Weak-Form Recovery of Stochastic Generators and Dynamical Invariants

TL;DR

This paper introduces a bias-free method for recovering stochastic generator operators from data by projecting onto spatial kernels, enabling accurate identification of dynamical invariants and spectral properties.

Contribution

It proposes a novel spatial kernel projection approach that eliminates endogeneity bias in stochastic generator estimation from sparse data.

Findings

Coefficient errors below 5% in benchmark tests

Stationary-density total-variation distances below 0.01

Faithful reproduction of true relaxation timescales

Abstract

Spectral gaps, Kramers escape rates, and position-dependent relaxation timescales are dynamical invariants encoded in the infinitesimal generator $\Lop$ of a stochastic flow. We show that weak projection of the governing It\^{o} SDE onto temporal test functions produces an endogeneity bias of order $O(T\,\dt^{3/2})$ that grows with the observation window and cannot be eliminated by additional data. Projecting instead onto spatial Gaussian kernels removes the bias exactly: $\mathcal{F}_{t_n}$-measurability and the tower property guarantee unbiased regression rows at every step. The resulting framework jointly identifies the drift $b(x)$ and diffusion $a(x)$ from a single sparse regression, producing an explicit symbolic enerator amenable to spectral analysis. Validation on three benchmark systems yields coefficient errors below 5%, stationary-density total-variation distances below 0.01, and autocorrelation functions that faithfully reproduce true relaxation timescales.