[Paper Review] Distribution Dependent SDEs with Singular Coefficients

Xing Huang, Feng‐Yu Wang|arXiv (Cornell University)|May 4, 2018

Stochastic processes and financial applications13 references3 citations

TL;DR

This paper establishes existence and uniqueness for distribution-dependent SDEs with singular coefficients under integrability conditions, extending Krylov's estimate and Zvonkin's transform to the McKean-Vlasov setting. It derives gradient estimates and log-Harnack inequalities under Dini continuity, generalizing classical results to the distribution-dependent case with new quantitative bounds on Wasserstein distance and time-dependent coefficients.

ABSTRACT

Under integrability conditions on distribution dependent coefficients, existence and uniqueness are proved for McKean-Vlasov type SDEs with non-degenerate noise. When the coefficients are Dini continuous in the space variable, gradient estimates and Harnack type inequalities are derived. These generalize the corresponding results derived for classical SDEs, and are new in the distribution dependent setting.

Motivation & Objective

• To extend Krylov’s estimate and Zvonkin’s transform to distribution-dependent SDEs with singular coefficients.
• To establish existence and pathwise uniqueness for McKean-Vlasov SDEs under integrability and Dini continuity conditions.
• To derive gradient estimates and log-Harnack inequalities for the associated Fokker-Planck equations in the distribution-dependent setting.
• To generalize classical results on SDEs to the non-Markovian, distribution-dependent framework with weaker regularity assumptions.

Proposed method

• Introduces a new Krylov-type estimate for distribution-dependent SDEs, proving moment bounds under integrability conditions on drift and diffusion coefficients.
• Uses an approximation argument and weak convergence of stochastic processes to construct weak solutions.
• Applies Zvonkin’s transform to identify laws of two solutions, enabling pathwise uniqueness via classical SDE techniques.
• Employs Girsanov’s theorem and martingale representation to derive log-Harnack inequalities via time-regularized coupling and entropy bounds.
• Derives a key estimate involving the Wasserstein distance $\mathbb{W}_2(\mu_0, \nu_0)$ and a time-dependent weight $\zeta_s$, leading to entropy control.
• Uses Young’s inequality and exponential moment bounds to control the Radon-Nikodym derivative $R_t$ in the Girsanov transformation.

Experimental results

Research questions

• RQ1 Can Krylov’s estimate be extended to distribution-dependent SDEs with singular coefficients?
• RQ2 Under what conditions does a weak solution to a distribution-dependent SDE become a strong solution?
• RQ3 Can gradient estimates and Harnack inequalities be derived when the drift is only Dini continuous rather than Lipschitz?
• RQ4 How does the Wasserstein distance between initial distributions affect the long-time behavior of the solution semigroup?
• RQ5 What is the role of Zvonkin’s transform in proving pathwise uniqueness for distribution-dependent SDEs?

Key findings

• Existence and uniqueness of strong solutions are established for distribution-dependent SDEs under integrability conditions on the coefficients and non-degenerate noise.
• A new Krylov-type estimate is derived, ensuring moment bounds of the form $\mathbb{E}\left[\int_s^t f_r(X_r)\,dr \mid \mathscr{F}_s\right] \leq C(t-s)^\delta \|f\|_{L^q_p(T)}$ for $ (p,q) \in \mathscr{K} $.
• Log-Harnack inequality is proven: $ (P_{t_0}\log f)(\nu_0) \leq \log(P_{t_0}f)(\mu_0) + \frac{C}{t_0} \mathbb{W}_2(\mu_0, \nu_0)^2 $, with explicit dependence on initial distance.
• Gradient estimates are obtained under Dini continuity of the drift, improving upon previous results that required Lipschitz conditions.
• The Harnack inequality with power is derived via coupling and entropy control, with the constant depending on $ \|\sigma_t^{-1}\|_\infty $ and a Dini modulus $ \phi $.
• The Radon-Nikodym derivative $ R_t $ satisfies $ \mathbb{E}[R_t \log R_t] \leq \frac{C}{t_0} \mathbb{W}_2(\mu_0, \nu_0)^2 $, ensuring uniform integrability and pathwise uniqueness.