Operators on nonlinear metric measure spaces I: A New Laplacian comparison theorem on Finsler manifolds and a generalized Li-Yau approach to gradient estimates of Finslerian Schr\"odinger equation

This paper is the first in a series of paper where we describe the differential operators on general nonlinear metric measure spaces, namely, the Finsler spaces. We try to propose a general method for gradient estimates of the positive solutions to nonlinear parabolic (or elliptic) equations on both compact and noncompact forward complete Finsler manifolds. The key of this approach is to find a valid comparison theorem, and particularly, to find related suitable curvature conditions. Actually, we define some non-Riemannian tensors and a generalization of the weighted Ricci curvature, called the mixed weighted Ricci curvature. With the assistance of these concepts, we prove a new kind of Laplacian comparison theorem. As an illustration of such method, we apply it to Finslerian Schr\"odinger equation. The Schr\"odinger equation is a typical class of parabolic equations, and the gradient estimation of their positive solution is representative. In this paper, based on the new Laplacian comparison theorem, we give global and local Li-Yau type gradient estimates for positive solutions to the Finslerian Schr\"odinger equation with respect to two different mixed weighted Ricci curvature conditions.