Regularity of Gaussian white noise on the d-dimensional torus

Abstract

Abstract. In this paper we prove that a Gaussian white noise on the ddimensional torus has paths in the Besov spacesThis result is shown to be optimal in several ways. We also show that Gaussian white noise on the d-dimensional torus has paths in a the Fourier-Besov spacê b

The $$n$$n-term Approximation of Periodic Generalized Lévy Processes

“…• In the Gaussian case, one has the exact value of κpsq. This is due to the fact that the converse result for the Besov regularity of Gaussian white noises are known [49]. A fundamental consequence is the following: If s " L´1w is a sparse process of order γ and s G " L´1w G is a Gaussian process corresponding to the same operator L, then we have almost surely that κpsq ě κps G q.…”

“…The Gaussian noise differs from the non-Gaussian Lévy noises in one main way: for p ě 2, the critical value for the non-Gaussian case is dp1{p´1q, while it is´d{2 for the Gaussian case. The Besov regularity of the Gaussian noise on the torus has been studied in detail in [49] using Fourier series techniques. We have re-obtain similar results in [2, Section 3] with waveletbased methods.…”

Multidimensional Lévy white noise in weighted Besov spaces

“…As for the Sobolev regularity, the Hölder regularity of a Lévy noise that we obtained is independent of the noise. However, the Gaussian white noise has a local Hölder regularity of (−τ ) for every τ > d 2 [48]. It means that our bounds for the regularity are suboptimal for the Gaussian white noise.…”

“…To the best of our knowledge, the Besov regularity of d-dimensional Lévy white noises has never been addressed in all generality. Kusuoka [25] estimated the weighted Sobolev regularity of the Gaussian white noise, while Veraar [48] obtained complete results on the local Besov regularity of the Gaussian white noise. However, these works are based on intrinsic Gaussian methods and are not easily extended to the non-Gaussian case.…”

The Cauchy Problem of Null Form Wave Equation on <b><i>T</i></b>^<i>d</i> with Random Initial Data

“…it is easy to see that, for b > 1=p 0 , we can reduce the proof of the above inequality to prove for the function in the form of f w 1 ¼ P n a n h n ðwÞe inÁx , where [21], all results on real Gaussian random variables have a complex version, and from now on, we will discuss based on this argument. However, for the complex case, the variance of N C ð0; On the other hand,…”