Hi,

I suggest you see the links and attached file on the topic.

Article Infinite-dimensional stochastic differential equations relat...

Article Infinite-dimensional stochastic differential equations and t...

Article Stochastic Differential Equations on Manifolds

Preprint Stochastic differential equations on noncompact manifolds: m...

Preprint Center Manifolds for Rough Partial Differential Equations

https://www.jstor.org/stable/2652756

Best regards

Years ago, we stumbled on some interesting phenomenon in a simple population dynamics model (finite dimensional): Article Features of Fast Living: On the Weak Selection for Longevity...

Yes, there has been significant work on the theory of central manifolds for stochastic differential equations (SDEs). Central manifolds are important in the study of SDEs because they provide a reduced-dimensional representation of the dynamics of the system near an equilibrium point. This can simplify the analysis of the system and provide insights into its behavior.

The theory of central manifolds for deterministic systems has been well-established for some time. However, the extension of this theory to SDEs has been an active area of research in recent years. Some of the key contributions in this area include the work of Kloeden and Neuenkirch on central manifolds for SDEs with additive noise and the work of Pavliotis and Stuart on central manifolds for SDEs with multiplicative noise.

The theory of central manifolds for SDEs has important applications in many areas of science and engineering, including the study of chemical reaction networks, climate modeling, and finance. Further research in this area is ongoing, with the aim of developing new techniques and applications for the theory of central manifolds in the context of SDEs.

Yes, there has been significant work on the theory of central manifolds for stochastic differential equations (SDEs). Central manifolds are important in the study of SDEs because they provide a reduced-dimensional representation of the dynamics of the system near an equilibrium point. This can simplify the analysis of the system and provide insights into its behavior.

The theory of central manifolds for deterministic systems has been well-established for some time. However, the extension of this theory to SDEs has been an active area of research in recent years. Some of the key contributions in this area include the work of Kloeden and Neuenkirch on central manifolds for SDEs with additive noise, and the work of Pavliotis and Stuart on central manifolds for SDEs with multiplicative noise.

The theory of central manifolds for SDEs has important applications in many areas of science and engineering, including the study of chemical reaction networks, climate modeling, and finance. Further research in this area is ongoing, with the aim of developing new techniques and applications for the theory of central manifolds in the context of SDEs.

Phil Hartman publish his seminal work in the late 1950 on the existence of stable and unstable manifolds. Concurrently Soviet mathematician D. M. Grobman published similar results independently. The results became known as the Hartman-Grobman theorem. This answered the question of topological conjugacy for hyperbolic stable points. When Charles Pugh, a student of Hartman, started working with Smale they reformulated the concept as the geometry of an integrated diffeomorphism on a manifold. This was the next logical step since in Hartman's proof he transformed the problem to one of an integrated local diffeomorphism on Euclidean space. Then Smale, Pugh and their students looked a non-hyperbolic stable points. This gave rise to the central manifold. However, while the stable and unstable manifolds are "stable" under "small" perturbations similar to exponential dichotomies for ODE's, the central manifold is not. This leads to the concept of bifurcations. There has been a lot of work to expand this theory to stochastic dynamical systems.

For example:

Article A Stochastic Version of Center Manifold Theory

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Article Limiting behavior of center manifolds for stochastic evoluti...

As Smale and Pugh determined the appropriate abstract setting is the study of such equations on a manifold where the invariant manifolds translate to invariant foliations of the underlying manifold. The following is along those lines.

https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=8362&context=etd