A Heat Kernel Expectation Approach to Boundary-Corrected Li--Yau Estimates for the Dirichlet Heat Equation

What the math gives to ML

The paper turns the reflected Dirichlet heat kernel into a normalized measure, so derivatives of a boundary-conditioned diffusion become expectations under an explicit kernel rather than unconstrained Gaussian derivatives. The transferable asset is the exact image-kernel correction: near an absorbing boundary, the score contains a hyperbolic term that diverges in the correct way and cannot be represented well by a whole-space Gaussian approximation. The most promising neural application is a diffusion model on a half-space or other positivity-constrained state space, using the killed-Brownian transition kernel as a training target or sampler instead of clipping unconstrained Gaussian trajectories.