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Random tensors, propagation of randomness, and nonlinear dispersive equations

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We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work (Deng et al. in: Invariant Gibbs measures and global strong solutions for the nonlinear Schrödinger equations in dimension two, arXiv:1910.08492), to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we establish almost-sure local well-posedness for semilinear Schrödinger equations in the full subcritical range relative to the probabilistic scaling (Theorem 1.1). The solution we construct has an explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients. As a byproduct we also obtain new results concerning regular data and long-time solutions, in particular Theorem 1.6, which provides long-time control for random homogeneous data, demonstrating the highly nontrivial fact that the first energy cascade happens at a much later time than in the deterministic setting. In the random setting, the probabilistic scaling is the natural scaling for dispersive equations, and is different from the natural scaling for parabolic equations. Our theory of random tensors can be viewed as the dispersive counterpart of the existing parabolic theories (regularity structures, para-controlled calculus and renormalization group techniques).

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  • 24 November 2021

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  1. See Remark 1.12 and the explanation in Sect. 9.1.

  2. Random data is a natural setting for dispersive equations (parallel to additive noise for parabolic ones) in view of the invariant measures. Of course one may also consider stochastic versions of (1.1), which are similar but correspond to different randomizations, see Sect. 9.2.

  3. We will assume \(N\ge 1\) throughout, and only “formally” need to replace N by 1/2 in a few places.

  4. Such structure lives on high frequencies and fine scales. It becomes more explicit when considering low regularity solutions, and may be obscured by the dominant coarse scale profile in high regularity solutions.

  5. In fact there is energy cascade in \(u_{\mathrm {ho}}\) at time \(N^{(p-1)(s-s_{cr})}\), dictated by the continuous resonance (CR) equation; see [17, 38]. Note that, the approximation leading to the CR equation does not work in our case as randomization destroys the differentiability in rescaled Fourier space; in fact the solution \(u_{\mathrm {ho}}\) in Theorem 1.6 has no energy cascade at this time. It is currently unknown whether the solution \(u_{\mathrm {ho}}\) in Theorem 1.6, or the corresponding ensemble average, satisfies some effective equation at time \(N^{(p-1)(s-s_{pr})}\).

  6. We remark that, after the submission of this manuscript, the justification of the wave kinetic equation has been done by the first author and Hani in [34] in certain regimes, including the case \(s=s_{pr}\) and \(\nu =0\) (the case \(s>s_{pr}\) still remains open). This is a probabilistically critical result and is not covered by Theorem 1.6.

  7. In comparison, when \(s>s_{pr}\), the proof of Theorem 1.1 easily implies the convergence of \(u_N\) (including the \(u_\lambda \) in Remark 1.4) in \(C_t^\iota H_x^{s-}\) for some \(\iota >0\) ultimately determined by (dps).

  8. In particular, and .

  9. Here \(\pi _>\) is the standard Bony paraproduct.

  10. In practice we will use \(\eta _k=|g_k|^{-1}g_k\), which is uniformly distributed on the unit circle, instead of \(g_k\).

  11. In practice we will also have a ± sign for each index of each tensor, for example a \(+\) sign for the index a in \(h_{abcde}\) or a − sign for the index x in \(h_{uvawx}'\). When precisely defining the merging operations, see Definition 3.6, we will restrict to the cases where for each repeated or paired index, its signs in the two tensors that it appears are the opposite (for example if the sign of b in \(h_{abcde}\) is \(+\) then the sign of u in \(h_{uvawx}'\) must be −, assuming \(b=u\)). In this section (including the simple case Definition 2.4) we will ignore this issue for simplicity.

  12. In practice this will have a small power loss \(M^\theta \) where \(\theta \) is an arbitrary small number and M is the size of \(k_A\in ({\mathbb {Z}}^d)^A\); see Propositions 4.144.15.

