# Exploring Finite-Sized Scale Invariance in Stochastic Variability with Toy Models: The Ornstein–Uhlenbeck Model

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## Abstract

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## 1. Introduction

## 2. Stochastic Processes: Ornstein–Uhlenbeck Model for Transport

## 3. Testing Stochastic Properties with OU Simulations

## 4. Discussions and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The figure shows realizations of the Ornstein Uhlenbeck process for the original time-series ${X}_{\mathrm{t}}$ (blue) and the rescaled one ${X}_{\mathrm{ff}\ast \mathrm{t}}$ (orange) for $\alpha =2$ and $\alpha =5$, respectively.

**Figure 2.**The histograms of the random processes or time-series, $X\left(t\right)$ (blue) and $X(\alpha \ast t)$ (light orange) for $\alpha =2,3,10$. We see that histograms for both the original and scaled processes appear to be similar.

**Figure 3.**The histogram of the original OU time-series, $X\left(t\right)$ with $\mu =3.0$, $\sigma =1.0$ and $\tau =0.1$ (blue gray) which is perfectly compatible with simulations of $X(\alpha \ast t)$ (red) for $\alpha =2,3,10$. This shows that statistically the scaling of time leaves the distribution compatible with the original one for $\alpha =2$ and $\alpha =3$, but shows deviations for $\alpha =10$.

**Figure 4.**The figure shows the histogram of the original OU time-series, $X\left(t\right)$ with $\mu =3.0$, $\sigma =1.0$ and values of $\tau =0.01$ (Left), $\tau =0.05$ (Center) and $\tau =0.5$ (Right) (blue grey) compared with simulations of $X(\alpha \ast t)$ (red) for $\alpha =3$. It shows that compatibility improves with decreasing decay times, $\tau $.

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**MDPI and ACS Style**

Chakraborty, N.
Exploring Finite-Sized Scale Invariance in Stochastic Variability with Toy Models: The Ornstein–Uhlenbeck Model. *Symmetry* **2020**, *12*, 1927.
https://doi.org/10.3390/sym12111927

**AMA Style**

Chakraborty N.
Exploring Finite-Sized Scale Invariance in Stochastic Variability with Toy Models: The Ornstein–Uhlenbeck Model. *Symmetry*. 2020; 12(11):1927.
https://doi.org/10.3390/sym12111927

**Chicago/Turabian Style**

Chakraborty, Nachiketa.
2020. "Exploring Finite-Sized Scale Invariance in Stochastic Variability with Toy Models: The Ornstein–Uhlenbeck Model" *Symmetry* 12, no. 11: 1927.
https://doi.org/10.3390/sym12111927