Gaussian Flow Dynamics: Stochastic Processes with Gaussian Marginal Laws

Gaussian marginal laws arise throughout stochastic modelling, but do not uniquely determine the underlying dynamics. We introduce Gaussian flow dynamics, a simple framework that constructs deterministic and stochastic dynamics directly from smoothly evolving Gaussian marginals while exposing their marginal-preserving degrees of freedom. The framework allows state-dependent diffusion coefficients and recovers every linear stochastic differential equation with Gaussian marginals.