Volterra parametrix correction ($H=P+P\ast {\sum }_{k\ge 1}(-1{)}^k{R}^{\ast k}$)

If $R=({\partial }_t+{\Delta }_x)P$ is smooth and bounded up to time zero on a closed manifold, the Volterra convolution of kernels solves $Q+R+R\ast Q=0$ by $Q={\sum }_{k\ge 1}(-1{)}^k{R}^{\ast k}$. Bounds by $C^k\mathrm{vol}(M{)}^{k-1}{t}^{k-1}/(k-1)!$ prove convergence. Then $H=P+P\ast Q$ is the exact Riemannian heat kernel and has the same initial delta limit.