Fourier Analysis of Fractional Diffusion Modeling Heavy-tailed Random Walks and comparison with α-Stable Distributions

We investigate the connection between fractional partial differential equations and heavy-tailed probability distributions. Using Fourier transform methods, we derive the governing equations whose solutions correspond to the Normal and Cauchy distributions and verify their closed-form solutions. Extending this approach to the L´evy case, we find that the characteristic function of the L´evy distribution does not satisfy the standard fractional diffusion equation. Instead, by analysing its Fourier representation, we identify a new fractional advection type equation whose solution reproduces the L´evy case. This result establishes a novel deterministic link between L´evy flight statistics and fractional transport dynamics. The analytical findings are supported by numerical evaluations, confirming that the proposed equation accurately generates the probability density function associated with the alpha-stable distribution.