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
References:
[1]

D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality,, Rev. Mat. Iberoam., 22 (2006), 683. doi: 10.4171/RMI/470. Google Scholar

[2]

D. Bakry and Z. Qian, Harnack inequalities on a manifold with positive or negative Ricci curvature,, Rev. Mat. Iberoamericana, 15 (1999), 143. doi: 10.4171/RMI/253. Google Scholar

[3]

P. Baxendale, Brownian motions in the diffeomorphism group. I,, Compositio Math., 53 (1984), 19. Google Scholar

[4]

J. M. Bismut, Théorie probabiliste du contrôle des diffusions,, Mem. Amer. Math. Soc., 4 (1976). Google Scholar

[5]

J. M. Bismut, Mécanique Aléatoire,, Lecture Notes in Mathematics, 866 (1981). Google Scholar

[6]

P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value,, Probab. Theory Related Fields, 136 (2006), 604. doi: 10.1007/s00440-006-0497-0. Google Scholar

[7]

F. Delbaen, Y. Hu and X. Bao, Backward SDEs with superquadratic growth,, Probab. Theory Related Fields, 150 (2011), 145. doi: 10.1007/s00440-010-0271-1. Google Scholar

[8]

H. Donnelly and P. Li, Lower bounds for the eigenvalues of Riemannian manifolds,, Michigan Math. J., 29 (1982), 149. doi: 10.1307/mmj/1029002668. Google Scholar

[9]

J. L. Doob, Stochastic Processes,, John Wiley & Sons, (1953). Google Scholar

[10]

J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Reprint of the 1984 edition,, Classics in Mathematics. Springer-Verlag, (2001). doi: 10.1007/978-3-642-56573-1. Google Scholar

[11]

R. Durrett, Brownian Motion and Martingales in Analysis. Wadsworth Mathematics Series,, Wadsworth International Group, (1984). Google Scholar

[12]

J. Eells and K. D. Elworthy, Stochastic dynamical systems,, Control theory and topics in functional analysis (I nternat. Sem., III (1976), 179. Google Scholar

[13]

S. Hamilton, A matrix Harnack estimate for the heat equation,, Comm. Anal. Geom., 1 (1993), 113. Google Scholar

[14]

L. Hörmander, Hypoelliptic second order differential equations,, Acta Math., 119 (1967), 147. doi: 10.1007/BF02392081. Google Scholar

[15]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland Mathematical Library, 24 (1981). Google Scholar

[16]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I. Reprint of the 1963 original,, Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, (1996). Google Scholar

[17]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth,, Ann. Probab., 28 (2000), 558. doi: 10.1214/aop/1019160253. Google Scholar

[18]

H. Kunita, On the representation of solutions of stochastic differential equations,, Seminar on Probability, 784 (1978), 282. Google Scholar

[19]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, (1968). Google Scholar

[20]

P. Li and S. T. Yau, On the parabolic kernel of the S chrödinger operator,, Acta Math., 156 (1986), 153. doi: 10.1007/BF02399203. Google Scholar

[21]

X. D. Li, Hamilton's Harnack inequality and the W-entropy formula on complete Riemannian manifolds,, , (2014). Google Scholar

[22]

G. M. Lieberman, Second Order Parabolic Differential Equations,, {World Scientific Publishing Co., (1996). doi: 10.1142/3302. Google Scholar

[23]

P. Malliavin, Géométrie Différentielle Stochastique. Séminaire de Mathéematiques Supérieures,, 64. Presses de l'Université de Montréal, 64 (1978). Google Scholar

[24]

M. A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem,, Finance Stoch., 13 (2009), 121. doi: 10.1007/s00780-008-0079-3. Google Scholar

[25]

É. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55. doi: 10.1016/0167-6911(90)90082-6. Google Scholar

[26]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,, Stochastics Stochastics Rep., 37 (1991), 61. doi: 10.1080/17442509108833727. Google Scholar

[27]

S. T. Yau, On the Harnack inequalities of partial differential equations,, Comm. Anal. Geom., 2 (1994), 431. Google Scholar

[28]

S. T. Yau, Harnack inequality for non-self-adjoint evolution equations,, Math. Res. Lett., 2 (1995), 387. doi: 10.4310/MRL.1995.v2.n4.a2. Google Scholar

show all references

References:
[1]

D. Bakry and M. Ledoux, A logarithmic Sobolev form of the Li-Yau parabolic inequality,, Rev. Mat. Iberoam., 22 (2006), 683. doi: 10.4171/RMI/470. Google Scholar

