A geometric formulation of GENERIC stochastic differential equations
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.
Cite
@article{arxiv.2509.09566,
title = {A geometric formulation of GENERIC stochastic differential equations},
author = {Mark A. Peletier and Marcello Seri},
journal= {arXiv preprint arXiv:2509.09566},
year = {2025}
}