Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be generalizable to construct a heat kernel for $D$. Has this already been done or tried?
Details of the Setup
The two sources I found most informative about the problem are "Eigenvalues in Riemannian Geometry" (Chavel), that constructs the heat kernel for the Laplace-Beltrami operator, and "Heat Kernels and Dirac Operators" (Berline, Getzler, Vergne), that constructs the heat kernel for generalized Laplacians in the compact case.
I assume $M$ to be a complete non-compact Riemannian manifold with Ricci curvature bounded from below. This is the setting of Chavel; the boundedness of the Ricci curvature is only used to prove uniqueness of the heat kernel, and is likely not to be enough for generalized Laplacians.
A generalized Laplacian is a linear second-order differential operator with smooth coefficients, whose second-order part equals the second-order part of the Laplace-Beltrami operator.
Overview of the construction and why I think it can be generalized
Chavel builds the heat kernel in the non-compact case as the limit of Dirichlet heat kernels on a sequence of regular domains exhausting $M$, and he constructs the heat kernels on regular domains by heat kernels on compact manifolds. The main ingredient that Chavel uses for the construction of the heat kernel on regular domains is that the heat kernel in the compact case is smooth and almost Euclidean, and the main ingredient he uses to pass from regular domains to non-compact manifolds is the strong maximum principle for solutions of the heat equation. But Berline-Getzer-Vergne construct a smooth heat kernel for generalized Laplacians, that I think to be also almost Euclidean, and the strong maximum principle holds in general for elliptic operators.
For me $M$ is a modular curve equipped with the hyperbolic metric, and $D$ is the Laplacian acting on the space of smooth modular forms of a fixed positive weight on $M$, i.e. smooth sections of powers of the Hodge bundle equipped with the Petersson metric. Thanks to a regularization procedure I would also be happy to have heat kernels for a generalized Laplacian with "mildly singular" first order coefficients on a compact Riemannian manifold.
A notable achievement is the construction of the heat kernel in this situation by Fay in "Fourier coefficients of the resolvent for a Fuchsian subgroup", while this is enough to indicate that such an heat kernel should in general exists, the construction as in Chavel would be better for me, since I aim to prove some convergence results on it via metric regularization.
Thank you for your help!