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

Let L=Δ−⟨∇ϕ,∇·⟩$L=\Delta -\langle \nabla \phi , \nabla \cdot \rangle$ be a symmetric diffusion operator with an invariant measure μ(dx)=e−ϕ(x)m(dx)$\mu (\mathrm{d}x)=e^{-\phi (x)}\mathfrak {m} (\mathrm{d}x)$ on a complete non‐compact smooth Riemannian manifold (M,g)$(M,g)$ with its volume element m=volg$\mathfrak {m} =\text{\rm vol}_g$, and ϕ∈C2(M)$\phi \in C^2(M)$ a potential function. In this paper, we prove a Laplacian comparison theorem on weighted complete Riemannian manifolds with CD(K,m)${\rm CD}(K, m)$‐condition for m⩽1$m\leqslant 1$ and a continuous function K$K$. As consequences, we give the optimal conditions on m$m$‐Bakry–Émery Ricci tensor for m⩽1$m\leqslant 1$ such that the (weighted) Myers' theorem, Bishop–Gromov volume comparison theorem, stochastic completeness and Feller property of L$L$‐diffusion processes hold on weighted complete Riemannian manifolds. Some of these results were well studied for m$m$‐Bakry–Émery Ricci curvature for m⩾n$m\geqslant n$ (Li, J. Math. Pures Appl. (9) 84 (2005), 1295–1361; Lott, Comment. Math. Helv. 78 (2003), 865–883; Qian, Q. J. Math. 48 (1987), 235–242; Wei and Wylie, J. Differential Geom. 83 (2009), 377–405) or m=1$m=1$ (Wylie, Trans. Amer. Math. Soc. 369 (2017), 6661–6681; Wylie and D. Yeroshkin, Preprint). When m