Variational Boundary Fluctuations as a First-Principles Origin of Langevin Noise

TL;DR

This paper derives Langevin noise from fundamental variational principles, showing how boundary fluctuations in Hamilton's framework naturally lead to stochastic forces with state-dependent characteristics.

Contribution

It introduces a first-principles derivation of Langevin noise from boundary fluctuations in Hamilton's principle, connecting variational methods with stochastic dynamics.

Findings

Langevin forces originate from boundary data fluctuations.

The amplitude of noise is filtered by the Hessian of Hamilton's principal function.

Additive Langevin forcing emerges as a Markovian coarse-grained limit.

Abstract

Stochastic forces are usually postulated or obtained by eliminating environmental degrees of freedom. Here we identify a variational origin: fluctuating endpoint data in Hamilton's principle induce fluctuations of the on-shell action. Hamilton--Jacobi propagation transports this boundary imprint, whose gradient generates an effective Langevin force inherited from boundary-action fluctuations. The resulting force is not freely specifiable: its amplitude is filtered by the Hessian of Hamilton's principal function, yielding multiplicative and state-dependent noise. Homogeneous additive Langevin forcing is recovered only as a Markovian coarse-grained limit.