  13. In [36] we used \(L=N^{1-\delta }\); here we need a smaller value of L which works better in the general setting.

  14. In addition, we remove edges connecting a node to its only child; they correspond to the random averaging operator in (2.11) and can be dealt with as in Sect. 2.4.1.

  15. Here \(h^{\circ }_{k}=(u_L)_k\) can be understood as a (0, 1) tensor which has no input variable.

  16. This corresponds to removing \(\circ \)’s and low-frequency \(\bullet \)’s from the iteration trees, or removing low-frequency leaves from the skeleton and flattened trees. In the main proof, in addition to this trimming after merging, we also need to trim the tensors before merging; see (5.23) and (5.24).

  17. See Proposition 5.1 for the full detailed version. In particular there are distinctions between \(h^{(*,0)}\) and \(h^{(*,1)}\) tensors, which we omit here.

  18. Roughly speaking \(\delta =\varepsilon ^{50}\), \(D=\delta ^{-50}\), \(\kappa =D^{50}\) and \(\theta \sim \kappa ^{-50}\) should suffice.

  19. Here a pairing \((k,k_j)\) means \(k=k_j\) and \(\zeta _j=+\).

  20. In this tensor \(k_{\mathcal {V}}\) and \(\lambda _{\mathcal {V}}\) appear as parameters. Note also that the definition does not involve \({\mathcal {P}}\) or \({\mathcal {Y}}\).

  21. If necessary we may replace the unions \(\cup \) by the disjoint unions \(\sqcup \) to avoid repetition of elements.

  22. Here we may also require \(N_{\mathfrak {l}}=N_{{\mathfrak {l}}'}\); whether we do so will not affect the result of this product.

  23. Here and below the phrase “linear combination” will refer to a linear combination with a fixed number of terms and fixed constant coefficients.

  24. In practice, the factor \(\chi _\tau \) will always come with a v which has the form \({\mathcal {I}}_\chi (\cdots )\), so we always have \(v(0)=0\).

  25. Strictly speaking this reduction may not preserve (4.25), but (4.25) is only used to guarantee \(|A_v|\le p-2\) in order to apply Lemma 4.8; after this (4.25) can be replaced by the more general versions where the linear combinations of \(k_j\) and k are fixed \({\mathbb {Z}}^d\) vectors instead of 0, which is preserved under the reduction.

  26. These over-paired variables include a pairing between \((k,k_{P_0})\) and \(k_{Q_0}\) as \(1\in Q_0\), and thus can be treated using the argument in the beginning of the proof.

  27. This matches Definition 3.4.

  28. Strictly speaking the sum over \({\mathscr {O}}\) should carry the coefficients in the linear combination of \({\widetilde{\Psi }}_k^{({\mathcal {S}})}\) that gives (5.22) as above; these are constants, and for simplicity we will treat them as 1.

  29. Note that \({\mathscr {O}}\) actually does not appear in the summation (5.28). But this is fine, since it can easily be checked that the size of \({\mathcal {S}}\) defined by (5.29) does not depend on \({\mathscr {O}}\).

  30. Note that, once we have (5.37), we automatically have the same bound for the norm with the weight in (5.36), with the right hand side multiplied by \(N^\kappa \), which is negligible as the right hand side is super-polynomial.

  31. The same applies to the \(k_E\) variables when measuring \(kk_B\rightarrow k_C\) norms, where \(E={\mathcal {U}}\backslash (B\cup C)\).

  32. i.e. which does not depend on \(\omega \).

  33. Recall that \({\mathcal {Q}}\) is defined in (3.10). Since \({\mathcal {Q}}\) contains the two-element pairings in \({\mathcal {A}}_i\), \(|{\mathcal {A}}_i\cap {\mathcal {Q}}|\) is even.

  34. Note this definition is only for \(j=1\) and not for \(2\le j\le r\), which we will define later.

  35. The support condition (6.13) for type \(R0^+\) can be immediately verified, since \({\mathcal {U}}_j^{\dagger \dagger } = \varnothing \).

