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September  2012, 11(5): 1587-1614. doi: 10.3934/cpaa.2012.11.1587

Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations

Andrea Bonfiglioli 1, and  Ermanno Lanconelli 1,

1. 

Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato, 5 - 40126 Bologna, Italy, Italy

Received  January 2011 Revised  January 2012 Published  March 2012

If $\mathcal{L}=\sum_{j=1}^m X_j^2+X_0$ is a Hörmander partial differential operator in $\mathbb{R}^N$, we give sufficient conditions on the $X_j$'s for the existence of a Lie group structure $\mathbb{G}=(\mathbb{R}^N,*)$, not necessarily nilpotent, such that $\mathcal{L}$ is left invariant on $\mathbb{G}$. We also investigate the existence of a global fundamental solution $\Gamma$ for $\mathcal{L}$, providing results ensuring a suitable left invariance property of $\Gamma$. Examples are given for operators $\mathcal{L}$ to which our results apply: some are new, some appear in recent literature, usually quoted as Kolmogorov-Fokker-Planck type operators.
Keywords: Kolmogorov-Fokker-Planck operator., fundamental solution, Hörmander operator.
Mathematics Subject Classification: Primary: 35J70, 43A80; Secondary: 37C1.
Citation: Andrea Bonfiglioli, Ermanno Lanconelli. Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1587-1614. doi: 10.3934/cpaa.2012.11.1587
References:
[1]

J. August and S. W. Zucker, Sketches with curvature: The curve indicator random field and Markov processes,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 387.  doi: 10.1109/TPAMI.2003.1190567.  Google Scholar

[2]

A. Bonfiglioli, Homogeneous Carnot groups related to sets of vector fields,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 7 (2004), 79.   Google Scholar

[3]

A. Bonfiglioli, An ODE's version of the formula of Baker, Campbell, Dynkin and Hausdorff and the construction of Lie groups with prescribed Lie algebra,, Mediterr. J. Math., 7 (2010), 387.  doi: 10.1007/s00009-010-0064-x.  Google Scholar

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A. Bonfiglioli and R. Fulci, "Topics in Noncommutative Algebra. The Theorem of Campbell, Baker, Hausdorff and Dynkin,", Lecture Notes in Mathematics, 2034 (2011).  doi: 10.1007/978-3-642-22597-0.  Google Scholar

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A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups,, Adv. Differ. Equ., 7 (2002), 1153.   Google Scholar

[6]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory for their sub-Laplacians,", Springer Monographs in Mathematics \textbf{26}, 26 (2007).   Google Scholar

[7]

J.-M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés,, Ann. Inst. Fourier (Grenoble), 19 (1969), 277.   Google Scholar

[8]

M. Bramanti, Singular integrals in nonhomogenoeus spaces: $L^2$ and $L^p$ continuity from Hölder Estimates,, Rev. Mat. Iberoamericana, 26 (2010), 347.   Google Scholar

[9]

M. Bramanti, G. Cupini, E. Lanconelli and E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators,, Math. Z., 266 (2010), 789.  doi: 10.1007/s00209-009-0599-3.  Google Scholar

[10]

G. Da Prato, "Kolmogorov Equations for Stochastic PDE's,", Advanced Courses in Mathematics, 9 (2004).   Google Scholar

[11]

G. Da Prato and A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients,, J. Evol. Equ., 7 (2007), 587.   Google Scholar

[12]

G. Da Prato and A. Lunardi, On a class of self-adjoint elliptic operators in $L^2$ spaces with respect to invariant measures,, J. Differ. Equations, 234 (2007), 54.   Google Scholar

[13]

A. Eggert, Extending the Campbell-Hausdorff multiplication,, Geom. Dedicata, 46 (1993), 35.  doi: 10.1007/BF01264092.  Google Scholar

[14]

B. Farkas and A. Lunardi, Maximal regularity for Kolmogorov operators in $L^2$ spaces with respect to invariant measures,, J. Math. Pures Appl., 86 (2006), 310.   Google Scholar

[15]

C. L. Fefferman and A. Sánchez-Calle, Fundamental solutions for second order subelliptic operators,, Ann. Math., 124 (1986), 247.  doi: 10.2307/1971278.  Google Scholar

[16]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups,, Ark. Mat., 13 (1975), 161.  doi: 10.1007/BF02386204.  Google Scholar

[17]

G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups,", Mathematical Notes, 28 (1982).   Google Scholar

[18]

B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients,, Ann. Sc. Norm. Super. Pisa, 4 (1983), 523.   Google Scholar

[19]

C. E. Gutiérrez and E. Lanconelli, Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for $X$-elliptic operators,, Commun. Partial Differ. Equations, 28 (2003), 1833.   Google Scholar

[20]

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

[21]

D. Jerison and A. Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields,, Indiana Univ. Math. J., 35 (1986), 835.   Google Scholar

[22]

