Abstract: We formally derive the path-kernel formula for the linear response, or the parameter-derivative of averaged statistics, of SDEs. The parameter controls the diffusion coefficients. Our formula combines and extends the path-perturbation and the kernel method (also called likelihood ratio or Cameron-Martin-Girsanov) by extending Bismut-Elworthy-Li's compensation idea to perturbation on dynamics. It tempers the unstableness by gradually moving the path-perturbation to hit the probability kernel. It does not assume hyperbolicity. We prove by direct comparison of bundles of paths across different parameters. Based on the new formula, we derive a pathwise sampling algorithm for linear responses and demonstrate it on the 40-dimensional Lorenz 96 system with noise; this numerical example is difficult for all other algorithms.

Biography: Angxiu Ni is an Assistant Professor in the Department of Mathematics at the University of California, Irvine. He received his Ph.D. in Mathematics from UC Berkeley, and subsequently held postdoctoral positions at BICMR, Peking University, and later served as an Assistant Professor at YMSC and Qiuzhen College, Tsinghua University. His research interests lie in applied dynamical systems and probability, with broad applications across various fields including fluid dynamics and machine learning. In particular, he studies how to take derivatives of averaged statistics.