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Stochastic differential equations with critically irregular drift coefficients
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...
| Published in: | Journal of Differential Equations |
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| ISSN: | 0022-0396 1090-2732 |
| Published: |
Elsevier BV
2023
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa63726 |
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| 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 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. |
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| Keywords: |
SDEs with irregular drifts, Existence, Uniqueness, Weak/strong solutions, The strong Feller property |
| College: |
Faculty of Science and Engineering |
| Funders: |
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. |
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30 |