Fractional Brownian motion with complex variance via random walk in the complex plane and applications |
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| Institution: | 1. University of Turku, Department of Biochemistry, Food Chemistry and Food Development, FI-20014 Turku, Finland;2. University of Turku, Functional Foods Forum, FI-20014 Turku, Finland;3. Monell Chemical Senses Center, Philadelphia, PA, USA;1. Pharmacognosy Department, Faculty of Pharmacy, Helwan University, Cairo, Egypt;2. Pharmacognosy Department, Faculty of Pharmacy, Umm Al-Qura University, Makkah, Saudi Arabia;3. Pharmacognosy Department, Faculty of Pharmacy, Cairo University, Cairo, Egypt;4. Pharmaceutical Biology Department, Faculty of Pharmacy and Biotechnology, German University in Cairo (GUC), Cairo, Egypt |
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| Abstract: | A model of complex-valued fractional Brownian motion has been built up recently as the limit of a random walk in the complex plane, but this model involves radial steps only. It is shown that, by using non-radial steps, this model can be easily extended to define a fractional Brownian motion with complex-valued variance. The relations between complex-valued Brownian motion and the heat equation of order n is clarified and mainly one obtains the general expression of the probability density functions for these processes. One shows that the maximum entropy principle (MPE) provides the probability density of the complex-valued fractional Brownian motion, exactly like for the standard Brownian motion. And lastly, one shows that the heat equation of order 2n (which is the Fokker–Planck equation (FPE) of the complex-valued Brownian motion) has a solution which is similar to that of the FPE of fractional order introduced before by the author, therefore, to some extent, an identification between the complex-valued model via random walk in the complex plane and the model involving a derivative of fractional order. |
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