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October 20, 2025Open Access

A critical stochastic heat equation with long-range noise

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Analysis reveals pointwise statistics approximated by FBSDE in semilinear stochastic heat equations, highlighting long-range noise effects.

The pointwise statistics of the solution to the stochastic heat equation can be approximated by a forward-backward stochastic differential equation (FBSDE).
Long-range correlations of the noise differentiate the resulting FBSDE from previous models based on space-time white noise.
The analysis considers a semilinear stochastic heat equation defined in three or more spatial dimensions.
A logarithmic attenuation of the noise is critical for achieving appropriate diffusive scaling limits.

Dunlap et al. (2025) studied this question.

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Analysis reveals pointwise statistics approximated by FBSDE in semilinear stochastic heat equations, highlighting long-range noise effects.