Stochastic partial differential equations



Placeholder, for the multidimensional PDE version of SDEs.

This picture of ice floes on the Bering shelf looks like it might be some kinda stochastic PDE thing, right?

So, how do I handle these?

As SDEs taking values in a Banach space

Keywords: Q-Wiener process, Cylindrical wiener-process, Banach-space-valued process.

A textbook that was recommended to me by Thomas Scheckter was Liu and Röckner (2015), which seems to have same blurb as the older @Prévôt and Röckner (2007). There are many free texts, e.g.

Kovács and Larsson Introduction to stochastic partial differential equations, or Michael Scheutzow’s, or Sam Punshon Smith’s.

References

Adler, Robert J. 2010. The Geometry of Random Fields. SIAM ed. Philadelphia: Society for Industrial and Applied Mathematics.
Adler, Robert J., and Jonathan E. Taylor. 2007. Random Fields and Geometry. Springer Monographs in Mathematics 115. New York: Springer. https://doi.org/10.1007/978-0-387-48116-6.
Adler, Robert J, Jonathan E Taylor, and Keith J Worsley. 2016. Applications of Random Fields and Geometry Draft. https://robert.net.technion.ac.il/files/2016/08/hrf1.pdf.
Bolin, David, and Kristin Kirchner. 2020. “The Rational SPDE Approach for Gaussian Random Fields With General Smoothness.” Journal of Computational and Graphical Statistics 29 (2): 274–85. https://doi.org/10.1080/10618600.2019.1665537.
Bréhier, Charles-Edouard. 2014. “A Short Introduction to Stochastic PDEs.” https://hal.archives-ouvertes.fr/hal-00973887v2.
Dalang, Robert C., Davar Khoshnevisan, and Firas Rassoul-Agha, eds. 2009. A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics 1962. Berlin: Springer. https://people.math.rochester.edu/faculty/cmlr/Preprints/Utah-Summer-School.pdf.
Dalang, Robert C., and Lluís Quer-Sardanyons. 2011. “Stochastic Integrals for SPDEs: A Comparison.” Expositiones Mathematicae 29 (1): 67–109. https://doi.org/10.1016/j.exmath.2010.09.005.
Gawarecki, Leszek, and Vidyadhar Mandrekar. 2011. Stochastic Differential Equations in Infinite Dimensions: With Applications to Stochastic Partial Differential Equations. Probability and Its Applications. Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-16194-0.
Hairer, Martin. 2009. “An Introduction to Stochastic PDEs,” July. https://arxiv.org/abs/0907.4178v1.
Holden, Helge, Bernt Øksendal, Jan Ubøe, and Tusheng Zhang. 1996. Stochastic Partial Differential Equations. Boston, MA: Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-9215-6.
Hu, Xiangping, and Ingelin Steinsland. 2016. “Spatial Modeling with System of Stochastic Partial Differential Equations.” WIREs Computational Statistics 8 (2): 112–25. https://doi.org/10.1002/wics.1378.
Kallenberg, Olav. 2002. Foundations of Modern Probability. 2nd ed. Probability and Its Applications. New York: Springer-Verlag. https://doi.org/10.1007/978-1-4757-4015-8.
Khoshnevisan, Davar. 2009. “A Primer on Stochastic Partial Differential Equations.” In A Minicourse on Stochastic Partial Differential Equations, edited by Davar Khoshnevisan and Firas Rassoul-Agha, 1962:1–38. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-85994-9_1.
Kotelenez, Peter. 2007. Stochastic Ordinary and Stochastic Partial Differential Equations: Transition From Microscopic to Macroscopic Equations. Springer Science & Business Media.
Kovacs, Mihaly, and Stig Larsson. 2008. “Introduction to Stochastic Partial Differential Equations.”
Lang, Annika. 2014. “Stochastic Partial Differential Equations.” In Computer Vision: A Reference Guide, edited by Katsushi Ikeuchi, 770–75. Boston, MA: Springer US. https://doi.org/10.1007/978-0-387-31439-6_681.
Lindgren, Finn, Håvard Rue, and Johan Lindström. 2011. “An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 73 (4): 423–98. https://doi.org/10.1111/j.1467-9868.2011.00777.x.
Liu, Wei, and Michael Röckner. 2015. Stochastic Partial Differential Equations: An Introduction. Springer. http://books.google.com?id=2QS0CgAAQBAJ.
Miller, David L., Richard Glennie, and Andrew E. Seaton. 2020. “Understanding the Stochastic Partial Differential Equation Approach to Smoothing.” Journal of Agricultural, Biological and Environmental Statistics 25 (1): 1–16. https://doi.org/10.1007/s13253-019-00377-z.
Peszat, S., and Professor J. Zabczyk. 2007. Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach. 1 edition. Cambridge ; New York: Cambridge University Press.
Prévôt, Claudia, and Michael Röckner. 2007. A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics. Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-540-70781-3.
Sigrist, Fabio, Hans R. Künsch, and Werner A. Stahel. 2015. “Stochastic Partial Differential Equation Based Modelling of Large Space-Time Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 (1): 3–33. https://doi.org/10.1111/rssb.12061.
Walsh, John B. 1986. “An Introduction to Stochastic Partial Differential Equations.” In École d’Été de Probabilités de Saint Flour XIV - 1984, edited by P. L. Hennequin, 1180:265–439. Lecture Notes in Mathematics. Springer Berlin Heidelberg. https://doi.org/10.1007/BFb0074920.
Whittle, P. 1954. “On Stationary Processes in the Plane.” Biometrika 41 (3/4): 434–49.

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