Recently, I am studying heat semigroup for noncompact manifolds with boundary.

In Issac Chavel's book "eigenvalues in Riemannian geometry". "Given a noncompact Riemannian manifold, it need not be complete. He construct heat kernel as follows: Pick an exhaustion $\Omega_1, \Omega_2,\cdots$, of M by regular domains. namely, $$\overline{\Omega_j} \subset \Omega_{j+1} \mbox{and} \mathop \cup \limits_{j = 1}^\infty {\Omega _j} = M $$ Let $q_j$ be the Dirichlet heat kernel of $\Omega _j$. We think $q_j$ as defined on $M \times M \times (0,\infty)$, vanishing identically whenever at least one of the space variables is in $M\ \Omega_j$. $q_j$ is an inceasing sequence, we may define $q=\mathop {\lim }\limits_{j \to \infty }q_j$.

We can prove q is a heat kernel for M. And when M is complete with Ricci curvature from bounded below, we can prove the uniqueness and M is stochastically complete."

What if M is a non-compact Riemannian manifold with boundary? (Since the construction above does not concern the boundary of M). Suppose in addition M has Ricci curvature bounded below, then M is stochastically complete?

I read Yau's book for the heat kernel estimate for manifolds with Ricci curvature bounded below. He only gives the estimates for compact manifolds with boundary and noncompact manifolds without boundary, why does he not deal with non-compact manifolds with boundary?