关于由杀死Feller过程生成的半群的Chernoff近似以及时间分数阶Fokker-Planck-Kolmogorov方程的Feynman公式
Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker-Planck-Kolmogorov equations
AI总结：
本文研究了由离开给定领域时被杀死的Feller过程生成的半群，这些半群对应于一类演化方程的Cauchy-Dirichlet型初始-外部值问题。通过Chernoff定理对这些半群进行近似，并将构造的Chernoff近似转化为Feynman公式，即n重积分的极限。此外，提出了一种近似时间分数阶（包括分布阶时间分数阶）演化方程解的方法，该方法基于时间分数阶与时间非分数阶演化方程之间的联系以及Chernoff近似。进一步，该方法导致时间分数阶演化方程解的Feynman公式。通过考虑一类分布阶时间分数阶扩散方程，获得了对应的Cauchy和Cauchy-Dirichlet问题的Feynman公式。
摘要：
考虑由离开给定领域时被杀死的Feller过程生成的半群。这些半群对应于该领域内一类演化方程的Cauchy-Dirichlet型初始-外部值问题。通过Chernoff定理对这些半群进行近似。对于一类被杀死的Feller过程，所构造的Chernoff近似转化为Feynman公式，即某些函数的n重积分随着n趋向于无穷大时的极限。利用Feynman公式表示演化方程的解可用于直接计算和模拟底层的随机过程。此外，提出了一种近似时间分数阶（包括分布阶时间分数阶）演化方程解的方法。该方法基于时间分数阶与时间非分数阶演化方程之间的联系以及Chernoff近似。此外，该方法导致时间分数阶演化方程解的Feynman公式。为了说明该方法，考虑了一类分布阶时间分数阶扩散方程；获得了对应Cauchy和Cauchy-Dirichlet问题的Feynman公式。
英文摘要：
Semigroups, generated by Feller processes killed upon leaving a given domain, are considered. These semigroups correspond to Cauchy-Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation converts into a Feynman formula, i.e. into a limit of $n$-fold iterated integrals of certain functions as $n\to\infty$. Representations of solutions of evolution equations by Feynman formulae can be used for direct calculations and simulation of underlying stochasstic processes. Further, a method to approximate solutions of time-fractional (including distributed order time-fractional) evolution equations is suggested. This method is based on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. Moreover, this method leads to Feynman formulae for solutions of time-fractional evolution equations. To illustrate the method, a class of distributed order time-fractional diffusion equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy-Dirichlet problems are obtained.
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