Homogeneous Dirichlet Forms on p.c.f. Fractals and their Spectral Asymptotics

We formulate a class of “homogeneous” Dirichlet forms (DF) that aims to explore those forms that do not satisfy the conventional energy self-similar identity (degenerate DFs). This class of DFs has been studied in Hambly Jones (J. Theoret. Probab., 15, 285–322 2002), Hambly and Kumagai (Potential Anal., 8, 359–397 1998), Hambly and Yang (J. Fractal Geom., 6, 1–51 2019) and Hattori et al. (Probab. Theory Related Fields, 100, 85–116 1994) in connection with the asymptotically one-dimensional diffusions on the Sierpinski gaskets (SG) and their generalizations. In this paper, we give a systematic study of such DFs and their spectral properties. We also emphasize the construction of some new homogeneous DFs. Moreover, a basic assumption on the resistance growth that was required in Hambly and Kumagai (Potential Anal., 8, 359–397 1998) to investigate the heat kernel and the existence of the “non-fixed point” limiting diffusion is verified analytically.