An effective computation of the kernels of the chaos decomposition of Lévy functionals is used, to prove, among other things, a chain- and product-rule of the Malliavin derivative for a large class of Lévy processes. In case of finite and infinite-dimensional Brownian motion, the well-known rules are obtained, but for Poisson processes, the results are new. The kernels of a Lévy functional can be computed by taking the expected value of the product of this functional and multiple white noise of the Lévy process.
Horst Osswald. "Computation of the kernels of Lévy functionals and applications." Illinois J. Math. 55 (3) 815 - 833, Fall 2011. https://doi.org/10.1215/ijm/1369841786