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Heat kernels for generalized Ornstein–Uhlenbeck operators

In this article, we use the Hamiltonian and Lagrangian formalism to study the -dimensional extended Ornstein–Uhlenbeck operator where is an real matrix, is a real parameter and means the gradient. Given the boundary conditions, we find the solutions of the associated Hamiltonian system of . 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.