{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T02:06:56Z","timestamp":1775527616052,"version":"3.50.1"},"reference-count":39,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T00:00:00Z","timestamp":1762473600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,3]]},"abstract":"Abstract<\/jats:title>\n \n In this paper we investigate large-scale linear systems driven by a fractional Brownian motion (fBm) with Hurst parameter\n \n \n \n $H\\in [1\/2, 1)$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We interpret these equations either in the sense of Young (\n \n \n \n $H>1\/2$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ) or Stratonovich (\n \n \n \n $H=1\/2$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ). In particular, fractional Young differential equations are well suited to modeling real-world phenomena as they capture memory effects, unlike other frameworks. Although it is very complex to solve them in high dimensions, model reduction schemes for Young or Stratonovich settings have not yet been much studied. To address this gap, we analyze important features of fundamental solutions associated with the underlying systems. We prove a weak type of semigroup property which is the foundation of studying system Gramians. From the Gramians introduced, a dominant subspace can be identified, which is shown in this paper as well. The difficulty for fractional drivers with\n \n \n \n $H>1\/2$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is that there is no link between the corresponding Gramians and algebraic equations, making the computation very difficult. Therefore we further propose empirical Gramians that can be learned from simulation data. Subsequently, we introduce projection-based reduced-order models using the dominant subspace information. We point out that such projections are not always optimal for Stratonovich equations, as stability might not be preserved and since the error might be larger than expected. Therefore an improved reduced-order model is proposed for\n \n \n \n $H=1\/2$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We validate our techniques conducting numerical experiments on some large-scale stochastic differential equations driven by fBm resulting from spatial discretizations of fractional stochastic PDEs. Overall, our study provides useful insights into the applicability and effectiveness of reduced-order methods for stochastic systems with fractional noise, which can potentially aid in the development of more efficient computational strategies for practical applications.\n <\/jats:p>","DOI":"10.1017\/jpr.2025.10039","type":"journal-article","created":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T06:53:57Z","timestamp":1762498437000},"page":"401-432","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["(Empirical) Gramian-based dimension reduction for stochastic differential equations driven by fractional Brownian motion"],"prefix":"10.1017","volume":"63","author":[{"given":"Nahid","family":"Jamshidi","sequence":"first","affiliation":[{"name":"Martin Luther University Halle-Wittenberg"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Redmann","sequence":"additional","affiliation":[{"name":"University of Rostock"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,11,7]]},"reference":[{"key":"S0021900225100399_ref21","first-page":"219","article-title":"Fat tails, long memory, and the stock market since the 1960\u2019s","volume":"26","author":"Lo","year":"1997","journal-title":"Economic Notes"},{"key":"S0021900225100399_ref17","doi-asserted-by":"publisher","DOI":"10.1109\/MIC.2004.46"},{"key":"S0021900225100399_ref35","unstructured":"[35] Shmatkov, A. (2005). Rate of convergence of Wong\u2013Zakai approximations for SDEs and SPDEs. PhD thesis, The University of Edinburgh."},{"key":"S0021900225100399_ref15","doi-asserted-by":"publisher","DOI":"10.1214\/15-AAP1114"},{"key":"S0021900225100399_ref37","first-page":"4325","article-title":"Long-memory processes in global ozone and temperature variations","volume":"6","author":"Varotsos","year":"2006","journal-title":"Atmos. Chem. Phys. Discuss."},{"key":"S0021900225100399_ref2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0510673103"},{"key":"S0021900225100399_ref5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974829"},{"key":"S0021900225100399_ref3","volume-title":"Vol. 6 of Adv. Des. 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Invited paper at the American Geophysical Union 1998 Fall Meeting."},{"key":"S0021900225100399_ref31","doi-asserted-by":"publisher","DOI":"10.1007\/s00498-022-00328-z"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225100399","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:26:07Z","timestamp":1775525167000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225100399\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,7]]},"references-count":39,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0021900225100399"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.10039","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,7]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}