Path properties of the solution to the stochastic heat equation with Lévy noise
Path properties of the solution to the stochastic heat equation with Lévy noise
复制标题
带 Lévy 噪声的随机热方程解的路径性质
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
T. Humeau
中科院分区：
文献类型：
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作者：
Carsten Chong;R. Dalang;T. Humeau
我们考虑随机热方程解的样本路径属性，在 $${mathbb {R}}^d$$Rd 或 $${mathbb {R}}^d$$Rd 的有界域中，由 Lévy 时空白噪声驱动。当将其视为具有无限维空间中的值的时间随机过程时，该解在索引小于 $$-frac{d}{2}$$-d2 的分数 Sobolev 空间中具有 càdlàg 修改。考虑到当其他变量固定时解在时间或空间上的部分规律性，我们确定 Lévy 噪声的 Blumenthal–Getoor 指数的临界值，使得具有较小指数的噪声需要连续的样本路径，而具有较大指数的 Lévy 噪声需要在任何非空开子集上无界的样本路径。我们的结果适用于加法和乘法 Lévy 噪声，以及轻尾和重尾跳跃。
We consider sample path properties of the solution to the stochastic heat equation, in $${mathbb {R}}^d$$Rd or bounded domains of $${mathbb {R}}^d$$Rd, driven by a Lévy space–time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a càdlàg modification in fractional Sobolev spaces of index less than $$-frac{d}{2}$$-d2. Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal–Getoor index of the Lévy noise such that noises with a smaller index entail continuous sample paths, while Lévy noises with a larger index entail sample paths that are unbounded on any non-empty open subset. Our results apply to additive as well as multiplicative Lévy noises, and to light- as well as heavy-tailed jumps.