Image-dependent conditional McKean–Vlasov SDEs for measure-valued diffusion processes


We consider a special class of mean field SDEs with common noise which depends on the image of the solution (i.e., the conditional distribution given noise). The strong well-posedness is derived under a monotone condition which is weaker than those used in the literature of mean field games; the Feynman–Kac formula is established to solve Schrördinegr type PDEs on \(\mathscr {P}_2\), and the ergodicity is proved for a class of measure-valued diffusion processes.

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The author is grateful to the referee for valuable suggestions and to Professor Renming Song for helpful conversations.

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Correspondence to Feng-Yu Wang.

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Supported in part by NNSFC (11771326, 11831014) and the DFG through the CRC 1283.

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Wang, FY. Image-dependent conditional McKean–Vlasov SDEs for measure-valued diffusion processes. J. Evol. Equ. (2021).

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Mathematics Subject Classification

  • 60J60
  • 58J65


  • Image dependent SDE
  • Measure-valued diffusion process
  • Ergodicity
  • Feynman–Kac formula
  • Intrinsic/Lions derivative