随机Langevin方程的分裂AVF格式的密度收敛性
Convergence in Density of Splitting AVF Scheme for Stochastic Langevin Equation
AI总结：
本文研究随机Langevin方程分裂AVF格式数值解的密度函数，证明其存在性与光滑性，并利用Malliavin微积分得出密度收敛阶等于最优强收敛阶。
摘要：
本文研究了随机Langevin方程的分裂平均向量场（AVF）格式的数值解的密度函数。为了处理所考虑方程中的非全局单调系数，我们首先给出了精确解和数值解的指数可积性性质。然后，通过证明其在Malliavin意义下的均匀非退化性，我们展示了数值解密度函数的存在性和光滑性。为了分析精确解与数值解密度函数之间的近似误差，我们利用Malliavin微积分推导了数值格式在每个Malliavin--Sobolev范数下的最优强收敛阶。结合Donsker delta函数的逼近结果和密度函数的光滑性，我们证明了密度的收敛阶与数值格式的最优强收敛阶一致。
英文摘要：
In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. To deal with the non-globally monotone coefficient in the considered equation, we first present the exponential integrability properties of the exact and numerical solutions. Then we show the existence and smoothness of the density function of the numerical solution by proving its uniform non-degeneracy in Malliavin sense. In order to analyze the approximate error between the density function of the exact solution and that of the numerical solution, we derive the optimal strong convergence rate in every Malliavin--Sobolev norm of the numerical scheme via Malliavin calculus. Combining the approximation result of Donsker's delta function and the smoothness of the density functions, we prove that the convergence rate in density coincides with the optimal strong convergence rate of the numerical scheme.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
