Extended Wiener Process in Nonstandard Analysis

Standing on a different view point from Anderson, we prove that the extended Wiener process defined by Anderson satisfies the definition of the Wiener process in standard analysis, for example the Wiener process at time t obeys the normal distribution ( ) 0, N t by showing the central limit theorem. The essential theory used in the proof is the extended convolution property in nonstandard analysis which is shown by Kanagawa, Nishiyama and Tchizawa (2018). When processing the extension by non-standardization, we have al-ready pointed out that it is needed to proceed the second extension for the convolution, not only to do the first extension for the delta function. In Section 2, we shall introduce again the extended convolution as prelimi-naries described in our previous paper. In Section 3, we shall provide the extended stochastic process using a hyper number N, and it satisfies the conditions being Wiener process. In Section 4, we shall give a new proof for the non-differentiability in the Wiener process.


Introduction
Anderson [1] provided stochastic processes in nonstandard analysis to show that for : f Ω → R the following condition (a) is equivalent with (b), (a) f is Loeb  for almost all ω ∈Ω (with respect to the Loeb measure L µ ), where * R is a lifted space of R constructed from ω -incomplete ultrafilter.
He proved that the measure on * R according to the process ( ) W t defined by Definition 3.2 satisfies the above condition (b). Therefore, there exists an extended Wiener measure on * R which is of Loeb measure (Loeb [2], Hurd and Loeb [3]). The existence of the extended Winer measure means that ( ) W t is an extended Wiener process in * R from Kolmogorov's extension theorem. Benot [4] [5] provided some applications of the extended Wiener measure to the stochastic analysis. Cutland [6] also introduced relevant applications of Loeb measure to the stochastic analysis.
On the other hand, in general, it is not easy to show the existence of the measurable function : F * Ω → R in (b) for stochastic processes. Therefore, we consider a scheme using extended convolution to show the existence of stochastic processes in R * without using the above result. As the main theorem in this paper, we shall provide the extended stochastic process ( ) W t described newly in * R . It satisfies the conditions being the Wiener process in nonstandard analysis. It will be proved in Section 3.
The extended Wiener process can be applied to construct some physical models, for example quantum mechanics, using the ω -incomplete ultrafilter. We consider that our observation is done through the ultrafilter, since the nature is originally described by nonstandard numbers. Because of the above reasons, we need a nonstandard analysis for the convolution by the hyperfunction.

Nonstandard Convolution by Hyperfunction
Let  be the space of all locally integrable functions on R . Define the space of all rapidly decreasing C ∞ functions  by holds for each , k m + ∈ Z , where + Z means the space of all positive integers.
From the above description, we can easily obtain that the convolution of f and g is defined by The definition of the convolution can be extended to the k-th convolution by From the above definitions, we can easily obtain the next result.
Let H  be the set of rapidly decreasing functions satisfying Figure 1 gives a typical example of hyperfunction at around the origin.

Extended Wiener Process in Nonstandard Analysis
Why does the extended stochastic process in Definition 3.2 satisfy the conditions We shall prove that the extended process Note that, in standard analysis, the last term in (2) is shown by the following

Proof of the Non-Differentiability of Wiener Process
In this section, we give the proof that the Wiener process has the property of non-differentiability. It is well known that Wiener process in standard analysis is non-differentiable a.s. though. The proof of the non-differentiability was shown by Dvoretski, Erdös and Kakutani [8]. (See e.g. Theorem 12.25 in Breiman [9].) On the other hand, the law of the iterated logarithm holds for the extended Wiener process as following (3) and (4). As to the law of the iterated logarithm in standard analysis, see e.g. Karatzas and Shreve [10], the Section 2.9. The original proof for i.i.d. random variables is due to Khintchine [11].  (3) and (4). Since the above proof in the sense of nonstandard analysis cannot be translated to the proof in standard analysis, another proof using standard one in [8] is needed to show the non-differentiability. This is a typical example of the advantage of the extended Wiener process.

Conclusion
Anderson [1] showed that the process ( ) W t defined by Definition 3.2 satisfies the conditions (1)-(4) in Definition 3.3 of Wiener process from the equivalence of (a) and (b) due to Loeb [2]. On the other hand, we showed the extended Wiener process for ( ) W t satisfies the conditions (1)-(4) directly by the nonstandardization of the convolution. When we extend the time space and obtain extended Wiener process, the nonstandard number N is efficient enough to describe the precise structure. It is also reminded that the N is applied to "extended Wiener measure" described in [12] [13]. The hyperfinite time line T is a key word. Notice that the delta function is described by the normal distribution in this state. Furthermore, we provided a new proof of the non-differentiability on the Wiener process using the extended law of the iterated logarithm for the Wiener process in nonstandard analysis.

Supported
The first author is supported in part by Grant-in-Aid Scientific Research (C), No. 18K03431, Ministry of Education, Science and Culture, Japan.