Placeholder, for the multidimensional PDE version of SDEs.
[Thispicture of ice floes on the Bering shelf looks like it might be some kinda stochastic PDE thing, right? No further metadata because Internet archive took their content off Flickr, sorry.
So, how do I handle these?
Intros that were recommended to me
(Dalang, Khoshnevisan, and Rassoul-Agha 2009a; Dalang and Quer-Sardanyons 2011; Da Prato and Zabczyk 2002, 2014; Holden et al. 1996; Kotelenez 2007; Liu and Röckner 2015; Lord, Powell, and Shardlow 2014)
As SDEs taking values in a Banach space
Keywords: Q-Wiener process, Cylindrical wiener-process, Banach-space-valued process.
Da Prato and Zabczyk (2014) is the classic reference. 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.
Adler, Robert J, Jonathan E Taylor, and Keith J Worsley. 2016. Applications of Random Fields and Geometry Draft.
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.
Bréhier, Charles-Edouard. 2014. “A Short Introduction to Stochastic PDEs.”
Curtain, Ruth F. 1975. “Infinite-Dimensional Filtering.” SIAM Journal on Control 13 (1): 89–104.
Da Prato, Giuseppe, and Jerzy Zabczyk. 2002. Second Order Partial Differential Equations in Hilbert Spaces. Cambridge, UK; New York: Cambridge University Press.
———. 2014. Stochastic Equations in Infinite Dimensions. Cambridge University Press.
Dalang, Robert C., Davar Khoshnevisan, and Firas Rassoul-Agha, eds. 2009a. A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics 1962. Berlin: Springer.
———. 2009b. A minicourse on stochastic partial differential equations. Vol. 1962. Lecture notes in mathematics 1962. Berlin: Springer.
Dalang, Robert C., and Lluís Quer-Sardanyons. 2011. “Stochastic Integrals for SPDEs: A Comparison.” Expositiones Mathematicae 29 (1): 67–109.
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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.
Hairer, Martin. 2009. “An Introduction to Stochastic PDEs,” July.
Hairer, M., A. M. Stuart, and J. Voss. 2007. “Analysis of SPDEs Arising in Path Sampling Part II: The Nonlinear Case.” The Annals of Applied Probability 17 (5-6): 1657–1706.
Hairer, M., A. M. Stuart, J. Voss, and P. Wiberg. 2005. “Analysis of SPDEs Arising in Path Sampling. Part I: The Gaussian Case.” Communications in Mathematical Sciences 3 (4): 587–603.
Holden, Helge, Bernt Øksendal, Jan Ubøe, and Tusheng Zhang. 1996. Stochastic Partial Differential Equations. Boston, MA: Birkhäuser Boston.
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Liu, Wei, and Michael Röckner. 2015. Stochastic Partial Differential Equations: An Introduction. Springer.
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McAllester, David. 2023. “On the Mathematics of Diffusion Models.” arXiv.
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.
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.
Preston, Leiph, and Christian Poppeliers. 2021. “LDRD #218329: Uncertainty Quantification of Geophysical Inversion Using Stochastic Partial Differential Equations.” SAND2021-10885. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States).
Prévôt, Claudia, and Michael Röckner. 2007. A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics. Berlin Heidelberg: Springer-Verlag.
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