KdV Evolution from General Ergodic Random Fields

Definitions and mathematical statement

The spatial domain is ${\mathbb{R}}$. A conventional normalization of the Korteweg--de Vries equation is \[ \partial_t u+\partial_x^3u+6u\partial_xu=0,\qquad u(0,\cdot)=u_0. \] The supplied record does not specify a normalization, the range or regularity of $u$, or how the nonlinear term is to be interpreted for rough data.

To explain the intended probabilistic terminology, let $E$ denote a measurable space of functions or distributions on ${\mathbb{R}}$ admitting measurable spatial translations $\tau_a f=f(\cdot+a)$, and let $\mu$ be the law of the initial random field. Under the usual stationary interpretation, $\mu$ is translation invariant if $\mu(\tau_a^{-1}A)=\mu(A)$ for every measurable $A\subseteq E$ and every $a\in{\mathbb{R}}$. It is ergodic if every measurable event invariant under all translations, modulo $\mu$-null sets, has probability zero or one. The source does not specify $E$ or explicitly define its use of ergodicity; this interpretation therefore requires confirmation.

Question, with unresolved qualifications. For general initial random fields on ${\mathbb{R}}$ whose laws are stationary and ergodic in this sense, does KdV admit an almost surely well-posed evolution beyond the white-noise initial-data case identified in the supplied record? Schematically, the intended assertion would require, for every admissible law $\mu$, a measurable set $E_\mu\subseteq E$ with $\mu(E_\mu)=1$, such that each $u_0\in E_\mu$ generates an evolution satisfying appropriate existence, uniqueness, and continuous-dependence requirements.

This is an incomplete formulation: the admissible laws, moment or growth assumptions, initial-data topology, solution space, uniqueness class, time interval, and meaning of continuous dependence are unspecified. The source also supplies no conditions at spatial infinity. Consequently, the displayed schematic assertion cannot be treated as a precise universal conjecture without further hypotheses. In particular, local versus global well-posedness and the meaning of almost-sure uniqueness remain unresolved.

Short English statement

The question asks whether KdV on ${\mathbb{R}}$ has an almost surely well-posed evolution from general spatially ergodic random initial data, extending the white-noise case described in the source. The allowed random fields and the required notion of well-posedness are unspecified, so the intended mathematical assertion needs further hypotheses.

Sources

• Local UnsolvedMath registry: permanent ID $30004246$, title "KdV Evolution from General Ergodic Random Fields," source problem code catalogue entry, collection "Oberwolfach Reports Open Problems." The supplied proposed year is 2019; no proposer is named.
• Supplied report citation: Large Scale Stochastic Dynamics (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/42, pp. 2605--2669. https://doi.org/10.4171/owr/2019/42. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or passage is supplied.
• Supplied literature source, labeled "KdV well-posedness paper": https://arxiv.org/abs/1802.04851. The record supplies no authors, title, or theorem reference for this link.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and reports that KdV is well-posed in $H^{-1}$, that white-noise dynamics are invariant, and that almost-sure well-posedness for a general ergodic random field on ${\mathbb{R}}$ remains unresolved. This is supplied dated curation, not an independently verified assessment; the record expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its accompanying verification note says that notation, typography, and sentence structure were repaired while preserving mathematical content. This editorial label does not establish mathematical correctness or independently verify the literature assessment.