6,095 research outputs found
Intrinsic Ultracontractivity of Feynman-Kac Semigroups for Symmetric Jump Processes
Consider the symmetric non-local Dirichlet form (D,\D(D)) given by with
\D(D) the closure of the set of functions on with compact
support under the norm , where and is a nonnegative symmetric measurable function on
. Suppose that there is a Hunt process on
corresponding to (D,\D(D)), and that (L,\D(L)) is its infinitesimal
generator. We study the intrinsic ultracontractivity for the Feynman-Kac
semigroup generated by , where is a
non-negative locally bounded measurable function such that Lebesgue measure of
the set is finite for every . By using
intrinsic super Poincar\'{e} inequalities and establishing an explicit lower
bound estimate for the ground state, we present general criteria for the
intrinsic ultracontractivity of . In particular, if
J(x,y)\asymp|x-y|^{-d-\alpha}\I_{\{|x-y|\le
1\}}+e^{-|x-y|^\gamma}\I_{\{|x-y|> 1\}} for some and
, and the potential function for some
, then is intrinsically ultracontractive if and
only if . When , we have the following explicit estimates
for the ground state where are
constants. We stress that, our method efficiently applies to the Hunt process
with finite range jumps, and some irregular potential
function such that .Comment: 31 page
Rank-dependent deactivation in network evolution
A rank-dependent deactivation mechanism is introduced to network evolution.
The growth dynamics of the network is based on a finite memory of individuals,
which is implemented by deactivating one site at each time step. The model
shows striking features of a wide range of real-world networks: power-law
degree distribution, high clustering coefficient, and disassortative degree
correlation.Comment: 5 pages, 5 figures, RevTex
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