Bismut formula for Lions derivative of distribution dependent SDEs and applications

Panpan Ren; Feng-Yu Wang

doi:10.1016/j.jde.2019.05.016

Bismut formula for Lions derivative of distribution dependent SDEs and applications

Panpan Ren, Feng-Yu Wang^*

^*Corresponding author for this work

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

50 Citations (Scopus)

Abstract

By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of P_tf (μ): = Ef (X_t^μ), where t > 0, f is a bounded measurable function, and X_t^μ solves a distribution dependent SDE with initial distribution μ. As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semigroup property and the invalidity of the formula P_tf (μ) = ∫P_tf (x) μ (dx), essential difficulties are overcome in the study.

© 2019 Elsevier Inc. All rights reserved.

Original language	English
Pages (from-to)	4745-4777
Journal	Journal of Differential Equations
Volume	267
Issue number	8
Online published	15 May 2019
DOIs	https://doi.org/10.1016/j.jde.2019.05.016
Publication status	Published - 5 Oct 2019
Externally published	Yes

Funding

Financial support by the DFG through the CRC 1283 “Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications” is acknowledged. The authors would like to thank the referee and Professor Yulin Song for helpful comments.

Research Keywords

Bismut formula
Distribution dependent SDEs
L-derivative
Wasserstein distance

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title = "Bismut formula for Lions derivative of distribution dependent SDEs and applications",

abstract = "By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of Ptf (μ): = Ef (Xtμ), where t > 0, f is a bounded measurable function, and Xtμ solves a distribution dependent SDE with initial distribution μ. As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semigroup property and the invalidity of the formula Ptf (μ) = ∫Ptf (x) μ (dx), essential difficulties are overcome in the study.{\textcopyright} 2019 Elsevier Inc. All rights reserved.",

keywords = "Bismut formula, Distribution dependent SDEs, L-derivative, Wasserstein distance",

author = "Panpan Ren and Feng-Yu Wang",

year = "2019",

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day = "5",

doi = "10.1016/j.jde.2019.05.016",

language = "English",

volume = "267",

pages = "4745--4777",

journal = "Journal of Differential Equations",

issn = "0022-0396",

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TY - JOUR

T1 - Bismut formula for Lions derivative of distribution dependent SDEs and applications

AU - Ren, Panpan

AU - Wang, Feng-Yu

PY - 2019/10/5

Y1 - 2019/10/5

N2 - By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of Ptf (μ): = Ef (Xtμ), where t > 0, f is a bounded measurable function, and Xtμ solves a distribution dependent SDE with initial distribution μ. As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semigroup property and the invalidity of the formula Ptf (μ) = ∫Ptf (x) μ (dx), essential difficulties are overcome in the study.© 2019 Elsevier Inc. All rights reserved.

AB - By using Malliavin calculus, Bismut type formulas are established for the Lions derivative of Ptf (μ): = Ef (Xtμ), where t > 0, f is a bounded measurable function, and Xtμ solves a distribution dependent SDE with initial distribution μ. As applications, explicit estimates are derived for the Lions derivative and the total variational distance between distributions of solutions with different initial data. Both degenerate and non-degenerate situations are considered. Due to the lack of the semigroup property and the invalidity of the formula Ptf (μ) = ∫Ptf (x) μ (dx), essential difficulties are overcome in the study.© 2019 Elsevier Inc. All rights reserved.

KW - Bismut formula

KW - Distribution dependent SDEs

KW - L-derivative

KW - Wasserstein distance

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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85065560281&origin=recordpage

U2 - 10.1016/j.jde.2019.05.016

DO - 10.1016/j.jde.2019.05.016

M3 - RGC 21 - Publication in refereed journal

SN - 0022-0396

VL - 267

SP - 4745

EP - 4777

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 8

ER -

Bismut formula for Lions derivative of distribution dependent SDEs and applications

Abstract

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