Some Relations Between Bounded Below Elliptic Operators and Stochastic Analysis

Abstract

We apply Malliavin Calculus tools to the case of a bounded below elliptic right-invariant pseudo-differential operators on a Lie group. We give examples of bounded below pseudo-differential elliptic operators on \(\mathbb {R}^d\) by using the theory of Poisson process and the Garding inequality. In the two cases, there are no stochastic processes because the considered semi-groups do not preserve positivity.