# Functional stochastic differential equations

SDEs taking values in some function space

July 10, 2024 — July 10, 2024

Banach space

dynamical systems

Hilbert space

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. 2011. “Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods.” In.

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*.