  In this paper, we develop a new approach to establish gradient estimates for
positive solutions to the heat equation of elliptic or subelliptic operators on
Euclidean spaces or on Riemannian manifolds. More precisely, we give some
estimates of the gradient of logarithm of a positive solution via the uniform
bound of the logarithm of the solution. Moreover, we give a generalized version
of Li-Yau's estimate. Our proof is based on the link between PDE and quadratic
BSDE. Our method might be useful to study some (nonlinear) PDEs.
