On the Asymptotic Behavior of the Second Moment of the Fourier Transform of a Random Measure

The behavior at infinity of the Fourier transform of the random measures that appear in the theory of multiplicative chaos of Mandelbrot, Peyrière, and Kahane is an area quite unexplored. For context and further reference, we first present an overview of this theory and then the result, which is the main objective of this work, generalizing a result previously announced by Kahane. We establish an estimate for the asymptotic behavior of the second moment of the Fourier transform of the limit random measure in the theory of multiplicative chaos. After looking at the behavior at infinity of the Fourier transform of some remarkable functions and measures, we prove a formula essentially due to Frostman, involving the Riesz kernels. 1. Introduction. The problem considered in the second section of this work admits a general formulation that can be stated as follows. A random measure is defined in the sense of a random object (see [10, page 9]) by the action of a random operator on a usual Borel measure in such a way that its Fourier transform is almost surely a uniformly continuous and bounded function. A natural conjecture to be made is that the almost sure behavior, at infinity, of the Fourier transform of the random measure is somehow related to the behavior at infinity of the Fourier transform of the Borel measure used to build this random measure. A technique that has given good results in problems such as the one presented here goes as follows (see [9, pages 253–255, 265–267]). One first gets good estimates on the behavior of the moments of the random functions and then, by an accumulation argument, the almost sure behavior is obtained. The study of the asymptotic behavior of the second moment, besides the instrumental usefulness for the technique described, can give an idea of what to expect on the almost sure behavior.

In the first chapter, a class of random periodic Schwartz distributions is introduced, some examples, elementary properties and a characterization result are studied and three applications are presented. A random Schwartz periodic distribution is, for us, just a function defined in a complete probability space and taking values in the space of Schwartz distributions over the line, that are left invariant by an integer translation, endowed with the natural algebraic and topological structures. The second chapter deals, primarily, with an extension of the methods of Kahane, as applied to the Brownian sheet, in what concerns analogs of the rapid points. After presenting the Brownian sheet process, by way of Gaussian white noise, some results, on the local behavior of this process and for some other processes associated with the sheet, are derived using the Schauder series representation. In the third chapter, we prove a formula essentially due to Frostman, we look at the behavior at infinity of the Fourier transform of some remarkable functions and measures and, finally, we study the asymptotic behavior of the second moment of the Fourier transform of a random measure that appears in the theory of multiplicative chaos. In the last chapter, a class of random tempered distributions on the line is introduced by considering random series, in the usual Hermite functions, having as coefficients random variables which satisfy certain growth conditions. This class is shown to be exactly the class of random Schwartz distributions having a mean. We present also a study on a possible converse of a result on Brownian distributions, that leads to a moment problem.
Keywords : random Schwartz periodic distribution; Brownian sheet; Fourier transform of a random measure; moment problem Classification : * 60G20 Generalized stochastic processes 60G17 Sample path properties 42B10 Fourier type transforms, several variables 46F25 Generalized functions on infinite-dimensional spaces Cited in ...
In his seminal book "Some random series of functions" (1985; Zbl 0571.60002), J.-P. Kahane has shown, in a systematic way, how to take advantage of Paul Lévy's construction of the Brownian process, using the Haar functions, in order to study the local behavior of this process. To reach this goal Kahane looks at the Haar's interpolation of the Brownian process done by Lévy, as a series expansion in the Schauder system, having Gaussian random variables as coefficients, and exploits this series representation with sharp estimates of the distribution function of the maximum of a finite subfamily of a normal sequence. With this method Kahane gets easily the results corresponding to the existence of rapid points and slow points [which were first discovered by S. Orey and S. J. Taylor, Proc. Lond. Math. Soc., III. Ser. 28, 174-192 (1974 The representation of the Brownian sheet as a sum of a series, which converges uniformly almost surely, of Schauder functions having as coefficients normal random variables, is a simple consequence of the definition of the Brownian sheet using Gaussian white noise. Some results on the local behavior of the Brownian sheet and for some other processes associated with the sheet, can be derived by using this representation. Namely, a uniform modulus of continuity, nondifferentiability results and at some points, faster oscillation than the one prescribed by the laws of iterated logarithm. In previous work [the author, "On the local behavior of the Brownian sheet", in: Isr. Math. Conf. Proc., AMS 1994] rapid points and almost sure everywhere nondifferentiability for the location homogeneous part of the Fourier-Schauder series representation were presented. Here we show the existence of rapid points for the independent increments of the Brownian sheet, using the same method. This method, first used by J.-P. Kahane ["Some random series of functions" (1968; Zbl 0192.53801)] to deal with similar properties of the Brownian unidimensional time process, consists on exploiting the Fourier-Schauder representation with sharp estimates of the distribution function of the maximum of a finite subfamily of a normal sequence. Some results on the usual increments behavior are also presented. This is an expository article. First some simple properties of convex sets in n-space are given and then some theorems on the separation of convex sets from affine subspaces and other convex sets are proved.

