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2021 The Φ34 measure via Girsanov’s theorem
Nikolay Barashkov, Massimiliano Gubinelli
Author Affiliations +
Electron. J. Probab. 26: 1-29 (2021). DOI: 10.1214/21-EJP635


We construct the Φ34 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 Φ34 measure is singular wrt. the Gaussian free field.

Funding Statement

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 Φ34. 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 (


Download Citation

Nikolay Barashkov. Massimiliano Gubinelli. "The Φ34 measure via Girsanov’s theorem." Electron. J. Probab. 26 1 - 29, 2021.


Received: 20 May 2020; Accepted: 27 April 2021; Published: 2021
First available in Project Euclid: 4 June 2021

Digital Object Identifier: 10.1214/21-EJP635

Primary: 81T08
Secondary: 60H30, 60L40


Vol.26 • 2021
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