Explicit Stochastic Averaging Errors in Higher Dimensions

Definitions and mathematical statement

Let $n\in{\mathbb{N}}$ satisfy $n>1$, and let $X(t)$ be the slow process, taking values in ${\mathbb{R}}^n$. The supplied source describes the stochastic-averaging expression \[ I_{\varepsilon}(T):=\int_0^T X(\tau)Z(\tau\varepsilon^{-2})^q\,{\,\mathrm{d}}\beta(\tau), \] whenever this integral is well defined. Here $\varepsilon$ denotes the averaging scale, $T$ the integration endpoint, and $Z$, $q$, and $\beta$ retain their source notation; their mathematical specifications are absent from the supplied record.

Source question. Can one obtain explicit error estimates for the stochastic-averaging limit associated with $I_{\varepsilon}(T)$ when the slow process takes values in ${\mathbb{R}}^n$ with $n>1$?

Unresolved formulation. The supplied record does not define the averaging limit or its dependence on $X$, $Z$, $q$, and $\beta$. It also does not specify the stochastic integral convention, the admissible parameters, the process hypotheses, the error metric, or the desired rate and uniformity. Consequently, the displayed question preserves the supplied target but does not determine a fully quantified error-bound problem. These missing qualifications require source clarification and human mathematical review.

Short English statement

The question asks for explicit quantitative errors in stochastic averaging for the displayed integral when the slow process has dimension greater than one. The supplied record leaves the processes, limiting object, assumptions, and meaning of error insufficiently specified to state a precise bound.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001866$, source problem code catalogue entry, titled Explicit Stochastic Averaging Errors in Higher Dimensions. No individual proposer is supplied.
• Supplied report citation: Mini-Workshop: Dynamics of Stochastic Systems and their Approximation (2012), Oberwolfach Reports, pp. 2283--2326, DOI 10.4171/owr/2011/40. Original OWR report: https://doi.org/10.4171/owr/2011/40.
• Supplied report-text reference: https://oa.tib.eu/renate/server/api/core/bitstreams/7ee09859-9440-461d-a8c5-84ec5410930a/content.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that the report leaves multidimensional stochastic-averaging error estimates open, and asserts that later work does not supply a general resolution. This is supplied dated curation, not an independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified. Its verification note says that the official report was checked and that the integral and higher-dimensional restriction were restored. This is a supplied verification claim, not verification performed for this draft.
• The record gives proposed year 2011 and an owr-year:2011 tag, while its report citation contains 2012. This dating discrepancy is retained without resolution.