On hitting time distributions of Markov processes with sub-Gaussian heat kernel bounds

What the math gives to ML

The paper supplies explicit sub-Gaussian first-hitting-time tails for Markov processes whose spatial propagation has walk dimension \(\beta\), including processes on irregular metric spaces and processes with polar states removed from the effective distance. The transferable asset is an exponential bound on the probability that a finite-time propagation process reaches a distant region, rather than merely a pointwise heat-kernel estimate. This can be used to turn diffusion-like attention or graph message passing into a principled sparse operator: choose a radius from a target tail probability and discard interactions outside that radius. The resulting architecture is falsifiable through FLOP reduction and approximation-error measurements, although the guarantee applies only when the learned transition matrix is a reasonable sub-Gaussian Markov kernel.