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Stochastic differential equations with critically irregular drift coefficients

Jinlong Wei, Guangying Lv, Jiang-lun Wu

Journal of Differential Equations, Volume: 371, Pages: 1 - 30

Swansea University Author: Jiang-lun Wu

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DOI (Published version): 10.1016/j.jde.2023.06.029

Abstract

This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a n...

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Published in: Journal of Differential Equations
ISSN: 0022-0396 1090-2732
Published: Elsevier BV 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa63726
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spelling v2 63726 2023-06-27 Stochastic differential equations with critically irregular drift coefficients dbd67e30d59b0f32592b15b5705af885 Jiang-lun Wu Jiang-lun Wu true false 2023-06-27 FGSEN This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:[0, T] ×Rd→Rdbe Borel measurable, where T>0is arbitrarily fixed and d⩾1. We consider the following SDE Xt=x+ t 0 b(s,Xs)ds+Wt,t∈[0,T],x∈Rd, where {Wt}t∈[0,T]is a d-dimensional standard Wiener process. For p, q∈[1, +∞), we denote by C[q]([0, T]; Lp(Rd))the space of all Borel measurable functions fsuch that t1 qf(t) ∈C([0, T]; Lp(Rd)). If b=b1+b2such that |b1(T−·)| ∈C[q]([0, T]; Lp(Rd))with 2/q+d/p=1and b1(T− ·)C[q]([0,T];Lp(Rd))is sufficiently small, and that b2is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0t1 qb(T−t)Lp(Rd)=0, the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with Lq(0, T; Lp(Rd))coefficients to the case of parabolic PDEs with L∞ [q](0, T; Lp(Rd))coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem3.1). Our results extend Krylov-Röckner and Krylov’s profound results of SDEs with singular time dependent drift coefficients [20,23]to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space. Journal Article Journal of Differential Equations 371 1 30 Elsevier BV 0022-0396 1090-2732 SDEs with irregular drifts, Existence, Uniqueness, Weak/strong solutions, The strong Feller property 1 10 2023 2023-10-01 10.1016/j.jde.2023.06.029 http://dx.doi.org/10.1016/j.jde.2023.06.029 COLLEGE NANME Science and Engineering - Faculty COLLEGE CODE FGSEN Swansea University This research was partly supported by the NSF of China grants 11771123 and 12171247, the fundamental research funds for the central universities of Zhongnan University of Economics and Law grant 112/31513111213. 2023-07-18T17:11:49.3342759 2023-06-27T15:58:03.8462947 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Jinlong Wei 1 Guangying Lv 2 Jiang-lun Wu 3 Under embargo Under embargo 2023-06-30T14:45:59.5617845 Output 367748 application/pdf Accepted Manuscript true 2024-06-26T00:00:00.0000000 Distributed under the terms of a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licence (CC BY-NC-ND 4.0). true eng http://creativecommons.org/licenses/by-nc-nd/4.0/
title Stochastic differential equations with critically irregular drift coefficients
spellingShingle Stochastic differential equations with critically irregular drift coefficients
Jiang-lun Wu
title_short Stochastic differential equations with critically irregular drift coefficients
title_full Stochastic differential equations with critically irregular drift coefficients
title_fullStr Stochastic differential equations with critically irregular drift coefficients
title_full_unstemmed Stochastic differential equations with critically irregular drift coefficients
title_sort Stochastic differential equations with critically irregular 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
container_title Journal of Differential Equations
container_volume 371
container_start_page 1
publishDate 2023
institution Swansea University
issn 0022-0396
1090-2732
doi_str_mv 10.1016/j.jde.2023.06.029
publisher Elsevier BV
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 http://dx.doi.org/10.1016/j.jde.2023.06.029
document_store_str 0
active_str 0
description This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:[0, T] ×Rd→Rdbe Borel measurable, where T>0is arbitrarily fixed and d⩾1. We consider the following SDE Xt=x+ t 0 b(s,Xs)ds+Wt,t∈[0,T],x∈Rd, where {Wt}t∈[0,T]is a d-dimensional standard Wiener process. For p, q∈[1, +∞), we denote by C[q]([0, T]; Lp(Rd))the space of all Borel measurable functions fsuch that t1 qf(t) ∈C([0, T]; Lp(Rd)). If b=b1+b2such that |b1(T−·)| ∈C[q]([0, T]; Lp(Rd))with 2/q+d/p=1and b1(T− ·)C[q]([0,T];Lp(Rd))is sufficiently small, and that b2is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0t1 qb(T−t)Lp(Rd)=0, the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with Lq(0, T; Lp(Rd))coefficients to the case of parabolic PDEs with L∞ [q](0, T; Lp(Rd))coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem3.1). Our results extend Krylov-Röckner and Krylov’s profound results of SDEs with singular time dependent drift coefficients [20,23]to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.
published_date 2023-10-01T17:11:44Z
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score 11.013148

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