[{"status":"public","author":[{"id":"ECEBF480-9E4F-11EA-B557-B0823DDC885E","first_name":"Lorenzo","orcid":"0000-0002-9881-6870","last_name":"Dello Schiavo","full_name":"Dello Schiavo, Lorenzo"},{"last_name":"Kopfer","full_name":"Kopfer, Eva","first_name":"Eva"},{"first_name":"Karl Theodor","last_name":"Sturm","full_name":"Sturm, Karl Theodor"}],"abstract":[{"text":"We study random perturbations of a Riemannian manifold (M, g) by means of so-called\r\nFractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields\r\nh• : ω \u0002→ hω will act on the manifold via the conformal transformation g \u0002→ gω := e2hω g.\r\nOur focus will be on the regular case with Hurst parameter H > 0, the critical case H = 0\r\nbeing the celebrated Liouville geometry in two dimensions. We want to understand how basic\r\ngeometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian\r\nmotion, spectral bound, or spectral gap change under the influence of the noise. And if so, is\r\nit possible to quantify these dependencies in terms of key parameters of the noise? Another\r\ngoal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian\r\nmanifold, a fascinating object of independent interest.","lang":"eng"}],"type":"journal_article","publication_status":"epub_ahead","citation":{"chicago":"Dello Schiavo, Lorenzo, Eva Kopfer, and Karl Theodor Sturm. “A Discovery Tour in Random Riemannian Geometry.” <i>Potential Analysis</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s11118-023-10118-0\">https://doi.org/10.1007/s11118-023-10118-0</a>.","ieee":"L. Dello Schiavo, E. Kopfer, and K. T. Sturm, “A discovery tour in random Riemannian geometry,” <i>Potential Analysis</i>. Springer Nature, 2024.","apa":"Dello Schiavo, L., Kopfer, E., &#38; Sturm, K. T. (2024). A discovery tour in random Riemannian geometry. <i>Potential Analysis</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11118-023-10118-0\">https://doi.org/10.1007/s11118-023-10118-0</a>","ista":"Dello Schiavo L, Kopfer E, Sturm KT. 2024. A discovery tour in random Riemannian geometry. Potential Analysis.","short":"L. Dello Schiavo, E. Kopfer, K.T. Sturm, Potential Analysis (2024).","mla":"Dello Schiavo, Lorenzo, et al. “A Discovery Tour in Random Riemannian Geometry.” <i>Potential Analysis</i>, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s11118-023-10118-0\">10.1007/s11118-023-10118-0</a>.","ama":"Dello Schiavo L, Kopfer E, Sturm KT. A discovery tour in random Riemannian geometry. <i>Potential Analysis</i>. 2024. doi:<a href=\"https://doi.org/10.1007/s11118-023-10118-0\">10.1007/s11118-023-10118-0</a>"},"day":"26","acknowledgement":"The authors would like to thank Matthias Erbar and Ronan Herry for valuable discussions on this project. They are also grateful to Nathanaël Berestycki, and Fabrice Baudoin for respectively pointing out the references [7], and [6, 24], and to Julien Fageot and Thomas Letendre for pointing out a mistake in a previous version of the proof of Proposition 3.10. The authors feel very much indebted to an anonymous reviewer for his/her careful reading and the many valuable suggestions that have significantly contributed to the improvement of the paper. L.D.S. gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft through CRC 1060 as well as through SPP 2265, and by the Austrian Science Fund (FWF) grant F65 at Institute of Science and Technology Austria. This research was funded in whole or in part by the Austrian Science Fund (FWF) ESPRIT 208. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission. E.K. and K.-T.S. gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft through the Hausdorff Center for Mathematics and through CRC 1060 as well as through SPP 2265.\r\nOpen Access funding enabled and organized by Projekt DEAL.