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We consider a class of single-particle one-dimensional stochastic equationswhich include external field, additive and multiplicative noises. We use aparameter $\theta \in [0,1]$ which enables the unification of the traditionalIt\^o and Stratonovich approaches, now recovered respectively as the $\theta=0$and $\theta=1/2$ particular cases to derive the associated Fokker-Planckequation (FPE). These FPE is a {\it linear} one, and its stationary state isgiven by a $q$-Gaussian distribution with $q = \frac{\tau + 2M (2 -\theta)}{\tau + 2M (1 - \theta)}<3$, where $\tau \ge 0$ characterizes thestrength of the confining external field, and $M \ge 0$ is the (normalized)amplitude of the multiplicative noise. We also calculate the standard kurtosis$\kappa_1$ and the $q$-generalized kurtosis $\kappa_q$ (i.e., the standardkurtosis but using the escort distribution instead of the direct one). Throughthese two quantities we numerically follow the time evolution of thedistributions. Finally, we exhibit how these quantities can be used asconvenient calibrations for determining the index $q$ from numerical dataobtained through experiments, observations or numerical computations.

Time evolution towards q-Gaussian stationary states through unified Ito-Stratonovich stochastic equation