Well-posedness of distribution dependent SDEs with singular drifts

Michael Röckner; Xicheng Zhang

doi:10.3150/20-BEJ1268

Well-posedness of distribution dependent SDEs with singular drifts

Michael Röckner, Xicheng Zhang

Research output: Contribution to journal › Article › peer-review

66 Citations (Scopus)

Abstract

Consider the following distribution dependent SDE: dX_t = σ_t(X_t, μ_Xt )dW_t + b_t(X_t, μ_Xt )dt, where μ_Xt stands for the distribution of X_t. In this paper for non-degenerate σ, we show the strong well-posedness of the above SDE under some integrability assumptions in the spatial variable and Lipschitz continuity in μ about b and σ. In particular, we extend the results of Krylov–Röckner (Probab. Theory Related Fields 131 (2005) 154–196) to the distribution dependent case.

Original language	English
Pages (from-to)	1131-1158
Number of pages	28
Journal	Bernoulli
Volume	27
Issue number	2
DOIs	https://doi.org/10.3150/20-BEJ1268
Publication status	Published - May 2021
Externally published	Yes

Keywords

Distribution dependent SDEs
McKean–Vlasov system
Singular drifts
Superposition principle
Zvonkin’s transformation

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10.3150/20-BEJ1268

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abstract = "Consider the following distribution dependent SDE: dXt = σt(Xt, μXt )dWt + bt(Xt, μXt )dt, where μXt stands for the distribution of Xt. In this paper for non-degenerate σ, we show the strong well-posedness of the above SDE under some integrability assumptions in the spatial variable and Lipschitz continuity in μ about b and σ. In particular, we extend the results of Krylov–R{\"o}ckner (Probab. Theory Related Fields 131 (2005) 154–196) to the distribution dependent case.",

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T1 - Well-posedness of distribution dependent SDEs with singular drifts

AU - Röckner, Michael

AU - Zhang, Xicheng

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N2 - Consider the following distribution dependent SDE: dXt = σt(Xt, μXt )dWt + bt(Xt, μXt )dt, where μXt stands for the distribution of Xt. In this paper for non-degenerate σ, we show the strong well-posedness of the above SDE under some integrability assumptions in the spatial variable and Lipschitz continuity in μ about b and σ. In particular, we extend the results of Krylov–Röckner (Probab. Theory Related Fields 131 (2005) 154–196) to the distribution dependent case.

AB - Consider the following distribution dependent SDE: dXt = σt(Xt, μXt )dWt + bt(Xt, μXt )dt, where μXt stands for the distribution of Xt. In this paper for non-degenerate σ, we show the strong well-posedness of the above SDE under some integrability assumptions in the spatial variable and Lipschitz continuity in μ about b and σ. In particular, we extend the results of Krylov–Röckner (Probab. Theory Related Fields 131 (2005) 154–196) to the distribution dependent case.

KW - Distribution dependent SDEs

KW - McKean–Vlasov system

KW - Singular drifts

KW - Superposition principle

KW - Zvonkin’s transformation

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VL - 27

SP - 1131

EP - 1158

JO - Bernoulli

JF - Bernoulli

IS - 2

ER -

Well-posedness of distribution dependent SDEs with singular drifts

Abstract

Keywords

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