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Quantitative approximation of stochastic kinetic equations: from discrete to continuum

Zimo Hao, Khoa Lê, Chengcheng Ling

Abstract

We study the convergence of a generic tamed Euler-Maruyama (EM) scheme for the kinetic type stochastic differential equations (SDEs) (also known as second order SDEs) with singular coefficients in both weak and strong probabilistic senses. We show that when the drift exhibits a relatively low regularity compared to the state of the art, the singular system is well-defined both in the weak and strong probabilistic senses. Meanwhile, the corresponding tamed EM scheme is shown to converge at the rate of 1/2 in both the weak and the strong senses.

Quantitative approximation of stochastic kinetic equations: from discrete to continuum

Abstract

We study the convergence of a generic tamed Euler-Maruyama (EM) scheme for the kinetic type stochastic differential equations (SDEs) (also known as second order SDEs) with singular coefficients in both weak and strong probabilistic senses. We show that when the drift exhibits a relatively low regularity compared to the state of the art, the singular system is well-defined both in the weak and strong probabilistic senses. Meanwhile, the corresponding tamed EM scheme is shown to converge at the rate of 1/2 in both the weak and the strong senses.
Paper Structure (10 sections, 18 theorems, 169 equations)

This paper contains 10 sections, 18 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Notations
  4. Main results
  5. Tools
  6. Stochastic sewing
  7. Calculus on Anisotropic Besov spaces
  8. Density estimates
  9. Strong and weak well-posedness and stability of SDEs
  10. Quantitative Regularization of the singular functionals

Key Result

Proposition 2.4

For $s>0$, $q\in[1,\infty]$ and ${\boldsymbol{p}}\in[1,\infty]^2$, an equivalent norm of ${\mathbb B}^{s,q}_{{\boldsymbol{p}};{\boldsymbol{a}}}$ is given by where $[s]$ is the integer part of $s$. In particular, ${\mathbb C}^s_{{\boldsymbol{a}}}:=\mathbf{B}^{s,\infty}_{\infty;{\boldsymbol{a}}}$ is the anisotropic Hölder-Zygmund space, and for $s\in(0,1)$, there is a constant $C=C(\alpha,d,s)>0$ s

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.8
  • Lemma 3.1
  • Lemma 3.2: Bernstein's inequality
  • Lemma 3.3
  • Remark 3.4
  • ...and 27 more