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arXiv · 2509.09566

A geometric formulation of GENERIC stochastic differential equations

Mark A. Peletier·Marcello Seri
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Abstract

We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction builds on the classical GENERIC or metriplectic formalism, extending it to manifolds by introducing a degenerate Poisson structure, a degenerate co-metric, and a volume form satisfying a unimodularity condition. The resulting equation preserves a particular Boltzmann-type measure, ensures almost-sure conservation of energy, and reduces to the deterministic GENERIC/metriplectic formulation in the zero-noise limit. This geometrization separates system-specific quantities from the ambient space, clarifies the roles of the underlying structures, and provides a foundation for analytic and numerical methods, as well as future extensions to quantum and coarse-grained systems.

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Mark A. Peletier, Marcello Seri. 2025-09-11. A geometric formulation of GENERIC stochastic differential equations. https://arxiv.org/abs/2509.09566

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