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Degenerate SDE with Hölder--Dini Drift and Non-Lipschitz Noise Coefficient

Feng-yu Wang Orcid Logo, Xicheng Zhang

SIAM Journal on Mathematical Analysis, Volume: 48, Issue: 3, Pages: 2189 - 2226

Swansea University Author: Feng-yu Wang Orcid Logo

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DOI (Published version): 10.1137/15M1023671

Abstract

The existence-uniqueness and stability of strong solutions are proved for a class of degenerate stochastic differential equations, where the noise coefficient might be non-Lipschitz, and the drift is locally Dini continuous in the component with noise (i.e., the second component) and locally Hölder-...

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Published in: SIAM Journal on Mathematical Analysis
Published: 2016
URI: https://cronfa.swan.ac.uk/Record/cronfa29740
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first_indexed 2016-09-04T18:44:11Z
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spelling 2017-11-15T09:54:44.9575136 v2 29740 2016-09-04 Degenerate SDE with Hölder--Dini Drift and Non-Lipschitz Noise Coefficient 6734caa6d9a388bd3bd8eb0a1131d0de 0000-0003-0950-1672 Feng-yu Wang Feng-yu Wang true false 2016-09-04 SMA The existence-uniqueness and stability of strong solutions are proved for a class of degenerate stochastic differential equations, where the noise coefficient might be non-Lipschitz, and the drift is locally Dini continuous in the component with noise (i.e., the second component) and locally Hölder--Dini continuous of order $\frac{2}{3}$ in the first component. Moreover, the weak uniqueness is proved under weaker conditions on the noise coefficient. Furthermore, if the noise coefficient is $C^{1+\varepsilon}$ for some ${\varepsilon}>0$ and the drift is Hölder continuous of order ${\alpha}{\in} (\frac{2}{3},1)$ in the first component and order ${\beta\in}(0,1) $ in the second, the solution forms a $C^1$-stochastic diffeormorphism flow. To prove these results, we present some new characterizations of Hölder--Dini space by using the heat semigroup and slowly varying functions. Journal Article SIAM Journal on Mathematical Analysis 48 3 2189 2226 31 12 2016 2016-12-31 10.1137/15M1023671 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2017-11-15T09:54:44.9575136 2016-09-04T17:47:45.1751909 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Feng-yu Wang 0000-0003-0950-1672 1 Xicheng Zhang 2
title Degenerate SDE with Hölder--Dini Drift and Non-Lipschitz Noise Coefficient
spellingShingle Degenerate SDE with Hölder--Dini Drift and Non-Lipschitz Noise Coefficient
Feng-yu Wang
title_short Degenerate SDE with Hölder--Dini Drift and Non-Lipschitz Noise Coefficient
title_full Degenerate SDE with Hölder--Dini Drift and Non-Lipschitz Noise Coefficient
title_fullStr Degenerate SDE with Hölder--Dini Drift and Non-Lipschitz Noise Coefficient
title_full_unstemmed Degenerate SDE with Hölder--Dini Drift and Non-Lipschitz Noise Coefficient
title_sort Degenerate SDE with Hölder--Dini Drift and Non-Lipschitz Noise Coefficient
author_id_str_mv 6734caa6d9a388bd3bd8eb0a1131d0de
author_id_fullname_str_mv 6734caa6d9a388bd3bd8eb0a1131d0de_***_Feng-yu Wang
author Feng-yu Wang
author2 Feng-yu Wang
Xicheng Zhang
format Journal article
container_title SIAM Journal on Mathematical Analysis
container_volume 48
container_issue 3
container_start_page 2189
publishDate 2016
institution Swansea University
doi_str_mv 10.1137/15M1023671
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 0
active_str 0
description The existence-uniqueness and stability of strong solutions are proved for a class of degenerate stochastic differential equations, where the noise coefficient might be non-Lipschitz, and the drift is locally Dini continuous in the component with noise (i.e., the second component) and locally Hölder--Dini continuous of order $\frac{2}{3}$ in the first component. Moreover, the weak uniqueness is proved under weaker conditions on the noise coefficient. Furthermore, if the noise coefficient is $C^{1+\varepsilon}$ for some ${\varepsilon}>0$ and the drift is Hölder continuous of order ${\alpha}{\in} (\frac{2}{3},1)$ in the first component and order ${\beta\in}(0,1) $ in the second, the solution forms a $C^1$-stochastic diffeormorphism flow. To prove these results, we present some new characterizations of Hölder--Dini space by using the heat semigroup and slowly varying functions.
published_date 2016-12-31T03:36:13Z
_version_ 1763751570677170176
score 11.016235

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