We describe a stochastic model for complex networks possessing three
qualitative features: power-law degree distributions, local clustering, and
slowly-growing diameter.
The model is mathematically natural, permits a wide variety
of explicit calculations, has the desired three qualitative features,
and fits the complete range of degree scaling exponents and clustering parameters.
Write-ups exist as a

short version
and as a
long version

We consider the rate of spread of a body of passive tracers moving under the influence of a random evolving vector field.
The vector field is of a type used as a model for ocean currents and was introduced by Kolmogorov. The rate of growth of the diameter of the body is of interest for practical reasons (such as in problems of pollution control) and we specify its rate of growth.

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.