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Quasi-invariance of the Stochastic Flow Associated to Itô’s SDE with Singular Time-Dependent Drift

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  • Dejun Luo

    (Chinese Academy of Sciences)

Abstract

In this paper, we consider the Itô SDE $$\begin{aligned} d X_t=d W_t+b(t,X_t)\,d t, \quad X_0=x\in \mathbb {R}^d, \end{aligned}$$ d X t = d W t + b ( t , X t ) d t , X 0 = x ∈ R d , where $$W_t$$ W t is a $$d$$ d -dimensional standard Wiener process and the drift coefficient $$b:[0,T]\times \mathbb {R}^d\rightarrow \mathbb {R}^d$$ b : [ 0 , T ] × R d → R d belongs to $$L^q(0,T;L^p(\mathbb {R}^d))$$ L q ( 0 , T ; L p ( R d ) ) with $$p\ge 2, q>2$$ p ≥ 2 , q > 2 and $$\frac{d}{p} +\frac{2}{q}

Suggested Citation

  • Dejun Luo, 2015. "Quasi-invariance of the Stochastic Flow Associated to Itô’s SDE with Singular Time-Dependent Drift," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1743-1762, December.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0554-z
    DOI: 10.1007/s10959-014-0554-z
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    References listed on IDEAS

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    1. Luo, Dejun, 2011. "Absolute continuity under flows generated by SDE with measurable drift coefficients," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2393-2415, October.
    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.
    3. Zhang, Xicheng, 2005. "Homeomorphic flows for multi-dimensional SDEs with non-Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(3), pages 435-448, March.
    4. Ma, Yutao, 2010. "Transportation inequalities for stochastic differential equations with jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(1), pages 2-21, January.
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