" This volume contains the expanded lecture notes of courses taught at the Emile Borel
Centre of the Henri Poincaré Institute (Paris). In the book, leading experts introduce recent …

The main theme of these lecture notes is to analyze heat conduction on disordered media
such as fractals and percolation clusters by means of both probabilistic and analytic …

We consider divergence form elliptic operators in dimension n≥ 2 with L∞ coefficients.
Although solutions of these operators are only Hölder-continuous, we show that they are …

Assume that there is some analytic structure, a differential equation or a stochastic process
for example, on a metric space. To describe asymptotic behaviors of analytic objects, the …

Let (X, d, µ) be a metric measure space with a local regular Dirichlet form. We give
necessary and sufficient conditions for a parabolic Harnack inequality with global space …

We consider the bond percolation problem on a transient weighted graph induced by the
excursion sets of the Gaussian free field on the corresponding cable system. Owing to the …

Let $\mathcal {G} $ be the incipient infinite cluster (IIC) for percolation on a homogeneous
tree of degree $ n_0+ 1$. We obtain estimates for the transition density of the continuous …

For a large class of amenable transient weighted graphs G, we prove that the sign clusters of
the Gaussian free field on G fall into a regime of strong supercriticality, in which two infinite …

We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets,
which are natural extension of planar Sierpinski carpets by allowing the small cells to live off …

We study the percolative properties of random interlacements on G× ℤ, where G is a
weighted graph satisfying certain sub-Gaussian estimates attached to the parameters α> 1 …