  In complex systems such as turbulent flows and financial markets, the
dynamics in long and short time-lags, signaled by Gaussian and fat-tailed
statistics, respectively, calls for a unified description. To address this
issue we analyze a real dataset, namely, price fluctuations, in a wide range of
temporal scales to embrace both regimes. By means of Kramers-Moyal (KM)
coefficients evaluated from empirical time series, we obtain the evolution
equation for the probability density function (PDF) of price returns. We also
present consistent asymptotic solutions for the timescale dependent equation
that emerges from the empirical analysis. From these solutions, new
relationships connecting PDF characteristics, such as tail exponents, to
parameters of KM coefficients arise. The results reveal a dynamical path that
leads from Gaussian to fat-tailed statistics, furnishing insights on other
complex systems where akin crossover is observed.
