A Complete Decomposition of Stochastic Differential Equations

2026-01-12

AI Summary

This work proposes a novel structured decomposition for stochastic differential equations (SDEs) with prescribed time-varying marginal distributions. By integrating probability flow analysis, SDE theory, and matrix decomposition techniques, the authors uniquely decompose any such SDE into three distinct components: a scalar potential field governing the evolution of the marginal distribution, a symmetric positive semi-definite diffusion matrix field, and a skew-symmetric matrix field. This tripartite decomposition provides the first complete characterization of the intrinsic geometric and probabilistic structure of SDEs under given marginal constraints. The decomposition is not only mathematically unique but also establishes a rigorous theoretical foundation for modeling, analyzing, and controlling complex stochastic systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• stochastic differential equations
• marginal distributions
• decomposition
• diffusion matrix
• skew-symmetric matrix

Innovation

• stochastic differential equations
• marginal distributions
• complete decomposition
• diffusion matrix
• skew-symmetric field