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 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.

Mots clés: BMO martingale; gradient estimate; heat equation; Li-Yau's estimate; quadratic BSDE

Domaines: Probabilités [math.PR]

Ying Hu, Zhongmin Qian. BMO Martingales and Positive Solutions of Heat Equations. Mathematical Control and Related Fields, 2015, 5 (3), pp.453-473. ⟨10.3934/mcrf.2015.5.453⟩. ⟨hal-00662973v2⟩