Fluctuations of eigenvalues and second order Poincaré inequalities

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, thus extending a formula recently used by Chatterjee [4]. We also explain how to derive concentration inequalities for Z in our framework.

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“…. , n (compare with Lemma 5.3 in Chatterjee [3]). In particular, Corollary 3.5 combined with Proposition 3.7 yields the following.…”

Section: First Example: Monotone Gaussian Functional Nite Casementioning

confidence: 78%

Density Formula and Concentration Inequalities with Malliavin Calculus

Nourdin¹,

Viens

2009

Electron. J. Probab.

108

206

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“…Similarly, (11.5) was shown to hold jointly over e, see [30,Remark 1.4]. In these works, the space is the discrete lattice Z d , d ≥ 3, and a key assumption is that the probability measure has an underlying product structure which makes available tools such as concentration inequalities, the Chatterjee-Stein [9,10] method of normal approximation and the Helffer-Sjöstrand representation of correlations [26,39,32]. Each of these papers makes essential use of, and refines, the optimal quantitative estimates first proved in [19,20,16].…”

Section: Informal Heuristics and Statement Of Main Resultsmentioning

confidence: 99%

The additive structure of elliptic homogenization

2016

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Abstract. One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in [2]: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

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“…We wish to prove (1.5) and (1.6). Our proof will be based on the following proposition, which is a version of Theorem 2.2 in [4] and Theorem 3.1 (and Remark 3.6) of [12], stated in a form that is convenient for our purpose.…”

Section: Proof Of Proposition 12mentioning

confidence: 99%