Lower Gaussian heat kernel bounds for the random conductance model in a degenerate ergodic environment

Quenched Invariance Principles for the Random Conductance Model on a Random Graph with Degenerate Ergodic Weights

We consider a stationary and ergodic random field {ω(e) : e ∈ Ed} that is parameterized by the edge set of the Euclidean lattice Z, d ≥ 2. The random variable ω(e), taking values in [0,∞) and satisfying certain moment bounds, is thought of as the conductance of the edge e. Assuming that the set of edges with positive conductances give rise to a unique infinite cluster C∞(ω), we prove a quenched...

Anomalous heat kernel behaviour for the dynamic random conductance model

We introduce the time dynamic random conductance model and consider the heat kernel for the random walk on this environment. In the case where conductances are bounded above, an example environment is presented which exhibits heat kernel decay that is asymptotically slower than in the well studied time homogeneous case being close to O ( n−1 ) as opposed to O ( n−2 ) . The example environment g...

Gaussian Upper Bounds for the Heat Kernel on Arbitrary Manifolds

The history of the heat kernel Gaussian estimates started with the works of Nash [25] and Aronson [2] where the double-sided Gaussian estimates were obtained for the heat kernel of a uniformly parabolic equation in IR in a divergence form (see also [15] for improvement of the original Nash’s argument and [26] for a consistent account of the Aronson’s results and related topics). In particular, ...

Degenerate Hypoelliptic Operators and the Gaussian Kernel

The Solution of the Kato Problem for Divergence Form Elliptic Operators with Gaussian Heat Kernel Bounds

We solve the Kato problem for divergence form elliptic operators whose heat kernels satisfy a pointwise Gaussian upper bound. More precisely, given the Gaussian hypothesis, we establish that the domain of the square root of a complex uniformly elliptic operator L = −div(A∇) with bounded measurable coefficients in Rn is the Sobolev space H1(Rn) in any dimension with the estimate ‖ √ Lf‖2 ∼ ‖∇f‖2...