Finite-time transition to intermittency for a stochastic heat equation driven by the square of a Gaussian field
Abstract
In this paper, we study the spatial behavior of the solution ψ ( x , t ) to the stochastic heat equation ∂ t ψ ( x , t ) − 1 2 ∂ x 2 2 ψ ( x , t ) = g S ( x , t ) 2 ψ ( x , t ) , with 0 ⩽ t ⩽ T , x ∈ R , and ψ ( x , 0 ) = 1 . Here, g > 0 is a coupling constant and S ( x , t ) is a stationary, homogeneous, and ergodic Gaussian field. Focusing on E ( x , g ) ≡ ψ ( x , T ) at a finite time T > 0 , we identify the critical coupling g c ( T ) above which the average of E ( 0 , g ) diverges. We show that in the subcritical regime g < g c ( T ) , E ( x , g ) is spatially ergodic, with no intermittency, while in the supercritical regime g > g c ( T ) it becomes spatially intermittent and loses ergodicity. Our results differ from the extensively studied case where S ( x , t ) 2 is replaced by S ( x , t ) , in which intermittency appears only asymptotically as T → + ∞ , with no finite-time intermittency.
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BibTeX
@article{mounaix2026finitetime,
title = {{Finite-time transition to intermittency for a stochastic heat equation driven by the square of a Gaussian field}},
author = {Philippe Mounaix},
journal = {Journal of Physics A: Mathematical and Theoretical},
year = {2026},
month = sep,
doi = {10.1088/1751-8121/aea45e},
eprint = {2601.18561},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05755222v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Finite-time transition to intermittency for a stochastic heat equation driven by the square of a Gaussian field
AU - Philippe Mounaix
PY - 2026
DA - 2026/09/18
JO - Journal of Physics A: Mathematical and Theoretical
DO - 10.1088/1751-8121/aea45e
UR - https://hal.science/hal-05755222v1
AB - In this paper, we study the spatial behavior of the solution ψ ( x , t ) to the stochastic heat equation ∂ t ψ ( x , t ) − 1 2 ∂ x 2 2 ψ ( x , t ) = g S ( x , t ) 2 ψ ( x , t ) , with 0 ⩽ t ⩽ T , x ∈ R , and ψ ( x , 0 ) = 1 . Here, g > 0 is a coupling constant and S ( x , t ) is a stationary, homogeneous, and ergodic Gaussian field. Focusing on E ( x , g ) ≡ ψ ( x , T ) at a finite time T > 0 , we identify the critical coupling g c ( T ) above which the average of E ( 0 , g ) diverges. We show that in the subcritical regime g < g c ( T ) , E ( x , g ) is spatially ergodic, with no intermittency, while in the supercritical regime g > g c ( T ) it becomes spatially intermittent and loses ergodicity. Our results differ from the extensively studied case where S ( x , t ) 2 is replaced by S ( x , t ) , in which intermittency appears only asymptotically as T → + ∞ , with no finite-time intermittency.
SN - 1751-8113
SN - 1751-8121
ER -