\Gamma-convergence of Onsager-Machlup functionals. Part II: Infinite product measures on Banach spaces

TL;DR

This paper develops the theory of Onsager-Machlup functionals for infinite product measures on Banach spaces, establishing convergence properties crucial for Bayesian inverse problems and transition path analysis.

Contribution

It extends Onsager-Machlup functional analysis to infinite-dimensional product measures on Banach spaces, including Gaussian, Cauchy, and Besov measures.

Findings

Established $ ext{Γ}$-convergence of Onsager-Machlup functionals for product measures.

Proved equicoercivity and convergence results for sequences of measures.

Provided a framework for analyzing MAP estimators and transition paths in infinite dimensions.

Abstract

We derive Onsager-Machlup functionals for countable product measures on weighted $\ell^p$ subspaces of the sequence space $\mathbb{R}^{\mathbb{N}}$. Each measure in the product is a shifted and scaled copy of a reference probability measure on $\mathbb{R}$ that admits a sufficiently regular Lebesgue density. We study the equicoercivity and $\Gamma$-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter $1 \leq p \leq 2$. Together with Part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.