查看“︁随机偏微分方程”︁的源代码

'''随机偏微分方程'''（英文：Stochastic partial differential equation，SPDE）为[[偏微分方程]]引入了[[随机]][[项]]和随机[[系数]]，类似于[[随机微分方程]]之于[[常微分方程]]。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。<ref>{{Cite book|last1=Prévôt|first1=Claudia|url=https://www.springer.com/gp/book/9783540707806|title=A Concise Course on Stochastic Partial Differential Equations|last2=Röckner|first2=Michael|date=2007|publisher=Springer-Verlag|isbn=978-3-540-70780-6|series=Lecture Notes in Mathematics|location=Berlin Heidelberg|language=en|access-date=2023-10-29|archive-date=2020-03-29|archive-url=https://web.archive.org/web/20200329191744/https://www.springer.com/gp/book/9783540707806|dead-url=no}}</ref><ref>{{Cite book|last1=Krainski|first1=Elias T.|url=https://www.crcpress.com/Advanced-Spatial-Modeling-with-Stochastic-Partial-Differential-Equations/Krainski-Gomez-Rubio-Bakka-Lenzi-Castro-Camilo-Simpson-Lindgren-Rue/p/book/9781138369856|title=Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA|last2=Gómez-Rubio|first2=Virgilio|last3=Bakka|first3=Haakon|last4=Lenzi|first4=Amanda|last5=Castro-Camilo|first5=Daniela|last6=Simpson|first6=Daniel|last7=Lindgren|first7=Finn|last8=Rue|first8=Håvard|publisher=Chapman and Hall/CRC Press|year=2018|isbn=978-1-138-36985-6|location=Boca Raton, FL|access-date=2023-10-29|archive-date=2020-03-29|archive-url=https://web.archive.org/web/20200329191743/https://www.crcpress.com/Advanced-Spatial-Modeling-with-Stochastic-Partial-Differential-Equations/Krainski-Gomez-Rubio-Bakka-Lenzi-Castro-Camilo-Simpson-Lindgren-Rue/p/book/9781138369856|dead-url=no}}</ref>

== 示例 ==
最常见的SPDE之一是随机[[热传导方程]]<ref>{{Cite journal |last=Edwards |first=S.F. |last2=Wilkinson |first2=D.R. |date=1982-05-08 |title=The Surface Statistics of a Granular Aggregate |url=https://www.jstor.org/stable/2397363 |journal=Proc. R. Soc. Lond. A |language=en |volume=381 |issue=1780 |pages=17–31 |doi=10.1098/rspa.1982.0056 |access-date=2023-10-29 |archive-date=2023-12-29 |archive-url=https://web.archive.org/web/20231229102637/https://www.jstor.org/stable/2397363 |dead-url=no }}</ref> ，形式上可以写作

:<math>
\partial_t u = \Delta u + \xi\;,
</math>

其中<math>\Delta</math>是[[拉普拉斯算子]]，<math>\xi</math>表示时空[[白噪声]]。其他例子还有知名方程的随机版本，如[[波动方程]]<ref>{{Cite journal |last=Dalang |first=Robert C. |last2=Frangos |first2=N. E. |date=1998 |title=The Stochastic Wave Equation in Two Spatial Dimensions |url=https://www.jstor.org/stable/2652898 |journal=The Annals of Probability |volume=26 |issue=1 |pages=187–212 |issn=0091-1798 |access-date=2023-10-29 |archive-date=2023-05-09 |archive-url=https://web.archive.org/web/20230509002004/https://www.jstor.org/stable/2652898 |dead-url=no }}</ref>和[[薛定谔方程]]。<ref>{{Cite journal |last=Diósi |first=Lajos |last2=Strunz |first2=Walter T. |date=1997-11-24 |title=The non-Markovian stochastic Schrödinger equation for open systems |url=https://www.sciencedirect.com/science/article/pii/S0375960197007172 |journal=Physics Letters A |language=en |volume=235 |issue=6 |pages=569–573 |doi=10.1016/S0375-9601(97)00717-2 |issn=0375-9601|arxiv=quant-ph/9706050 }}</ref>

== 讨论 ==
一个困难是缺乏正规性。在一个空间维度中，随机热传导方程的解在空间上几乎只有1/2-[[赫尔德条件|赫尔德连续]]，在时间上则只有1/4-赫尔德连续。对于二维及更高维度，解甚至不是函数值，但可以理解为随机[[分布 (数学分析)|分布]]。

