This talk will discuss random walks on percolation clusters.

The first case is supercritical ($p>p_c$) bond percolation in
$Z^d$. Here one can obtain Aronsen type bounds on the transition
probabilities, using analytic methods based on ideas of Nash.

For the critical case ($p=p_c$) one needs to study the incipient
infinite cluster (IIC). The easiest situation is the IIC on trees -
where the methods described above lead to an alternative approach to
results of Kesten (1986). (This case is joint work with T. Kumagai).

I will talk on a generalization of classical Calabi's strong maximum (1957) in the framework of Dirichlet forms associated with strong Feller diffusion processes.
The proof is stochastic and the result can be applicable to a singular geometric space appeared in the measured Gromov-Hausdorff convergence (precisely in the convergence by spectral distance by Kasue Kumura) of compact Riemannian manifolds with uniform lower Ricci curvature and uniform upper diameter.