Alexander Grigor’yan

Heat Kernel and Analysis on Manifolds

Laplace operator and the heat equation in

Estimates of heat kernels for non-local regular Dirichlet forms

\mathbb{R}^n

Heat Kernels on Metric Spaces with Doubling Measure

Function spaces in

Heat Kernels on Metric Measure Spaces

\mathbb{R}^n

On stochastic completeness of jump processes

Laplace operator on a Riemannian manifold Laplace operator and heat equation in

Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces

L^{2}(M)

Heat kernel on manifolds with ends

Weak maximum principle and related topics Regularity theory in

Cohomology of digraphs and (undirected) graphs

\mathbb{R}^n

Negative Eigenvalues of Two-Dimensional Schrödinger Operators

The heat kernel on a manifold Positive solutions Heat kernel as a fundamental solution Spectral properties Distance function and completeness Gaussian estimates in the integrated form Green function and Green operator Ultracontractive estimates and eigenvalues Pointwise Gaussian estimates I Pointwise Gaussian estimates II Reference material Bibliography Some notation Index

Obtaining upper bounds of heat kernels from lower bounds

In this paper we present new heat kernel upper bounds for a certain class of non-local regular Dirichlet forms on metric measure spaces, including fractal spaces. We use a new purely analytic method where one of the main tools is the parabolic maximum principle. We deduce off-diagonal upper bound of the heat kernel from the on-diagonal one under the volume regularity hypothesis, restriction of the jump kernel and the survival hypothesis. As an application, we obtain two-sided estimates of heat kernels for non-local regular Dirichlet forms with finite effective resistance, including settings with the walk dimension greater than 2.