BMO martingales and positive solutions of heat equations
Ying Hu Zhongmin Qian
Mathematical Control & Related Fields 2015, 5(3): 453-473 doi: 10.3934/mcrf.2015.5.453
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.
keywords: quadratic BSDE. BMO martingale heat equation gradient estimate Li-Yau's estimate

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