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The covariant Langevin equation of diffusion on Riemannian manifolds

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arxiv 2310.17314 v2 pith:2AQZNMRN submitted 2023-10-26 cond-mat.stat-mech gr-qcmath-phmath.MP

The covariant Langevin equation of diffusion on Riemannian manifolds

classification cond-mat.stat-mech gr-qcmath-phmath.MP
keywords covariantequationlangevindifferentialformframestochasticadditional
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The covariant form of the multivariable diffusion-drift process is described by the covariant Fokker--Planck equation using the standard toolbox of Riemann geometry. The covariant form of the equivalent Langevin stochastic differential equation is long sought after in both physics and mathematics. We show that the simplest covariant Stratonovich stochastic differential equation depending on the local orthogonal frame (cf. vielbein) becomes the desired covariant Langevin equation provided we impose an additional covariant constraint: the vectors of the frame must be divergence-free.

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  1. On the use of the Belopol'skaya-Daletskii representation of a diffusion on a Riemann manifold to construct path integrals

    cond-mat.stat-mech 2026-07 conditional novelty 4.0

    A Belopol'skaya-Daletskii (exponential-map) formulation yields the known scalar-curvature term R/6 in finite-dimensional path integrals for diffusions on Riemannian manifolds.