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Solving McKean-Vlasov SDEs via relative entropy

Yi Han

TL;DR

The paper develops a robust relative-entropy framework to establish weak well-posedness for McKean–Vlasov dynamics under path-dependent linear growth and Krylov-type singular drifts, extending beyond prior bounded/remediable cases. It unifies Brownian and fractional Brownian settings by employing Girsanov transforms for $B^H$ and combining averaging with truncation to obtain fixed-point solutions, including mixed drift structures and sublinear perturbations. The framework is then transported to infinite-dimensional mean-field SPDEs (heat, wave, and boundary-noise) to construct self-consistent SPDEs and prove propagation of chaos with entropy-based rates. Overall, the work broadens the solvability landscape for density-dependent SDEs/SPDEs, providing new probabilistic tools for non-Markovian and singular interactions with potential applications in physics and applied probability.

Abstract

In this paper we explore the merit of relative entropy in proving weak well-posedness of McKean-Vlasov SDEs and SPDEs, extending the technique introduced in Lacker arxiv:2105.02983. In the SDE setting, we prove weak existence and uniqueness when the interaction is path dependent and only assumed to have linear growth. Meanwhile, we recover and extend the current results when the interaction has Krylov's $L_t^q-L_x^p$ type singularity for $\frac{d}{p}+\frac{2}{q}<1$, where $d$ is the dimension of space. We connect the aforementioned two cases which are traditionally disparate, and form a solution theory that is sufficiently robust to allow perturbations of sublinear growth at the presence of singularity, giving rise to the well-posedness of a new family of McKean-Vlasov SDEs. Our strategy naturally extends to the cases of a fractional Brownian driving noise $B^H$ for all $H\in\left(0,1\right)$, obtaining new results in each separate case $H\in\left(0,\frac{1}{2}\right)$ and $H\in\left(\frac{1}{2},1\right)$. In the SPDE setting, we construct McKean-Vlasov type SPDEs with bounded measurable coefficients from the prototype of stochastic heat equation in spatial dimension one, and we do the same construction for the stochastic wave equation and a SPDE with white noise acting only on the boundary. In addition, we generalize some quantitative propagation of chaos results for SDEs into the SPDE setting.

Solving McKean-Vlasov SDEs via relative entropy

TL;DR

The paper develops a robust relative-entropy framework to establish weak well-posedness for McKean–Vlasov dynamics under path-dependent linear growth and Krylov-type singular drifts, extending beyond prior bounded/remediable cases. It unifies Brownian and fractional Brownian settings by employing Girsanov transforms for and combining averaging with truncation to obtain fixed-point solutions, including mixed drift structures and sublinear perturbations. The framework is then transported to infinite-dimensional mean-field SPDEs (heat, wave, and boundary-noise) to construct self-consistent SPDEs and prove propagation of chaos with entropy-based rates. Overall, the work broadens the solvability landscape for density-dependent SDEs/SPDEs, providing new probabilistic tools for non-Markovian and singular interactions with potential applications in physics and applied probability.

Abstract

In this paper we explore the merit of relative entropy in proving weak well-posedness of McKean-Vlasov SDEs and SPDEs, extending the technique introduced in Lacker arxiv:2105.02983. In the SDE setting, we prove weak existence and uniqueness when the interaction is path dependent and only assumed to have linear growth. Meanwhile, we recover and extend the current results when the interaction has Krylov's type singularity for , where is the dimension of space. We connect the aforementioned two cases which are traditionally disparate, and form a solution theory that is sufficiently robust to allow perturbations of sublinear growth at the presence of singularity, giving rise to the well-posedness of a new family of McKean-Vlasov SDEs. Our strategy naturally extends to the cases of a fractional Brownian driving noise for all , obtaining new results in each separate case and . In the SPDE setting, we construct McKean-Vlasov type SPDEs with bounded measurable coefficients from the prototype of stochastic heat equation in spatial dimension one, and we do the same construction for the stochastic wave equation and a SPDE with white noise acting only on the boundary. In addition, we generalize some quantitative propagation of chaos results for SDEs into the SPDE setting.
Paper Structure (33 sections, 19 theorems, 212 equations)

This paper contains 33 sections, 19 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Interactions of linear growth
  3. Singular drifts
  4. Non-Markovian processes
  5. Stochastic PDEs
  6. List of main results
  7. Results for Brownian motion
  8. Results for fractional Brownian motion
  9. Application to mean field stochastic PDEs
  10. A list of remarks
  11. Linear growth interactions, proof of Theorem \ref{['theorem01']}
  12. Interactions with Krylov's integrability condition
  13. Krylov's estimate
  14. Proof of Theorem \ref{['lplqfirst']}
  15. Proof of Theorem \ref{['wes']}
  16. ...and 18 more sections

Key Result

Theorem 1.1

Assume that $(b_0,b)$ are path dependent, progressively measurable and Then the McKean-Vlasov SDEIn this example we only consider first order interactions of the form $\langle b(t,x,\cdot),\mu_t\rangle$, but not the nonlinear ones. A very detailed explanation of this confinement is given in Remark linearmore. admits a unique weak solution $\mu$ from $\mu_{0}$ satisfying $\mathbb{E}\left[\left\|X\

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8: Multiplicative noise
  • Remark 1.9: Second order systems and degenerate noise
  • Remark 1.10: Density dependent diffusion coefficient
  • ...and 32 more