Annals of Probability

Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients

Longjie Xie and Xicheng Zhang

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By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility to the associated diffusion semigroup.

Article information

Ann. Probab., Volume 44, Number 6 (2016), 3661-3687.

Received: April 2015
Revised: August 2015
First available in Project Euclid: 14 November 2016

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Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Weak differentiability Krylov’s estimate Zvonkin’s transformation strong Feller property irreducibility


Xie, Longjie; Zhang, Xicheng. Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients. Ann. Probab. 44 (2016), no. 6, 3661--3687. doi:10.1214/15-AOP1057.

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