Invariance of the white-noise ($\sigma=0$) weighted Gaussian measure for the periodic Benjamin–Ono equation

The periodic Benjamin–Ono equation $\partial_tu+\mathcal{H}\partial_x^2u+u\,\partial_xu=0$ is a completely integrable Hamiltonian PDE whose conserved energies $E_{\sigma/2}$, $\sigma=0,1,2,\dots$, form a ladder from the $L^2$ mass ($\sigma=0$) through the Hamiltonian ($\sigma=1$) up to higher regularities. Tzvetkov–Visciglia, Deng–Tzvetkov–Visciglia, and Deng constructed and proved the invariance of the weighted Gaussian measures $\rho_{\sigma/2}$ associated with these laws for every integer $\sigma\ge1$; the base Gaussians are supported on $H^s$ only for $s<(\sigma-1)/2$, so the proven results descend to just below $L^2$ and stop. The endpoint $\sigma=0$ is white noise itself (the density degenerates because $E_0$ is the pure $L^2$ mass), and it was left open in Deng's 2015 paper as "out of reach with current techniques". The obstruction is purely dynamical: frequency-truncated flows preserve the truncated white-noise measures exactly, but white noise is supported strictly below the sharp deterministic well-posedness threshold $H^{-1/2}$, so no flow is known on its support. Tzvetkov (2024) constructed renormalized almost-sure dynamics in Birkhoff coordinates preserving the white-noise-type measure, and stated the transfer to physical space — solving Benjamin–Ono with white-noise initial data — as an open problem. The problem: construct renormalized almost-sure global dynamics for the periodic Benjamin–Ono equation from white-noise data and prove white-noise invariance.

Background

The Benjamin–Ono (BO) equation on the one-dimensional torus $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$ (i.e., with periodic boundary conditions) is

$$\partial_tu+\mathcal{H}\partial_x^2u+u\,\partial_xu=0,$$

where $\mathcal{H}$ is the Hilbert transform on periodic functions, defined on Fourier modes by $\mathcal{H}(e^{inx})=-i\,\operatorname{sgn}(n)e^{inx}$. It models the propagation of long internal gravity waves in a two-layer fluid of infinite depth, and it is a basic example of a completely integrable Hamiltonian PDE: the equation is an infinite-dimensional Hamiltonian system (here $\partial_tu=-\tfrac12\,\partial_x\nabla E_{1/2}(u)$ for the Hamiltonian functional $E_{1/2}$) possessing an infinite hierarchy of independent conserved quantities. On the torus these form a ladder of conservation laws $E_{\sigma/2}$, $\sigma=0,1,2,\dots$, of the form

$$E_{\sigma/2}(u)=\|u\|^2_{\dot H^{\sigma/2}}+R_{\sigma/2}(u),$$

where $\|u\|^2_{\dot H^{s}}=\sum_{n\neq0}|n|^{2s}|\hat u(n)|^2$ is the homogeneous Sobolev norm on mean-zero periodic functions and the remainder $R_{\sigma/2}$ is a finite sum of integrals of polynomials in $u$ and its derivatives, of total degree at least three. The first instances are $E_0(u)=\|u\|^2_{L^2}$ (the mass, the unique law with vanishing remainder) and $E_{1/2}(u)=\|u\|^2_{\dot H^{1/2}}+\frac13\int_{\mathbb{T}}u^3\,dx$ (the Hamiltonian). The mean $\hat u(0)$ is also conserved, so one works on mean-zero functions.

