在這篇文章，我們使用漢米爾頓與拉格朗日形式來研究這個二
維廣義Ornstein-Uhlenbeck 算子
給定邊界條件後，我們解出這個算子所對應的漢米爾頓系統。然後
利用拉格朗日函數來建構動能函數，利用范夫來克公式得到熱核的
量能項。最後，我們討論此算子的正則與奇異區域。

In this thesis, we use the Hamiltonian and Lagrangian formalism to study the two-dimensional generalized Ornstein-Uhlenbeck operator Given the boundary conditions, we find the solutions of the associated Hamiltonian system of this operator. Then we construct the action function by the Lagrangian function and use the van Vleck's formula to obtain the volume element of the heat kernel. Finally, we discuss the regular and singular regions of this operator.

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