## State-density flows of non-degenerate density-dependent mean field SDEs and associated PDEs

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Ziyu Huang and Shanjian Tang
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## Abstract:

In this paper, we study a combined system of a Fokker-Planck (FP) equation for $m^{t,\mu }$ with initial $(t,\mu )\in [0,T]\times L^2(\mathbb {R}^d)$, and a stochastic differential equation for $X^{t,x,\mu }$ with initial $(t,x)\in [0,T]\times \mathbb {R}^d$, whose coefficients depend on the solution of FP equation. We develop a combined probabilistic and analytical method to explore the regularity of the functional $V(t,x,\mu )=\mathbb {E}[\Phi (X^{t,x,\mu }_T,m^{t,\mu }(T,\cdot ))]$. Our main result states that, under a non-degenerate condition and appropriate regularity assumptions on the coefficients, the function $V$ is the unique classical solution of a nonlocal partial differential equation of mean-field type. The proof depends heavily on the differential properties of the flow $\mu \mapsto (m^{t,\mu }, X^{t,x,\mu })$ over $\mu \in L^2(\mathbb {R}^d)$. We also give an example to illustrate the role of our main result. Finally, we give a discussion on the $L^1$ choice case in the initial $\mu$.## References

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## Additional Information

**Ziyu Huang**- Affiliation: School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s Republic of China
- ORCID: 0000-0002-2370-9202
- Email: zyhuang19@fudan.edu.cn
**Shanjian Tang**- Affiliation: Department of Finance and Control Sciences, School of Mathematical Sciences, and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, People’s Republic of China
- ORCID: 0000-0003-3884-042X
- Email: sjtang@fudan.edu.cn
- Received by editor(s): December 12, 2021
- Received by editor(s) in revised form: March 12, 2022, and July 2, 2022
- Published electronically: December 15, 2022
- Additional Notes: This research was partially supported by National Nature Science Foundation of China (Grants No. 11631004 and No. 12031009) and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Handan Road 220, Shanghai 200433, China
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 1553-1599 - MSC (2020): Primary 60H30, 60H10; Secondary 35K55
- DOI: https://doi.org/10.1090/tran/8809
- MathSciNet review: 4549685