The Itô-Nisio theorem, quadratic Wiener functionals, and 1-solitons

Among Professor Kiyosi Ito's achievements, there is the Ito-Nisio theorem, a completely general theorem relative to the Fourier series decomposition of Brownian motion. In this paper, some of its applications will be reviewed, and new applications to 1-soliton solutions to the Korteweg-de Vries (KdV for short) equation and Eulerian polynomials will be given.

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