# Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Montroll-Weiss Formalism

#### 2.2. Generalised Diffusion Equation

## 3. Main Results

#### 3.1. Exponential Memory-Kernel and Non-Gaussian Solutions

#### 3.2. Mittag-Leffler Memory–Kernel and Non-Gaussian Solutions

#### 3.3. Diffusive Aspects of Non-Singular Diffusion Equations

#### 3.4. Random Walk Process with Stochastic Resetting and Memory

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**This curves illustrate the changes of distribution $\rho (x,t)$ caused by the exponential kernel (Equation (3) in Equation (1)), and the usual random walk (Brownian motion) case $\mathcal{K}\left(t\right)=\delta \left(t\right)$ (or $\tau =0$ in Equation (3)), considering $\phi \left(x\right)=\delta \left(x\right)$, $\mathcal{D}{f}^{-1}=1$, $t=1$, and a family of characteristic times $\{{10}^{-2},{10}^{-1},0.5,1,2,4\}$ (in context Caputo-Fabrizio operator $\tau =(1-\alpha ){\alpha}^{-1}$ and $\mathcal{D}{f}^{-1}=1$).

**Figure 2.**This curves illustrate the changes of distribution $\rho (x,t)$ caused by the exponential kernel (Equation (3) in Equation (1)), considering $\phi \left(x\right)=\delta \left(x\right)$, $\mathcal{D}{f}^{-1}=1$ and $\tau =4$ to different times $\{0.1,1,2,4,6,8\}$ (in context of Caputo-Fabrizio operator $\alpha =0.2$ and $\mathcal{D}{f}^{-1}=1$).

**Figure 3.**This curves illustrate the changes of distribution $\rho (x,t)$ caused by the Mittag-Leffler kernel (Equation (4) in Equation (1)), and the usual (Brownian motion) case $\mathcal{K}\left(t\right)=\delta \left(t\right)$ (or $\alpha =1$ and $\tau \to 0$ in Equation (4)), considering $\phi \left(x\right)=\delta \left(x\right)$, $\mathcal{D}{b}^{-1}=1$, $t=1$, $\tau ={10}^{-2}$ and different values to index $\alpha $.

**Figure 4.**This curves illustrate the changes of distribution $\rho (x,t)$ caused by the Mittag-Leffler kernel (Equation (4) in Equation (1)), and the usual (Brownian motion) case $\mathcal{K}\left(t\right)=\delta \left(t\right)$ (or $\alpha =1$ and $\tau \to 0$ in Equation (4)). Considering ,$\phi \left(x\right)=\delta \left(x\right)$, $\mathcal{D}{b}^{-1}=1$, $t=1$, $\alpha =0.25$ and different values of parameter $\tau $.

**Figure 5.**These curves illustrate the changes of MSD caused by a set of fractional $\alpha $-index in Mittag-Leffler kernel, considering $\tau ={10}^{-4}$, $\mathcal{D}{b}^{-1}=0.5$ and $r=1$.

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Dos Santos, M.A.F. Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels. *Fractal Fract.* **2018**, *2*, 20.
https://doi.org/10.3390/fractalfract2030020

**AMA Style**

Dos Santos MAF. Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels. *Fractal and Fractional*. 2018; 2(3):20.
https://doi.org/10.3390/fractalfract2030020

**Chicago/Turabian Style**

Dos Santos, Maike A. F. 2018. "Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels" *Fractal and Fractional* 2, no. 3: 20.
https://doi.org/10.3390/fractalfract2030020