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Quantitative weak propagation of chaos for McKean--Vlasov branching diffusion processes

Wenjing Cao, Zhenjie Ren, Xiaolu Tan

TL;DR

This work analyzes weak propagation of chaos for McKean--Vlasov branching diffusions where marginal measures are nonnegative finite (not necessarily probabilistic). The authors lift the branching dynamics to a non-branching framework on an extended space, formulate a functional Itô calculus on measure spaces, and derive a backward Kolmogorov equation for a value function $U(t,\mu)$. Under regularity and nondegeneracy assumptions, they prove a quantitative rate of convergence for the empirical measures: $|G(\mu_T)-\mathbb{E}[G(\mu_T^N)]|\le C/N$ for smooth terminal functionals $G$, establishing a practical $1/N$ accuracy for finite-population approximations. The methodology combines a lifted non-branching diffusion, flow properties of the environment measure, and intrinsic/linear functional derivatives to achieve the result, with potential impact on Monte Carlo methods for mean-field-type systems with branching.

Abstract

We study in this paper the weak propagation of chaos for McKean--Vlasov diffusions with branching, whose induced marginal measures are nonnegative finite measures but not necessary probability measures. The flow of marginal measures satisfies a non-linear Fokker--Planck equation, along which we provide a functional Itô's formula. We then consider a functional of the terminal marginal measure of the branching process, whose conditional value is solution to a Kolmogorov backward master equation. By using Itô's formula and based on the estimates of second-order linear and intrinsic functional derivatives of the value function, we finally derive a quantitative weak convergence rate for the empirical measures of the branching diffusion processes with finite population.

Quantitative weak propagation of chaos for McKean--Vlasov branching diffusion processes

TL;DR

This work analyzes weak propagation of chaos for McKean--Vlasov branching diffusions where marginal measures are nonnegative finite (not necessarily probabilistic). The authors lift the branching dynamics to a non-branching framework on an extended space, formulate a functional Itô calculus on measure spaces, and derive a backward Kolmogorov equation for a value function . Under regularity and nondegeneracy assumptions, they prove a quantitative rate of convergence for the empirical measures: for smooth terminal functionals , establishing a practical accuracy for finite-population approximations. The methodology combines a lifted non-branching diffusion, flow properties of the environment measure, and intrinsic/linear functional derivatives to achieve the result, with potential impact on Monte Carlo methods for mean-field-type systems with branching.

Abstract

We study in this paper the weak propagation of chaos for McKean--Vlasov diffusions with branching, whose induced marginal measures are nonnegative finite measures but not necessary probability measures. The flow of marginal measures satisfies a non-linear Fokker--Planck equation, along which we provide a functional Itô's formula. We then consider a functional of the terminal marginal measure of the branching process, whose conditional value is solution to a Kolmogorov backward master equation. By using Itô's formula and based on the estimates of second-order linear and intrinsic functional derivatives of the value function, we finally derive a quantitative weak convergence rate for the empirical measures of the branching diffusion processes with finite population.
Paper Structure (20 sections, 17 theorems, 210 equations)

This paper contains 20 sections, 17 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Functionals on the space of measures
  5. The McKean--Vlasov branching diffusion
  6. Finite-population branching process
  7. Quantitative weak propagation of chaos
  8. Proofs
  9. A non-branching diffusion process and Kolmogorov backward equation
  10. Proof of Proposition \ref{['prop_bKol']}
  11. Proof of Theorem \ref{['thm_WPoC']}
  12. Proof of Theorem \ref{['ito_formula']}
  13. Itô's formula for $\mu_t$
  14. Proof of the equation \ref{['ito_environment']}.
  15. Itô's formula for $\mu_t^N$
  16. ...and 5 more sections

Key Result

Proposition 2.12

Let Assumptions assum:coefficients and assum:initial hold true. Then both branching SDE SDE_E_MeanField and SDE_E_particle have a unique strong solution denoted respectively by $\bar{Z}$ and $\{ Z^i \}^N_{i=1}$.

Theorems & Definitions (39)

  • Definition 2.1: Bounded Lipschitz distance
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4: Linear and Intrinsic derivatives
  • Definition 2.5: $\mathcal{C}^2$ and $\mathcal{C}_b^2$ functionals
  • Definition 2.6: $\mathcal{C}^{1,2}$ functional
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Proposition 2.12
  • ...and 29 more