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Construction of a complex-valued fractional Brownian motion of order \(N\). (English) Zbl 0762.60073
A Brownian motion of order \(n\) \((n>2)\) is defined by a probabilistic approach different from Hochberg’s and Mandelbrot’s. This process is constructed from sums of independent \(\mathbb{R}^{1/n}_ +\)-valued random variables (rv) (where \(\mathbb{R}^{1/n}_ +=\{z\in\mathbb{C}; z^ n\in\mathbb{R}_ +\})\). Many properties of the real standard Brownian motion are generalized at order \(n\), but in the case \(n>2\), it is interesting to describe the Brownian motion of order \(n\) on the \(\sigma\) algebra \(\otimes[B(\mathbb{R}^{1/n}_ +)]^{\mathbb{R}_ +}\) [where \(B(\mathbb{R}^{1/n}_ +)\) is the \(\sigma\) algebra generated by sets of type \(A(0,h)=\{z\in\mathbb{C}; z^ n\in[0;h^ n[\}, (h\in\mathbb{R}^*_ +)]\). This \(\sigma\) algebra is totally different from \(\otimes[B(\mathbb{R})]^{\mathbb{R}_ +}\). Thus this study shows the fractal nature of the Brownian motion of order \(n\), and gives invariance scale (self-similarity) properties.

MSC:
60J65 Brownian motion
60K40 Other physical applications of random processes
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