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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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).

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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).

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

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