2009
DOI: 10.1214/ejp.v14-707
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Density Formula and Concentration Inequalities with Malliavin Calculus

Abstract: We show how to use the Malliavin calculus to obtain a new exact formula for the density ρ of the law of any random variable Z which is measurable and dierentiable with respect to a given isonormal Gaussian process. The main advantage of this formula is that it does not refer to the divergence operator (dual of the Malliavin derivative). In particular, density lower bounds can be obtained in some instances. Among several examples, we provide an application to the (centered) maximum of a general Gaussian process… Show more

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Cited by 108 publications
(215 citation statements)
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“…They then have the same density ρ * so 1 and 2 immediately follow. We next prove that g X Law = g Z , imitating the technique Nourdin and Viens used to prove (12) (see Theorem 3.1 [18]). Let f be a continuous function with compact support, and F any antiderivative.…”
Section: Np Bound In Wiener Spacementioning
confidence: 99%
See 4 more Smart Citations
“…They then have the same density ρ * so 1 and 2 immediately follow. We next prove that g X Law = g Z , imitating the technique Nourdin and Viens used to prove (12) (see Theorem 3.1 [18]). Let f be a continuous function with compact support, and F any antiderivative.…”
Section: Np Bound In Wiener Spacementioning
confidence: 99%
“…In this case, we have the same assumptions (and corresponding functionals, defined below) for each X n . Note that for Z ∈ D 1,2 , the support necessarily has to be an interval (see Theorem 3.1 [18], Proposition 2.1.7 [19]), a consequence that carries over to Wiener-Poisson space. The continuity assumption of the density ρ * is not strong at all, since general processes like solutions of stochastic differential equations driven by Brownian motion or (under mild conditions) fractional Brownian motion (for example see [2]) have continuous densities.…”
Section: Remarkmentioning
confidence: 99%
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