Infinite-dimensional Gaussian change of variables’ formula

Abstract

In this paper, we study the Banach space \(\ell _{\infty }\) of the bounded real sequences, and a measure \(N(a,\Gamma )\) over \(\left( \textbf{R}^{\infty },\mathcal {B}^{\infty }\right) \) analogous to the finite-dimensional Gaussian law. The main result of our paper is a change of variables’ formula for the integration, with respect to \(N(a,\Gamma )\), of the measurable real functions on \(\left( E_{\infty },\mathcal {B}^{\infty }\left( E_{\infty }\right) \right) \), where \(E_{\infty }\) is the separable Banach space of the convergent real sequences. This change of variables is given by some \(\left( m,\sigma \right) \) functions, defined over a subset of \(E_{\infty }\), with values on \(E_{\infty }\), with properties that generalize the analogous ones of the finite-dimensional diffeomorphisms.