Integrability of Functionals of Dirichlet Processes, Probabilistic Representations of Semigroups, and Estimates of Heat Kernels

This paper consists of three parts. In Part I, we obtain results on the integrability of functional (of exponential type) of Dirichlet processes. In Part II, we give a striking probabilistic representation of semigroup (probably non-Markovian) associated with a non-divergence operator. Part III is devoted to perturbation bounds on the operator norm of semigroups and a new (short) proof of the off-diagonal estimates of the heat kernel associated with a divergence operator. The theory of Dirichlet forms and forward, backward martingales decompositions play a central role in the whole paper.