Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions
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DOI：
10.5802/afst.1442
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发表时间：
2014-04
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影响因子：
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通讯作者：
Martin Hairer;H. Weber
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作者：
Martin Hairer;H. Weber
我们研究了在二维和三维空间中由强度为$\sqrt{\varepsilon}$和相关长度为$\delta$的噪声项驱动的随机Allen-Cahn方程。我们研究了对角线极限$\delta, \varepsilon \to 0$，并充分描述了依赖于$\delta$和$\varepsilon$之间关系的大偏差行为。最近发展的规则结构理论允许充分分析消失的相关长度$\delta$和固定噪声强度$\varepsilon$的解决方案的行为。一个关键的事实是，为了得到非平凡极限$\delta \to 0$，有必要引入发散反项。正则结构理论允许对一些有趣的方程严格地分析这种重整化过程。我们的主要结果是这些重整化解的大偏差原理。这个结果的一个有趣的特征是发散的重整化常数在大偏差率函数的水平上消失了。我们应用这一结果，在$\delta, \varepsilon$上推导了一个尖锐的条件，该条件保证对角线格式$\varepsilon, \delta \to 0$的大偏差原理，无需重整化。
We study the stochastic Allen-Cahn equation driven by a noise term with intensity $\sqrt{\varepsilon}$ and correlation length $\delta$ in two and three spatial dimensions. We study diagonal limits $\delta, \varepsilon \to 0$ and describe fully the large deviation behaviour depending on the relationship between $\delta$ and $\varepsilon$. The recently developed theory of regularity structures allows to fully analyse the behaviour of solutions for vanishing correlation length $\delta$ and fixed noise intensity $\varepsilon$. One key fact is that in order to get non-trivial limits as $\delta \to 0$, it is necessary to introduce diverging counterterms. The theory of regularity structures allows to rigorously analyse this renormalisation procedure for a number of interesting equations. Our main result is a large deviation principle for these renormalised solutions. One interesting feature of this result is that the diverging renormalisation constants disappear at the level of the large deviations rate function. We apply this result to derive a sharp condition on $\delta, \varepsilon$ that guarantees a large deviation principle for diagonal schemes $\varepsilon, \delta \to 0$ for the equation without renormalisation.