On Hölder continuity of the solution of stochastic wave equations in dimension three
On Hölder continuity of the solution of stochastic wave equations in dimension three
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DOI：
10.1007/s40072-014-0035-5
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发表时间：
2014-07
期刊：
影响因子：
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通讯作者：
Yaozhong Hu;Jingyu Huang;D. Nualart
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文献类型：
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作者：
Yaozhong Hu;Jingyu Huang;D. Nualart
在本文中，我们研究了由时间上白、空间上相关的高斯噪声驱动的三个空间维度的随机波动方程。我们主要关心的是解在时间变量和空间变量上的样本路径 Hölder 连续性。条件以协方差函数的平均霍尔德连续性或其谱测量的形式给出。协方差函数的一些例子被证明满足我们的条件，其中包括 Dalang 和 Sanz-Solé 的著作（Hölder–Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three. Memoirs of the American Mathematical Society Number 931 2009）。特别是，我们获得了由（空间非均匀）分数布朗噪声驱动的随机波动方程解的霍尔德连续性结果。对于这种特定的噪声，还讨论了所获得的 Hölder 指数的最优性。
In this paper, we study the stochastic wave equations in the three spatial dimensions driven by a Gaussian noise which is white in time and correlated in space. Our main concern is the sample path Hölder continuity of the solution both in time variable and in space variables. The conditions are given either in terms of the mean Hölder continuity of the covariance function or in terms of its spectral measure. Some examples of the covariance functions are proved to satisfy our conditions, which include the case of the work Dalang and Sanz-Solé (Hölder–Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three. Memoirs of the American Mathematical Society Number 931 2009). In particular, we obtain the Hölder continuity results for the solution of the stochastic wave equations driven by (space inhomogeneous) fractional Brownian noises. For this particular noise, the optimality of the obtained Hölder exponents is also discussed.