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Random tensors, propagation of randomness, and nonlinear dispersive equations
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Random tensors, propagation of randomness, and nonlinear dispersive equations
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Abstract. The purpose of this paper is twofold. We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work arXiv:1910.08492, to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we also solve Conjecture 1.7 in arXiv:1910.08492, and establish almost-sure local well-posedness for semilinear Schr\"{o}dinger equations in spaces that are subcritical in the probabilistic scaling. The solution we find has an explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients. 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, which covers the full subcritical regime in the probabilistic scaling, can be viewed as the dispersive counterpart of the existing parabolic theories (regularity structure, para-controlled calculus and renormalization group techniques).
Forward citations
Cited by 4 Pith papers
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Schatten-Class Wick Multipliers for Gaussian Fields on Closed Riemannian Manifolds: Oriented Tensor Estimates and Covariance-Profile Transfer
Establishes that the L^p(Ω; S_r) norm of a finite-order decoupled homogeneous Gaussian chaos operator is bounded by C_m (p+r)^{m/2} times the maximum oriented Schatten flattening norm of its kernel.
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Schatten-Class Wick Multipliers for Gaussian Fields on Closed Riemannian Manifolds: Oriented Tensor Estimates and Covariance-Profile Transfer
Fixed-order Hilbertian Gaussian chaos operators satisfy ||T_K^{(m)}||_{L^p S_r} ≤ C_m (p+r)^{m/2} max_S ||F_S(K)||_{S_r}, with constants independent of Hilbert dimensions and cutoff ranks.
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Schatten-Class Wick Multipliers for Gaussian Fields on Closed Riemannian Manifolds: Oriented Tensor Estimates and Covariance-Profile Transfer
Fixed-order Hilbertian Gaussian chaoses satisfy ||T_K^{(m)}||_{L^p S_r} ≤ C_m (p+r)^{m/2} max_S ||F_S(K)||_{S_r}, with completion, Wick, and cutoff-Cauchy consequences.
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Schatten-Class Wick Multipliers for Gaussian Fields on Closed Riemannian Manifolds: Oriented Tensor Estimates and Covariance-Profile Transfer
A finite-order Gaussian chaos estimate converts deterministic Schatten flattening bounds into dimension-free moment, tail, and convergence estimates for operator-valued random tensors.
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