  We address data-driven learning of the infinitesimal generator of stochastic
diffusion processes, essential for understanding numerical simulations of
natural and physical systems. The unbounded nature of the generator poses
significant challenges, rendering conventional analysis techniques for
Hilbert-Schmidt operators ineffective. To overcome this, we introduce a novel
framework based on the energy functional for these stochastic processes. Our
approach integrates physical priors through an energy-based risk metric in both
full and partial knowledge settings. We evaluate the statistical performance of
a reduced-rank estimator in reproducing kernel Hilbert spaces (RKHS) in the
partial knowledge setting. Notably, our approach provides learning bounds
independent of the state space dimension and ensures non-spurious spectral
estimation. Additionally, we elucidate how the distortion between the intrinsic
energy-induced metric of the stochastic diffusion and the RKHS metric used for
generator estimation impacts the spectral learning bounds.
