We construct the measure on a periodic three dimensional box as an absolutely continuous perturbation of a random translation of the Gaussian free field. The shifted measure is constructed via Girsanov’s theorem and the relevant filtration is the one generated by a scale parameter. As a byproduct we give a self-contained proof that the measure is singular wrt. the Gaussian free field.
This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the Hausdorff Center for Mathematics under Germany’s Excellence Strategy – EXC-2047/1 – 390685813 and through CRC 1060 - project number 211504053 and by EPSRC via Grant Number EP/R014604/1.
M.G. would like to thank S. Albeverio, D. Brydges, C. Garban and M. Hairer for interesting discussions on the topic of singularity of . N.B would like to thank B. Bringmann for some helpful comments and for pointing out a mistake in an earlier version of the paper. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program SRQ: Scaling limits, Rough paths, Quantum field theory during which part of the work on this paper was undertaken. This paper has been written with TeXmacs (www.texmacs.org).
"The measure via Girsanov’s theorem." Electron. J. Probab. 26 1 - 29, 2021. https://doi.org/10.1214/21-EJP635