Local Linearization and Exact Oscillations for the Nonlinear Stochastic Heat Equation with Spatially Rough Noise

Abstract

Consider the one-dimensional nonlinear stochastic heat equation $$ \frac{\partial u(t,x)}{\partial t} = \frac{\partial^2 u(t,x)}{\partial x^2} + \sigma(u(t,x))\dot W(t,x), \qquad t>0,\ x\in\mathbb R, $$ where $\dot W$ is a Gaussian noise that is white in time and fractional in space with Hurst parameter $H\in(1/4,1/2)$. Hu et al. ({\it Ann. Probab.} {\bf 45} (2017), 4561--4616) established existence and uniqueness of the solution under the assumptions that $\sigma$ is differentiable, both $\sigma$ and $\sigma'$ are Lipschitz continuous, and $\sigma(0)=0$. Hu and Wang ({\it Ann. Inst. Henri Poincar\'e Probab. Stat.} {\bf 58} (2022), 379--423) subsequently removed the technical condition $\sigma(0)=0$ by developing the well-posedness theory in a suitable weighted space associated with the weight $\lambda(x)= c_H (1 + |x|^2)^{H-1}$. In this weighted framework, which allows $\sigma(0)\ne0$, 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 $\sigma(u(t,x))$. 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 L\'evy-type uniform moduli of continuity in both time and space, together with Khinchin and Chung laws of the iterated logarithm.