对于线性方程，通常可以通过[[半群]]手段找到温和解（mild solution）。<ref>{{Cite journal|last=Walsh|first=John B.|date=1986|editor-last=Carmona|editor-first=René|editor2-last=Kesten|editor2-first=Harry|editor3-last=Walsh|editor3-first=John B.|editor4-last=Hennequin|editor4-first=P. L.|title=An introduction to stochastic partial differential equations|journal=École d'Été de Probabilités de Saint Flour XIV - 1984|series=Lecture Notes in Mathematics|volume=1180|language=en|publisher=Springer Berlin Heidelberg|pages=265–439|doi=10.1007/bfb0074920|isbn=978-3-540-39781-6}}</ref>
然而，当考虑非线性方程时，问题就开始出现了。例如

:<math>
\partial_t u = \Delta u + P(u) + \xi,
</math>
其中<math>P</math>是多项式。在这种情况下，我们甚至不知道该如何理解这个方程。这样的方程在多维情形下也不会有数值解，因此也没有点。众所周知，[[分布 (数学分析)|分布]]空间没有积结构。这是此类理论的核心问题。这就需要某种形式的[[重整化]]。

为规避某些特定方程的此类问题，早期的尝试是所谓的“普拉托-德布斯切技巧”（da Prato–Debussche trick），即把此类非线性方程作为线性方程的扰动来研究。<ref>{{cite journal |first=Giuseppe |last=Da Prato |first2=Arnaud |last2=Debussche |title=Strong Solutions to the Stochastic Quantization Equations |journal=Annals of Probability |volume=31 |issue=4 |year=2003 |pages=1900–1916 |jstor=3481533 }}</ref>然而，这只能在非常受限的环境中使用，因为它既取决于非线性因子，也取决于驱动噪声项的正规性。近年来，这一领域急剧扩大，现在已有大型机制可以保证各种亚临界SPDE的局部存在性。<ref>{{cite journal |first=Ivan |last=Corwin |first2=Hao |last2=Shen |title=Some recent progress in singular stochastic partial differential equations |journal=Bull. Amer. Math. Soc. |volume=57 |year=2020 |issue=3 |pages=409–454 |doi=10.1090/bull/1670 |doi-access=free }}</ref>
== 另见 ==
*[[布朗面]]
*[[KPZ方程]]
*[[库什纳方程]]
*[[威克积]]
==参考文献==
{{Reflist}}

== 阅读更多 ==
*{{cite book |last1=Bain |first1=A. |last2=Crisan |first2=D. |year=2009 |title=Fundamentals of Stochastic Filtering |series=Stochastic Modelling and Applied Probability |publisher=Springer |volume=60 |location=New York |edition= |isbn=978-0387768953 |doi=}}
*{{cite book |last1=Holden |first1=H. |last2=Øksendal |first2=B. |last3=Ubøe |first3=J. |last4=Zhang |first4=T. |year=2010 |title=Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach |series=Universitext |publisher=Springer |location=New York |edition=2nd |isbn=978-0-387-89487-4 |doi=10.1007/978-0-387-89488-1 }}
*{{cite journal |last1=Lindgren |first1=F. |last2=Rue |first2=H. |last3=Lindström |first3=J. |date=2011 |title=An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach |url=https://academic.oup.com/jrsssb/article/73/4/423/7034732 |journal=Journal of the Royal Statistical Society Series B: Statistical Methodology |volume=73 |issue=4 |pages=423–498 |arxiv= |bibcode= |doi=10.1111/j.1467-9868.2011.00777.x |issn=1369-7412 |s2cid= |hdl=20.500.11820/1084d335-e5b4-4867-9245-ec9c4f6f4645 |hdl-access=free |access-date=2023-10-29 |archive-date=2024-04-27 |archive-url=https://web.archive.org/web/20240427060224/https://academic.oup.com/jrsssb/article/73/4/423/7034732 |dead-url=no }}
*{{cite book |last1=Xiu |first1=D. |year=2010 |title=Numerical Methods for Stochastic Computations: A Spectral Method Approach |series= |publisher=Princeton University Press |location= |edition= |isbn=978-0-691-14212-8 |doi= |volume=}}

== 外部链接 ==
* {{cite web |url=https://web.math.rochester.edu/people/faculty/cmlr/Preprints/Utah-Summer-School.pdf |title=A Minicourse on Stochastic Partial Differential Equations |date=2006 }}
* {{cite arXiv |title=An Introduction to Stochastic PDEs |first=Martin |last=Hairer |author-link=Martin Hairer |year=2009 |class=math.PR |eprint=0907.4178 }}
[[Category:随机微分方程]]
[[Category:偏微分方程]]
[[Category:金融数学| ]]