Diffusions on fractals often exhibit anomalous behavior, characterized by sub-Gaussian heat kernel estimates. Associated with such diffusions are several notions of dimension, including spectral dimension, which can be interpreted in terms of eigenvalue asymptotics of the generator of the diffusion, and martingale dimension, which measures the number of independent martingale directions of the diffusion.
In 2013, Hino established an upper bound on martingale dimension in terms of spectral dimension for certain self-similar fractals and conjectured that this relationship holds in greater generality. In this talk, I will introduce these notions of dimension, discuss our resolution of Hino's conjecture, and outline some of the ideas behind the proof.
This is joint work with Aobo Chen (Aarhus University).