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The fluctuation dissipation theorem relates the linear response of a system to Gaussian fluctuations. The natural question that comes to my mind is the possible derivation of an analogous FDT in presence of non-Gaussian noise. I would like to know if such an attempt has already been made with links to relevant literature. If not, then what are the difficulties (conceptual or technical) towards such a derivation?

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You might be interested in the nonlinear fluctuation dissipation theorem. Here are some sources:

1) A review by Lucarini and Colangeli

2) The classic book by Stratonovich on nonlinear fluctuations (very advanced)

3) One of the original papers by Efremov

4) Application of the nonlinear FDT to classical plasmas

5) A more readable paper (with references) on the application of nonlinear FDT to plasmas that I wrote a few years ago

6) Kubo's second book on statistical physics (might be one of the best sources if you can get a copy)

EDIT: Here are some possibly more relevant references for you on a linear system driven by non-Gaussian noise:

A paper by Kanazawa et. al.

A presentation by Kanazawa

A paper by Dubkov et. al.

A paper by Field et. al.

A proof by Grafov

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    $\begingroup$ Thanks for the references. I will take a look at them. I do have the Kubo's two volume book and it's a great reference. I just wanted some clarification on FDTs for linear Langevin systems driven by non-Gaussian noise. Would appreciate any comments on that. I am not explicitly asking about non-linearity in responses here. $\endgroup$ – noisyoscillator Dec 31 '18 at 14:42
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    $\begingroup$ @noisyoscillator Added a few additional references that might be more what you're looking for. $\endgroup$ – Joshuah Heath Dec 31 '18 at 16:31
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    $\begingroup$ Thanks a lot Joshuah. These latest additions are more in line with what I was looking for. I was going through Kanazawa's work and his Springer thesis while you posted the links. I appreciate your help in this regard. Cheers and a happy new year! $\endgroup$ – noisyoscillator Dec 31 '18 at 19:33
  • $\begingroup$ Still it looks like a general FDR for linear systems driven by non-Gaussian sources is missing. I found a paper discussing an FDR when the system is driven by Lévy type noise. $\endgroup$ – noisyoscillator Dec 31 '18 at 19:59

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