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

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