Abstract: A major problem in PDE, solved independently in the late 1950s by de Giorgi, Moser and Nash, was the regularity of solutions to second order divergence form equations. Moser's approach was to use a Harnack inequality, which can be understood in probabilistic terms.
These PDE methods are very versatile, and have been extended to manifolds, general metric spaces, and graphs. They give continuity results for harmonic functions, and lead to estimates of the transition density of Markov processes. Of particular importance is the fact that the PDE approach is robust enough to able to handle small local perturbations of the space.
These lectures will review this progress, and will present applications of Harnack inequalities to random walks on random graphs.