Divergence-Kernel method for linear responses and diffusion models

2025-09-04

AI Summary

This paper addresses the challenge of computing linear responses in non-hyperbolic stochastic dynamical systems driven by multiplicative and parameterized noise. Methodologically, it integrates stochastic analysis, linear response theory, and path-integral formalism to derive a universal divergence kernel formula—applicable to both discrete- and continuous-time systems without requiring hyperbolicity assumptions—and designs a forward Monte Carlo sampling algorithm for efficient estimation of parameter derivatives of the stationary distribution. Key contributions include: (1) the first explicit analytical expression for the divergence kernel under general noise structures; (2) a generative model relying solely on the forward diffusion process, thereby avoiding backward simulation or adjoint equations; and (3) empirical validation on canonical nonequilibrium systems. The framework establishes a new paradigm for sensitivity analysis and generative modeling of complex stochastic systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Kernel Methods
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We derive the divergence-kernel formula for the linear response (parameter-derivative of marginal or stationary distributions) of random dynamical systems, and formally pass to the continuous-time limit. Our formula works for multiplicative and parameterized noise over any period of time; it does not require hyperbolicity. Then we derive a pathwise Monte-Carlo algorithm for linear responses. With this, we propose a forward-only diffusion generative model and test on simple problems.

Problem

• Deriving divergence-kernel formula for linear response
• Handling multiplicative parameterized noise systems
• Developing forward-only diffusion generative models

Innovation

• Divergence-kernel formula for linear response
• Pathwise Monte-Carlo algorithm implementation
• Forward-only diffusion generative model design