Fluctuating boundary data in Hamilton's principle propagate via Hamilton-Jacobi to produce state-dependent multiplicative Langevin forces, with additive noise recovered only after Markovian coarse-graining.
Mori, Progress of Theoretical Physics33, 423 (1965)
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Variational Boundary Fluctuations as a First-Principles Origin of Langevin Noise
Fluctuating boundary data in Hamilton's principle propagate via Hamilton-Jacobi to produce state-dependent multiplicative Langevin forces, with additive noise recovered only after Markovian coarse-graining.