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Let $L=\Delta-\nabla\phi\cdot \nabla$ be a symmetric diffusion operator withan invariant measure $\mu({\rm} d x)=e^{-\phi(x)}{\mathfrak m}({\rm d} x)$ on acomplete non-compact smooth Riemannian manifold $(M,g)$ with its volume element${\mathfrak m}={\rm vol}_g$, and $\phi\in C^2(M)$ a potential function. In thispaper, we prove a Laplacian comparison theorem on weighted complete Riemannianmanifolds with ${\rm CD}(K, m)$-condition for $m\leq 1$ and a continuousfunction $K$. As consequences, we give the optimal conditions on$m$-Bakry-\'Emery Ricci tensor for $m\leq1$ such that the (weighted) Myers'theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, andthe Cheeger-Gromoll type splitting theorem, stochastic completeness and Fellerproperty of $L$-diffusion processes hold on weighted complete Riemannianmanifolds. Some of these results were well-studied for $m$-Bakry-\'Emery Riccicurvature for $m\geq n$ (\!\!\cite{Lot,Qian,XDLi05, WeiWylie}) or $m=1$(\!\!\cite{Wylie:WarpedSplitting, WylieYeroshkin}). When $m<1$, our results arenew in the literature.

New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds