On Some Results of the Nonuniqueness of Solutions Obtained by the Feynman–Kac Formula

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors obtained an interesting result of the nonuniqueness of the Cauchy problem solution for the backward heat equation with terminal condition. In particular, the obtained counterexample, which is the result of the application of the Feynman-Kac formula.

Regarding the essence of the presented result.

1. It is well known that for boundary value problems for linear evolutionary PDEs (in particular, parabolic ones), the results of uniqueness depend significantly on the functional spaces in which the solutions are considered. In particular, this fact is known for the Cauchy problem (see, for example, A. Tychonoff, “Théorèmes d'unicité pour l'équation de la chaleur”, Mat. Sb., 42:2 (1935), 199–216). Therefore, in my opinion, the authors got into such a situation by formulating a counterexample. Therefore, I would advise the authors to change the title of the paper. A formula is a formula. Formula cannot be nonunique. The solutions of the problem obtained by this formula can fall into the spaces of unity or can go beyond such classes. In my opinion, the title "On some results of the nonuniqueness of solutions obtained by the Feynman-Kac formula" more accurately reflects the essence of the work.

2. In this regard, a deeper comparative analysis of the problem considered in the article from the point of view of the general theory of linear PDEs (in particular, issues of correctness of solutions) makes sense. Authors announce future research on this matter in the Conclusions. I believe that such a comment would be appropriate in the form of a remark in Section 3.

3. The interpretation of the obtained result is an interesting practical application, which the authors emphasize. It is precisely here that the essence of the concerns expressed in the Conclusions should be revealed (such as "This indicates that when using the Feynman-Kac formula, albeit a useful and elegant tool, care should be taken" etc.). And from this point of view, the presented work, in my opinion, should be continued.

I did not find any substantive remark in the technical comments. I can only express my desire to more clearly outline (highlight clearly) the received counterexample, for example, in the form of a statement.

I suggest that the authors take these comments into account. Their purpose is not to criticize, but to try to improve the presentability of the work.

Comments on the Quality of English Language

Satisfactory

Author Response

We are grateful to the reviewer for comments which are very helpful in improving the manuscript. 

1. Following the suggestion, we have changed the title to “On some results of the non-uniqueness of solutions obtained by the Feynman-Kac formula”.
2. With appreciation, we have added a remark in Sec. 3.
3. We thank the reviewer for the suggestion that this work should be continued. This has been addressed in Sec. 4.

Reviewer 2 Report

Comments and Suggestions for Authors

In this paper, based on a counterexample, the Feynman-Kac formula does not roduce a unique solution but carry infinitely many solutions of the corresponding partial ifferential equation. Some specific comments are given as follows. 

1) Some background about the Feynman-Kac formula should be given in the introduction part. 

2) The main motivation and contributions of this paper should be clarified. 

3) The main results should be presented as the style of theorems. 

4) Numerical examples should be given to show the effectiveness of the main results as a separate section. 

5) Pls provide a clear and concise conclusion.

Comments on the Quality of English Language

In this paper, based on a counterexample, the Feynman-Kac formula does not roduce a unique solution but carry infinitely many solutions of the corresponding partial ifferential equation. Some specific comments are given as follows. 

1) Some background about the Feynman-Kac formula should be given in the introduction part. 

2) The main motivation and contributions of this paper should be clarified. 

3) The main results should be presented as the style of theorems. 

4) Numerical examples should be given to show the effectiveness of the main results as a separate section. 

5) Pls provide a clear and concise conclusion.

Author Response

We sincerely appreciate the time and effort spent reviewing our manuscript. Appreciating the valuable comments, we have revised the manuscript accordingly.

1. Some background has been given in Sec. 1.
2. Section 1 has been revised for more clarity.
3. In consideration of the comment, we have revised the corresponding part slightly. Please note that the main result is presented as a counter-example rather than a theorem regarding the uniqueness of the Feynman-Kac formula.
4. The main result is presented by a closed-form solution of a deterministic partial differential equation. For instance, equation (9) provides a simple exemplary solution. We have thus plotted Eq. (9) and included the result in Fig. 2, compared with the conventional solution in Fig. 1.
5. We have revised Sec. 4, providing a clear and concise conclusion.

