In this talk, we discuss the heat kernel for the second-order operator \[ \Delta=\frac{1}{2}\sum_{k=1}^n (\frac{\partial}{\partial x_k})^2 +\frac{1}{2}\sum_{k=1}^n(x_k^{m_k}\frac{\partial}{\partial y_k})^2 \] with \(m_k\in\mathbb{N}\), which is a degenerate elliptic operator. Obviously, this operator is closed related to the Grushin operator \[ L_G=\frac{1}{2}(\frac{\partial}{\partial x})^2 +\frac{1}{2}(x^m\frac{\partial}{\partial y})^2 \] with \(m\in\mathbb{N}\). In this talk, we study the formula for the heat kernel of the diffusion operator \(\frac{\partial}{\partial t}-\Delta\). The formula involves an integral of a product between the volume function and an exponential term. We also discuss the small time asymptotic expansions of the heat kernel.

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