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Coupling by Change of Measure for Conditional McKean-Vlasov SDEs and Applications

Xing Huang

Abstract

The couplings by change of measure are applied to establish log-Harnack inequality(equivalently the entropy-cost estimate) for conditional McKean-Vlasov SDEs and derive the quantitative conditional propagation of chaos in relative entropy for mean field interacting particle system with common noise. For the log-Harnack inequality, two different types of couplings will be constructed for non-degenerate conditional McKean-Vlasov SDEs with multiplicative noise. As to the quantitative conditional propagation of chaos in relative entropy, the initial distribution of interacting particle system is allowed to be singular with that of limit equation. The above results are also extended to conditional distribution dependent stochastic Hamiltonian system.

Coupling by Change of Measure for Conditional McKean-Vlasov SDEs and Applications

Abstract

The couplings by change of measure are applied to establish log-Harnack inequality(equivalently the entropy-cost estimate) for conditional McKean-Vlasov SDEs and derive the quantitative conditional propagation of chaos in relative entropy for mean field interacting particle system with common noise. For the log-Harnack inequality, two different types of couplings will be constructed for non-degenerate conditional McKean-Vlasov SDEs with multiplicative noise. As to the quantitative conditional propagation of chaos in relative entropy, the initial distribution of interacting particle system is allowed to be singular with that of limit equation. The above results are also extended to conditional distribution dependent stochastic Hamiltonian system.
Paper Structure (9 sections, 6 theorems, 128 equations)

This paper contains 9 sections, 6 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Non-degenerate case
  3. Log-Harnack inequality
  4. Case 1: $\tilde{\sigma}_t(x,\mu)=\tilde{\sigma}_t(x)$
  5. Case 2: $\tilde{\sigma}_t(x,\mu)=\tilde{\sigma}_t(\mu)$
  6. Conditional propagation of chaos
  7. Conditional distribution dependent stochastic Hamiltonian system
  8. Log-Haranck inequality
  9. Conditional propagation of chaos

Key Result

Lemma 1.1

Assume (H). Then E0 is well-posed and $\mathscr L_{X_t^\xi|\mathscr F_t^B}=\mathscr L_{X_t^{\tilde{\xi}}|\mathscr F_t^B}$ for any initial values $\xi,\tilde{\xi}\in L^2(\Omega^1\rightarrow\mathbb R^d,\mathscr F^1_0,\mathbb P^1)$ with $\mathscr L_{\xi}=\mathscr L_{\tilde{\xi}}$. Moreover, there exis

Theorems & Definitions (13)

  • Definition 1.1
  • Lemma 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 3.1
  • ...and 3 more