Quantitative approximation of stochastic kinetic equations: from discrete to continuum

TL;DR

This paper demonstrates the convergence of a tamed Euler-Maruyama scheme for kinetic stochastic differential equations with low-regularity drift, establishing a convergence rate of 1/2 in both weak and strong senses.

Contribution

It introduces a tamed EM scheme for singular kinetic SDEs and proves its convergence with explicit rates under low regularity conditions.

Findings

Convergence rate of 1/2 in weak sense

Convergence rate of 1/2 in strong sense

Well-posedness of singular kinetic SDEs with low regularity

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.