Reviewer 3 Report

Comments and Suggestions for Authors

The paper discusses the non-uniqueness of PDE solution using the Feynman-Kac formula by presenting a counterexample. The Authors show that additional solutions arise, which, along with the possible compromising of the true solution, can be of some interest to investigate. The obtained results may improve the precision of PDE simulation using the Feynman-Kac approach. The paper is well-written and relatively easy to perceive for a math paper. However, I have several recommendations and questions for the Authors.

1. Please, indicate more directly the source that claims that using the Feynman-Kac formula provides a unique solution for any arbitrary boundary-value problem. In the case of the PDE solution, it is not as easy to say as in the ODE case. My concern here is some sort of "too loud statements from a very modest research".

2. The Authors correctly note, that the "uniqueness of a heat transfer boundary problem is not a trivial question as sometimes claimed". Indeed, it is not. The chosen example is illustrative enough to show it. However, it is a known fact. Actually, scholars who apply the Feynman-Kac formula in practice, are usually aware of these conditions. What is the problem statement then? To prove this fact rigorously? In my opinion, the relevance of the study should be highlighted further, for example, for some cases when the existence of another solution may cause a real mistake in calculations.

3. How can this effect appear in numerical simulations? In other words, how can one distinguish a "wrong" solution from a "correct" one in practical cases? Solving PDEs is mostly of practical interest, and usually applying analytical transformations is the preliminary stage of the whole thing. Can the numerical solutions, obtained through simulations, occasionally switch between two "real" trajectories of such a system? Possibly leading to some nonlinear effects?

4. The paper is short, dry, and lacks some life, e.g. illustrations. It will certainly benefit from some topological interpretations, practical examples, graphs, plots, etc. If everything is correct and the discovery is that important, please show more examples where it can have an impact.

5. Please, double-check Eqs. 29. and 30.

6. The Conclusions section may be enriched with practical consequences of the obtained important results, recommendations to scholars, and a clearer description of future works, which "are currently under investigation". The recommendation that "care should be taken" is certainly nice, but what exactly should researchers do to avoid mistakes using the Feynman-Kac formula in their research?

Nevertheless, I believe the study is valuable to the mathematical community and can be published after some revisions. I wish the Authors all the luck during the revision process.

Comments on the Quality of English Language

The quality of English language is generally fine. I recommend proofreading, especially the abstract and the Conclusions section.

Author Response

We are grateful to the reviewer for recognizing the merits of the work and giving valuable comments to improve the manuscript. With appreciation, we have revised the manuscript accordingly.

1. Following the comment, we have indicated the source addressing the uniqueness explicitly and rephrased the corresponding part modestly.
2. With appreciation, we have added a paragraph in Sec. 3, explaining the meaning and relevance of the study.
3. In consideration of the comment, we have added a paragraph in Sec. 3, discussing the generalized solution according to the initial conditions. Please note that the PDE considered here is deterministic and its generalized solution is given in the closed form.
4. Accommodating the comment, we have added Figures 1 and 2, which exhibit a simple instance of the generalized solution as well as the conventional solution.
5. In accordance with the suggestion, we have rechecked Eqs. (29) and (30) and confirmed their validity.
6. Taking into account the comment, we have expanded Sec. 4, adding an explanation of future works. Specific directions in detail are under current investigation and beyond the scope of this study.

Accommodating the suggestion for the English language, we have proofread carefully and improved expressions.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Basically, the authors took into account my comments. I propose to accept the article

Comments on the Quality of English Language

Satisfactory

Reviewer 2 Report

Comments and Suggestions for Authors

The authors answered my concerned questions in the revised version. Therefore, I recommend it accepted.

Reviewer 3 Report

Comments and Suggestions for Authors

Thank you very much for providing a revised version of your manuscript. I am generally satisfied with the point-by-point reply letter and the revisions made, and therefore, can recommend the manuscript for publication. I wish the authors good luck in their future studies.

Comments on the Quality of English Language

"is used widely to compute efficiently solutions" - to efficiently compute?

 

ref. [10] - "stochastic proesses" - processes