Stochastic partial differential equation

One of the most studied SPDEs is the stochastic heat equation,{{Cite journal |last1=Edwards |first1=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|jstor=2397363 |bibcode=1982RSPSA.381...17E }} which may formally be written as

:$$

\partial_t u = \Delta u + \xi\;,

where $\backslash Delta$ is the Laplacian and $\backslash xi$ denotes space-time white noise. Other examples also include stochastic versions of famous linear equations, such as the wave equation{{Cite journal |last1=Dalang |first1=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 |doi=10.1214/aop/1022855416 |jstor=2652898 |issn=0091-1798}} and the Schrödinger equation.{{Cite journal |last1=Diósi |first1=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 |bibcode=1997PhLA..235..569D }}

One difficulty is their lack of regularity. In one dimensional space, 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.

For linear equations, one can usually find a mild solution via semigroup techniques.{{Cite book|last=Walsh|first=John B.|chapter=An introduction to stochastic partial differential equations |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=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}}

However, problems start to appear when considering non-linear equations. For example

:$$

\partial_t u = \Delta u + P(u) + \xi,

where $P$ is a polynomial. In this case it is not even clear how one should make sense of the equation. Such an equation will also not have a function-valued solution in dimension larger than one, and hence no pointwise meaning. It is well known that the space of distributions has no product structure. This is the core problem of such a theory. This leads to the need of some form of renormalization.

An early attempt to circumvent such problems for some specific equations was the so called da Prato–Debussche trick which involved studying such non-linear equations as perturbations of linear ones.{{cite journal |first1=Giuseppe |last1=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 }} However, this can only be used in very restrictive settings, as it depends on both the non-linear factor and on the regularity of the driving noise term. In recent years, the field has drastically expanded, and now there exists a large machinery to guarantee local existence for a variety of sub-critical SPDEs.{{cite journal |first1=Ivan |last1=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 }}

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• Brownian surface
• Kardar–Parisi–Zhang equation
• Kushner equation
• Malliavin calculus
• Polynomial chaos
• Wick product
• Zakai equation

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{{Reflist}}

• {{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 }}
• {{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=}}

• {{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 }}

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Category:Stochastic differential equations

Category:Partial differential equations