# Functional stochastic differential equations

SDEs taking values in some function space

July 10, 2024 — July 10, 2024

Banach space

Hilbert space

dynamical systems

Lévy processes

probability

SDEs

signal processing

spatial

stochastic processes

time series

Placeholder, for the infinite-dimensional version of SDEs.

## 1 Infinite-dimensional Wiener processes

Point of confusion: Cylindrical versus q-Wiener processes.

## 2 Incoming

## 3 References

Curtain, Ruth F. 1975. “Infinite-Dimensional Filtering.”

*SIAM Journal on Control*.
Curtain, Ruth F, and Falb. 1971. “Stochastic Differential Equations in Hilbert Space.”

*Journal of Differential Equations*.
Curtain, Ruth F., and Pritchard, eds. 1978.

*Infinite Dimensional Linear Systems Theory*. Lecture Notes in Control and Information Sciences.
Fox. 1986. “Functional-Calculus Approach to Stochastic Differential Equations.”

*Physical Review A*.
Fuhrman. 2003. “A Class of Stochastic Optimal Control Problems in Hilbert Spaces: BSDEs and Optimal Control Laws, State Constraints, Conditioned Processes.”

*Stochastic Processes and Their Applications*.
Hairer, Stuart, and Voss. n.d. “Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods.”

Kotelenez, and Curtain. 1982. “Local Behaviour of Hilbert Space Valued Stochastic Integrals and the Continuity of Mild Solutions of Stochastic Evolution Equations.”

*Stochastics*.
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)*.
McAllester. 2023. “On the Mathematics of Diffusion Models.”

Ocone. 1988. “Stochastic Calculus of Variations for Stochastic Partial Differential Equations.”

*Journal of Functional Analysis*.
Owhadi, and Scovel. 2015. “Conditioning Gaussian Measure on Hilbert Space.”

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*.