The Link between Stochastic Differential Equations with Non-Markovian Coefficients and Backward Stochastic Partial Differential Equations

In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework.Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.

两比特海森堡自旋链体系中几何量子失协在非马尔可夫环境下的演化

利用量子态扩散方法,研究了在非马尔可夫环境下海森堡XXZ自旋模型中几何量子失协(GQD)的演化特性。分析了环境关联参数γ、xy平面上的耦合常数J_(xy)、z方向上的耦合常数J_(z)、外加磁场B以及DzyaloshinskiiMoriya(DM)相互作用对GQD动力学演化特性的影响。结果表明:γ越短,即环境的非马尔可夫效应越明显,越有利于有效提高系统的GQD;选取的初始态不同,J_(xy)和J_(z)对GQD起到的作用也不同;在非马尔可夫环境下,适当地增大B可提高GQD,而DM相互作用的引入会导致GQD衰减振荡。值得一提的是:可以从无纠缠初态系统有效地诱导出GQD。