[2]

D. Bakry and Z. Qian, Harnack inequalities on a manifold with positive or negative Ricci curvature,, Rev. Mat. Iberoamericana, 15 (1999), 143. doi: 10.4171/RMI/253. Google Scholar

[3]

P. Baxendale, Brownian motions in the diffeomorphism group. I,, Compositio Math., 53 (1984), 19. Google Scholar

[4]

J. M. Bismut, Théorie probabiliste du contrôle des diffusions,, Mem. Amer. Math. Soc., 4 (1976). Google Scholar

[5]

J. M. Bismut, Mécanique Aléatoire,, Lecture Notes in Mathematics, 866 (1981). Google Scholar

[6]

P. Briand and Y. Hu, BSDE with quadratic growth and unbounded terminal value,, Probab. Theory Related Fields, 136 (2006), 604. doi: 10.1007/s00440-006-0497-0. Google Scholar

[7]

F. Delbaen, Y. Hu and X. Bao, Backward SDEs with superquadratic growth,, Probab. Theory Related Fields, 150 (2011), 145. doi: 10.1007/s00440-010-0271-1. Google Scholar

[8]

H. Donnelly and P. Li, Lower bounds for the eigenvalues of Riemannian manifolds,, Michigan Math. J., 29 (1982), 149. doi: 10.1307/mmj/1029002668. Google Scholar

[9]

J. L. Doob, Stochastic Processes,, John Wiley & Sons, (1953). Google Scholar

[10]

J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Reprint of the 1984 edition,, Classics in Mathematics. Springer-Verlag, (2001). doi: 10.1007/978-3-642-56573-1. Google Scholar

[11]

R. Durrett, Brownian Motion and Martingales in Analysis. Wadsworth Mathematics Series,, Wadsworth International Group, (1984). Google Scholar

[12]

J. Eells and K. D. Elworthy, Stochastic dynamical systems,, Control theory and topics in functional analysis (I nternat. Sem., III (1976), 179. Google Scholar

[13]

S. Hamilton, A matrix Harnack estimate for the heat equation,, Comm. Anal. Geom., 1 (1993), 113. Google Scholar

[14]

L. Hörmander, Hypoelliptic second order differential equations,, Acta Math., 119 (1967), 147. doi: 10.1007/BF02392081. Google Scholar

[15]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes,, North-Holland Mathematical Library, 24 (1981). Google Scholar

[16]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I. Reprint of the 1963 original,, Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, (1996). Google Scholar

[17]

M. Kobylanski, Backward stochastic differential equations and partial differential equations with quadratic growth,, Ann. Probab., 28 (2000), 558. doi: 10.1214/aop/1019160253. Google Scholar

[18]

H. Kunita, On the representation of solutions of stochastic differential equations,, Seminar on Probability, 784 (1978), 282. Google Scholar

[19]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, (1968). Google Scholar

[20]

P. Li and S. T. Yau, On the parabolic kernel of the S chrödinger operator,, Acta Math., 156 (1986), 153. doi: 10.1007/BF02399203. Google Scholar

[21]

X. D. Li, Hamilton's Harnack inequality and the W-entropy formula on complete Riemannian manifolds,, , (2014). Google Scholar

[22]

G. M. Lieberman, Second Order Parabolic Differential Equations,, {World Scientific Publishing Co., (1996). doi: 10.1142/3302. Google Scholar

[23]

P. Malliavin, Géométrie Différentielle Stochastique. Séminaire de Mathéematiques Supérieures,, 64. Presses de l'Université de Montréal, 64 (1978). Google Scholar

[24]

M. A. Morlais, Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem,, Finance Stoch., 13 (2009), 121. doi: 10.1007/s00780-008-0079-3. Google Scholar

[25]

É. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation,, Systems Control Lett., 14 (1990), 55. doi: 10.1016/0167-6911(90)90082-6. Google Scholar

[26]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations,, Stochastics Stochastics Rep., 37 (1991), 61. doi: 10.1080/17442509108833727. Google Scholar

[27]

S. T. Yau, On the Harnack inequalities of partial differential equations,, Comm. Anal. Geom., 2 (1994), 431. Google Scholar

[28]

S. T. Yau, Harnack inequality for non-self-adjoint evolution equations,, Math. Res. Lett., 2 (1995), 387. doi: 10.4310/MRL.1995.v2.n4.a2. Google Scholar

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