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Ouvrages Année : 2017

Kac-Rice formulas for random fields and theirs applications in: random geometry, roots of random polynomials and some engineering problems

Résumé

There exist two variants of the change of variables formula for multiple integrals very useful in integral geometry. The first one corresponds to smooth, locally bijective functions G from Rd to Rd and the second applies to smooth functions G from Rd to Rj with d>j, having a differential with maximal rank. These formulas are called "area formula" and "coarea formula" respectively. Applying these formulas to trajectories of random fields and taking expectation afterwards, one obtains the well-known Kac-Rice formulas. In recent times and fundamentally due to the appearance of two excellent books (Adler and Taylor, 2007) and (Azaïs and Wschebor, 2009), there has been a growing interest in the application of these formulas in such varied domains as: random algebraic geometry, algorithm complexity for solving large systems of equations, study of zeros of random polynomial systems and finally, engineering applications. The present work is divided in three parts. 1. In the first part, we give an analytical proof of the area and coarea formulas. Such a proof, originally attributed to Banach and Federer (1969), will be made by using elementary tools of vector calculus and measure theory in Rd. 2. The above formulas form the basis for establishing the validity of Kac-Rice formulas for random fields. They allow computing the expectation of the measure of the level sets C_{Q,X}(y)={t in Q \subset Rd: X(t)=y}, where X: Omega x Rd to Rj is a random fields and d \ge j. We must point out that one can obtain a Kac-Rice formula for almost sure all level by using the area and coarea formulas, Fubini theorem and duality. However, in applications the interest is directed to a fixed level y. For instance, the zeros in the study of the roots of a random polynomial. This precision leads us to a delicate study for generalizing the classical inverse function and implicit function theorems. For this part we based our approach in two seminal works: firstly an article of E.Cabana (1985), published in the conference proceedings of the II CLAPEM and secondly in the Lecture Notes of Mathematics of M.Wshebor (1985). The method we use also makes it possible to obtain the Kac-Rice formula for the upper moments of the level measurement. 3. The work ends with several applications. First, we show examples where the hypothesis can be checked and then we use the Kac-Rice formulas for obtaining conditions about the finiteness of the first and second moment of the measure of the level sets. The very important case of the Gaussian random fields leads us to explicitly computations. Afterwards, we address the study of the number of roots of algebraic and trigonometric random polynomials. We emphasize the asymptotic behavior of the expectation and the variance of the number of roots. Particular attention is devoted to systems of random polynomials of several variables that are invariant under the action of the group of rotations in Rd. Another theme we consider is the nodal curves of the system of random waves considered by Berry and Dennis (2000). These curves are called dislocations in physics and correspond to lines of darkness in light propagation, or threads of silence in sound propagation. We also study the application of the Kac-Rice formula to sea modeling and to random gravitational lenses.
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hal-01953070 , version 1 (22-01-2019)

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  • HAL Id : hal-01953070 , version 1

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Corinne Berzin, Alain Latour, José R. León. Kac-Rice formulas for random fields and theirs applications in: random geometry, roots of random polynomials and some engineering problems. Ediciones IVIC, 2017, Escuela Venezolana de Matematicas, Eloy Sira, 978-980-261-180-5. ⟨hal-01953070⟩
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