## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 5, 11 pp.

### Stability and instability of Gaussian heat kernel estimates for random walks among time-dependent conductances

Ruojun Huang and Takashi Kumagai

#### Abstract

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.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 5, 11 pp.

**Dates**

Received: 6 June 2015

Accepted: 18 December 2015

First available in Project Euclid: 3 February 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1454514625

**Digital Object Identifier**

doi:10.1214/15-ECP4347

**Mathematical Reviews number (MathSciNet)**

MR3485374

**Zentralblatt MATH identifier**

1338.60196

**Subjects**

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

Secondary: 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

**Keywords**

heat kernel estimates recurrence stability time-dependent random walks transience

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Huang, Ruojun; Kumagai, Takashi. Stability and instability of Gaussian heat kernel estimates for random walks among time-dependent conductances. Electron. Commun. Probab. 21 (2016), paper no. 5, 11 pp. doi:10.1214/15-ECP4347. https://projecteuclid.org/euclid.ecp/1454514625