Stochastic Resetting and Large Deviations

These Lecture Notes are concise and complement the recent review "J. Phys. A: Math.Theor. 53 , 193001 (2020)" by Evans, Majumdar, and Schehr. The interest and literature in stochastic has been growing rapidly, and these notes will help bring researchers up to date.

Since these are meant to be lecture notes (probably to introduce young researchers to the field), it would have been helpful to provide a few intermediate mathematical steps (throughout the paper).

Overall, the lecture notes are well-written and fulfils the acceptance criteria. I recommend it for publication.

I found a few typos/errors in these notes, which the authors should correct.

1. The factor $\sqrt{s}$ in the prefactor of the expression in equation (13) should be in the numerator, i.e., equation (13) should read
$\widetilde{P}(x,s|x_0)=\frac{1}{2}\, \sqrt{\frac{s}{D}}\, e^{-\sqrt{\frac{s}{D}}\, |x-x_0| }$

2. In line 143, "using" should be "Using"

3. With the scaling variable $z=x_0/\sqrt{4 D t}$, equation (21) should read
$-2z\frac{dg}{dz} = \frac{d^2 g}{dz^2}$. If the authors want to keep equation (21) unchanged, the correct scaling variable should be $z=x_0/\sqrt{D t}$.

4. The sentence after equation (55) ends with a mathematical expression, and the following sentence again starts with a mathematical variable $y$. It might be better to rephrase the first part of the sentence. For example, "The dimensionless parameter $y$ is the ratio..."

5. In line 289, expression (ii), both $s_0$ and $s$ appear. Is there a typo? Otherwise, give details.

6. In equation (77) and line 370, perhaps $x_i$ in the argument of the theta function should be replaced by $x$.

Publish (surpasses expectations and criteria for this Journal; among top 10%)

These Lecture Notes are concise and complement the recent review "J. Phys. A: Math.Theor. 53 , 193001 (2020)" by Evans, Majumdar, and Schehr. The interest and literature in stochastic has been growing rapidly, and these notes will help bring researchers up to date.

Since these are meant to be lecture notes (probably to introduce young researchers to the field), it would have been helpful to provide a few intermediate mathematical steps (throughout the paper).

Overall, the lecture notes are well-written and fulfils the acceptance criteria. I recommend it for publication.

I found a few typos/errors in these notes, which the authors should correct.

1. The factor $\sqrt{s}$ in the prefactor of the expression in equation (13) should be in the numerator, i.e., equation (13) should read
$\widetilde{P}(x,s|x_0)=\frac{1}{2}\, \sqrt{\frac{s}{D}}\, e^{-\sqrt{\frac{s}{D}}\, |x-x_0| }$

2. In line 143, "using" should be "Using"

3. With the scaling variable $z=x_0/\sqrt{4 D t}$, equation (21) should read
$-2z\frac{dg}{dz} = \frac{d^2 g}{dz^2}$. If the authors want to keep equation (21) unchanged, the correct scaling variable should be $z=x_0/\sqrt{D t}$.

4. The sentence after equation (55) ends with a mathematical expression, and the following sentence again starts with a mathematical variable $y$. It might be better to rephrase the first part of the sentence. For example, "The dimensionless parameter $y$ is the ratio..."

5. In line 289, expression (ii), both $s_0$ and $s$ appear. Is there a typo? Otherwise, give details.

6. In equation (77) and line 370, perhaps $x_i$ in the argument of the theta function should be replaced by $x$.

Publish (surpasses expectations and criteria for this Journal; among top 10%)

The present manuscript represents an interesting lecture notes on stochastic resetting and the large deviation in stochastic resetting, which is of interest to analyze the transition to the nonequilibrium stationary state of the system in the long time limit. It also deals with the cost of resetting. The topic of stochastic resetting has been considered as a hot topic in nonequilibrium statistical physics in the last decade and it is still a mechanism which is used to explain different phenomena, for example, in economy. Therefore, the manuscript is appropriate for publication in SciPost Physics Lecture Notes.

Here are some additional comment to the manuscript.

- It would be nice if Authors mention that the nonequilibrium stationary state (34) is the Laplace distribution. Graphical representation of it would be useful, from where the cusp at the origin could be visible.

- It could be useful if the Authors give the renewal equation in Laplace domain, fro $x_0=x_r$, which will yield
$$\tilde{P}_r(x,s|x_r)=\frac{s+r}{s} P_0(x,s+r|x_r),$$
from where one can easily find the nonequilibrium stationary state reached in the long time limit, which is actually eq. (40), $$P^{\ast}(x)=r\hat{P}_0(x,r|x_r),$$
by using the final value theorem of the Laplace transform,
$$\lim_{t→\infty}f(t) = \lim_{s→0}s\hat{f}(s).$$

- Graphical representation of the MFPT vs resetting rate could be useful in order to see that there is an optimal resetting rate which minimizes the MFPT.

- When talking about general resetting distribution, useful reference could be [Phys. Rev. E 99, 012141 (2019)].

- Optional, it could be useful if the Authors show the transition of the mean squared displacement from normal diffusion in the short time limit to saturation in the long time limit due to the resetting of the particle.

Publish (surpasses expectations and criteria for this Journal; among top 10%)