0809.46033
Esquível, Manuel L. Sur une classe de distributions aléatoires périodiques. (On a class of periodic random distributions). (French. Extended English abstract) Ann. Sci. Math. Qué. 17, No.2, 169-186 (1993).  http://www.lacim.uqam.ca/ annales/volumes/17-2/169.html The theoretical foundations for the study and application of periodic random distributions have been established since at least the sixties. In the context of the fractal geometry of Benoit Mandelbrot, mathematical models of irregular surfaces that can be obtained by computer have led us to consider stochastic processes arising from Fourier series with random coefficients. In this article, we introduce a class of such periodic random distributions. Three examples of this case are: random mass on the unit circle, Brownian motion (classical and fractional) and classical Schwartz distributions that are randomized by translations. We begin with a result giving conditions which characterize the distributions of this class. These conditions are easy to verify and this is done for the three previous examples. Two important questions in harmonic analysis are considered: uniqueness of the representation by Fourier series and differentiability. Also, we examine the statistical problem of the existence of a generalized first moment for these random distributions. As an application, a classical result in Fourier analysis, useful for constructing particular solutions of ordinary differential equations with constant coefficients, is generalized to this class of periodic random distributions. This last result is applied to obtain the Fourier-Wiener-Schwartz series of a particular solution of a generalized Langevin equation. We conclude with a comment on the regularity of the solution.
Keywords : periodic random distributions; fractal geometry of Benoit Mandelbrot; Fourier series with random coefficients; random mass on the unit circle; Brownian motion; classical Schwartz distributions that are randomized by translations; existence of a generalized first moment; periodic random distributions; Fourier-Wiener-Schwartz series; Langevin equation; regularity Classification : * 46F10 Operations with distributions (generalized functions) 60E99 Distribution theory in probability theory 28A80 Fractals Cited in ...

0676.35011
Esquível, Manuel L. Sur la méthode des séries de Fourier dans leséquations différentiellesà coefficients constants. (On the method of Fourier series for differential equations with constant coefficients). The author extends the method of Fourier series to obtain solutions of the differential equation P (D)u = f (P(D) = differential polynomial with constant coefficients) to the case where f does not satisfy the so called compatibility conditions P (n) = 0 ⇒f (n) = 0 (f(n) = coefficient of the Fourier-Schwartz transform).

R.Salvi
Keywords : Fourier series; differential polynomial; constant coefficients; Fourier-Schwartz transform Classification : * 35E20 General theory of PDE with constant coefficients 35C10 Series solutions of PDE Cited in ...
In this note we explicitly enunciate and prove an easy and perhaps known condition on the Radon-Nikodým derivative of one measure relative to another, in order to get set inclusion of their respective L 1 spaces. Let λ and µ be two σ-finite measures over a measure space and let dµ = hdλ + dµ 1 be the Lebesgue Radon-Nikodým decomposition of µ with respect to λ. A necessary and sufficient condition for L 1 µ ⊆ L 1 λ is that: ∃K > 0λ({h < K}) = 0. No priority research about this subject has been done by the author.
Keywords : L 1 inclusions; Radon-Nikodým derivative; Lebesgue Radon-Nikodým decomposition Classification : * 28A15 Differentiation of set functions 28A25 Integration with respect to measures and other set functions 46E30 Spaces of measurable functions 46E35 Sobolev spaces and generalizations Cited in ...