BMO martingales and positive solutions of heat equations

Ying  Hu; Zhongmin  Qian

doi:10.3934/mcrf.2015.5.453

Article Contents

2015, Volume 5, Issue 3: 453-473. Doi: 10.3934/mcrf.2015.5.453

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BMO martingales and positive solutions of heat equations

Ying Hu^1, and
Zhongmin Qian^2,

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

Received: May 31, 2014

Revised: February 28, 2015

Published: August 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 60H10.

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