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Regularity of Gaussian white noise on the \(d\)-dimensional torus. (English) Zbl 1252.60035

Nawrocki, Marek (ed.) et al., Marcinkiewicz centenary volume. Proceedings of the Józef Marcinkiewicz centenary conference, June 28–July 2, 2010. Warszawa: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-14-0/pbk). Banach Center Publications 95, 385-398 (2011).
It was shown by J. Bourgain [Commun. Math. Phys. 166, No. 1, 1–26 (1994; Zbl 0822.35126)] that Gibbs measures are invariant for the nonlinear Schrödinger equation. Building on these ideas it has been shown by T. Oh [Commun. Math. Phys. 292, No. 1, 217–236 (2009; Zbl 1185.35237)] that the mean zero Gaussian white noise on the torus \(T\) is invariant for the periodic Korteweg-de Vries equation (KdV). Here, it is used that the KdV equation is well-posed for initial conditions from function spaces with a negative smoothness index, such as Sobolev spaces \(H^{s,p}(T)\) with \(s< 0\) and \(1\leq p<\infty\). A negative smoothness index \(s\) is needed here, since it is well-known that Gaussian white noise is supported on \(\bigcap_{s<-1/2} H^{s,p}\setminus H^{-1/2,p}\). Apparently, the first results on the support of Gaussian white noise into this direction were obtained in the case \(p\neq 2\) by S. Kusuoka [J. Fac. Sci., Univ. Tokyo, Sect. I A 29, 387–400 (1982; Zbl 0515.60068)]. Using an equivalent wavelet definition of Besov spaces, B. Roynette [Stochastics Stochastics Rep. 43, No. 3–4, 221–260 (1993; Zbl 0808.60071)] showed for the Brownian motion \(B: [0,1]\times\Omega\to \mathbb{R}\) that \[ \text{P}(B\in B^{1/2}_{p,\infty}(0,1))= 1. \] The main result in the present paper is that for the \(d\)-dimensional Gaussian white noise \(W: \Omega\to T^d,\) one has for all \(1\leq p<\infty\) that \[ W\in B^{-d/2}_{p,\infty}(T^d)\quad\text{almost surely}. \] Furthermore, it is proved that this result is optimal in several ways.
For the entire collection see [Zbl 1234.00021].

MSC:

60G15 Gaussian processes
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
60H40 White noise theory
60G17 Sample path properties
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