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Weak approximation of kinetic SDEs: closing the criticality gap

Zimo Hao, Khoa Lê, Chengcheng Ling

Abstract

We study the weak convergence of a generic tamed Euler-Maruyama scheme for kinetic stochastic differential equations (SDEs) with integrable drifts. We show that the marginal density of the considered scheme converges at rate 1/2 to the corresponding marginal density of the SDE. The convergence rate is independent from the criticality gap, which is new compared to previous results.

Weak approximation of kinetic SDEs: closing the criticality gap

Abstract

We study the weak convergence of a generic tamed Euler-Maruyama scheme for kinetic stochastic differential equations (SDEs) with integrable drifts. We show that the marginal density of the considered scheme converges at rate 1/2 to the corresponding marginal density of the SDE. The convergence rate is independent from the criticality gap, which is new compared to previous results.
Paper Structure (13 sections, 19 theorems, 174 equations)

This paper contains 13 sections, 19 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Notations and examples
  3. Notations
  4. Examples
  5. Preliminaries
  6. Estimates on Markov semigroup
  7. Exponential estimates
  8. Proof of main results
  9. Taming error
  10. Quadrature error
  11. Proof of \ref{['thm-weak']}
  12. Some technical lemmas
  13. Paraproduct estimates in anisotropic Besov spaces

Key Result

Theorem 1.3

Assume that ass:mainass.bn hold and initial data $(\xi,\eta)\in{{\mathbb R}^d}\times{{\mathbb R}^d}$ is given. Then for any ${\boldsymbol{q}}\in[2,\infty)^2$ with ${\boldsymbol{q}}\geqslant{\boldsymbol{p}}$, there is a constant $C=C(d,{\boldsymbol{p}},{\boldsymbol{q}},\zeta,\vartheta,\delta,\kappa_b where ${\boldsymbol{q}}'=(\frac{q_x}{q_x-1},\frac{q_v}{q_v-1})$ is the Hölder conjugate of ${\bolds

Theorems & Definitions (42)

  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3: Convolution
  • Example 2.4: Cut-off
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 32 more