A Complete Decomposition of Stochastic Differential Equations

TL;DR

The paper addresses constructing stochastic dynamics that reproduce prescribed time-dependent marginals $p(x,t)$ by proving a complete decomposition into a unique scalar field $\phi(x,t)$ that governs marginal evolution and diffusion fields $D(x,t)$ (symmetric PSD) and $Q(x,t)$ (skew-symmetric) that preserve marginals. The explicit SDE form is $dx = \phi \nabla_x \log p \,dt + \nabla_x \phi \,dt + [D+Q]\nabla_x \log p \,dt + \nabla_x \cdot [D+Q] \,dt + \sqrt{2D}\,dw$, with $D=D^T \succeq 0$ and $Q^T = -Q$, and the FP equation reduces to $\partial_t p = -\Delta_x [\phi p]$, linking marginal dynamics to a Poisson equation. This work generalizes autonomous diffusion results (e.g., Ma et al.) through a Helmholtz decomposition, providing a complete characterization of all SDEs compatible with a given marginal flow and clarifying implications for time-reversal and generative denoising models. The framework offers practical avenues to optimize path-wise behavior (via $D$ and $Q$) without altering marginals, with potential applications to entropy-regularized sampling and non-autonomous dynamics.

Abstract

We show that any stochastic differential equation with prescribed time-dependent marginal distributions admits a decomposition into three components: a unique scalar field governing marginal evolution, a symmetric positive-semidefinite diffusion matrix field and a skew-symmetric matrix field.

Theorems & Definitions (12)

• Theorem 1
• proof : Proof: \ref{['eq:decomp']} $\implies p(x,t)$
• proof : Proof: \ref{['eq:decomp']} $\impliedby p(x,t)$
• Theorem 2
• proof : Proof: Existence
• proof : Proof: Uniqueness
• Theorem 1
• proof : Proof: \ref{['eq:decomp']} $\implies p(x,t)$
• proof : Proof: \ref{['eq:decomp_2']} $\impliedby p(x,t)$
• Theorem 2