Consider random conductances that allow long range jumps. In particular we consider conductances $C_{xy} = w_{xy}|x − y|^{−d−\alpha}$ for distinct $x, y \in Z^d$ and $0 < \alpha < 2$, where $\lbrace w_{xy} = w_{yx} : x, y \in Z^d\rbrace$ are non-negative independent random variables with mean 1. We prove that under some moment conditions for $w$, suitably rescaled Markov chains among the random conductances converge to a rotationally symmetric $\alpha$-stable process almost surely w.r.t. the randomness of the environments. The proof is a combination of analytic and probabilistic methods based on the recently established de Giorgi-Nash-Moser theory for processes with long range jumps. If time permits, we also discuss quenched heat kernel estimates as well. This is a joint work with Xin Chen (Shanghai) and Jian Wang (Fuzhou).[-]

In recent years, interest in time changes of stochastic processes according to irregular measures has arisen from various sources. Fundamental examples of such time-changed processes include the so-called Fontes-Isopi-Newman (FIN) diffusion and fractional kinetics (FK) processes, the introduction of which were partly motivated by the study of the localization and aging properties of physical spin systems, and the two- dimensional Liouville Brownian motion, which is the diffusion naturally associated with planar Liouville quantum gravity.
This FIN diffusions and FK processes are known to be the scaling limits of the Bouchaud trap models, and the two-dimensional Liouville Brownian motion is conjectured to be the scaling limit of simple random walks on random planar maps.
In the first part of my talk, I will provide a general framework for studying such time changed processes and their discrete approximations in the case when the underlying stochastic process is strongly recurrent, in the sense that it can be described by a resistance form, as introduced by J. Kigami. In particular, this includes the case of Brownian motion on tree-like spaces and low-dimensional self-similar fractals.
In the second part of my talk, I will discuss heat kernel estimates for (generalized) FIN diffusions and FK processes on metric measure spaces.
This talk is based on joint works with D. Croydon (Warwick) and B.M. Hambly (Oxford) and with Z.-Q. Chen (Seattle), P. Kim (Seoul) and J. Wang (Fuzhou).[-]