# A Semi-Deterministic Random Walk with Resetting

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## Abstract

**:**

## 1. Preliminaries

## 2. The Model

#### 2.1. Reset Times

#### 2.2. Reset Averse and Reset-Inclined Systems

#### 2.3. Comments on Markovianness

## 3. Equilibrium Distribution

## 4. Escape Probabilities

## 5. Escape Times

#### 5.1. Symmetry Properties of First Passage Times

- If ${l}_{a}$ is defined in Equation (32) and ${l}_{a}+{l}_{b}=1$ and $\mathbb{E}X\equiv <X>$ indicates the expected value of the random variable X, we have$$\mathbb{E}{\tau}_{a,b}^{\rho}={l}_{a}\mathbb{E}{\tau}_{a,a}^{\rho}+{l}_{b}\mathbb{E}{\tau}_{b,b}^{\rho}\phantom{\rule{4.pt}{0ex}}$$
- ${\tau}_{a,a}^{\rho}$ is independent of $\rho $. Besides, the distributions in the symmetric case and one-sided case are equal, namely, for any b$${\tau}_{a,a}^{\rho}={\tau}_{a,b}^{1}={\tau}_{a,\infty}^{1}\equiv {\tau}_{a}^{1}\phantom{\rule{4.pt}{0ex}};{\tau}_{a,\infty}^{\rho}={\tau}_{a}^{\rho}\phantom{\rule{4.pt}{0ex}}$$

#### 5.2. Mean Exit Time

- (S1) ${x}_{1}>0$ and ${\mathbf{t}}_{1}>a$;
- (S2) ${x}_{1}<0$ and ${\mathbf{t}}_{1}>b$;
- (S3) ${x}_{1}>0$ and ${\mathbf{t}}_{1}\le a$;
- (S4) corresponds to having ${x}_{1}<0$ and ${\mathbf{t}}_{1}\le b$;
- (S5) corresponds to ${x}_{1}=0$.

#### 5.3. Distribution of the Exit Time

#### 5.4. FPT under the Model (11)

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A typical sample paths of the process where ${\mathbf{t}}_{1}=5,{\mathbf{t}}_{2}=7,\dots $ and ${x}_{1}={x}_{6}=1$.

**Table 1.**The table summarizes the propensity to resetting in terms of the decay of ${p}_{n}:=Pr({\mathbf{t}}_{1}=n)$ and the equilibrium distribution. In all cases $\lambda >0$.

${\mathit{p}}_{\mathit{n}\to \mathit{\infty}}$ | ${\mathit{F}}_{{\mathbf{t}}_{1}}\left(\mathit{n}\right)$ | ${\mathit{q}}_{\mathit{n}\to \mathit{\infty}}$ | Propensity | Tails | $\mathbb{E}{\mathbf{t}}_{1}$ | ${\mathit{\pi}}_{\mathit{n}\to \mathit{\infty}}$ |
---|---|---|---|---|---|---|

$O\left({e}^{-\lambda {n}^{\alpha}}\right),\alpha >1$ | $O({e}^{-\lambda {n}^{\alpha}})$ | 0 | $\phantom{\rule{4.pt}{0ex}}\mathrm{inclined}$ | Super-exp. | <∞ | $O\left({e}^{-\lambda {n}^{\alpha}}\right)$ |

$O({(\frac{{e}^{-\lambda}}{n})}^{n},$ | $O({(\frac{{e}^{-\lambda}}{n})}^{n},$ | 0 | $\phantom{\rule{4.pt}{0ex}}\mathrm{inclined}$ | Super-exp. | <∞ | $O({(\frac{{e}^{-\lambda}}{n})}^{n}$ |

$O\left({e}^{-\lambda n}\right)$ | $O\left({e}^{-\lambda n}\right)$ | $\in (0,1)$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{neutral}$ | $\phantom{\rule{4.pt}{0ex}}\mathrm{exp}.$ | <∞ | $O\left({e}^{-\lambda n}\right)$ |

$O\left({e}^{-\lambda {n}^{\alpha}}\right),0<\alpha <1$ | $O\left({e}^{-\lambda {n}^{\alpha}}\right),0<\alpha <1$ | 1 | $\phantom{\rule{4.pt}{0ex}}\mathrm{averse}$ | Sub-exp. | <∞ | $O\left({e}^{-\lambda {n}^{\alpha}}\right)$ |

$O(1/{n}^{\alpha}),\phantom{\rule{4pt}{0ex}}\alpha >2$ | $O(1/{n}^{\alpha -1}),\phantom{\rule{4pt}{0ex}}\alpha >2$ | 1 | $\phantom{\rule{4.pt}{0ex}}\mathrm{averse}$ | Power-law | <∞ | $O\left({n}^{1-\alpha}\right)$ |

$O(1/{n}^{\alpha}),\phantom{\rule{4pt}{0ex}}1<\alpha \le 2$ | $O(1/{n}^{\alpha -1}),\phantom{\rule{4pt}{0ex}}1<\alpha \le 2$ | 1 | $\phantom{\rule{4.pt}{0ex}}\mathrm{averse}$ | Power-law | =∞ | —- |

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**MDPI and ACS Style**

Villarroel, J.; Montero, M.; Vega, J.A.
A Semi-Deterministic Random Walk with Resetting. *Entropy* **2021**, *23*, 825.
https://doi.org/10.3390/e23070825

**AMA Style**

Villarroel J, Montero M, Vega JA.
A Semi-Deterministic Random Walk with Resetting. *Entropy*. 2021; 23(7):825.
https://doi.org/10.3390/e23070825

**Chicago/Turabian Style**

Villarroel, Javier, Miquel Montero, and Juan Antonio Vega.
2021. "A Semi-Deterministic Random Walk with Resetting" *Entropy* 23, no. 7: 825.
https://doi.org/10.3390/e23070825