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

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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

Abstract
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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 and 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), 387-400. doi: 10.1109/TPAMI.2003.1190567.
[2]	A. Bonfiglioli, Homogeneous Carnot groups related to sets of vector fields, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 7 (2004), 79-107.
[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), 387-414. doi: 10.1007/s00009-010-0064-x.
[4]	A. Bonfiglioli and R. Fulci, "Topics in Noncommutative Algebra. The Theorem of Campbell, Baker, Hausdorff and Dynkin," Lecture Notes in Mathematics, 2034, Springer-Verlag, 2011. doi: 10.1007/978-3-642-22597-0.
[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), 1153-1192.
[6]	A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory for their sub-Laplacians," Springer Monographs in Mathematics 26, New York, NY, Springer, 2007.
[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), 277-304.
[8]	M. Bramanti, Singular integrals in nonhomogenoeus spaces: $L^2$ and $L^p$ continuity from Hölder Estimates, Rev. Mat. Iberoamericana, 26 (2010), 347-366.
[9]	M. Bramanti, G. Cupini, E. Lanconelli and E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators, Math. Z., 266 (2010), 789-816. doi: 10.1007/s00209-009-0599-3.
[10]	G. Da Prato, "Kolmogorov Equations for Stochastic PDE's," Advanced Courses in Mathematics, CRM Barcelona, Basel: Birkhäuser, 9 (2004).
[11]	G. Da Prato and A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients, J. Evol. Equ., 7 (2007), 587-614.
[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), 54-79.
[13]	A. Eggert, Extending the Campbell-Hausdorff multiplication, Geom. Dedicata, 46 (1993), 35-45. doi: 10.1007/BF01264092.
[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), 310-321.
[15]	C. L. Fefferman and A. Sánchez-Calle, Fundamental solutions for second order subelliptic operators, Ann. Math., 124 (1986), 247-272. doi: 10.2307/1971278.
[16]	G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204.
[17]	G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups," Mathematical Notes, 28, Princeton University Press, Princeton, N.J. 1982.
[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, Cl. Sci., 4 (1983), 523-541.
[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), 1833-1862.
[20]	L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.
[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), 835-854.
[22]	A. E. Kogoj and E. Lanconelli, An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations, Mediterr. J. Math., 1 (2004), 51-80. doi: 10.1007/s00009-004-0004-8.
[23]	A. E. Kogoj and E. Lanconelli, Link of groups and homogeneous Hörmander operators, Proc. Am. Math. Soc., 135 (2007), 2019-2030. doi: 10.1090/S0002-9939-07-08646-7.
[24]	A. N. Kolmogorov, Zufällige Bewegungen, Ann. of Math., 35 (1934), 116-117. doi: 10.2307/1968123.
[25]	S. Kusuoka and D. Stroock, The partial Malliavin calculus and its application to nonlinear filtering, Stochastics, 12 (1984), 83-142.
[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), 165-189. doi: 10.2307/1971418.
[27]	E. Lanconelli and A. E. Kogoj, $X$-elliptic operators and $X$-control distances, Ricerche Mat., 49 (2000), 223-243.
[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), 81-106.
[29]	E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Semin. Mat. Torino, 52 (1994), 29-63.
[30]	A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $\mathbb{R}^{N}$, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 24 (1997), 133-164.
[31]	D. Mumford, Elastica and computer vision, in "Algebraic Geometry and its Applications" (eds. Bajaj, Chandrajit) Springer-Verlag, New-York, 491-506 (1994). doi: 10.1007/978-1-4612-2628-4.
[32]	A. Nagel, F. Ricci and E. M. Stein, Fundamental solutions and harmonic analysis on nilpotent groups, Bull. Am. Math. Soc., New Ser., 23 (1990), 139-144. doi: 10.1090/S0273-0979-1990-15920-8.
[33]	F. Nazarov, S. Treil and A. Volberg, The $Tb$-theorem on non-homogeneous spaces, Acta Math., 190 (2003), 151-239. doi: 10.1007/BF02392690.
[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., VI. Ser., C, Anal. Funz. Appl., 3 (1984), 85-107.
[35]	L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320. doi: 10.1007/BF02392419.
[36]	V. S. Varadarajan, "Lie Groups, Lie Algebras and their Representations," Graduate Texts in Mathematics, Springer-Verlag, New York, 1984.
[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), 868-877.
[38]	W. Wojtyński, Quasinilpotent Banach-Lie algebras are Baker-Campbell-Hausdorff, J. Funct. Anal., 153 (1998), 405-413. doi: 10.1006/jfan.1997.3202.

