One of the most studied SPDEs is the stochastic heat equation, which may formally be written as
where is the Laplacian and denotes space-time white noise.
Discussion
One difficulty is their lack of regularity. In one space dimension, solutions to the stochastic heat equation are only almost 1/2-Hölder continuous in space and 1/4-Hölder continuous in time. For dimensions two and higher, solutions are not even function-valued, but can be made sense of as random distributions.
参见
Brownian surface
Kardar–Parisi–Zhang equation
Kushner equation
Wick product
Zakai equation
延伸阅读
Holden, H., Øksendal, B., Ubøe, J, Zhang, T., 2010. Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach (页面存档备份,存于互联网档案馆). Universitext, Springer, New York, NY, ed. 2.
随机偏微分方程, 英文, stochastic, partial, differential, equations, spdes, 类似于一般的随机微分方程, 其本质上是带有随机项和随机系数的偏微分方程, 随机微分方程在量子场论, 统计力学, 金融数学中有着广泛的应用, 目录, examples, discussion, 参见, 延伸阅读examples, 编辑one, most, studied, spdes, stochastic, heat, equation, which, formally, writte. 随机偏微分方程 英文 Stochastic partial differential equations SPDEs 类似于一般的随机微分方程 其本质上是带有随机项和随机系数的偏微分方程 随机微分方程在量子场论 统计力学 金融数学中有着广泛的应用 目录 1 Examples 2 Discussion 3 参见 4 延伸阅读Examples 编辑One of the most studied SPDEs is the stochastic heat equation which may formally be written as t u D u 3 displaystyle partial t u Delta u xi where D displaystyle Delta is the Laplacian and 3 displaystyle xi denotes space time white noise Discussion 编辑One difficulty is their lack of regularity In one space dimension solutions to the stochastic heat equation are only almost 1 2 Holder continuous in space and 1 4 Holder continuous in time For dimensions two and higher solutions are not even function valued but can be made sense of as random distributions 参见 编辑Brownian surface Kardar Parisi Zhang equation Kushner equation Wick product Zakai equation延伸阅读 编辑Holden H Oksendal B Uboe J Zhang T 2010 Stochastic Partial Differential Equations A Modeling White Noise Functional Approach 页面存档备份 存于互联网档案馆 Universitext Springer New York NY ed 2 这是一篇与应用数学相关的小作品 你可以通过编辑或修订扩充其内容 查论编 取自 https zh wikipedia org w index php title 随机偏微分方程 amp oldid 72951995, 维基百科,wiki,书籍,书籍,图书馆,