## 4 Discussion
The super-statistical approach states that the interevent times of an aftershock sequence may be described by a local Poisson process \(p(T|\beta)=\beta e^{-\beta T}\), with \(\beta\) an intensive fluctuatingparameter. On a long-time scale, \(\beta\) is distributed with a possibility density \(f(\beta)\)[25; 44; 45; 46; 47]. Then the probability distribution \(p(T)\) is given as:  
\[p(T)=\int_{0}^{\infty}f(\beta)\beta e^{-\beta T}d\beta, \tag{14}\]  
In the scenario where a Gamma distribution provides the probability density of \(\beta\)[43; 44; 45; 46]:  
\[f(\beta)=\frac{1}{\Gamma(n/2)}\left(\frac{n}{2\beta_{0}}\right)^{n/2}\beta^{ \frac{n}{2}-1}exp\left(-\frac{n\beta}{2\beta_{0}}\right), \tag{15}\]  
Integration of Equation (14) is analytically calculated [48], obtained \(p(T)\approx C(1+B(q-1)T)^{1/(1-q)}\), which is exactly the result in term of NESP, where \(q=1+\left([2(n+2)]\right)\) and \(B=2\beta_{0}/(2-q)\)[14]. Since in all the analyzed cases q is in the range 1.61-1.71, we conclude that the corresponding number of degrees of freedom is close to one (\(n=1\)). The latter implies that the evolution of an aftershock sequence could be influenced by a stochastic mechanism with memory effects. In accordance with [49; 50], the stochastic differential equation for the evolution with time \(t\), of interevent times \(T\) of an aftershock sequence:  
\[dT=-\gamma(T-<T>)dt+\varphi\sqrt{T}W_{t}, \tag{16}\]  
This stochastic equation is made up of two parts that control how the seismicity evolves. The primary goal of the first deterministic term is to restore the seismic rate \(R\) to its usual value of \(R=1/\zeta T>\) based on a constant \(\gamma\) which expresses the rate of relaxation to the mean waiting time \(<T>\). Memory effects in seismicity's development are depicted in the second stochastic part. The stochastic term \(W_{t}\) describes a Wiener process following a Gaussian distribution with zero mean and unitary variance that could follow the macroscopic effects in the evolution of interevent times in the aftershock sequence. \(W_{t}\)' random sign causes an increase (\(W_{t}>0\)) or decrease (\(W_{t}<0\)) of \(T\). The construction of this term operates in a way that large values of \(T\) cause large amplitude of the stochastic term, which leads to an increase or decrease in \(T\) depending on the sign of \(W_{t}\). The parameter \(\varphi\) introduces a noise component to the process and can be expressed as \(\varphi=\sqrt{2\gamma<T>}\)[50]. Equation (16) is a stochastic differential equation that represents a multiplicative noise example, known as the Feller process [48; 49; 50]. We write the corresponding Fokker-Planck equation for Equation (16), to ascertain the evolution of the interevent time series \(T\) after some time \(t\), given the probability distribution \(f(T,t)\), as [51]:  
\[\frac{\partial f(T,t)}{\partial t}=\frac{\partial}{\partial T}[\gamma(T-<T>) f(T,t)]+\frac{\partial^{2}}{\partial T^{2}}[T<T>\gamma f(T,t)], \tag{17}\]  
The latter Fokker-Planck equation's stationary solution, Equation (17), is the distribution [48]:  
\[p(T/<T>)=f(T)=\frac{1}{<T>}e^{-\frac{2}{<T>}T}, \tag{18}\]  
where Equation (18) presents the conditional probability of \(T\) given \(<\)T\(>\). It is necessary to account for local variations in the seismic rate \(R=1/<\)T\(>\) associated with non-stationarities in the evolution of the earthquake activity over time scales significantly larger than \(1/\gamma\) in order to achieve stationarity in Equation (16).