How is the definition of heat kernels generalized to operators acting on $\mathbf{R}^n$-valued functions?

Question

If we consider an operator $A\colon C^\infty(\mathbf{R}^n,\mathbf{R})\to C^\infty(\mathbf{R}^n,\mathbf{R})$, the heat kernel is a function $$k\colon (0,\infty)\times\mathbf{R}^n\times\mathbf{R}^n\to\mathbf{R}$$ satisfying \begin{equation}\tag{1} \frac{\partial}{\partial t}k+Ak=0 \end{equation} and \begin{equation}\tag{2} \lim_{t \to 0}\int_{\mathbf{R}^n} k(t,x,y)\phi(y)\,dy = \phi(x) \end{equation} for all test functions $\phi\colon \mathbf{R}^n\to\mathbf{R}$.$^1$

Now consider an operator $H\colon C^\infty(\mathbf{R}^m,\mathbf{R}^n)\to C^\infty(\mathbf{R}^m,\mathbf{R}^n)$. To generalize ($2$), the heat kernel needs to be a function $$Q\colon (0,\infty)\times\mathbf{R}^m\times\mathbf{R}^m\to\mathbf{R}^{n\times n}$$ but then it is not clear how to generalize ($1$) since the expression $HQ$ is not defined.

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$^1$ In ($1$), $Ak\colon (0,\infty)\times\mathbf{R}^n\times\mathbf{R}^n\to\mathbf{R}$ is defined by $(Ak)(t,x,y):=(Ak(t,\,\cdot\,,y))(x)$

Answer 1

The generalization is straightforward if one knows the correct definition of heat kernels. Equation $(1)$ was supposed to generalize the definition of the heat kernel of the Laplacian, but the heat kernel is actually defined by the following property (in the case of the Laplacian, it can be shown that $(1)$ implies this property):

Consider some test function $\phi_0\in C^\infty(\mathbf{R}^n,\mathbf{R})$. Then the family $$\phi\colon(0,\infty)\to C^\infty(\mathbf{R}^n,\mathbf{R})$$ defined by $$\phi_t(x)=\int k(t,x,y)\phi(y) dy$$ is the solution to the heat equation with initial state $\phi_0$: \begin{equation} \dot{\phi}+A\phi=0\land \lim_{t\to 0}\phi_t=\phi_0 \end{equation}