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On weak solutions of stochastic differential equations with sharp drift coefficients

Jinlong Wei , Guangying Lv , Jiang-lun Wu Orcid Logo

Swansea University Author: Jiang-lun Wu Orcid Logo

Abstract

We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of stochastic differential equations (SDEs for short). To be more precise, let $b: [0,T]\times{\mathbb R}^d\rightar...

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Published: 2017
Online Access: https://arxiv.org/abs/1711.05058
URI: https://cronfa.swan.ac.uk/Record/cronfa36768
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fullrecord <?xml version="1.0"?><rfc1807><datestamp>2020-10-27T14:41:52.2170123</datestamp><bib-version>v2</bib-version><id>36768</id><entry>2017-11-14</entry><title>On weak solutions of stochastic differential equations with sharp drift coefficients</title><swanseaauthors><author><sid>dbd67e30d59b0f32592b15b5705af885</sid><ORCID>0000-0003-4568-7013</ORCID><firstname>Jiang-lun</firstname><surname>Wu</surname><name>Jiang-lun Wu</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2017-11-14</date><deptcode>SMA</deptcode><abstract>We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of stochastic differential equations (SDEs for short). To be more precise, let $b: [0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$ be Borel measurable, where $T&gt;0$ is arbitrarily fixed. Consider $$X_t=x+\int_0^tb(s,X_s)ds+W_t,\quad t\in[0,T], \, x\in\mathbb{R}^d,$$where $\{W_t\}_{t\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\cdot)\in\cC_q^0((0,T];L^p(\mR^d))$ with $2/q+d/p=1$ for $p,q\ge1$ and $\|b_1(T-\cdot)\|_{\cC_q((0,T];L^p(\mR^d))}$ is sufficiently small, and that $b_2$ is bounded and Borel measurable, then there exits a unique weak solution to the above equation. Furthermore, we obtain the strong Feller property of the semi-group and existence of density associated with above SDE. Besides, we extend the classical partial differential equations (PDEs) results for $L^q(0,T;L^p(\mR^d))$ coefficients to $L^\infty_q(0,T;L^p(\mR^d))$ ones, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Lemma \ref{lem2.1}).</abstract><type>Journal Article</type><journal/><volume/><journalNumber/><paginationStart/><paginationEnd/><publisher/><placeOfPublication/><isbnPrint/><isbnElectronic/><issnPrint/><issnElectronic/><keywords>Existence, uniqueness, weak solution, SDEs with irregular drifts, the strong Feller property.</keywords><publishedDay>14</publishedDay><publishedMonth>11</publishedMonth><publishedYear>2017</publishedYear><publishedDate>2017-11-14</publishedDate><doi/><url>https://arxiv.org/abs/1711.05058</url><notes/><college>COLLEGE NANME</college><department>Mathematics</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>SMA</DepartmentCode><institution>Swansea University</institution><apcterm/><lastEdited>2020-10-27T14:41:52.2170123</lastEdited><Created>2017-11-14T14:21:23.7296301</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>Jinlong Wei</firstname><surname/><order>1</order></author><author><firstname>Guangying Lv</firstname><surname/><order>2</order></author><author><firstname>Jiang-lun</firstname><surname>Wu</surname><orcid>0000-0003-4568-7013</orcid><order>3</order></author></authors><documents/><OutputDurs/></rfc1807>
spelling 2020-10-27T14:41:52.2170123 v2 36768 2017-11-14 On weak solutions of stochastic differential equations with sharp drift coefficients dbd67e30d59b0f32592b15b5705af885 0000-0003-4568-7013 Jiang-lun Wu Jiang-lun Wu true false 2017-11-14 SMA We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of stochastic differential equations (SDEs for short). To be more precise, let $b: [0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\int_0^tb(s,X_s)ds+W_t,\quad t\in[0,T], \, x\in\mathbb{R}^d,$$where $\{W_t\}_{t\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\cdot)\in\cC_q^0((0,T];L^p(\mR^d))$ with $2/q+d/p=1$ for $p,q\ge1$ and $\|b_1(T-\cdot)\|_{\cC_q((0,T];L^p(\mR^d))}$ is sufficiently small, and that $b_2$ is bounded and Borel measurable, then there exits a unique weak solution to the above equation. Furthermore, we obtain the strong Feller property of the semi-group and existence of density associated with above SDE. Besides, we extend the classical partial differential equations (PDEs) results for $L^q(0,T;L^p(\mR^d))$ coefficients to $L^\infty_q(0,T;L^p(\mR^d))$ ones, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Lemma \ref{lem2.1}). Journal Article Existence, uniqueness, weak solution, SDEs with irregular drifts, the strong Feller property. 14 11 2017 2017-11-14 https://arxiv.org/abs/1711.05058 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2020-10-27T14:41:52.2170123 2017-11-14T14:21:23.7296301 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Jinlong Wei 1 Guangying Lv 2 Jiang-lun Wu 0000-0003-4568-7013 3
title On weak solutions of stochastic differential equations with sharp drift coefficients
spellingShingle On weak solutions of stochastic differential equations with sharp drift coefficients
Jiang-lun Wu
title_short On weak solutions of stochastic differential equations with sharp drift coefficients
title_full On weak solutions of stochastic differential equations with sharp drift coefficients
title_fullStr On weak solutions of stochastic differential equations with sharp drift coefficients
title_full_unstemmed On weak solutions of stochastic differential equations with sharp drift coefficients
title_sort On weak solutions of stochastic differential equations with sharp drift coefficients
author_id_str_mv dbd67e30d59b0f32592b15b5705af885
author_id_fullname_str_mv dbd67e30d59b0f32592b15b5705af885_***_Jiang-lun Wu
author Jiang-lun Wu
author2 Jinlong Wei
Guangying Lv
Jiang-lun Wu
format Journal article
publishDate 2017
institution Swansea University
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
url https://arxiv.org/abs/1711.05058
document_store_str 0
active_str 0
description We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of stochastic differential equations (SDEs for short). To be more precise, let $b: [0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\int_0^tb(s,X_s)ds+W_t,\quad t\in[0,T], \, x\in\mathbb{R}^d,$$where $\{W_t\}_{t\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\cdot)\in\cC_q^0((0,T];L^p(\mR^d))$ with $2/q+d/p=1$ for $p,q\ge1$ and $\|b_1(T-\cdot)\|_{\cC_q((0,T];L^p(\mR^d))}$ is sufficiently small, and that $b_2$ is bounded and Borel measurable, then there exits a unique weak solution to the above equation. Furthermore, we obtain the strong Feller property of the semi-group and existence of density associated with above SDE. Besides, we extend the classical partial differential equations (PDEs) results for $L^q(0,T;L^p(\mR^d))$ coefficients to $L^\infty_q(0,T;L^p(\mR^d))$ ones, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Lemma \ref{lem2.1}).
published_date 2017-11-14T03:46:07Z
_version_ 1763752193264975872
score 11.013731

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