Publication

Periodic space-time homogenisation of the $φ^4_2$ equation

2023

Abstract

We consider the homogenisation problem for the ϕ42 equation on the torus 𝕋2, namely the behaviour as ε→0 of the solutions to the equation \textit{suggestively} written as ∂tuε−∇⋅A(x/ε,t/ε2)∇uε=−u3ε+ξ where ξ denotes space-time white noise and A:𝕋2×ℝ is uniformly elliptic, periodic and Hölder continuous. When the noise is regularised at scale δ≪1 we show that any joint limit ε,δ→0 recovers the classical dynamical ϕ42 model. In certain regimes or if the regularisation is chosen in a specific way adapted to the problem, we show that the counterterms can be chosen as explicit local functions of A.

Official source

https://infoscience.epfl.ch/entities/publication/0bd328e4-dffb-46fc-a478-8f8535d9ebdd

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2023

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/0bd328e4-dffb-46fc-a478-8f8535d9ebdd

About this result

Related concepts (31)

Noise (electronics)

In electronics, noise is an unwanted disturbance in an electrical signal. Noise generated by electronic devices varies greatly as it is produced by several different effects. In particular, noise is inherent in physics and central to thermodynamics. Any conductor with electrical resistance will generate thermal noise inherently. The final elimination of thermal noise in electronics can only be achieved cryogenically, and even then quantum noise would remain inherent. Electronic noise is a common component of noise in signal processing.

Noise reduction

Noise reduction is the process of removing noise from a signal. Noise reduction techniques exist for audio and images. Noise reduction algorithms may distort the signal to some degree. Noise rejection is the ability of a circuit to isolate an undesired signal component from the desired signal component, as with common-mode rejection ratio. All signal processing devices, both analog and digital, have traits that make them susceptible to noise.

Noise

Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound. Acoustic noise is any sound in the acoustic domain, either deliberate (e.g., music or speech) or unintended. In contrast, noise in electronics may not be audible to the human ear and may require instruments for detection.

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Periodic space-time homogenisation of the $φ^4_2$ equation

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