Bismut formula for Lions derivative of distribution dependent SDEs and applications

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Original languageEnglish
Pages (from-to)4745-4777
Journal / PublicationJournal of Differential Equations
Issue number8
Online published15 May 2019
Publication statusPublished - 5 Oct 2019
Externally publishedYes


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.

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Research Area(s)

  • Bismut formula, Distribution dependent SDEs, L-derivative, Wasserstein distance