Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise
We provide a complete elaboration of the L 2 -Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogoro...
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Veröffentlicht in: | Journal of evolution equations 2022-03, Vol.22 (1), Article 11 |
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creator | Bertram, Alexander Grothaus, Martin |
description | We provide a complete elaboration of the
L
2
-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way. |
doi_str_mv | 10.1007/s00028-022-00773-y |
format | Article |
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L
2
-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.</description><identifier>ISSN: 1424-3199</identifier><identifier>EISSN: 1424-3202</identifier><identifier>DOI: 10.1007/s00028-022-00773-y</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Analysis ; Cauchy problems ; Differential equations ; Diffusion ; Diffusion coefficient ; Dirichlet problem ; Hilbert space ; Martingales ; Mathematics ; Mathematics and Statistics ; Noise ; Operators (mathematics) ; Perturbation theory ; Semigroups</subject><ispartof>Journal of evolution equations, 2022-03, Vol.22 (1), Article 11</ispartof><rights>The Author(s) 2022. corrected publication 2022</rights><rights>The Author(s) 2022. corrected publication 2022. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c363t-e66977f0a8bab1e0383d234662a2f44857938380a41daac13c82a67d20de43a13</citedby><cites>FETCH-LOGICAL-c363t-e66977f0a8bab1e0383d234662a2f44857938380a41daac13c82a67d20de43a13</cites><orcidid>0000-0002-9609-6996 ; 0000-0002-6560-8223</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00028-022-00773-y$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00028-022-00773-y$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Bertram, Alexander</creatorcontrib><creatorcontrib>Grothaus, Martin</creatorcontrib><title>Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise</title><title>Journal of evolution equations</title><addtitle>J. Evol. Equ</addtitle><description>We provide a complete elaboration of the
L
2
-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.</description><subject>Analysis</subject><subject>Cauchy problems</subject><subject>Differential equations</subject><subject>Diffusion</subject><subject>Diffusion coefficient</subject><subject>Dirichlet problem</subject><subject>Hilbert space</subject><subject>Martingales</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Noise</subject><subject>Operators (mathematics)</subject><subject>Perturbation theory</subject><subject>Semigroups</subject><issn>1424-3199</issn><issn>1424-3202</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kMtOwzAQRS0EEuXxA6wssTaM7cROl6gqD6kSG1ixsKaOQ13lhZ0W5e9xCYgdq5m5umdGcwm54nDDAfRtBABRMBCCpVFLNh6RGc9ExqQAcfzb8_n8lJzFuAXgOi_yGXlbxujawWNNG1b6GH2Pg9_7YaTYlnQz9p3tXLCT1FV0he272_uWlmOLjbeRfvphQ5tdPfi-9vZAO9p2ProLclJhHd3lTz0nr_fLl8UjWz0_PC3uVsxKJQfmlJprXQEWa1xzB7KQpZCZUgJFlWVFrudJKgAzXiJaLm0hUOlSQOkyiVyek-tpbx-6j52Lg9l2u9Cmk0YomYs8V1Inl5hcNnQxBleZPvgGw2g4mEOIZgrRpBDNd4hmTJCcoJjM6fHwt_of6gtWRHZa</recordid><startdate>20220301</startdate><enddate>20220301</enddate><creator>Bertram, Alexander</creator><creator>Grothaus, Martin</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-9609-6996</orcidid><orcidid>https://orcid.org/0000-0002-6560-8223</orcidid></search><sort><creationdate>20220301</creationdate><title>Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise</title><author>Bertram, Alexander ; Grothaus, Martin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c363t-e66977f0a8bab1e0383d234662a2f44857938380a41daac13c82a67d20de43a13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Analysis</topic><topic>Cauchy problems</topic><topic>Differential equations</topic><topic>Diffusion</topic><topic>Diffusion coefficient</topic><topic>Dirichlet problem</topic><topic>Hilbert space</topic><topic>Martingales</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Noise</topic><topic>Operators (mathematics)</topic><topic>Perturbation theory</topic><topic>Semigroups</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bertram, Alexander</creatorcontrib><creatorcontrib>Grothaus, Martin</creatorcontrib><collection>Springer Open Access</collection><collection>CrossRef</collection><jtitle>Journal of evolution equations</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bertram, Alexander</au><au>Grothaus, Martin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise</atitle><jtitle>Journal of evolution equations</jtitle><stitle>J. Evol. Equ</stitle><date>2022-03-01</date><risdate>2022</risdate><volume>22</volume><issue>1</issue><artnum>11</artnum><issn>1424-3199</issn><eissn>1424-3202</eissn><abstract>We provide a complete elaboration of the
L
2
-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00028-022-00773-y</doi><orcidid>https://orcid.org/0000-0002-9609-6996</orcidid><orcidid>https://orcid.org/0000-0002-6560-8223</orcidid><oa>free_for_read</oa></addata></record> |
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subjects | Analysis Cauchy problems Differential equations Diffusion Diffusion coefficient Dirichlet problem Hilbert space Martingales Mathematics Mathematics and Statistics Noise Operators (mathematics) Perturbation theory Semigroups |
title | Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise |
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