The fractional Brownian motion property of the turbulent refractive index and the Fermat's Extremal Principle

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We introduce fractional Brownian motion processes (fBm) as an alternative model for the turbulent index of refraction. These processes allow to reconstruct most of the index properties, but they are not differentiable. We overcome the apparent impossibility of their use in the variational equation coming from the Fermat's Principle with the introduction of a Stochastic Calculus. Afterwards, we successfully provide a solution for the stochastic ray equation; moreover, its implications in the statistical analysis of experimental data is discussed.