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The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations

Yue Wu Orcid Logo, Chenggui Yuan Orcid Logo

Journal of Theoretical Probability, Volume: 36, Issue: 1

Swansea University Author: Chenggui Yuan Orcid Logo

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DOI (Published version): 10.1007/s10959-023-01236-x

Abstract

In this paper, we study the numerical method for approximating the random periodic solution of semilinear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating infinite-dimensional objects. We first show the existence and uniq...

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Published in: Journal of Theoretical Probability
ISSN: 0894-9840 1572-9230
Published: Springer Science and Business Media LLC 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa62486
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spelling 2023-03-02T14:37:30.7093064 v2 62486 2023-02-03 The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations 22b571d1cba717a58e526805bd9abea0 0000-0003-0486-5450 Chenggui Yuan Chenggui Yuan true false 2023-02-03 SMA In this paper, we study the numerical method for approximating the random periodic solution of semilinear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating infinite-dimensional objects. We first show the existence and uniqueness of the random periodic solution to the equation as the limit of the pull-back flows of the equation, and observe that its mild form is well defined in the intersection of a family of decreasing Hilbert spaces. Then, we propose a Galerkin-type exponential integrator scheme and establish its convergence rate of the strong error to the mild solution, where the order of convergence directly depends on the space (among the family of Hilbert spaces) for the initial point to live. We finally conclude with a best order of convergence that is arbitrarily close to 0.5. Journal Article Journal of Theoretical Probability 36 1 Springer Science and Business Media LLC 0894-9840 1572-9230 Random periodic solution; Stochastic evolution equations; Galerkin method; Discrete exponential integrator scheme 25 1 2023 2023-01-25 10.1007/s10959-023-01236-x COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University This work is supported by the Alan Turing Institute for funding this work under EPSRC grant EP/N510129/1 and EPSRC for funding though the project EP/S026347/1, titled ‘Unparameterised multi-modal data, high order signatures, and the mathematics of data science’. 2023-03-02T14:37:30.7093064 2023-02-03T11:15:41.1189009 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Yue Wu 0000-0002-6281-2229 1 Chenggui Yuan 0000-0003-0486-5450 2 62486__26458__32df025af16340438999a2c950b35075.pdf 62486.pdf 2023-02-03T11:32:08.5986175 Output 455354 application/pdf Version of Record true This article is licensed under a Creative Commons Attribution 4.0 International License true eng http://creativecommons.org/licenses/by/4.0/
title The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations
spellingShingle The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations
Chenggui Yuan
title_short The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations
title_full The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations
title_fullStr The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations
title_full_unstemmed The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations
title_sort The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations
author_id_str_mv 22b571d1cba717a58e526805bd9abea0
author_id_fullname_str_mv 22b571d1cba717a58e526805bd9abea0_***_Chenggui Yuan
author Chenggui Yuan
author2 Yue Wu
Chenggui Yuan
format Journal article
container_title Journal of Theoretical Probability
container_volume 36
container_issue 1
publishDate 2023
institution Swansea University
issn 0894-9840
1572-9230
doi_str_mv 10.1007/s10959-023-01236-x
publisher Springer Science and Business Media LLC
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description In this paper, we study the numerical method for approximating the random periodic solution of semilinear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating infinite-dimensional objects. We first show the existence and uniqueness of the random periodic solution to the equation as the limit of the pull-back flows of the equation, and observe that its mild form is well defined in the intersection of a family of decreasing Hilbert spaces. Then, we propose a Galerkin-type exponential integrator scheme and establish its convergence rate of the strong error to the mild solution, where the order of convergence directly depends on the space (among the family of Hilbert spaces) for the initial point to live. We finally conclude with a best order of convergence that is arbitrarily close to 0.5.
published_date 2023-01-25T04:22:08Z
_version_ 1763754459794505728
score 11.016235

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