Heat kernel on manifold with boundary

Question

Let $M$ be a Riemannian manifold with boundary, let $\mathcal{V}$ be a metric vector bundle over $M$ and let $L$ be a formally self-adjoint Laplace type operator, acting on sections of $\mathcal{V}$. Suppose that $L$ is endowed with local elliptic boundary conditions $B$.

I would like to have a reference for the following statement: If $p_t(x, y)$ is a time-dependent section of the bundle $\mathcal{V} \boxtimes \mathcal{V}^*$ that is $C^2$ in the space variables and $C^1$ in the time variable such that

1. $p_t(x, y)$ satisfies the heat equation with respect to the $x$ variable, for each $y$ fixed.
2. We have $\int_M p_t(x, y) u(y) \mathrm{d} y \longrightarrow u(x)$ for each smooth section $u$ of $\mathcal{V}$, as $t \rightarrow 0$.
3. $p_t(x, y)$ satisfies the boundary condition in the $x$ variable for each fixed $y$.

Then $p_t(x, y)$ is the heat kernel of $(L, B)$.

I can't find a reference...