Publication

A critical stochastic heat equation with long-range noise

2025

Abstract

We consider a semilinear stochastic heat equation in spatial dimension at least 3, forced by a noise that is white in time with a covariance kernel that decays like |𝑥 | −2 as |𝑥 | → ∞. We show that in an appropriate diffusive scaling limit with a logarithmic attenuation of the noise, the pointwise statistics of the solution can be approximated by the solution to a forward-backward stochastic differential equation (FBSDE). The scaling and structure of the problem is similar to that of the two-dimensional stochastic heat equation forced by an approximation of space-time white noise considered by the first author and Gu (Ann. Probab., 2022). However the resulting FBSDE is different due to the long-range correlations of the noise.

Official source

https://infoscience.epfl.ch/entities/publication/4d0affe4-05d4-40bc-9c2f-d0b2f537d5b1

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2025

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/4d0affe4-05d4-40bc-9c2f-d0b2f537d5b1

About this result

Related concepts (32)

White noise

In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, including physics, acoustical engineering, telecommunications, and statistical forecasting. White noise refers to a statistical model for signals and signal sources, rather than to any specific signal.

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.

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.

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