{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T14:59:41Z","timestamp":1776869981357,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"303","license":[{"start":{"date-parts":[[2017,5,25]],"date-time":"2017-05-25T00:00:00Z","timestamp":1495670400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We prove that a finite-difference centered approximation for the Kolmogorov equation in the whole space preserves the decay properties of continuous solutions as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t right-arrow normal infinity\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t \\to \\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , independently of the mesh-size parameters. This is a manifestation of the property of numerical hypo-coercivity, and it holds both for semi-discrete and fully discrete approximations. The method of proof is based on the energy methods developed by Herau and Villani, employing well-balanced Lyapunov functionals mixing different energies, suitably weighted and equilibrated by multiplicative powers in time. The decreasing character of this Lyapunov functional leads to the optimal decay of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -norms of solutions and partial derivatives, which are of different order because of the anisotropy of the model.\n                  <\/p>","DOI":"10.1090\/mcom\/3157","type":"journal-article","created":{"date-parts":[[2016,3,16]],"date-time":"2016-03-16T15:43:06Z","timestamp":1458142986000},"page":"97-119","source":"Crossref","is-referenced-by-count":16,"title":["Numerical hypocoercivity for the Kolmogorov equation"],"prefix":"10.1090","volume":"86","author":[{"given":"Alessio","family":"Porretta","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrique","family":"Zuazua","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,5,25]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s00205-010-0321-y","article-title":"Large time asymptotics for partially dissipative hyperbolic systems","volume":"199","author":"Beauchard, Karine","year":"2011","journal-title":"Arch. Ration. Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1006\/jfan.1993.1011","article-title":"Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions","volume":"111","author":"Bouchut, Fran\u00e7ois","year":"1993","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"issue":"11","key":"3","doi-asserted-by":"publisher","first-page":"985","DOI":"10.1002\/(SICI)1099-1476(19980725)21:11<985::AID-MMA919>3.0.CO;2-B","article-title":"Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation","volume":"21","author":"Carpio, Ana","year":"1998","journal-title":"Math. Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0170-4214","issn-type":"print"},{"key":"4","unstructured":"E. L. Foster, J. Loh\u00e9ac, and M.-B. Tran, A structure preserving scheme for the Kolmogorov equation, preprint 2014 (arXiv:1411.1019v3)."},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/j.jfa.2006.11.013","article-title":"Short and long time behavior of the Fokker-Planck equation in a confining potential and applications","volume":"244","author":"H\u00e9rau, Fr\u00e9d\u00e9ric","year":"2007","journal-title":"J. Funct. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1236","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s00205-003-0276-3","article-title":"Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential","volume":"171","author":"H\u00e9rau, Fr\u00e9d\u00e9ric","year":"2004","journal-title":"Arch. Ration. Mech. 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Nauk SSSR","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"key":"10","first-page":"466","article-title":"On the equations of Brownian motion","volume":"9","author":"Il\u2032in, A. M.","year":"1964","journal-title":"Teor. Verojatnost. i Primenen."},{"issue":"1","key":"11","doi-asserted-by":"publisher","first-page":"116","DOI":"10.2307\/1968123","article-title":"Zuf\u00e4llige Bewegungen (zur Theorie der Brownschen Bewegung)","volume":"35","author":"Kolmogoroff, A.","year":"1934","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"12","unstructured":"J. Loh\u00e9ac, A splitting method for Kolmogorov-type equations, in preparation."},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"1571","DOI":"10.1007\/s10208-014-9232-x","article-title":"Propagation of 1D waves in regular discrete heterogeneous media: a Wigner measure approach","volume":"15","author":"Marica, Aurora","year":"2015","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"key":"14","unstructured":"A. Porretta, A note on the Sobolev and Gagliardo-Nirenberg inequality when \ud835\udc5d>\ud835\udc41, in preparation."},{"issue":"950","key":"15","isbn-type":"print","doi-asserted-by":"publisher","first-page":"iv+141","DOI":"10.1090\/S0065-9266-09-00567-5","article-title":"Hypocoercivity","volume":"202","author":"Villani, C\u00e9dric","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821844984","journal-title":"Mem. Amer. Math. 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