随机标量反应扩散方程的定量不稳定性
Quantitative instability for stochastic scalar reaction-diffusion equations
AI总结：
本文研究乘性噪声驱动的随机标量反应扩散方程在零附近的不稳定性，利用随机均匀化、校正子估计和拟基本解型线性化方法构造Lyapunov泛函，证明正Lyapunov指数下非零解流具有小负矩一致界。
摘要：
本文研究随机标量反应扩散方程在零附近的不稳定性，该方程由时间白噪声、空间光滑的乘性噪声驱动，并假设零是该方程的一个不动点。我们证明，如果零处的Lyapunov指数为正，则非零解的流在小负矩上具有一致界。证明建立在随机均匀化的思想上。我们需要对涉及无穷维投影过程的Poisson问题解的适当校正子估计，以及依赖于定量拟基本解型论证的线性化步骤。总体而言，我们能够为非线性动力学构造一个合适的Lyapunov泛函，并解决文献中遗留的一些问题。
英文摘要：
This work studies the instability of stochastic scalar reaction diffusion equations, driven by a multiplicative noise that is white in time and smooth in space, near to zero, which is assumed to be a fixed point for the equation. We prove that if the Lyapunov exponent at zero is positive, then the flow of non-zero solutions admits uniform bounds on small negative moments. The proof builds on ideas from stochastic homogenisation. We require suitable corrector estimates for the solution to a Poisson problem involving an infinite-dimensional projective process, together with a linearisation step that hinges on quantitative parametrix-like arguments. Overall, we are able to construct an appropriate Lyapunov functional for the nonlinear dynamics and address some problems left open in the literature.
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