problem, in this paper we study the occupation time of one-dimensional Brownian motion. In particular, we show that occupation times of one-dimensional Brownian motion is a white noise distribution in the sense of Hida.
Although it is possible to study the problem by classical probabilistic method, we use a white noise approach to generalize the concept also to higher dimensions in later research.
Moreover, in future work an extension to more general process (e.g. with fractional Gaussian noise) is planed. In the next section we provide neccesary background on the white noise theory. The main result together with its proof are given afterward.

White Noise Analysis
In this section we give background on the white noise theory used throughout this paper. For a more comprehensive discussions including various applications of white noise theory we refer to [8,9,12,13] and references therein. We start with the Gelfand where C is the Borel σ-algebra generated by cylinder sets on S ′ (R) and the unique probability measure µ is established through the Bochner-Minlos theorem by fixing the characteristic function Here |·| 0 denotes the usual norm in the L 2 (R), and ⟨·, ·⟩ denotes the dual pairing between S ′ (R) and S(R). The dual pairing is considered as the bilinear extension of the inner product on L 2 (R), i.e.
for all g ∈ L 2 (R) and f ∈ S(R). This probability space is known as the real-valued white noise space since it contains the sample paths of the one-dimensional Gaussian white noise. In this setting a one-dimensional Brownian motion can be represented by a continuous modification of the stochastic process B = (B t ) t≥0 with where 1 A denotes the indicator function of a set A ⊂ R.

If a stochastic distribution process
In the following we state a sufficient condition on the Bochner integrability of a family of Hida distributions which depend on an additional parameter.
The corresponding Donsker's delta distribution is given by It has been proved that δ 0 (B t − x) ∈ (S) * . Furthermore, its S-transform is given by for any φ ∈ S(R). For details and proofs see e.g. [8,12]. The Donsker delta distribution is an important research object in the Gaussian analysis. For example, it can be used to study local times, self-intersection local times, stochastic current and Feynman integrals, see e.g. [2,3,6,14,15,18]. The derivatives of Donsker's delta distribution has been also studied in [16]. In [17] Donsker's delta distribution is analyzed in the context of stochastic processes with memory.

Main Result
Now we are ready to prove the main finding of the paper. Proof: It is apparent, at least formally, that occupation times can be obtained by integrating Donsker's delta distribution with respect to the product measure on [a, b] × [0, T ]. In this regard we will use Kondratiev-Streit integration theorem (Theorem 2.2) to prove the statement. Observe that for any φ ∈ S(R) is a measurable function with respect to the product measure on [0, T ] × [a, b]. Now for any z ∈ C and φ ∈ S(R) we have The first factor

Conclusions
We give a mathematically rigorous meaning to the occupation times of a standard Brownian motion as a Hida distribution. An expression for the expected value for the occupation times is also obtained. For the application point of view it is desirable to have a more explicit form for the expected value. This will be done in the future work. We would like also to mention that our present result is limited to one-dimensional setting.
For further research we plan to generalize the result to higher spatial dimensions.