Stochastic partial differential equations

SDEs taking values in some function space

January 27, 2021 — May 25, 2022

Banach space
Hilber space
dynamical systems
Lévy processes
signal processing
stochastic processes
time series

Placeholder, for the multidimensional PDE version of SDEs. An important special case of functional stochastic DEs.

Figure 1: This picture 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?

2 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.

3 Incoming

4 References

Adler, Robert J. 2010. The Geometry of Random Fields.
Adler, Robert J., and Taylor. 2007. Random Fields and Geometry. Springer Monographs in Mathematics 115.
Adler, Robert J, Taylor, and Worsley. 2016. Applications of Random Fields and Geometry Draft.
Bolin, and Kirchner. 2020. The Rational SPDE Approach for Gaussian Random Fields With General Smoothness.” Journal of Computational and Graphical Statistics.
Bréhier. 2014. A Short Introduction to Stochastic PDEs.”
Curtain. 1975. Infinite-Dimensional Filtering.” SIAM Journal on Control.
Da Prato, and Zabczyk. 2002. Second Order Partial Differential Equations in Hilbert Spaces.
———. 2014. Stochastic Equations in Infinite Dimensions.
Dalang, Khoshnevisan, and Rassoul-Agha, eds. 2009a. A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics 1962.
———. 2009b. A minicourse on stochastic partial differential equations. Lecture notes in mathematics 1962.
Dalang, and Quer-Sardanyons. 2011. Stochastic Integrals for SPDEs: A Comparison.” Expositiones Mathematicae.
Duffin, Cripps, Stemler, et al. 2021. Statistical Finite Elements for Misspecified Models.” Proceedings of the National Academy of Sciences.
Gawarecki, and Mandrekar. 2011. Stochastic Differential Equations in Infinite Dimensions: With Applications to Stochastic Partial Differential Equations. Probability and Its Applications.
Hairer, Martin. 2009. An Introduction to Stochastic PDEs.”
Hairer, M., Stuart, Voss, et al. 2005. Analysis of SPDEs Arising in Path Sampling. Part I: The Gaussian Case.” Communications in Mathematical Sciences.
Hairer, M., Stuart, and Voss. 2007. Analysis of SPDEs Arising in Path Sampling Part II: The Nonlinear Case.” The Annals of Applied Probability.
Holden, Øksendal, Ubøe, et al. 1996. Stochastic Partial Differential Equations.
Hu, and Steinsland. 2016. Spatial Modeling with System of Stochastic Partial Differential Equations.” WIREs Computational Statistics.
Hu, Steinsland, Simpson, et al. 2013. Spatial Modelling of Temperature and Humidity Using Systems of Stochastic Partial Differential Equations.”
Kallenberg. 2002. Foundations of Modern Probability. Probability and Its Applications.
Khoshnevisan. 2009. A Primer on Stochastic Partial Differential Equations.” In A Minicourse on Stochastic Partial Differential Equations.
Kotelenez. 2007. Stochastic Ordinary and Stochastic Partial Differential Equations: Transition From Microscopic to Macroscopic Equations.
Kovacs, and Larsson. 2008. “Introduction to Stochastic Partial Differential Equations.”
Lang. 2014. Stochastic Partial Differential Equations.” In Computer Vision: A Reference Guide.
Lindgren, Rue, and 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).
Liu, and Röckner. 2015. Stochastic Partial Differential Equations: An Introduction.
Lord, Powell, and Shardlow. 2014. An Introduction to Computational Stochastic PDEs. Cambridge Texts in Applied Mathematics.
McAllester. 2023. On the Mathematics of Diffusion Models.”
Miller, Glennie, and Seaton. 2020. Understanding the Stochastic Partial Differential Equation Approach to Smoothing.” Journal of Agricultural, Biological and Environmental Statistics.
Peszat, and Zabczyk. 2007. Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach.
Preston, and Poppeliers. 2021. LDRD #218329: Uncertainty Quantification of Geophysical Inversion Using Stochastic Partial Differential Equations. SAND2021-10885.
Prévôt, and Röckner. 2007. A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics.
Roosta-Khorasani, Doel, and Ascher. 2014. Data Completion and Stochastic Algorithms for PDE Inversion Problems with Many Measurements.”
Särkkä. 2011. Linear Operators and Stochastic Partial Differential Equations in Gaussian Process Regression.” In Artificial Neural Networks and Machine Learning – ICANN 2011. Lecture Notes in Computer Science.
Sigrist, Künsch, and Stahel. 2015a. Spate : An R Package for Spatio-Temporal Modeling with a Stochastic Advection-Diffusion Process.” Application/pdf. Journal of Statistical Software.
———. 2015b. Stochastic Partial Differential Equation Based Modelling of Large Space-Time Data Sets.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Solin. 2016. Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.”
Solin, and Särkkä. 2013. Infinite-Dimensional Bayesian Filtering for Detection of Quasiperiodic Phenomena in Spatiotemporal Data.” Physical Review E.
Tsyrulnikov, and 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.
Walsh. 1986. An Introduction to Stochastic Partial Differential Equations.” In École d’Été de Probabilités de Saint Flour XIV - 1984.
Whittle. 1954. “On Stationary Processes in the Plane.” Biometrika.
Zhang, and Karniadakis. 2017. Numerical Methods for Stochastic Partial Differential Equations with White Noise. Applied Mathematical Sciences.