Improving Monte Carlo simulations by Dirichlet forms

Equipping the probability space with a local Dirichlet form with square field operator \Gamma and generator A allows to improve Monte Carlo simulations of expectations and densities as soon as we are able to simulate a random variable X together with \Gamma[X] and A[X]. We give examples on the Wiener space, on the Poisson space and on the Monte Carlo space. When X is real-valued we give an explicit formula yielding the density at the speed of the law of large numbers.