一类随机heat equations的非线性噪声激发性的注记
Remarks on non-linear noise excitability of some stochastic heat equations
摘要：
考虑了在区间$(0, L)$上的形式为$\partial_t u=Δu + λσ(u)\dot w$的非线性抛物型随机偏微分方程，其中$\dot w$表示时空白噪声，$σ$是Lipschitz连续的.在Dirichlet边界条件和$σ$的线性增长条件下，我们证明了$L^2$ -能量的阶数为$\exp[\text{const}\timesλ^4]$ as $λ\rightarrow \infty$ .这大大改进了Khoshnevisan和Kim最近的结果。我们的方法与他们的方法有很大的不同，它允许我们对同一方程得出相同的结论，但具有Neumann边界条件。这改进了Khoshnevisan和Kim的另一个结果。
英文摘要：
We consider nonlinear parabolic SPDEs of the form $\partial_t u=Δu + λσ(u)\dot w$ on the interval $(0, L)$, where $\dot w$ denotes space-time white noise, $σ$ is Lipschitz continuous. Under Dirichlet boundary conditions and a linear growth condition on $σ$, we show that the expected $L^2$-energy is of order $\exp[\text{const}\timesλ^4]$ as $λ\rightarrow \infty$. This significantly improves a recent result of Khoshnevisan and Kim. Our method is very different from theirs and it allows us to arrive at the same conclusion for the same equation but with Neumann boundary condition. This improves over another result of Khoshnevisan and Kim.
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