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On the distances between probability density functions. (English) Zbl 1307.60072
Summary: We give estimates of the distance between the densities of the laws of two functionals $$F$$ and $$G$$ on the Wiener space in terms of the Malliavin-Sobolev norm of $$F-G$$. We actually consider a more general framework which allows one to treat with similar (Malliavin type) methods functionals of a Poisson point measure (solutions of jump type stochastic equations). We use the above estimates in order to obtain a criterion which ensures that convergence in distribution implies convergence in total variation distance; in particular, if the functionals at hand are absolutely continuous, this implies convergence in $$L^{1}$$ of the densities.

##### MSC:
 60H07 Stochastic calculus of variations and the Malliavin calculus 60H30 Applications of stochastic analysis (to PDEs, etc.)
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