Heat kernel estimates for random walks with degenerate weights

@article{Andres2014HeatKE,
  title={Heat kernel estimates for random walks with degenerate weights},
  author={Sebastian Andres and Jean-Dominique Deuschel and Martin Slowik},
  journal={Electronic Journal of Probability},
  year={2014},
  volume={21}
}
We establish Gaussian-type upper bounds on the heat kernel for a continuous-time random walk on a graph with unbounded weights under an ergodicity assumption. For the proof we use Davies' perturbation method, where we show a maximal inequality for the perturbed heat kernel via Moser iteration. 
View PDF on arXiv
34 Citations
Highly Influential Citations
1
Background Citations
22
Methods Citations
13
Results Citations
1

34 Citations

Heat kernel estimates and intrinsic metric for random walks with general speed measure under degenerate conductances

We establish heat kernel upper bounds for a continuous-time random walk under unbounded conductances satisfying an integrability assumption, where we correct and extend recent results by the authors…

Lower Gaussian heat kernel bounds for the Random Conductance Model in a degenerate ergodic environment.

Rate functions for random walks on Random conuctance Models and related topics

We consider laws of the iterated logarithm and the rate function for sample paths of random walks on random conductance models under the assumption that the random walks enjoy long time sub-Gaussian…

Heat kernel upper bounds for interacting particle systems

We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in…

Random conductance models with stable-like jumps: Heat kernel estimates and Harnack inequalities

Harnack inequalities on weighted graphs and some applications to the random conductance model

We establish elliptic and parabolic Harnack inequalities on graphs with unbounded weights. As an application we prove a local limit theorem for a continuous time random walk $$X$$X in an environment…

Off-Diagonal Heat Kernel Estimates for Symmetric Diffusions in a Degenerate Ergodic Environment

We study a symmetric diffusion process on $\mathbb {R}^{d}$ ℝ d , d ≥ 2, in divergence form in a stationary and ergodic random environment. The coefficients are assumed to be…

Green kernel asymptotics for two-dimensional random walks under random conductances

We consider random walks among random conductances on $\mathbb{Z}^2$ and establish precise asymptotics for the associated potential kernel and the Green function of the walk killed upon exiting…

Anchored heat kernel upper bounds on graphs with unbounded geometry and anti-trees

We derive Gaussian heat kernel bounds on graphs with respect to a fixed origin for large times under the assumption of a Sobolev inequality and volume doubling on large balls. The upper bound from our…

Local Central Limit Theorem for diffusions in a degenerate and unbounded Random Medium

We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients. We prove a quenched local…

References

SHOWING 1-10 OF 19 REFERENCES

Harnack Inequalities and Local Central Limit Theorem for the Polynomial Lower Tail Random Conductance Model

We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near $0$. We consider both constant and variable speed models. Our…

Harnack inequalities on weighted graphs and some applications to the random conductance model

We establish elliptic and parabolic Harnack inequalities on graphs with unbounded weights. As an application we prove a local limit theorem for a continuous time random walk $$X$$X in an environment…

Gaussian Upper Bounds for Heat Kernels of Continuous Time Simple Random Walks

  • M. Folz
  • Mathematics, Computer Science
  • 2011
The distance function which appears in this estimate is not in general the graph metric, but a new metric which is adapted to the random walk, and long-range non-Gaussian bounds in this new metric are established.

Invariance principle for the random conductance model with unbounded conductances.

We study a continuous time random walk X in an environment of i.i.d. random conductances μ e ∈ [1, ∞). We obtain heat kernel bounds and prove a quenched invariance principle for X. This holds even…

Parabolic Harnack inequality and estimates of Markov chains on graphs

On a graph, we give a characterization of a parabolic Harnack inequality and Gaussian estimates for reversible Markov chains by geometric properties (volume regularity and Poincare inequality).

Gaussian bounds and parabolic Harnack inequality on locally irregular graphs

A well known theorem of Delmotte is that Gaussian bounds, parabolic Harnack inequality, and the combination of volume doubling and Poincaré inequality are equivalent for graphs. In this paper we…

Invariance principle for the random conductance model in a degenerate ergodic environment

We study a continuous time random walk, X, on Zd in an environment of random conductances taking values in (0,∞). We assume that the law of the conductances is ergodic with respect to space shifts.…

Heat kernels and spectral theory

Preface 1. Introductory concepts 2. Logarithmic Sobolev inequalities 3. Gaussian bounds on heat kernels 4. Boundary behaviour 5. Riemannian manifolds References Notation index Index.

Random walks on supercritical percolation clusters

We obtain Gaussian upper and lower bounds on the transition density qt(x;y) of the continuous time simple random walk on a supercritical percolation cluster C1 in the Euclidean lattice. The bounds,…

Upper bounds for symmetric Markov transition functions

Abstract : A large number of properties which are peculiar to symmetric Markov semigroups stem from the fact that such semigroups can be analyzed simultaneously by Hilbert space techniques as well as…