A. E. Kogoj and E. Lanconelli, An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations,, Mediterr. J. Math., 1 (2004), 51.  doi: 10.1007/s00009-004-0004-8.  Google Scholar

[23]

A. E. Kogoj and E. Lanconelli, Link of groups and homogeneous Hörmander operators,, Proc. Am. Math. Soc., 135 (2007), 2019.  doi: 10.1090/S0002-9939-07-08646-7.  Google Scholar

[24]

A. N. Kolmogorov, Zufällige Bewegungen,, Ann. of Math., 35 (1934), 116.  doi: 10.2307/1968123.  Google Scholar

[25]

S. Kusuoka and D. Stroock, The partial Malliavin calculus and its application to nonlinear filtering,, Stochastics, 12 (1984), 83.   Google Scholar

[26]

S. Kusuoka and D. Stroock, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator,, Ann. of Math., 127 (1988), 165.  doi: 10.2307/1971418.  Google Scholar

[27]

E. Lanconelli and A. E. Kogoj, $X$-elliptic operators and $X$-control distances,, Ricerche Mat., 49 (2000), 223.   Google Scholar

[28]

E. Lanconelli and A. Pascucci, On the fundamental solution for hypoelliptic second order partial differential equations with non-negative characteristic form,, Ricerche Mat., 48 (1999), 81.   Google Scholar

[29]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators,, Rend. Semin. Mat. Torino, 52 (1994), 29.   Google Scholar

[30]

A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $\mathbbR^n$,, Ann. Sc. Norm. Super. Pisa, 24 (1997), 133.   Google Scholar

[31]

D. Mumford, Elastica and computer vision,, in, (1994), 491.  doi: 10.1007/978-1-4612-2628-4.  Google Scholar

[32]

A. Nagel, F. Ricci and E. M. Stein, Fundamental solutions and harmonic analysis on nilpotent groups,, Bull. Am. Math. Soc., 23 (1990), 139.  doi: 10.1090/S0273-0979-1990-15920-8.  Google Scholar

[33]

F. Nazarov, S. Treil and A. Volberg, The $Tb$-theorem on non-homogeneous spaces,, Acta Math., 190 (2003), 151.  doi: 10.1007/BF02392690.  Google Scholar

[34]

P. Negrini and V. Scornazzani, Superharmonic functions and regularity of boundary points for a class of elliptic-parabolic partial differential operators,, Boll. Unione Mat. Ital., 3 (1984), 85.   Google Scholar

[35]

L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups,, Acta Math., 137 (1976), 247.  doi: 10.1007/BF02392419.  Google Scholar

[36]

V. S. Varadarajan, "Lie Groups, Lie Algebras and their Representations,", Graduate Texts in Mathematics, (1984).   Google Scholar

[37]

Y. Wang, Y. Zhou, D. K. Maslen and G. S. Chirikjian, Solving phase-noise Fokker-Planck equations using the motion-group Fourier transform,, IEEE Transactions on Communications, 54 (2006), 868.   Google Scholar

[38]

W. Wojtyński, Quasinilpotent Banach-Lie algebras are Baker-Campbell-Hausdorff,, J. Funct. Anal., 153 (1998), 405.  doi: 10.1006/jfan.1997.3202.  Google Scholar

show all references

References:
[1]

J. August and S. W. Zucker, Sketches with curvature: The curve indicator random field and Markov processes,, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 387.  doi: 10.1109/TPAMI.2003.1190567.  Google Scholar

[2]

A. Bonfiglioli, Homogeneous Carnot groups related to sets of vector fields,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 7 (2004), 79.   Google Scholar

[3]

A. Bonfiglioli, An ODE's version of the formula of Baker, Campbell, Dynkin and Hausdorff and the construction of Lie groups with prescribed Lie algebra,, Mediterr. J. Math., 7 (2010), 387.  doi: 10.1007/s00009-010-0064-x.  Google Scholar

[4]

A. Bonfiglioli and R. Fulci, "Topics in Noncommutative Algebra. The Theorem of Campbell, Baker, Hausdorff and Dynkin,", Lecture Notes in Mathematics, 2034 (2011).  doi: 10.1007/978-3-642-22597-0.  Google Scholar

[5]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups,, Adv. Differ. Equ., 7 (2002), 1153.   Google Scholar

[6]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory for their sub-Laplacians,", Springer Monographs in Mathematics \textbf{26}, 26 (2007).   Google Scholar

[7]

J.-M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés,, Ann. Inst. Fourier (Grenoble), 19 (1969), 277.   Google Scholar

[8]

M. Bramanti, Singular integrals in nonhomogenoeus spaces: $L^2$ and $L^p$ continuity from Hölder Estimates,, Rev. Mat. Iberoamericana, 26 (2010), 347.   Google Scholar

[9]

M. Bramanti, G. Cupini, E. Lanconelli and E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators,, Math. Z., 266 (2010), 789.  doi: 10.1007/s00209-009-0599-3.  Google Scholar

[10]

