BMO Martingales and Positive Solutions of Heat Equations

Ying Hu (1) , Zhongmin Qian (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 MI - Mathematical Institute [Oxford]

In this paper, we establish two new estimates concerning positive solutions of a linear parabolic equation. More precisely, we give explicit 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.

Domaines: Probabilités [math.PR]

Ying Hu, Zhongmin Qian. BMO Martingales and Positive Solutions of Heat Equations. 2012. ⟨hal-00662973v1⟩