# American Institute of Mathematical Sciences

September  2015, 5(3): 453-473. doi: 10.3934/mcrf.2015.5.453

## BMO martingales and positive solutions of heat equations

 1 IRMAR, Université Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex 2 Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

Received  June 2014 Revised  March 2015 Published  July 2015

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.
Citation: Ying Hu, Zhongmin Qian. BMO martingales and positive solutions of heat equations. Mathematical Control & Related Fields, 2015, 5 (3) : 453-473. doi: 10.3934/mcrf.2015.5.453
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