### 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).