Print Email Facebook Twitter Hamilton–Jacobi equations for controlled gradient flows Title Hamilton–Jacobi equations for controlled gradient flows: The comparison principle Author Conforti, G. (Route de Saclay) Kraaij, R.C. (TU Delft Delft Institute of Applied Mathematics; TU Delft Applied Probability) Tonon, D. (Università degli Studi di Padova) Date 2023 Abstract Motivated by recent developments in the fields of large deviations for interacting particle systems and mean field control, we establish a comparison principle for the Hamilton–Jacobi equation corresponding to linearly controlled gradient flows of an energy function E defined on a metric space (E,d). Our analysis is based on a systematic use of the regularizing properties of gradient flows in evolutional variational inequality (EVI) formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian at hand and, in combination with the use of the Tataru's distance, for establishing the key estimates needed to bound the difference of the Hamiltonians in the proof of the comparison principle. Our abstract results apply to a large class of examples only partially covered by the existing theory, including gradient flows on Hilbert spaces and the Wasserstein space equipped with a displacement convex energy functional E satisfying McCann's condition. Subject Comparison principleEVI gradient flowsHamilton Jacobi equationsOptimal transportTataru distance To reference this document use: http://resolver.tudelft.nl/uuid:f9b2bc59-52c7-40f2-bfd1-b1cd69b9af61 DOI https://doi.org/10.1016/j.jfa.2023.109853 Embargo date 2023-07-01 ISSN 0022-1236 Source Journal of Functional Analysis, 284 (9) Bibliographical note Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public. Part of collection Institutional Repository Document type journal article Rights © 2023 G. Conforti, R.C. Kraaij, D. Tonon Files PDF 1_s2.0_S0022123623000101_main.pdf 942 KB Close viewer /islandora/object/uuid:f9b2bc59-52c7-40f2-bfd1-b1cd69b9af61/datastream/OBJ/view