Heriot-Watt Mathematics Report Series
HWM99-4, 22 Jan 1999
Upper bounds for heat kernels associated with Dirichlet forms
L Notarantonio
Abstract
Let $\Omega$ be a topological space, $dx$ a Borel measure on
$X$ and let $ p_t(x,y)$ be the heat kernel of an ultracontractive
(sub-)Markovian semigroup $e^{-tH}$ on $L^2(\Omega,dx)$. We prove upper
bounds on \( p_t(x,y)\) which either suggest a large deviation character
or have gaussian behavior of the type
\[
p_t(x,y) \le A(t, d(x,y)) \exp\left[-
\frac{(d(x,y))^2}{4t}\right], \ \ t>0,\ \ x,y\in \Omega.
\]
The function \( d(x,y)\) is a (pseudo-)distance defined in term of
the Dirichlet form associated with \( e^{-tH}\). Examples will include
both some non-local operators (``powers of the Laplacian'' and some random
walks on the integers) and local differential operators (among others
Hörmander sum of squares on a Lie group).
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