Reduction of the Pauli–Fierz double stochastic integral to an ordinary integral

Develop an Itô-formula or related reduction of the path-dependent stochastic double integral arising in the full Pauli–Fierz model to an ordinary Riemann integral, in order to facilitate control of its exponential moments and the analysis of ground-state spatial decay.

Background

The Feynman–Kac representation of the full Pauli–Fierz model produces a path-dependent double stochastic integral whose integrand is not adapted in the sense required for ordinary Itô stochastic integration. The paper develops exponential-moment estimates by decomposing the expression into martingale and non-martingale components and applying exponential-martingale methods, Jensen’s inequality, and auxiliary bounds.

A direct reduction to an ordinary Riemann integral would provide a more transparent way to control this stochastic quantity. The authors state that such a reduction is unavailable for the full Pauli–Fierz model they study, identifying the development of such a technique as an unresolved methodological problem relevant to obtaining sharper Agmon-distance lower bounds.