Local Linearization and Exact Oscillations for the Nonlinear Stochastic Heat Equation with Spatially Rough Noise
Abstract
Consider the one-dimensional nonlinear stochastic heat equation (∂ u(t,x))/(∂ t) = (∂2 u(t,x))/(∂ x2) + σ(u(t,x))Ẇ(t,x), t>0, x∈ℝ, where Ẇ is a Gaussian noise that is white in time and fractional in space with Hurst parameter H∈(1/4,1/2). Hu et al. ({\it Ann. Probab.} {\bf 45} (2017), 4561--4616) established existence and uniqueness of the solution under the assumptions that σ is differentiable, both σ and σ' are Lipschitz continuous, and σ(0)=0. Hu and Wang ({\it Ann. Inst. Henri Poincaré Probab. Stat.} {\bf 58} (2022), 379--423) subsequently removed the technical condition σ(0)=0 by developing the well-posedness theory in a suitable weighted space associated with the weight λ(x)= cH (1 + |x|2)H−1. In this weighted framework, which allows σ(0)≠0, we establish temporal and spatial local linearizations: small increments of the nonlinear solution are approximated by the corresponding increments of the solution to the additive equation, multiplied by the local coefficient $σ(u(t,x))$. The main difficulty lies in handling the nonlocal fractional-difference structure induced by the rough spatial noise. We address this difficulty using weighted heat-kernel increment estimates. These approximations yield exact Lévy-type uniform moduli of continuity in both time and space, together with Khinchin and Chung laws of the iterated logarithm.
Related
- Bi-parameter local linearization for the stochastic wave equation with rough noise
- Inverse Problems for the Monge-Ampère Equation: Linearization and Nonlinear Recovery
- Central Limit Theorem for Stochastic Nonlinear Heat Equation with Pure-Jump Lévy White Noise
- Polynomial mixing for the weakly damped stochastic nonlinear Schrödinger equation on the whole space
- Chance-Constrained Nonlinear Covariance Control via Robust Linearization Remainder Bounds
BibTeX
@misc{qian2026local,
title = {{Local Linearization and Exact Oscillations for the Nonlinear Stochastic Heat Equation with Spatially Rough Noise}},
author = {Bin Qian and Mengxin Tong and Ran Wang and Yimin Xiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19535},
eprint = {2610.19535},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19535},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local Linearization and Exact Oscillations for the Nonlinear Stochastic Heat Equation with Spatially Rough Noise
AU - Bin Qian
AU - Mengxin Tong
AU - Ran Wang
AU - Yimin Xiao
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19535
UR - https://arxiv.org/abs/2610.19535
AB - Consider the one-dimensional nonlinear stochastic heat equation (∂ u(t,x))/(∂ t) = (∂^2 u(t,x))/(∂ x^2) + σ(u(t,x))Ẇ(t,x), t>0, x∈ℝ, where Ẇ is a Gaussian noise that is white in time and fractional in space with Hurst parameter H∈(1/4,1/2). Hu et al. ({\it Ann. Probab.} {\bf 45} (2017), 4561--4616) established existence and uniqueness of the solution under the assumptions that σ is differentiable, both σ and σ' are Lipschitz continuous, and σ(0)=0. Hu and Wang ({\it Ann. Inst. Henri Poincaré Probab. Stat.} {\bf 58} (2022), 379--423) subsequently removed the technical condition σ(0)=0 by developing the well-posedness theory in a suitable weighted space associated with the weight λ(x)= c_H (1 + |x|^2)^H−1. In this weighted framework, which allows σ(0)≠0, we establish temporal and spatial local linearizations: small increments of the nonlinear solution are approximated by the corresponding increments of the solution to the additive equation, multiplied by the local coefficient $σ(u(t,x))$. The main difficulty lies in handling the nonlocal fractional-difference structure induced by the rough spatial noise. We address this difficulty using weighted heat-kernel increment estimates. These approximations yield exact Lévy-type uniform moduli of continuity in both time and space, together with Khinchin and Chung laws of the iterated logarithm.
ER -