# Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

## TL;DR

This paper extends geometric control concepts of scaling and saturation to infinite-dimensional systems, applying them to nonlinear PDEs and SPDEs to analyze control, support properties, and ergodic behavior of stochastic systems.

## Contribution

It introduces the dual notions of scaling and saturation in an infinite-dimensional context and applies them to control and stochastic analysis of PDEs and SPDEs.

## Key findings

- Established dual notions of scaling and saturation in infinite dimensions.
- Analyzed support properties of probability laws for SPDEs.
- Provided insights into ergodic and mixing properties of invariant measures.

## Abstract

We establish the dual notions of scaling and saturation from geometric control theory in an infinite-dimensional setting. This generalization is applied to the low-mode control problem in a number of concrete nonlinear partial differential equations. We also develop applications concerning associated classes of stochastic partial differential equations (SPDEs). In particular, we study the support properties of probability laws corresponding to these SPDEs as well as provide applications concerning the ergodic and mixing properties of invariant measures for these stochastic systems.