[Paper] A Complete Decomposition of Stochastic Differential Equations

Source: arXiv - 2601.07834v1

Overview

The paper proves that any stochastic differential equation (SDE) whose marginal distributions are known at each time can be uniquely split into three building blocks: a scalar field that drives the evolution of the marginals, a symmetric positive‑semidefinite diffusion matrix, and a skew‑symmetric matrix that captures “rotational” dynamics. This decomposition offers a clean, mathematically rigorous way to separate the deterministic, diffusive, and conservative parts of a stochastic system, opening new avenues for analysis, simulation, and control.

Key Contributions

• Universal decomposition theorem for SDEs with prescribed time‑dependent marginals.
• Uniqueness proof showing that the three components (scalar field, symmetric diffusion, skew‑symmetric drift) are uniquely determined by the marginal law.
• Constructive algorithm to recover the three fields from a given marginal trajectory, based on solving a linear PDE and a matrix factorisation.
• Link to optimal transport: the scalar field coincides with the optimal transport potential that pushes the initial distribution to the marginal at time t.
• Illustrative examples (Gaussian, Ornstein‑Uhlenbeck, and nonlinear diffusion) that demonstrate how the decomposition simplifies both analytical insight and numerical implementation.

Methodology

1. Marginal specification – Start with a family of probability densities ({p_t(x)}_{t\ge0}) that are smooth in time and space.
2. Continuity equation – Express the evolution of (p_t) via the continuity equation (\partial_t p_t + \nabla!\cdot (v_t p_t)=0), where (v_t) is an unknown velocity field.
3. Helmholtz‑Hodge split – Decompose (v_t) into a gradient part (\nabla \phi_t) (the scalar field) and a divergence‑free part (J_t).
4. Diffusion matrix extraction – Impose that the stochastic dynamics be driven by a diffusion matrix (a_t(x)) that is symmetric positive‑semidefinite, satisfying the Fokker‑Planck equation (\partial_t p_t = -\nabla!\cdot (b_t p_t) + \tfrac12 \nabla!\cdot (a_t \nabla p_t)).
5. Skew‑symmetric drift – The remaining part of the drift, after accounting for (\nabla \phi_t) and the diffusion term, must be a skew‑symmetric matrix field (K_t(x)) (i.e., (K_t^\top = -K_t)).
6. Uniqueness argument – By comparing two possible decompositions and using the positivity of the diffusion term, the authors show the three fields cannot differ, establishing uniqueness.

The construction reduces to solving a Poisson‑type equation for (\phi_t) and a matrix factorisation for (a_t), both of which have well‑established numerical solvers.

Results & Findings

• Existence: For any smooth marginal trajectory ({p_t}), there exists at least one SDE whose solution matches those marginals, and it can be expressed in the three‑component form.
• Uniqueness: The scalar field (\phi_t), diffusion matrix (a_t), and skew‑symmetric drift (K_t) are uniquely determined, eliminating ambiguity in model specification.
• Interpretability: The scalar field aligns with the optimal transport potential, the diffusion matrix captures stochastic spreading, and the skew‑symmetric part encodes “circulatory” forces that do not affect the marginal density.
• Numerical validation: Simulations on synthetic data confirm that reconstructing the SDE from the three fields reproduces the prescribed marginals to machine precision.

Practical Implications

• Model design for ML & finance – When a practitioner knows the desired distribution of a state variable (e.g., asset price, latent variable in a generative model) at each future time, they can now construct an SDE that guarantees those marginals, with clear control over diffusion and rotational dynamics.
• Improved simulation pipelines – Separating diffusion from skew‑symmetric drift allows developers to use specialized integrators (e.g., symplectic schemes for the skew part) that are more stable and efficient than generic SDE solvers.
• Interpretability in stochastic control – The decomposition isolates the part of the control that actually changes the marginal distribution (the scalar field) from the part that only re‑orients trajectories, aiding in the design of cost‑effective control policies.
• Connection to normalizing flows – The scalar field can be interpreted as a time‑dependent potential flow, suggesting new architectures for continuous normalizing flows that enforce exact marginal constraints.
• Robustness analysis – Since the diffusion matrix is explicitly positive‑semidefinite, developers can enforce numerical stability constraints (e.g., eigenvalue bounds) directly during model training or calibration.

Limitations & Future Work

• Smoothness requirement – The theorem assumes sufficiently smooth marginal densities; extending the result to distributions with jumps or singular components (e.g., Lévy processes) remains open.
• Computational cost – Solving the Poisson equation for (\phi_t) and factorising the diffusion matrix at each time step can be expensive in high dimensions; scalable approximations (e.g., neural PDE solvers) are a promising direction.
• Extension to manifold‑valued states – The current formulation works in Euclidean space; adapting the decomposition to manifolds (e.g., rotation groups, graphs) would broaden applicability.
• Learning the decomposition – Future work could explore end‑to‑end training of neural networks that directly output the three fields from data, enabling data‑driven discovery of SDEs with prescribed marginals.

Authors

• Samuel Duffield

Paper Information

• arXiv ID: 2601.07834v1
• Categories: math.PR, cs.LG, math.ST
• Published: January 12, 2026
• PDF: Download PDF