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504"}],"quality_controlled":"1","oa_version":"Published Version","_id":"14934","publication_identifier":{"eissn":["1572-929X"],"issn":["0926-2601"]},"date_updated":"2024-02-05T13:04:23Z","oa":1,"article_processing_charge":"Yes (via OA deal)","publication":"Potential Analysis","language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1","title":"A discovery tour in random Riemannian geometry","article_type":"original","date_published":"2024-01-26T00:00:00Z","doi":"10.1007/s11118-023-10118-0","year":"2024","month":"01","date_created":"2024-02-04T23:00:54Z","department":[{"_id":"JaMa"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s11118-023-10118-0"}]},{"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"03","date_published":"2023-03-01T00:00:00Z","article_type":"original","file":[{"file_name":"2023_PotentialAnalysis_DelloSchiavo.pdf","file_size":806391,"date_created":"2023-10-04T09:18:59Z","checksum":"625526482be300ca7281c91c30d41725","date_updated":"2023-10-04T09:18:59Z","access_level":"open_access","success":1,"content_type":"application/pdf","relation":"main_file","file_id":"14387","creator":"dernst"}],"date_created":"2021-10-17T22:01:17Z","has_accepted_license":"1","department":[{"_id":"JaMa"}],"intvolume":"        58","status":"public","day":"01","type":"journal_article","publication":"Potential Analysis","file_date_updated":"2023-10-04T09:18:59Z","page":"573-615","external_id":{"arxiv":["2003.01366"],"isi":["000704213400001"]},"title":"Ergodic decomposition of Dirichlet forms via direct integrals and applications","doi":"10.1007/s11118-021-09951-y","year":"2023","ec_funded":1,"ddc":["510"],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"isi":1,"abstract":[{"text":"We study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.","lang":"eng"}],"author":[{"full_name":"Dello Schiavo, Lorenzo","last_name":"Dello Schiavo","orcid":"0000-0002-9881-6870","first_name":"Lorenzo","id":"ECEBF480-9E4F-11EA-B557-B0823DDC885E"}],"publication_status":"published","citation":{"chicago":"Dello Schiavo, Lorenzo. “Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications.” <i>Potential Analysis</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s11118-021-09951-y\">https://doi.org/10.1007/s11118-021-09951-y</a>.","apa":"Dello Schiavo, L. (2023). Ergodic decomposition of Dirichlet forms via direct integrals and applications. <i>Potential Analysis</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11118-021-09951-y\">https://doi.org/10.1007/s11118-021-09951-y</a>","ieee":"L. Dello Schiavo, “Ergodic decomposition of Dirichlet forms via direct integrals and applications,” <i>Potential Analysis</i>, vol. 58. Springer Nature, pp. 573–615, 2023.","ista":"Dello Schiavo L. 2023. Ergodic decomposition of Dirichlet forms via direct integrals and applications. Potential Analysis. 58, 573–615.","short":"L. Dello Schiavo, Potential Analysis 58 (2023) 573–615.","mla":"Dello Schiavo, Lorenzo. “Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications.” <i>Potential Analysis</i>, vol. 58, Springer Nature, 2023, pp. 573–615, doi:<a href=\"https://doi.org/10.1007/s11118-021-09951-y\">10.1007/s11118-021-09951-y</a>.","ama":"Dello Schiavo L. Ergodic decomposition of Dirichlet forms via direct integrals and applications. <i>Potential Analysis</i>. 2023;58:573-615. doi:<a href=\"https://doi.org/10.1007/s11118-021-09951-y\">10.1007/s11118-021-09951-y</a>"},"_id":"10145","publication_identifier":{"eissn":["1572-929X"],"issn":["0926-2601"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"The author is grateful to Professors Sergio Albeverio and Andreas Eberle, and to Dr. Kohei Suzuki, for fruitful conversations on the subject of the present work, and for respectively pointing out the references [1, 13], and [3, 20]. Finally, he is especially grateful to an anonymous Reviewer for their very careful reading and their suggestions which improved the readability of the paper.","project":[{"name":"IST Austria Open Access Fund","_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854"},{"grant_number":"F6504","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","name":"Taming Complexity in Partial Differential Systems"},{"grant_number":"716117","name":"Optimal Transport and Stochastic Dynamics","_id":"256E75B8-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"oa_version":"Published Version","quality_controlled":"1","arxiv":1,"volume":58,"oa":1,"date_updated":"2023-10-04T09:19:12Z","article_processing_charge":"Yes (via OA deal)"}]