American Institute of Mathematical Sciences

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

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

 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.
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 [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

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
 [1] Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250 [2] Vladimir I. Bogachev, Stanislav V. Shaposhnikov, Alexander Yu. Veretennikov. Differentiability of solutions of stationary Fokker--Planck--Kolmogorov equations with respect to a parameter. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3519-3543. doi: 10.3934/dcds.2016.36.3519 [3] Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 [4] Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157 [5] María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637 [6] Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124 [7] Valerii Los, Vladimir A. Mikhailets, Aleksandr A. Murach. An isomorphism theorem for parabolic problems in Hörmander spaces and its applications. Communications on Pure & Applied Analysis, 2017, 16 (1) : 69-98. doi: 10.3934/cpaa.2017003 [8] Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215 [9] Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017 [10] Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485 [11] Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016 [12] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [13] Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165 [14] Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617 [15] Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911 [16] Luca Lorenzi. Optimal Hölder regularity for nonautonomous Kolmogorov equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 169-191. doi: 10.3934/dcdss.2011.4.169 [17] Peter C. Gibson. On the measurement operator for scattering in layered media. Inverse Problems & Imaging, 2017, 11 (1) : 87-97. doi: 10.3934/ipi.2017005 [18] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [19] Yunmei Chen, Xianqi Li, Yuyuan Ouyang, Eduardo Pasiliao. Accelerated bregman operator splitting with backtracking. Inverse Problems & Imaging, 2017, 11 (6) : 1047-1070. doi: 10.3934/ipi.2017048 [20] Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453

2018 Impact Factor: 0.925