A note on an integration by parts formula for the generators of uniform translations on configuration space

TL;DR

This paper derives an integration by parts formula for translation generators on configuration spaces with Gibbs measures, showing it holds under certain conditions like high temperature and low intensity regimes.

Contribution

It introduces a new integration by parts formula for translation operators on configuration spaces with Gibbs measures, linking measure invariance to the validity of the formula.

Findings

The formula holds if and only if translation invariance is preserved.

In high temperature and low intensity regimes, the formula is automatically valid.

The work connects measure invariance with the applicability of the integration by parts formula.

Abstract

An integration by parts formula is derived for the first order differential operator corresponding to the action of translations on the space of locally finite simple configurations of infinitely many points on R^d. As reference measures, tempered grand canonical Gibbs measures are considered corresponding to a non-constant non-smooth intensity (one-body potential) and translation invariant potentials fulfilling the usual conditions. It is proven that such Gibbs measures fulfill the intuitive integration by parts formula if and only if the action of the translation is not broken for this particular measure. The latter is automatically fulfilled in the high temperature and low intensity regime.