February 2023 SDEs with critical time dependent drifts: Weak solutions
Michael Röckner, Guohuan Zhao
Author Affiliations +
Bernoulli 29(1): 757-784 (February 2023). DOI: 10.3150/22-BEJ1478

Abstract

For d3, we prove that time-inhomogeneous stochastic differential equations driven by additive noises with drifts in critical Lebesgue space Lq([0,T];Lp(Rd)), where (p,q)(d,]×[2,) and dp+2q=1, or (p,q)=(d,) and divbL([0,T];Ld2+ε(Rd)), are well-posed. The weak uniqueness is obtained by solving corresponding Kolmogorov backward equations in some second-order Sobolev spaces, which is analytically interesting in itself.

Acknowledgements

Research of Michael and Guohuan is supported by the German Research Foundation (DFG) through the Collaborative Research Centre (CRC) 1283 Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications.

The second named author is very grateful to Nicolai Krylov and Xicheng Zhang who encouraged him to persist in studying this problem, and also Moritz Kassmann for providing him an excellent environment to work at Bielefeld University.

Citation

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Michael Röckner. Guohuan Zhao. "SDEs with critical time dependent drifts: Weak solutions." Bernoulli 29 (1) 757 - 784, February 2023. https://doi.org/10.3150/22-BEJ1478

Information

Received: 1 March 2021; Published: February 2023
First available in Project Euclid: 13 October 2022

MathSciNet: MR4497266
zbMATH: 07634411
Digital Object Identifier: 10.3150/22-BEJ1478

Keywords: De Giorgi’s method , Kolmogorov equations , Ladyzhenskaya–Prodi–Serrin condition , weak solutions

Vol.29 • No. 1 • February 2023
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