Stochastic partial differential equations

SDEs taking values in some function space

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?

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.


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.
Duffin, Connor, Edward Cripps, Thomas Stemler, and Mark Girolami. 2021. Statistical Finite Elements for Misspecified Models.” Proceedings of the National Academy of Sciences 118 (2).
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.
Hu, Xiangping, and Ingelin Steinsland. 2016. Spatial Modeling with System of Stochastic Partial Differential Equations.” WIREs Computational Statistics 8 (2): 112–25.
Hu, Xiangping, Ingelin Steinsland, Daniel Simpson, Sara Martino, and Håvard Rue. 2013. Spatial Modelling of Temperature and Humidity Using Systems of Stochastic Partial Differential Equations,” July.
Kallenberg, Olav. 2002. Foundations of Modern Probability. 2nd ed. Probability and Its Applications. New York: Springer-Verlag.
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. Berlin, Heidelberg: Springer Berlin Heidelberg.
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.
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.
Liu, Wei, and Michael Röckner. 2015. Stochastic Partial Differential Equations: An Introduction. Springer.
Lord, Gabriel J., Catherine E. Powell, and Tony Shardlow. 2014. An Introduction to Computational Stochastic PDEs. 1st edition. Cambridge Texts in Applied Mathematics. New York, NY, USA: Cambridge University Press.
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.
Roosta-Khorasani, Farbod, Kees van den Doel, and Uri Ascher. 2014. Data Completion and Stochastic Algorithms for PDE Inversion Problems with Many Measurements.” arXiv.
Särkkä, Simo. 2011. Linear Operators and Stochastic Partial Differential Equations in Gaussian Process Regression.” In Artificial Neural Networks and Machine Learning – ICANN 2011, edited by Timo Honkela, Włodzisław Duch, Mark Girolami, and Samuel Kaski, 6792:151–58. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer.
Sigrist, Fabio, Hans R. Künsch, and Werner A. Stahel. 2015a. Spate : An R Package for Spatio-Temporal Modeling with a Stochastic Advection-Diffusion Process.” Application/pdf. Journal of Statistical Software 63 (14).
———. 2015b. 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.
Solin, Arno. 2016. Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.” Aalto University.
Solin, Arno, and Simo Särkkä. 2013. Infinite-Dimensional Bayesian Filtering for Detection of Quasiperiodic Phenomena in Spatiotemporal Data.” Physical Review E 88 (5): 052909.
Tsyrulnikov, Michael, and Alexander Rakitko. 2019. Impact of Non-Stationarity on Hybrid Ensemble Filters: A Study with a Doubly Stochastic Advection-Diffusion-Decay Model.” Quarterly Journal of the Royal Meteorological Society 145 (722): 2255–71.
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. Springer Berlin Heidelberg.
Whittle, P. 1954. “On Stationary Processes in the Plane.” Biometrika 41 (3/4): 434–49.
Zhang, Zhongqiang, and George Em Karniadakis. 2017. Numerical Methods for Stochastic Partial Differential Equations with White Noise. Vol. 196. Applied Mathematical Sciences. Cham: Springer International Publishing.

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