A natural question for a Hamiltonian PDE is whether the formal Gibbs-type measure $\exp(-E(u))\,du$ can be given a rigorous meaning and proved invariant under the flow, yielding a canonical stationary statistical description of the long-time dynamics. For BO this program was carried out as follows (indexing by the level $\sigma$: the law $E_{\sigma/2}$ lives at regularity $H^{\sigma/2}$). Tzvetkov first constructed the Gibbs measure associated with the Hamiltonian $E_{1/2}$, i.e., the $\sigma=1$ level; its Gaussian base is the law $\mu_{1/2}$ of the random Fourier series $\sum_{n\neq0}g_ne^{inx}/|n|^{1/2}$ with independent standard complex Gaussians $g_n$, supported exactly on $H^s$ for $s<0$ — just below $L^2$. Tzvetkov and Visciglia then associated to every higher law $E_{\sigma/2}$, $\sigma\ge2$, a weighted Gaussian measure

$$d\rho_{\sigma/2,R}=F_{\sigma/2,R}(u)\,d\mu_{\sigma/2}, \qquad \mu_{\sigma/2}=\text{law of }\sum_{n\neq0}\frac{g_n}{|n|^{\sigma/2}}e^{inx},$$

where the density $F_{\sigma/2,R}$ is a product of cutoffs $\chi_R(E_{j/2})$ of the lower conservation laws times $e^{-R_{\sigma/2}}$, and the cutoff of the next-to-leading law $E_{(\sigma-1)/2}$ must be renormalized by the harmonic sum $\alpha_N=\sum_{n\le N}1/n\sim\log N$, because $E_{(\sigma-1)/2}$ diverges logarithmically on the support of $\mu_{\sigma/2}$. The base Gaussian $\mu_{\sigma/2}$ is supported exactly on $H^s$ for $s<(\sigma-1)/2$, so these measures live at increasingly rough regularity as $\sigma$ decreases. Invariance under the BO flow was proved by Tzvetkov–Visciglia for $\sigma\ge4$ and by Deng–Tzvetkov–Visciglia for $\sigma\in\{2,3\}$; Deng proved the $\sigma=1$ case, the Gibbs measure, answering a question posed in Tzvetkov's 2010 paper. With these results, every integer level $\sigma\ge1$ is settled.

The endpoint $\sigma=0$ is of a different nature. Since $R_0\equiv0$, the formal density disappears: renormalizing $\exp(-E_0(u))\,du$ yields precisely the Gaussian $\mu_0$, the law of the random Fourier series $\sum_{n\neq0}g_ne^{inx}$ with flat coefficients — that is, (mean-zero) white noise, supported exactly on $H^s$ for $s<-1/2$. Formal invariance is compelling: the $L^2$ mass is exactly conserved, and each frequency-truncated BO flow (a finite-dimensional Hamiltonian ODE obtained by projecting onto modes $|n|\le N$) preserves both the truncated $L^2$ norm and the truncated Gaussian measure, so the truncated white-noise measures are exactly invariant. The entire difficulty is dynamical, not static: one must pass to the limit and construct a flow on the white-noise support itself. The published version of Deng's Gibbs-measure paper records in Remark 1.1: "The case $\sigma=0$ still seems out of reach with current techniques." The earlier arXiv version of the same remark is more pessimistic, expressing that no positive result is expected at $\sigma=0$; any positive statement is thus expected to require a renormalization of the dynamics.

The obstruction is sharp. Gérard–Kappeler–Topalov proved that periodic BO is globally well-posed in $H^s$ exactly for $s>-1/2$ and ill-posed for $s\le-1/2$: the critical exponent $s_c=-1/2$ is the well-posedness threshold. Independently, and by a different method, Killip–Laurens–Vişan established well-posedness for $s>-1/2$ on both the line and the circle. White noise is almost surely in $H^s$ for every $s<-1/2$ and in no $H^s$ with $s\ge-1/2$: it lives strictly below the threshold, where no deterministic solution concept is available. This is the qualitative difference with the settled $\sigma=1$ case, where the measure sits just below $L^2$ and probabilistic well-posedness arguments could compensate for the loss of determinism. It is also the difference with the KdV (Korteweg–de Vries) equation, for which white-noise invariance is known on the circle (Oh) and on the line (Killip–Murphy–Vişan): KdV's weaker dispersion permits deterministic well-posedness below white-noise regularity, while for BO the deterministic threshold and the white-noise level coincide exactly.

