Path-Kernel Method for Differentiating Unstable Diffusions

We derive and prove the path-kernel formula for the linear response (parameter-derivative of averaged statistics) of SDEs. The parameter may affect the drift coefficient, the diffusion coefficient, and the initial condition. The formula tempers the unstableness by gradually moving the derivative from path-perturbation to kernel-differentiation, without assuming hyperbolicity. We prove it by direct comparison of bundles of paths across different parameter values. We also derive a pathwise Monte Carlo algorithm for estimating linear responses and demonstrate it on the 40-dimensional noisy Lorenz--96 system. Our result provides a new computational tool for optimization, and has already led to a follow-up application to data assimilation.