Publication

Stochastic nonlinear wave equation with rougher than white noise

2025

Abstract

We study the singular stochastic wave equation on T 2 , with a cubic nonlinearity and Gaussian rough 'Matérn' forcing (a Fourier multiplier of order α > 0 applied to space-time white noise) and establish local well-posedness for α < 3 8. This extends [GKO18] beyond white noise and strengthens the quadratic-case result [OO21] (α < 1 2). Our argument develops new trilinear estimates in Bourgain spaces together with case-specific cubic counting estimates.

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2025

Abstract

Official source

Related concepts (32)

Additive white Gaussian noise

Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics: Additive because it is added to any noise that might be intrinsic to the information system. White refers to the idea that it has uniform power spectral density across the frequency band for the information system. It is an analogy to the color white which may be realized by uniform emissions at all frequencies in the visible spectrum.

Colors of noise

In audio engineering, electronics, physics, and many other fields, the color of noise or noise spectrum refers to the power spectrum of a noise signal (a signal produced by a stochastic process). Different colors of noise have significantly different properties. For example, as audio signals they will sound differently to human ears, and as they will have a visibly different texture. Therefore, each application typically requires noise of a specific color.

Pink noise

Pink noise, noise or fractal noise is a signal or process with a frequency spectrum such that the power spectral density (power per frequency interval) is inversely proportional to the frequency of the signal. In pink noise, each octave interval (halving or doubling in frequency) carries an equal amount of noise energy. Pink noise sounds like a waterfall. It is often used to tune loudspeaker systems in professional audio. Pink noise is one of the most commonly observed signals in biological systems.

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Stochastic nonlinear wave equation with rougher than white noise

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