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), 387-400. doi: 10.1109/TPAMI.2003.1190567.
[2]	A. Bonfiglioli, Homogeneous Carnot groups related to sets of vector fields, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 7 (2004), 79-107.
[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), 387-414. doi: 10.1007/s00009-010-0064-x.
[4]	A. Bonfiglioli and R. Fulci, "Topics in Noncommutative Algebra. The Theorem of Campbell, Baker, Hausdorff and Dynkin," Lecture Notes in Mathematics, 2034, Springer-Verlag, 2011. doi: 10.1007/978-3-642-22597-0.
[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), 1153-1192.
[6]	A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory for their sub-Laplacians," Springer Monographs in Mathematics 26, New York, NY, Springer, 2007.
[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), 277-304.
[8]	M. Bramanti, Singular integrals in nonhomogenoeus spaces: $L^2$ and $L^p$ continuity from Hölder Estimates, Rev. Mat. Iberoamericana, 26 (2010), 347-366.
[9]	M. Bramanti, G. Cupini, E. Lanconelli and E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators, Math. Z., 266 (2010), 789-816. doi: 10.1007/s00209-009-0599-3.
[10]	G. Da Prato, "Kolmogorov Equations for Stochastic PDE's," Advanced Courses in Mathematics, CRM Barcelona, Basel: Birkhäuser, 9 (2004).
[11]	G. Da Prato and A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients, J. Evol. Equ., 7 (2007), 587-614.
[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), 54-79.
[13]	A. Eggert, Extending the Campbell-Hausdorff multiplication, Geom. Dedicata, 46 (1993), 35-45. doi: 10.1007/BF01264092.
[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), 310-321.
[15]	C. L. Fefferman and A. Sánchez-Calle, Fundamental solutions for second order subelliptic operators, Ann. Math., 124 (1986), 247-272. doi: 10.2307/1971278.
[16]	G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204.
[17]	G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups," Mathematical Notes, 28, Princeton University Press, Princeton, N.J. 1982.
[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, Cl. Sci., 4 (1983), 523-541.
[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), 1833-1862.
[20]	L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.
[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), 835-854.
[22]	A. E. Kogoj and E. Lanconelli, An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations, Mediterr. J. Math., 1 (2004), 51-80. doi: 10.1007/s00009-004-0004-8.
[23]	A. E. Kogoj and E. Lanconelli, Link of groups and homogeneous Hörmander operators, Proc. Am. Math. Soc., 135 (2007), 2019-2030. doi: 10.1090/S0002-9939-07-08646-7.
[24]	A. N. Kolmogorov, Zufällige Bewegungen, Ann. of Math., 35 (1934), 116-117. doi: 10.2307/1968123.
[25]	S. Kusuoka and D. Stroock, The partial Malliavin calculus and its application to nonlinear filtering, Stochastics, 12 (1984), 83-142.
[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), 165-189. doi: 10.2307/1971418.
[27]	E. Lanconelli and A. E. Kogoj, $X$-elliptic operators and $X$-control distances, Ricerche Mat., 49 (2000), 223-243.
[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), 81-106.
[29]	E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Semin. Mat. Torino, 52 (1994), 29-63.
[30]	A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $\mathbb{R}^{N}$, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 24 (1997), 133-164.
[31]	D. Mumford, Elastica and computer vision, in "Algebraic Geometry and its Applications" (eds. Bajaj, Chandrajit) Springer-Verlag, New-York, 491-506 (1994). doi: 10.1007/978-1-4612-2628-4.
[32]	A. Nagel, F. Ricci and E. M. Stein, Fundamental solutions and harmonic analysis on nilpotent groups, Bull. Am. Math. Soc., New Ser., 23 (1990), 139-144. doi: 10.1090/S0273-0979-1990-15920-8.
[33]	F. Nazarov, S. Treil and A. Volberg, The $Tb$-theorem on non-homogeneous spaces, Acta Math., 190 (2003), 151-239. doi: 10.1007/BF02392690.
[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., VI. Ser., C, Anal. Funz. Appl., 3 (1984), 85-107.
[35]	L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320. doi: 10.1007/BF02392419.
[36]	V. S. Varadarajan, "Lie Groups, Lie Algebras and their Representations," Graduate Texts in Mathematics, Springer-Verlag, New York, 1984.
[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), 868-877.
[38]	W. Wojtyński, Quasinilpotent Banach-Lie algebras are Baker-Campbell-Hausdorff, J. Funct. Anal., 153 (1998), 405-413. doi: 10.1006/jfan.1997.3202.

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