G. Da Prato, "Kolmogorov Equations for Stochastic PDE's,", Advanced Courses in Mathematics, 9 (2004).   Google Scholar

[11]

G. Da Prato and A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients,, J. Evol. Equ., 7 (2007), 587.   Google Scholar

[12]

G. Da Prato and A. Lunardi, On a class of self-adjoint elliptic operators in $L^2$ spaces with respect to invariant measures,, J. Differ. Equations, 234 (2007), 54.   Google Scholar

[13]

A. Eggert, Extending the Campbell-Hausdorff multiplication,, Geom. Dedicata, 46 (1993), 35.  doi: 10.1007/BF01264092.  Google Scholar

[14]

B. Farkas and A. Lunardi, Maximal regularity for Kolmogorov operators in $L^2$ spaces with respect to invariant measures,, J. Math. Pures Appl., 86 (2006), 310.   Google Scholar

[15]

C. L. Fefferman and A. Sánchez-Calle, Fundamental solutions for second order subelliptic operators,, Ann. Math., 124 (1986), 247.  doi: 10.2307/1971278.  Google Scholar

[16]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups,, Ark. Mat., 13 (1975), 161.  doi: 10.1007/BF02386204.  Google Scholar

[17]

G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups,", Mathematical Notes, 28 (1982).   Google Scholar

[18]

B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients,, Ann. Sc. Norm. Super. Pisa, 4 (1983), 523.   Google Scholar

[19]

C. E. Gutiérrez and E. Lanconelli, Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for $X$-elliptic operators,, Commun. Partial Differ. Equations, 28 (2003), 1833.   Google Scholar

[20]

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

[21]

D. Jerison and A. Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields,, Indiana Univ. Math. J., 35 (1986), 835.   Google Scholar

[22]

A. E. Kogoj and E. Lanconelli, An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations,, Mediterr. J. Math., 1 (2004), 51.  doi: 10.1007/s00009-004-0004-8.  Google Scholar

[23]

A. E. Kogoj and E. Lanconelli, Link of groups and homogeneous Hörmander operators,, Proc. Am. Math. Soc., 135 (2007), 2019.  doi: 10.1090/S0002-9939-07-08646-7.  Google Scholar

[24]

A. N. Kolmogorov, Zufällige Bewegungen,, Ann. of Math., 35 (1934), 116.  doi: 10.2307/1968123.  Google Scholar

[25]

S. Kusuoka and D. Stroock, The partial Malliavin calculus and its application to nonlinear filtering,, Stochastics, 12 (1984), 83.   Google Scholar

[26]

S. Kusuoka and D. Stroock, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator,, Ann. of Math., 127 (1988), 165.  doi: 10.2307/1971418.  Google Scholar

[27]

E. Lanconelli and A. E. Kogoj, $X$-elliptic operators and $X$-control distances,, Ricerche Mat., 49 (2000), 223.   Google Scholar

[28]

E. Lanconelli and A. Pascucci, On the fundamental solution for hypoelliptic second order partial differential equations with non-negative characteristic form,, Ricerche Mat., 48 (1999), 81.   Google Scholar

[29]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators,, Rend. Semin. Mat. Torino, 52 (1994), 29.   Google Scholar

[30]

A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $\mathbbR^n$,, Ann. Sc. Norm. Super. Pisa, 24 (1997), 133.   Google Scholar

[31]

D. Mumford, Elastica and computer vision,, in, (1994), 491.  doi: 10.1007/978-1-4612-2628-4.  Google Scholar

[32]

A. Nagel, F. Ricci and E. M. Stein, Fundamental solutions and harmonic analysis on nilpotent groups,, Bull. Am. Math. Soc., 23 (1990), 139.  doi: 10.1090/S0273-0979-1990-15920-8.  Google Scholar

[33]

F. Nazarov, S. Treil and A. Volberg, The $Tb$-theorem on non-homogeneous spaces,, Acta Math., 190 (2003), 151.  doi: 10.1007/BF02392690.  Google Scholar

[34]

P. Negrini and V. Scornazzani, Superharmonic functions and regularity of boundary points for a class of elliptic-parabolic partial differential operators,, Boll. Unione Mat. Ital., 3 (1984), 85.   Google Scholar

[35]

L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups,, Acta Math., 137 (1976), 247.  doi: 10.1007/BF02392419.  Google Scholar

[36]

V. S. Varadarajan, "Lie Groups, Lie Algebras and their Representations,", Graduate Texts in Mathematics, (1984).   Google Scholar

[37]

Y. Wang, Y. Zhou, D. K. Maslen and G. S. Chirikjian, Solving phase-noise Fokker-Planck equations using the motion-group Fourier transform,, IEEE Transactions on Communications, 54 (2006), 868.   Google Scholar

[38]

W. Wojtyński, Quasinilpotent Banach-Lie algebras are Baker-Campbell-Hausdorff,, J. Funct. Anal., 153 (1998), 405.  doi: 10.1006/jfan.1997.3202.  Google Scholar

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