{"oa_version":"Published Version","issue":"5","scopus_import":"1","department":[{"_id":"JaMa"}],"doi":"10.1112/jlms.70003","type":"journal_article","month":"11","acknowledgement":"The authors are grateful to Masha Gordina for helpful references, and to Nathanaël Berestycki, Baptiste Cerclé, and Ewain Gwynne for valuable comments on the first circulated version of this paper. They also would like to thank Sebastian Andres, Peter Friz, and Yizheng Yuan for pointing out an erroneous formulation in the previous version of Theorem 5.7. Moreover, KTS would liketo express his thanks to Sebastian Andres, Matthias Erbar, Martin Huesmann, and Jan Mass for stimulating discussions on previous attempts to this project. LDS gratefully acknowledges financial support from the European Research Council (grant agreement No 716117, awarded to J. Maas), from the Austrian Science Fund (FWF) project 10.55776/ESP208, and from the Austrian Science Fund (FWF) project 10.55776/F65.RH, EK, and KTS gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft through the project “Random Riemannian Geometry” within the SPP 2265 “Random Geomet-ric Systems,” through the Hausdorff Center for Mathematics (project ID 390685813), and through project B03 within the CRC 1060 (project ID 211504053). RH and KTS also gratefully acknowledge financial support from the European Research Council through the ERC AdG “RicciBounds”(grant agreement 694405).Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. Open access funding enabled and organized by Projekt DEAL.","volume":110,"_id":"18490","ec_funded":1,"OA_type":"hybrid","file":[{"date_updated":"2024-11-04T08:54:26Z","file_name":"2024_JourLondonMathSoc_Schiavo.pdf","file_size":911476,"success":1,"access_level":"open_access","date_created":"2024-11-04T08:54:26Z","checksum":"143816823b5f43bd3748da8e3e91cef5","file_id":"18497","relation":"main_file","creator":"dernst","content_type":"application/pdf"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"ieee":"L. Dello Schiavo, R. Herry, E. Kopfer, and K. T. Sturm, “Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension,” <i>Journal of the London Mathematical Society</i>, vol. 110, no. 5. London Mathematical Society, 2024.","ama":"Dello Schiavo L, Herry R, Kopfer E, Sturm KT. Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension. <i>Journal of the London Mathematical Society</i>. 2024;110(5). doi:<a href=\"https://doi.org/10.1112/jlms.70003\">10.1112/jlms.70003</a>","short":"L. Dello Schiavo, R. Herry, E. Kopfer, K.T. Sturm, Journal of the London Mathematical Society 110 (2024).","mla":"Dello Schiavo, Lorenzo, et al. “Conformally Invariant Random Fields, Liouville Quantum Gravity Measures, and Random Paneitz Operators on Riemannian Manifolds of Even Dimension.” <i>Journal of the London Mathematical Society</i>, vol. 110, no. 5, e70003, London Mathematical Society, 2024, doi:<a href=\"https://doi.org/10.1112/jlms.70003\">10.1112/jlms.70003</a>.","apa":"Dello Schiavo, L., Herry, R., Kopfer, E., &#38; Sturm, K. T. (2024). Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension. <i>Journal of the London Mathematical Society</i>. London Mathematical Society. <a href=\"https://doi.org/10.1112/jlms.70003\">https://doi.org/10.1112/jlms.70003</a>","ista":"Dello Schiavo L, Herry R, Kopfer E, Sturm KT. 2024. Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension. Journal of the London Mathematical Society. 110(5), e70003.","chicago":"Dello Schiavo, Lorenzo, Ronan Herry, Eva Kopfer, and Karl Theodor Sturm. “Conformally Invariant Random Fields, Liouville Quantum Gravity Measures, and Random Paneitz Operators on Riemannian Manifolds of Even Dimension.” <i>Journal of the London Mathematical Society</i>. London Mathematical Society, 2024. <a href=\"https://doi.org/10.1112/jlms.70003\">https://doi.org/10.1112/jlms.70003</a>."},"year":"2024","article_number":"e70003","day":"01","date_updated":"2024-11-04T08:59:19Z","OA_place":"publisher","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"For large classes of even-dimensional Riemannian manifolds (Formula presented.), we construct and analyze conformally invariant random fields. These centered Gaussian fields (Formula presented.), called co-polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: (Formula presented.). They share a fundamental quasi-invariance property under conformal transformations. In terms of the co-polyharmonic Gaussian field (Formula presented.), we define the Liouville Quantum Gravity measure, a random measure on (Formula presented.), heuristically given as (Formula presented.) and rigorously obtained as almost sure weak limit of the right-hand side with (Formula presented.) replaced by suitable regular approximations (Formula presented.). In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on (Formula presented.) and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's (Formula presented.) -curvature: we give a rigorous meaning to the Polyakov–Liouville measure (Formula presented.) and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2-dimensional Riemannian manifolds, all compact non-negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even-dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel (Formula presented.) rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds. "}],"publication":"Journal of the London Mathematical Society","article_type":"original","quality_controlled":"1","license":"https://creativecommons.org/licenses/by/4.0/","publication_identifier":{"eissn":["1469-7750"],"issn":["0024-6107"]},"intvolume":"       110","article_processing_charge":"Yes (via OA deal)","author":[{"first_name":"Lorenzo","id":"ECEBF480-9E4F-11EA-B557-B0823DDC885E","full_name":"Dello Schiavo, Lorenzo","last_name":"Dello Schiavo","orcid":"0000-0002-9881-6870"},{"first_name":"Ronan","last_name":"Herry","full_name":"Herry, Ronan"},{"full_name":"Kopfer, Eva","last_name":"Kopfer","first_name":"Eva"},{"first_name":"Karl Theodor","last_name":"Sturm","full_name":"Sturm, Karl Theodor"}],"project":[{"_id":"256E75B8-B435-11E9-9278-68D0E5697425","name":"Optimal Transport and Stochastic Dynamics","grant_number":"716117","call_identifier":"H2020"},{"grant_number":"E208","name":"Configuration Spaces over Non-Smooth Spaces","_id":"34dbf174-11ca-11ed-8bc3-afe9d43d4b9c"},{"_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems","grant_number":"F6504"}],"status":"public","oa":1,"has_accepted_license":"1","file_date_updated":"2024-11-04T08:54:26Z","publisher":"London Mathematical Society","date_created":"2024-11-03T23:01:44Z","ddc":["510"],"publication_status":"published","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"title":"Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension","date_published":"2024-11-01T00:00:00Z"}