Emergence of heavy-tailed skew distributions from the heat equation

It is well known that the symmetric Gaussian function, called the fundamental solution, serves as the Green's function of the heat equation. In reality, on the other hand, distribution functions obtained empirically often differ from the Gaussian function. This study presents a new solution of the heat equation, satisfying localized initial conditions like the Gaussian fundamental solution. The new solution corresponds to a hetero-mixture distribution, which generalizes the Gaussian distribution function to a skewed and heavy-tailed distribution, and thus provides a candidate for the empirical distribution functions.