Stability and instability of Gaussian heat kernel estimates for random walks among time-dependent conductances. (English) Zbl 1338.60196

Huang, Ruojun; Kumagai, Takashi

Electron. Commun. Probab. 21, Paper No. 5, 11 p. (2016).

Summary: We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is proved by giving an example of a ballistic and transient time-dependent random walk on \(\mathbb{Z}\) among uniformly elliptic time-dependent conductances. For continuous time random walks, we show the instability when the holding times are i.i.d. \(\exp(1)\), and in contrast, we prove the stability when the holding times change by sites in such a way that the base measure is a uniform measure.

MSC:

60J35	Transition functions, generators and resolvents
60J05	Discrete-time Markov processes on general state spaces
60J25	Continuous-time Markov processes on general state spaces
60J45	Probabilistic potential theory

Keywords:

heat kernel estimates; recurrence; stability; time-dependent random walks; transience