The Choquet–Deny Equation in a Banach Space
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Bibliographic record
Abstract
Abstract Let G be a locally compact group and π a representation of G by weakly* continuous isometries acting in a dual Banach space E . Given a probability measure μ on G , we study the Choquet–Deny equation π ( μ ) x = x , x ∈ E . We prove that the solutions of this equation form the range of a projection of norm 1 and can be represented by means of a “Poisson formula” on the same boundary space that is used to represent the bounded harmonic functions of the random walk of law μ . The relation between the space of solutions of the Choquet–Deny equation in E and the space of bounded harmonic functions can be understood in terms of a construction resembling the W *-crossed product and coinciding precisely with the crossed product in the special case of the Choquet–Deny equation in the space E = B ( L 2 ( G )) of bounded linear operators on L 2 ( G ). Other general properties of the Choquet–Deny equation in a Banach space are also discussed.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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