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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 |
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. |
[2] |
A. Bonfiglioli, Homogeneous Carnot groups related to sets of vector fields,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 7 (2004), 79.
|
[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. |
[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. |
[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.
|
[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).
|
[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.
|
[8] |
M. Bramanti, Singular integrals in nonhomogenoeus spaces: $L^2$ and $L^p$ continuity from Hölder Estimates,, Rev. Mat. Iberoamericana, 26 (2010), 347.
|
[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. |
[10] |
G. Da Prato, "Kolmogorov Equations for Stochastic PDE's,", Advanced Courses in Mathematics, 9 (2004).
|
[11] |
G. Da Prato and A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients,, J. Evol. Equ., 7 (2007), 587.
|
[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.
|
[13] |
A. Eggert, Extending the Campbell-Hausdorff multiplication,, Geom. Dedicata, 46 (1993), 35.
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.
|
[15] |
C. L. Fefferman and A. Sánchez-Calle, Fundamental solutions for second order subelliptic operators,, Ann. Math., 124 (1986), 247.
doi: 10.2307/1971278. |
[16] |
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups,, Ark. Mat., 13 (1975), 161.
doi: 10.1007/BF02386204. |
[17] |
G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups,", Mathematical Notes, 28 (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, 4 (1983), 523.
|
[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.
|
[20] |
L. Hörmander, Hypoelliptic second order differential equations,, Acta Math., 119 (1967), 147.
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.
|
[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. |
[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. |
[24] |
A. N. Kolmogorov, Zufällige Bewegungen,, Ann. of Math., 35 (1934), 116.
doi: 10.2307/1968123. |
[25] |
S. Kusuoka and D. Stroock, The partial Malliavin calculus and its application to nonlinear filtering,, Stochastics, 12 (1984), 83.
|
[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. |
[27] |
E. Lanconelli and A. E. Kogoj, $X$-elliptic operators and $X$-control distances,, Ricerche Mat., 49 (2000), 223.
|
[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.
|
[29] |
E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators,, Rend. Semin. Mat. Torino, 52 (1994), 29.
|
[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.
|
[31] |
D. Mumford, Elastica and computer vision,, in, (1994), 491.
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., 23 (1990), 139.
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.
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., 3 (1984), 85.
|
[35] |
L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups,, Acta Math., 137 (1976), 247.
doi: 10.1007/BF02392419. |
[36] |
V. S. Varadarajan, "Lie Groups, Lie Algebras and their Representations,", Graduate Texts in Mathematics, (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. 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. |
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. |
[2] |
A. Bonfiglioli, Homogeneous Carnot groups related to sets of vector fields,, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 7 (2004), 79.
|
[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. |
[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. |
[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.
|
[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).
|
[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.
|
[8] |
M. Bramanti, Singular integrals in nonhomogenoeus spaces: $L^2$ and $L^p$ continuity from Hölder Estimates,, Rev. Mat. Iberoamericana, 26 (2010), 347.
|
[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. |
[10] |
G. Da Prato, "Kolmogorov Equations for Stochastic PDE's,", Advanced Courses in Mathematics, 9 (2004).
|
[11] |
G. Da Prato and A. Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients,, J. Evol. Equ., 7 (2007), 587.
|
[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.
|
[13] |
A. Eggert, Extending the Campbell-Hausdorff multiplication,, Geom. Dedicata, 46 (1993), 35.
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.
|
[15] |
C. L. Fefferman and A. Sánchez-Calle, Fundamental solutions for second order subelliptic operators,, Ann. Math., 124 (1986), 247.
doi: 10.2307/1971278. |
[16] |
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups,, Ark. Mat., 13 (1975), 161.
doi: 10.1007/BF02386204. |
[17] |
G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups,", Mathematical Notes, 28 (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, 4 (1983), 523.
|
[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.
|
[20] |
L. Hörmander, Hypoelliptic second order differential equations,, Acta Math., 119 (1967), 147.
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.
|
[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. |
[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. |
[24] |
A. N. Kolmogorov, Zufällige Bewegungen,, Ann. of Math., 35 (1934), 116.
doi: 10.2307/1968123. |
[25] |
S. Kusuoka and D. Stroock, The partial Malliavin calculus and its application to nonlinear filtering,, Stochastics, 12 (1984), 83.
|
[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. |
[27] |
E. Lanconelli and A. E. Kogoj, $X$-elliptic operators and $X$-control distances,, Ricerche Mat., 49 (2000), 223.
|
[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.
|
[29] |
E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators,, Rend. Semin. Mat. Torino, 52 (1994), 29.
|
[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.
|
[31] |
D. Mumford, Elastica and computer vision,, in, (1994), 491.
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., 23 (1990), 139.
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.
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., 3 (1984), 85.
|
[35] |
L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups,, Acta Math., 137 (1976), 247.
doi: 10.1007/BF02392419. |
[36] |
V. S. Varadarajan, "Lie Groups, Lie Algebras and their Representations,", Graduate Texts in Mathematics, (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. 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. |
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