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We give a extensive account of a recent new way of applying the Dirichletform theory to random Poisson measures. The main application is to obtainexistence of density for thelaws of random functionals of L\'evy processes orsolutions of stochastic differential equations with jumps. As in the Wienercase the Dirichlet form approach weakens significantly theregularityassumptions. The main novelty is an explicit formula for the gradient or forthe "carr\'e du champ' on the Poisson space called the lent particle formulabecause based on adding a new particle to the system, computing the derivativeof the functional with respect to this new argument and taking back thisparticle before applying the Poisson measure. The article is expository in itsfirst part and based on Bouleau-Denis [12] with several new examples,applications to multiple Poisson integrals are gathered in the last part whichconcerns the relation with the Fock space and some aspects of the secondquantization.

The Lent Particle Method, Application to Multiple Poisson Integrals