On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations
On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations
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DOI：
10.1214/15-aihp719
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发表时间：
2014-10
期刊：
影响因子：
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通讯作者：
Le Chen;Kunwoo Kim
中科院分区：
文献类型：
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作者：
Le Chen;Kunwoo Kim
本文证明了$\mathbb{R}$上具有测度值初值的非线性随机分数阶热方程的样本路径比较原理.我们给出了定量估计如何接近零的解决方案可以。这些结果扩展了随机热方程的穆勒的比较原理，允许更一般的初始数据，如（狄拉克）三角洲措施和措施较重的尾巴比线性指数增长在$\pm\infty$。这些结果推广了Moreno弗洛雷斯[25]最近的工作，他证明了具有δ初始数据的随机热方程解的严格正性。作为一个应用，我们建立了方程的完全双曲性。作为中间步骤，我们从测度值初值出发，证明了解的H“older正则性，这在某种意义上推广了Chen和Dalang [6]最近的工作.
In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on $\mathbb{R}$ with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. These results extend Mueller's comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures with heavier tails than linear exponential growth at $\pm\infty$. These results generalize a recent work by Moreno Flores [25], who proves the strict positivity of the solution to the stochastic heat equation with the delta initial data. As one application, we establish the full intermittency for the equation. As an intermediate step, we prove the H\"older regularity of the solution starting from measure-valued initial data, which generalizes, in some sense, a recent work by Chen and Dalang [6].