Rigorous Noise-Induced Phenomena in Random Dynamical Systems

Definitions and mathematical statement

The supplied record concerns random dynamical systems, including independent and identically distributed random iteration of complex polynomials on the Riemann sphere \[ \widehat{{\mathbb{C}}}={\mathbb{C}}\cup\{\infty\}. \] For notation, if a measurable family of complex polynomial maps and a probability law on that family are specified, let $P_1,P_2,\ldots$ be independent random maps with that common law, each extended to $\widehat{{\mathbb{C}}}$. The corresponding random orbit from $z\in\widehat{{\mathbb{C}}}$ is \[ X_0(z)=z,\qquad X_n(z)=P_n\circ P_{n-1}\circ\cdots\circ P_1(z),\quad n\geq 1. \] This notation describes the example mentioned in the source; it does not impose a polynomial family, degree restriction, probability law, or initial condition absent from the record.

Source request. Give mathematically complete proofs for various noise-induced phenomena observed in numerical experiments for random dynamical systems, including i.i.d. random dynamics of complex polynomials on the Riemann sphere.

Unresolved formulation. The supplied record does not identify the phenomena, numerical experiments, dynamical systems, noise distributions, or conclusions to be proved. It does not define what constitutes a noise-induced phenomenon or specify a comparison with dynamics without noise. In the polynomial example, the family of maps, their degrees, their common distribution, and any conditions on starting points are unspecified. No universal or existential quantifier over these choices is supplied. Consequently, this is a broad research request rather than a uniquely specified mathematical proposition; the displayed framework alone does not determine a theorem to prove.

Short English statement

The request is to prove noise-induced effects observed numerically in random dynamical systems, with independent, identically distributed random polynomial iteration on the Riemann sphere given as an example. The supplied record does not identify particular effects or sufficient hypotheses and conclusions to state a definite theorem.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001959$, title: Rigorous Noise-Induced Phenomena in Random Dynamical Systems; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. The record gives proposed year 2012 and names no individual proposer.
• Supplied report citation: Mini-Workshop: Thermodynamic Formalism, Geometry and Stochastics (2012), Oberwolfach Reports, pp. 77--129, DOI 10.4171/owr/2012/03. Supplied source URL and literature reference labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/03. No problem-specific page or named author is supplied.
• Supplied literature assessment, checked on 2026-08-21: status labeled open; the assessment describes an explicitly broad request for rigorous proofs of noise-induced phenomena and states that the extracted statement supplies no resolved theorem or finite target. This is supplied dated curation, not an independently verified literature assessment. The background requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field reports manual review finding the exact extracted statement self-contained, mathematically coherent, and free of LaTeX or report-layout artifacts. That supplied verification does not resolve the missing mathematical targets and hypotheses identified here.