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The spectrum profile that emerges in molecular spectroscopy and atmosphericradiative transfer as the combined effect of Doppler and pressure broadeningsis known as the Voigt profile function. Because of its convolution integralrepresentation, the Voigt profile can be interpreted as the probability densityfunction of the sum of two independent random variables with Gaussian density(due to the Doppler effect) and Lorentzian density (due to the pressureeffect). Since these densities belong to the class of symmetric L\'evy stabledistributions, a probabilistic generalization is proposed as the convolution oftwo arbitrary symmetric L\'evy densities. We study the case when the widths ofthe considered distributions depend on a scale-factor $\tau$ that isrepresentative of spatial inhomogeneity or temporal non-stationarity. Theevolution equations for this probabilistic generalization of the Voigt functionare here introduced and interpreted as generalized diffusion equationscontaining two Riesz space-fractional derivatives, thus classified asspace-fractional diffusion equations of double order.

Evolution equations of the probabilistic generalization of the Voigt profile function