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The Probabilistic Scaling Paradigm

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Abstract

In this note we further discuss the probabilistic scaling introduced by the authors in (arXiv:1910.08492, 2019) and (Invent. Math. 228, 539–686, 2022). In particular we do a case study comparing the stochastic heat equation, the nonlinear wave equation and the nonlinear Schrödinger equation.

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Notes

  1. One may also ask about how their distribution evolve over time; at least how do the ensemble averages evolve. Such questions arise in wave turbulence; see [11, 13, 14, 17,18,19,20] and the references therein.

  2. Heuristically, one can check the following example [46]: \(\dot{x}= \sigma (x)\ \dot{w}\). At least when w is fractional Brownian motion, there is no canonical stochastic definition of iterated integrals below the threshold \(\gamma = \frac{1}{4}\) (say for \(\sigma (x) = x\)), where \(\gamma \) refers the regularity of w (which is of course also the regularity of the solution x). However, the regime \(\gamma >0\) is in principle subcritical in the parabolic setting.

  3. Here and also in the following parts of this paper we write \(C^\gamma \) for the Besov space \(B_{\infty ,\infty }^{\gamma }\).

  4. This square root cancellation is derived from a classic large deviation property for the sum of independent Gaussian variables, as referenced in Lemma 4.4 in [22].

  5. \(k_i \ne k_j\) if the corresponding signs \(\pm _i\) and \(\pm _j\) are the opposite.

  6. For example [36, 41] focus on such discrepancies.

  7. In wave turbulence theory it is customary to perform another reduction so the space scale becomes N and time scale becomes \(N^2\); in this setting the probabilistically critical problem would correspond to \(T_{\text {kin}}=N^2\).

  8. Thus, in particular, proving that with high probability, there is no energy cascade between Fourier modes (i.e. \(|\widehat{u}(t,k)|^2\approx |\widehat{u}(0,k)|^2\) with negligible error for large N).

  9. However, it is compatible with the wave equation due to the finite speed of propagation; specifically, the result of [10] is expected to hold also for the Gibbs measure on \(\mathbb {R}^3\). We refrain from discussing this here, but for more details, refer to [10].

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Acknowledgements

Y. D. is funded in part by NSF-DMS-2246908 and a Sloan Fellowship. A.N. is funded in part by NSF-DMS-2052740, NSF-DMS-2101381 and the Simons Foundation Collaboration Grant on Wave Turbulence (Nahmod’s Award ID 651469). H.Y. is funded in part by the Shanghai Technology Innovation Action Plan (No.22JC1402400) and a Chinese overseas high-level young talents program (2022).

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Yu Deng

  2. Department of Mathematics, University of Massachusetts, Amherst, MA, 01003, USA

    Andrea R. Nahmod

  3. Institute of Mathematical Sciences, ShanghaiTech University, Shanghai, 201210, China

    Haitian Yue

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  1. Yu Deng
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Correspondence to Andrea R. Nahmod.

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Deng, Y., Nahmod, A.R. & Yue, H. The Probabilistic Scaling Paradigm. Vietnam J. Math. (2024). https://doi.org/10.1007/s10013-023-00672-w

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