Rd中线性抛物型方程组族生成的Bernstein过程
On Bernstein processes generated by hierarchies of linear parabolic systems in Rd
摘要：
本文研究了定义在Rd中的线性确定性抛物型偏微分方程正倒向解的无穷族生成的Bernstein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算子的椭圆部分可以在标准L2-空间中实现为具有紧预解式的无界薛定谔算子。我们感兴趣的伯恩斯坦过程一般是非马尔可夫的，可能是平稳的或非平稳的，并产生的加权平均的措施自然与纯点谱的运营商。我们还引入了具有量子统计力学中密度算符的大部分属性的含时迹类算符，并证明了通常用这类算符计算的某些有界自伴观测量的统计平均值与相应过程函数的期望值一致.在特定的情况下，给定的抛物型方程涉及量子谐振子的各向同性系统的哈密顿量，我们证明了相关的过程之一是相同的规律与周期Ornstein-Uhlenbeck过程。
英文摘要：
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 Schrödinger 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.
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