Boundary Terms in Dynamical Large Deviations

Definitions and mathematical statement

A stochastic conservation law is a conservation equation incorporating randomness. A vanishing-viscosity perturbation introduces a viscosity parameter tending to zero. A dynamic large-deviation rate functional describes the exponential cost of deviations of trajectories in a specified asymptotic regime. Boundary terms here refer to contributions associated with the boundary in the corresponding large-deviation formulation; their precise meaning is not specified by the supplied record.

Source question. For the stochastic conservation law with vanishing-viscosity perturbation and dynamic large-deviation functional, can boundary conditions be added so that the corresponding boundary terms for dynamic large deviations are obtained?

Unresolved formulation. The supplied record does not specify the stochastic equation, spatial domain or dimension, time interval, unknown field, flux, viscosity operator, noise, initial conditions, admissible boundary conditions, or the dynamic large-deviation functional. It also does not specify the asymptotic scaling, trajectory space, topology, or the relation between the vanishing-viscosity limit and the large-deviation limit. Consequently, the question asks for a boundary extension of an unidentified underlying formulation. No particular equation, class of boundary data, formula for a boundary contribution, or universal quantifier over conservation laws can be supplied without additional source information. A precise theorem-level target therefore remains undetermined.

Short English statement

The question asks whether boundary conditions can be incorporated into the intended stochastic conservation law with vanishing viscosity, and whether the associated dynamic large-deviation boundary contributions can be derived. The supplied record does not identify the equation or large-deviation framework, so it supports this research question but does not determine a precise theorem statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001619$, title Boundary Terms in Dynamical Large Deviations, exact source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2010.
• Supplied report citation: Large Scale Stochastic Dynamics (2011), Oberwolfach Reports, pp. 2877--2975, DOI 10.4171/owr/2010/50. Supplied literature-source label: "original OWR report." https://doi.org/10.4171/owr/2010/50. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or passage is supplied.
• Supplied literature-source label: "Bertini--Landim--Mourragui." https://arxiv.org/abs/0804.2458. No paper title, theorem reference, or precise connection to the intended stochastic equation is supplied.
• Supplied literature-source label: "Franco--Goncalves--Landim--Neumann." https://arxiv.org/abs/2203.14417. No paper title, theorem reference, or precise connection to the intended stochastic equation is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and states that dynamic large-deviation principles with boundary conditions have been proved for several one-dimensional boundary-driven exclusion models, including weakly asymmetric and Robin-boundary cases, while the general stochastic-equation and boundary-term question remains open. This is supplied dated curation derived from the OWR record, not an independently verified mathematical or literature assessment; its time-sensitive status and cited claims require independent verification.