Extinction Criteria for the Derrida–Retaux Process

Definitions and mathematical statement

Let $\mu_0$ be an initial probability distribution on the state space denoted by $\mathbb{R}_+$ in the supplied source, and let $(X_t)_{t\geq 0}$ denote the continuous-time Derrida--Retaux process with $\mathcal{L}(X_0)=\mu_0$. The supplied record does not specify the process dynamics, the convention for the endpoint of $\mathbb{R}_+$, or the definition and domain of the quantity $F_\infty(\mu_0)$. Consequently, these symbols retain their source-specific meanings; the following is a conditional formulation requiring those definitions before it becomes self-contained.

Characterization problem. Determine a condition $\mathcal{C}$ on the initial law such that, for every initial law $\mu_0$ admissible for the source model, \[ F_\infty(\mu_0)=0 \quad\Longleftrightarrow\quad \mathcal{C}(\mu_0). \] The record describes the target as general initial laws and supplies no moment or integrability restrictions. Any restrictions required to define the process or $F_\infty$ remain to be recovered from the source.

Additional conjecture reported by the source. For every such admissible initial law, does \[ F_\infty(\mu_0)=0 \quad\Longrightarrow\quad \forall\varepsilon>0,\qquad \lim_{t\to\infty}\mathbb{P}(|X_t|>\varepsilon)=0 \] hold? The conclusion is precisely convergence of $X_t$ to zero in probability. Neither this implication nor the requested characterization is asserted here as a theorem.

Short English statement

Find an exact criterion on the initial distribution for the continuous-time Derrida--Retaux quantity $F_\infty$ to vanish. The source also conjectures that vanishing forces the process to converge to zero in probability. The supplied record omits the dynamics and the definition of $F_\infty$, so a self-contained formulation requires further source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004541$, title: Extinction Criteria for the Derrida--Retaux Process; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. No individual proposer is identified.
• Stochastic Processes under Constraints (2021), Oberwolfach Reports, pp. 1601--1656, DOI: https://doi.org/10.4171/owr/2020/32. This is also the supplied literature-sources link, labelled "OWR report." The registry records proposed year 2020 and the tag owr-year:2020; these are retained alongside the supplied 2021 citation.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, says that necessary and sufficient conditions for general initial laws are explicitly posed as an open question, and reports that only special initial distributions are solved. This is supplied dated curation, not an independently verified assessment.
• The registry assigns statement status corrected_verified and reports that the official report was checked, the question isolated, and notation repaired for self-contained publication. These are supplied provenance claims; no independent source visit or mathematical verification is claimed in this draft.