We consider Kolmogorov operators with constant diffusion matrices and linear drifts, i.e., Ornstein–Uhlenbeck operators, and show that all solutions to the corresponding stationary Fokker–Planck–Kolmogorov equations (including signed solutions) are invariant measures for the generated semigroups. This also gives a relatively explicit description of all solutions.

中文翻译：

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退化 Ornstein-Uhlenbeck 算子的柯尔莫哥洛夫方程

我们考虑具有恒定扩散矩阵和线性漂移的柯尔莫哥洛夫算子，即 Ornstein-Uhlenbeck 算子，并表明相应的平稳 Fokker-Planck-Kolmogorov 方程的所有解（包括符号解）都是生成的半群的不变测度。这也对所有解决方案给出了相对明确的描述。