Talagrand Concentration Inequalities for Stochastic Partial Differential Equations

TL;DR

This paper extends Talagrand concentration inequalities, which relate Wasserstein distances and relative entropy, to stochastic partial differential equations, broadening the scope of measure concentration results.

Contribution

It develops a new theoretical framework for Talagrand inequalities applicable to stochastic PDEs, expanding measure concentration theory beyond stochastic differential equations.

Findings

Established transportation-information inequalities for certain stochastic PDEs

Extended measure concentration results to infinite-dimensional settings

Provided foundational tools for analyzing stochastic PDEs using concentration inequalities

Abstract

One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely continuous measure with finite relative entropy. Such transportation-information inequalities were recently established for some stochastic differential equations. Here, we develop a similar theory for some stochastic partial differential equations.