AbstractWe apply the Davies method to prove that for any regular Dirichlet form on a
metric measure space, an off-diagonal stable-like upper bound of the heat
kernel is equivalent to the conjunction of the on-diagonal upper bound, a
cutoff inequality on any two concentric balls, and the jump kernel upper
bound, for any walk dimension. If in addition the jump kernel vanishes, that
is, if the Dirichlet form is strongly local, we obtain a
sub-Gaussian upper bound. This gives a unified approach to obtaining heat
kernel upper bounds for both the non-local and the local Dirichlet forms.
AbstractIn this paper we prove that sub-Gaussian estimates of heat kernels of regular Dirichlet forms are equivalent to the regularity of measures, two-sided bounds of effective resistances and the locality of semigroups, on strongly recurrent compact metric spaces. Upper bounds of effective resistances imply the compact embedding theorem for domains of Dirichlet forms, and give rise to the existence of Green functions with zero Dirichlet boundary conditions. Green functions play an important role in our analysis. Our emphasis in this paper is on the analytic aspects of deriving two-sided sub-Gaussian bounds of heat kernels. We also give the probabilistic interpretation for each of the main analytic steps.
The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.
In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.
The Davies Method for Heat Kernel Upper Bounds of Non-Local Dirichlet Forms on Ultra-Metric Spaces