Abstract In this article we define and investigate statistical operators and an entropy functional for Bernstein stochastic processes associated with hierarchies of forward-backward systems of decoupled deterministic linear parabolic partial differential equations. The systems under consideration are defined on open bounded domains of Euclidean space where is arbitrary, and are subject to Neumann boundary conditions. We assume that the elliptic part of the parabolic operator in the equations is a self-adjoint Schrödinger operator, bounded from below and with compact resolvent in The statistical operators we consider are then trace-class operators defined from sequences of probabilities associated with the point spectrum of the elliptic part in question, which allow the distinction between pure and mixed processes. We prove in particular that the Bernstein processes of maximal entropy are those for which the associated sequences of probabilities are of Gibbs type. We illustrate our results by considering processes associated with a specific hierarchy of forward-backward heat equations defined in a two-dimensional disk.

中文翻译：

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关于最大熵的伯恩斯坦过程

摘要 在本文中，我们定义并研究了与解耦确定性线性抛物线偏微分方程的前向后向系统的层次结构相关的伯恩斯坦随机过程的统计算子和熵函数。所考虑的系统定义在欧几里得空间的开放有界域上，其中是任意的，并受诺依曼边界条件的约束。我们假设方程中抛物线算子的椭圆部分是一个自伴随薛定谔算子，从下面有界并且在 中具有紧凑的解算子 我们考虑的统计算子是从与点谱相关的概率序列定义的迹类算子所讨论的椭圆部分，它允许区分纯过程和混合过程。我们特别证明了最大熵的伯恩斯坦过程是那些相关的概率序列是吉布斯类型的过程。我们通过考虑与二维圆盘中定义的前向后向热方程的特定层次结构相关的过程来说明我们的结果。