Divergence-Kernel method for linear responses and diffusion models

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.