Enhancement of stability of metastable states in the presence of Lévy noise

1. Obtained an exact analytical result for the nonlinear relaxation time
2. Found non-monotonic behaviour interpreted as enhanced stability of metastable states

1. Literature review in introduction is too generic
2. Numerical confirmation missing

The authors investigate the nonlinear relaxation time (NLRT) of a particle in a metastable potential driven by Levy noise. An exact expression for the NLRT is derived and further evaluated analytically for the special case of a cubic metastable potential. The solution reveals the interesting phenomenon of noise enhanced stability, previously observed for Gaussian noise.

The study is carefully performed and mathematically sound. I am quite intrigued by the fact that such an exact solution for the NLRT can be found, in particular since other studies mostly consider asymptotic treatments of similar escape problems for metastable systems. The manuscript is thus certainly suitable for publication in principle, but I recommend that the authors consider the following comments to put the work better into context as well as provide independent confirmation.

1. The introduction on the problem is too generic. The problem of escape from metastable states driven by Levy-type noise has attracted a considerable amount of theoretical research over the last two decades. The authors write that "there are a lot of numerical results and some analytical approximations [3,32,35]." But there are lot more than 3 articles that are relevant here (see, e.g., the references discussed in Ref.[35]). This does not have to be exhaustive, but it would be useful to know the different technical approaches used previously and the results found.

2. Likewise, I wonder if the discussion in Section 2 requires some more references. Has this approach been discussed previously for Gaussian noise?

3. The discussion of the ratio <\tau_NLRT>/\tau_d below Eqs.(28,29) is confusing, because \tau_d appears without any proper introduction. It would be clearer if first the behaviour of <\tau_NLRT> would be discussed by itself together with the relevant figures (Fig.2b and Figs.3a,b). Then, \tau_d could be introduced (this needs more explanations) and the ratio <\tau_NLRT>/\tau_d discussed.

4. The authors should clarify whether "nonlinear relaxation time" and "mean residence time" are the same quantity. If they are identical, I would recommend to use only one of the two terminologies throughout the manuscript.

5. The analytical results should be supplemented by numerical results. It is straightforward to simulate Eq.(2), thus numerical confirmation should be provided.

6. The authors mention that the enhancement of stability of metastable states has been observed previously for Gaussian noise. How does \tau_NLRT differ quantitatively from this case? It would be important to understand how the stability is affected by the non-Gaussianity of the noise.

See report

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1. First exact analytical derivation of NLRT with Levy noise.
2. Demonstration of significant Noise Enhanced Stabilization effect for Levy noise case.

1. Missing figure plots
2. Lack of proper citations of origial papers, describing basic definitions.

This manuscript presents the results of analytical derivation of a Nonlinear Relaxation Time (NLRT) in the case of Levy noise. In particular, the following important results are derived: the exact results of the NLRT for a particle moving in an arbitrary potential profile in the presence of Lévy noise; a closed expression written by quadratures for NLRT for the particular case of Cauchy noise in cubic metastable potential. To my knowledge, these are the first exact analytical results for any more complex than Markovian case, which have numerous applications in physics, but especially in biology. Besides that, using the obtained analytical results the authors investigate the Noise Enhanced Stabilization effect and show its significant amplification in comparison with white noise case. The paper is interesting and clearly written, so it can be recommended for publication after addressing minor comments listed below.

Comments:
1. In Fig. 3b the only empty frame without any curves is visible, please correct.
2. When describing the basic definition for NLRT (3),(4), the authors should cite the first paper
https://doi.org/10.1016/0378-4371(95)00395-9, where this definition was used for exact derivation of NLRT of a Brownian particle in a smooth potential. Later, this definition was generalized to arbitrary moments of transition times https://doi.org/10.1016/S0375-9601(97)00599-9. Also, it has been demonstrated there for smooth symmetric potentials, that the moments of the First Passage times to the point of symmetry completely coincide with the corresponding moments of transition times. As suggestion for future studies, the authors may try to extend the obtained results to the case of the standard deviation of transition time, which has important applications for description of switching errors of various electronic devices. Another important paper, where the NLRT for the first time has been expressed by quadratures for Markovian processes in smooth potentials, is https://doi.org/10.1016/0921-4534(96)00426-1. There, the NES effect, studied by the authors, was described analytically both using the exact expression and asymptotic series in the low noise limit, see the plots in Fig. 4, so this reference should be added to the list of papers, devoted to NES effect.

1. In Fig. 3b the only empty frame without any curves is visible, please correct.
2. When describing the basic definition for NLRT (3),(4), the authors should cite the first paper
https://doi.org/10.1016/0378-4371(95)00395-9, where this definition was used for exact derivation of NLRT of a Brownian particle in a smooth potential. Later, this definition was generalized to arbitrary moments of transition times https://doi.org/10.1016/S0375-9601(97)00599-9. Also, it has been demonstrated there for smooth symmetric potentials, that the moments of the First Passage times to the point of symmetry completely coincide with the corresponding moments of transition times. As suggestion for future studies, the authors may try to extend the obtained results to the case of the standard deviation of transition time, which has important applications for description of switching errors of various electronic devices. Another important paper, where the NLRT for the first time has been expressed by quadratures for Markovian processes in smooth potentials, is https://doi.org/10.1016/0921-4534(96)00426-1. There, the NES effect, studied by the authors, was described analytically both using the exact expression and asymptotic series in the low noise limit, see the plots in Fig. 4, so this reference should be added to the list of papers, devoted to NES effect.

Ask for minor revision