The Signed BAR Uniqueness Problem
Statement
For a multidimensional reflected diffusion, does the basic adjoint relationship uniquely characterize the stationary distribution? The question had stood unresolved for more than thirty-five years since the BAR approach was introduced. For stable Harrison-Reiman data with a nonsingular -matrix reflection matrix, the finite-signed uniqueness problem is settled, via pathwise differentiability of the reflected process; the nonsigned version is also shown unique within the Harrison-Reiman class.
Context
A foundational uniqueness question underpinning the BAR method for reflected diffusions, open since the approach was introduced and central to the heavy-traffic queueing literature that uses it.
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The paper names the model contribution in its title and states that the proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors, with the chat logs published. It also records a negative result worth having: on a harder related task both ChatGPT 5.5 Pro extended and Claude Opus 4.8 max failed.
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