  36. In the analogous sense that (6.55) is the modified version of (6.44).

  37. In fact in the cases we consider the bound in (6.79) is always better than that in (6.78) due to the \({\mathcal {X}}_{1,1}\) factor in (6.79), so below we will always assume that we are using (6.78).

  38. We may also need to multiply this h by functions \({\mathbf {1}}_{\langle k\rangle \ge M^2}\) or \({\mathbf {1}}_{\langle k_1\rangle \ge M^2}\), but they do not affect Proposition 4.15 (which can be easily checked), so the proof below will proceed in the same way.

  39. See Definition 3.1. Here simplicity implies that, if \(\zeta =+\) and \(k=k'\), then k must also equal some other \(k_j\), which has already been fixed.

  40. Here we have assumed \(1\le j\le r\). If \(r+1\le j\le q\), then this j would correspond to the input function \(z_{N_j}\) in (5.22), which gains a big power \(N_j^{-D_1}\). Thus the case where the maximum \(N_j\) occurs at this j will be strictly easier than the cases we actually treat in the proof.

  41. When \({\mathcal {S}}\) is fixed, the sum in \((N_j,{\mathcal {S}}_j)\) etc. for \(j\ge 2\) involves at most \((\log L)^{\kappa }\) terms, which is negligible in view of the \(L^{\varepsilon \delta }\) gain we will obtain below. The sum in \((N_1,{\mathcal {S}}_1)\) involves at most \(\kappa \) terms if \(N_1\sim M\) and \({\mathcal {S}}_1={\mathcal {S}}'\), and at most \((\log M)^{\kappa }\) terms otherwise; either way this is negligible in view of the gain from \(\Upsilon \), and the gain of at least \(M^{\delta ^5}\) coming from trimming assuming \({\mathcal {S}}_1\ne {\mathcal {S}}'\), which is evident from the proof of Proposition 6.1.

  42. If needed, we can always view an \({\mathbb {R}}\)-multilinear operator of degree \(q-1\) as one of degree q by adding a trivial input function.

  43. Because we do not need to distinguish the low-frequency inputs as there is only one scale.

  44. The choice of \(N^{1+\theta }\) is because \(\phi \) is not compactly supported and may have a Schwartz tail.

  45. The reader may notice the similarity with the construction of \(z_M\) in (5.28).

  46. Even with the \({\mathcal {N}}_{\mathrm {np}}\) nonlinearity, some renormalization may still be needed for higher order iterations, but not for the first nonlinear iteration which is discussed here. Also whether u is real or complex valued, and whether \({\mathcal {N}}_{\mathrm {np}}\) contains complex conjugates, does not affect the scaling heuristics.

  47. This distinction may be artificial from the regularity structure perspective, but is convenient in comparison with the dispersive case here.

  48. Put in another way, imagine one has all the input and output frequencies being the same in the nonlinearity. Then (9.2) is locally well-posed, without any renormalization or special arguments, if and only if \(s>s_{pa}\), while for (9.1) the same thing holds if and only if \(s>s_{pr}\).

  49. This should be compared to the stochastic quantization problem for (9.2) where the solution theory relies on the Gaussian noise instead of the non-Gaussian measure, as observed in [8].

  50. For example, invariance of Gibbs measure and the associated almost-sure global well-posedness result would imply that there is no stable blowup mechanism at the regularity of the support of the Gibbs measure.


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The authors would like to thank Hendrik Weber for helpful comments regarding the regularity structures theory and the reference [22]. They would also like to thank the referees for their useful comments and suggestions.

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Correspondence to Yu Deng.

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Y.D. is funded in part by NSF DMS-1900251 and Sloan Fellowship.

A.N. is funded in part by NSF DMS-1800852, DMS-2101381 and the Simons Foundation Collaboration Grant on Wave Turbulence (Nahmod’s Award ID 651469).

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Deng, Y., Nahmod, A.R. & Yue, H. Random tensors, propagation of randomness, and nonlinear dispersive equations. Invent. math. 228, 539–686 (2022).

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