In this article we investigate the properties of Bernstein processes generated by infinite hierarchies of forward-backward systems of decoupled linear deterministic parabolic partial differential equations defined in Rd, where d is arbitrary. An important feature of those systems is that the elliptic part of the parabolic operators may be realized as an unbounded Schr\"odinger operator with compact resolvent in standard L2-space. The Bernstein processes we are interested in are in general non-Markovian, may be stationary or non-stationary and are generated by weighted averages of measures naturally associated with the pure point spectrum of the operator. We also introduce time-dependent trace-class operators which possess most of the attributes of density operators in Quantum Statistical Mechanics, and prove that the statistical averages of certain bounded self-adjoint observables usually evaluated by means of such operators coincide with the expectation values of suitable functions of the underlying processes. In the particular case where the given parabolic equations involve the Hamiltonian of an isotropic system of quantum harmonic oscillators, we show that one of the associated processes is identical in law with the periodic Ornstein-Uhlenbeck process.

中文翻译：

---

关于由 Rd 中线性抛物线系统的层次结构生成的伯恩斯坦过程

在本文中，我们研究了由在 Rd 中定义的解耦线性确定性抛物线偏微分方程的无限层次结构产生的伯恩斯坦过程的性质，其中 d 是任意的。这些系统的一个重要特征是抛物线算子的椭圆部分可以实现为标准 L2 空间中具有紧解元的无界 Schr\"odinger 算子。我们感兴趣的 Bernstein 过程通常是非马尔可夫的，可以是平稳的或非平稳的，并且由与算子的纯点谱自然相关的度量的加权平均生成。我们还引入了具有量子统计力学中密度算子的大部分属性的时间相关迹类算子，并证明通常通过此类算子评估的某些有界自伴随观测值的统计平均值与底层过程的合适函数的期望值一致。在给定的抛物线方程涉及量子谐振子各向同性系统的哈密顿量的特殊情况下，我们表明相关过程之一与周期性 Ornstein-Uhlenbeck 过程在定律上是相同的。