## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 16 (2011), paper no. 62, 1693-1722.

### Gaussian Upper Bounds for Heat Kernels of Continuous Time Simple Random Walks

#### Abstract

We consider continuous time simple random walks with arbitrary speed measure $\theta$ on infinite weighted graphs. Write $p_t(x,y)$<em></em> for the heat kernel of this process. Given on-diagonal upper bounds for the heat kernel at two points $x_1,x_2$, we obtain a Gaussian upper bound for $p_t(x_1,x_2)$<em></em>. The distance function which appears in this estimate is not in general the graph metric, but a new metric which is adapted to the random walk. Long-range non-Gaussian bounds in this new metric are also established. Applications to heat kernel bounds for various models of random walks in random environments are discussed.

#### Article information

**Source**

Electron. J. Probab. Volume 16 (2011), paper no. 62, 1693-1722.

**Dates**

Accepted: 12 September 2011

First available in Project Euclid: 1 June 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1464820231

**Digital Object Identifier**

doi:10.1214/EJP.v16-926

**Mathematical Reviews number (MathSciNet)**

MR2835251

**Zentralblatt MATH identifier**

1244.60099

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 30K08 60K37: Processes in random environments

**Keywords**

random walk heat kernel Gaussian upper bound random walk in random environment

**Rights**

This work is licensed under a Creative Commons Attribution 3.0 License.

#### Citation

Folz, Matthew. Gaussian Upper Bounds for Heat Kernels of Continuous Time Simple Random Walks. Electron. J. Probab. 16 (2011), paper no. 62, 1693--1722. doi:10.1214/EJP.v16-926. https://projecteuclid.org/euclid.ejp/1464820231.