The problem has recently been recast in integrable-systems coordinates. Gérard and Kappeler proved that periodic BO is integrable in a strong sense: on mean-zero $L^2$ there is a nonlinear Fourier transform — Birkhoff coordinates $\zeta_n(u)$, $n\ge1$, a canonical change of variables adapted to the equation — in which the flow takes a simple form; Gérard–Kappeler–Topalov extended this map to all $H^s$, $s>-1/2$. Tzvetkov used it to construct new non-degenerate invariant measures on $H^s_0(\mathbb{T})$ for every $s>-1/2$, thereby filling all regularities strictly above the threshold without reaching it. In the same note he analyzed the white-noise level in coordinates: for data with Birkhoff coordinates $\zeta_n=g_n/\sqrt{n}$ (the white-noise image), the phases of the truncated coordinate flows diverge almost surely like $c_N=\sum_{n\le N}1/n$, but after renormalizing by the factor $e^{2itnc_N}$ the flows converge almost surely to a well-defined flow $\zeta_n\mapsto\zeta_ne^{ith_n(\zeta)}$ which leaves the white-noise-type measure invariant (his Theorem 7). The missing step is stated explicitly: "It is a challenging open problem to suitably extend the Birkhoff map of [7, 8] to this setting. This would in particular allow to solve (after a renormalization) the Benjamin-Ono equation with white noise initial data." His Remark 8 adds that such an extension is expected to require probabilistic arguments and would yield probabilistic well-posedness in the supercritical Fourier–Lebesgue space $FL^{-1/2,4}$ (data with $\langle n\rangle^{-1/2}\hat u_0(n)\in\ell^4$).

Problem Statement

Construct an almost-surely defined, global-in-time dynamics for the periodic Benjamin–Ono equation

$$\partial_tu+\mathcal{H}\partial_x^2u+u\,\partial_xu=0\qquad\text{on }\mathbb{T}\times\mathbb{R}$$

starting from white-noise initial data — the $\sigma=0$ member $\mu_0$ of the Tzvetkov–Visciglia family of weighted Gaussian measures, i.e., the centered Gaussian law of the random Fourier series $\sum_{n\neq0}g_ne^{inx}$ (mean-zero white noise, supported exactly on $H^s$ for $s<-1/2$) — and prove that $\mu_0$ is invariant under this dynamics.

Concretely, this means: (i) define the flow $\Phi_t$ $\mu_0$-almost surely for all $t\in\mathbb{R}$; (ii) state explicitly any renormalization of the dynamics and show it is well-defined, while the measure itself requires no renormalization since $R_0\equiv0$; (iii) prove the invariance identity $\mu_0(\Phi_t^{-1}(A))=\mu_0(A)$ for all Borel sets $A$ and all admissible $t$; and (iv) show that the constructed dynamics is the Benjamin–Ono dynamics with white-noise data rather than an arbitrary measure-preserving flow, consistent with the deterministic BO flow on regular initial data in a precise sense. The function-space framework (Sobolev, Fourier–Lebesgue, or Birkhoff-coordinate spaces) is not prescribed: any framework in which $\mu_0$ is meaningful and the above conditions hold is admissible.

This is the $\sigma=0$ case of the weighted Gaussian measures associated with the Benjamin–Ono conservation laws — the unique member of the family left open after Tzvetkov–Visciglia ($\sigma\ge4$), Deng–Tzvetkov–Visciglia ($\sigma\in\{2,3\}$), and Deng ($\sigma=1$, the Gibbs measure) — equivalently, in Tzvetkov's 2024 formulation, the extension of the Birkhoff map to the white-noise level that would "solve (after a renormalization) the Benjamin-Ono equation with white noise initial data".

References

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