Two-dimensional semilinear stochastic heat equations with critical long-range correlations
Abstract
We study a (2+1)-dimensional semilinear stochastic heat equation in which the noise is correlated like f(log |z|)/|z|2 as |z|→ ∞. Under an appropriate rescaling and logarithmic attenuation, we show that the limiting two-point statistics are given by the solution to a forward-backward stochastic differential equation. As shown in the work of Cosco, Cottini, and Donadini (2025), with whose results our study agrees in the linear case, the interesting feature of the two-dimensional, long-range problem is that one sees a varying influence on the limiting distribution from a range of spatial scales. We show that this phenomenon appears in the decoupling function for the limiting object and that the function f modulates the weight of the contribution of each spatial scale.
Related
- Weyl Pseudo Almost Periodic Type Solutions to Semilinear Stochastic Evolution Equations Driven by Fractional Brownian Motion
- Critical semilinear damped wave equations beyond moduli of continuity
- Nonequilibrium Thermodynamics of Semilinear Stochastic Partial Differential Equations
- The critical exponent for the two-dimensional semilinear wave equation with strong damping
- Grow-up rates of inhomogeneous semilinear heat equations in the critical dimension
BibTeX
@misc{castillo2026twodimensional,
title = {{Two-dimensional semilinear stochastic heat equations with critical long-range correlations}},
author = {Bryan Castillo and Alexander Dunlap},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26249},
eprint = {2610.26249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26249},
}
RIS
TY - UNPB
TI - Two-dimensional semilinear stochastic heat equations with critical long-range correlations
AU - Bryan Castillo
AU - Alexander Dunlap
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26249
UR - https://arxiv.org/abs/2610.26249
AB - We study a (2+1)-dimensional semilinear stochastic heat equation in which the noise is correlated like f(log |z|)/|z|^2 as |z|→ ∞. Under an appropriate rescaling and logarithmic attenuation, we show that the limiting two-point statistics are given by the solution to a forward-backward stochastic differential equation. As shown in the work of Cosco, Cottini, and Donadini (2025), with whose results our study agrees in the linear case, the interesting feature of the two-dimensional, long-range problem is that one sees a varying influence on the limiting distribution from a range of spatial scales. We show that this phenomenon appears in the decoupling function for the limiting object and that the function f modulates the weight of the contribution of each spatial scale.
ER -