## Revista Matemática Iberoamericana

### Random walks on graphs with volume and time doubling

András Telcs

#### Abstract

This paper studies the on- and off-diagonal upper estimate and the two-sided transition probability estimate of random walks on weighted graphs.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 22, Number 1 (2006), 17-54.

Dates
First available in Project Euclid: 24 May 2006

https://projecteuclid.org/euclid.rmi/1148492175

Mathematical Reviews number (MathSciNet)
MR2267312

Zentralblatt MATH identifier
1118.60062

#### Citation

Telcs , András. Random walks on graphs with volume and time doubling. Rev. Mat. Iberoamericana 22 (2006), no. 1, 17--54. https://projecteuclid.org/euclid.rmi/1148492175

#### References

• Barlow, M.T.: Diffusions on Fractals. In Lectures on Probability Theory and Statistics, Ecole d'été de Probabilités de Saint-flour XXV-1995, 1-121. Lecture Notes in Math. 1690, Springer, 1998.
• Barlow, M.T., Bass, R.: Stability of the parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356 (2004), no 4, 1501-1533.
• Barlow, M.T., Bass, R., Kumagai, T.: Stability of the parabolic Harnack inequalities on metric measure spaces. Preprint.
• Coulhon, T., Grigor'yan, A.: Random walks on graphs with regular volume growth. Geom. Funct. Anal. 8 (1998), 656-701.
• Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 (1999), 181-232.
• Grigor'yan, A.: Heat kernel upper bounds on fractal spaces. Preprint.
• Grigor'yan, A., Telcs, A.: Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109 (2001), no. 3, 452-510.
• Grigor'yan, A., Telcs, A.: Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324 (2002), 521-556.
• Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53 (1981), 57-73.
• Hambly, B., Kumagai, T.: Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries. In Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, 233-259. Proc. Sympos. Pure Math. 72, Part 2. Amer. Math. Soc., Providence, RI, 2004.
• Hebisch W., Saloff-Coste, L.: On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51 (2001), no. 5, 1437-1481.
• Jones, O. D.: Transition probabilities for the simple random walk on the Sierpiński graph. Stochastic Process. Appl. 61 (1996), no. 1, 45-69.
• Kumagai, T., Sturm, K-T.: Construction of diffusion processes on fractals, $d$-sets, and general metric measure spaces. J. Math. Kyoto Univ. 45 (2005), no. 2, 307-327.
• Li, P., Wang, J.: Mean value inequalities. Indiana Univ. Math. J. 48 (1999), no. 4, 1257-1283.
• Telcs, A.: Random walks on graphs, electric networks and fractals. Probab. Theory Related Fields 82 (1989), 435-449.
• Telcs, A.: Volume and time doubling of graphs and random walks: the strongly recurrent case. Comm. Pure Appl. Math. 54 (2001), 975-1018.
• Telcs, A.: Some notes on the Einstein relation. To appear in J. Stat. Phys.
• Telcs, A.: Upper bound for transition probabilities on graphs and isoperimetric inequalities. To appear in Markov Proc. and Rel. Fields.