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Stochastic Differential Equations with Local Growth Singular Drifts

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  • Wenjie Ye

    (Chinese Academy of Sciences)

Abstract

In this paper, we study the weak differentiability of global strong solution of stochastic differential equations, the strong Feller property of the associated diffusion semigroups and the global stochastic flow property in which the singular drift b and the weak gradient of Sobolev diffusion $$\sigma $$ σ are supposed to satisfy $$\left\| \left| b\right| \cdot \mathbbm {1}_{B(R)}\right\| _{p_1}\le O((\log R)^{{(p_1-d)^2}/{2p^2_1}})$$ b · 1 B ( R ) p 1 ≤ O ( ( log R ) ( p 1 - d ) 2 / 2 p 1 2 ) and $$\left\| \left\| \nabla \sigma \right\| \cdot \mathbbm {1}_{B(R)}\right\| _{p_1}\le O((\log ({R}/{3}))^{{(p_1-d)^2}/{2p^2_1}})$$ ∇ σ · 1 B ( R ) p 1 ≤ O ( ( log ( R / 3 ) ) ( p 1 - d ) 2 / 2 p 1 2 ) , respectively. The main tools for these results are the decomposition of global two-point motions in Fang et al. (Ann Probab 35(1):180–205, 2007), Krylov’s estimate, Khasminskii’s estimate, Zvonkin’s transformation and the characterization for Sobolev differentiability of random fields in Xie and Zhang (Ann Probab 44(6):3661–3687, 2016).

Suggested Citation

  • Wenjie Ye, 2024. "Stochastic Differential Equations with Local Growth Singular Drifts," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2576-2614, September.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:3:d:10.1007_s10959-024-01333-5
    DOI: 10.1007/s10959-024-01333-5
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    References listed on IDEAS

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    1. Krylov, N.V., 2021. "On stochastic Itô processes with drift in Ld," Stochastic Processes and their Applications, Elsevier, vol. 138(C), pages 1-25.
    2. Zhang, Xicheng, 2005. "Strong solutions of SDES with singular drift and Sobolev diffusion coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1805-1818, November.
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