-
Previous Article
Topological stability and spectral decomposition for homeomorphisms on noncompact spaces
- DCDS Home
- This Issue
-
Next Article
The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity
Wiener-Landis criterion for Kolmogorov-type operators
1. | Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino "Carlo Bo", Piazza della Repubblica, 13 - 61029 Urbino (PU), Italy |
2. | Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato, 5 - 40126 Bologna, Italy |
3. | Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5 - 00185 Roma, Italy |
We establish a necessary and sufficient condition for a boundary point to be regular for the Dirichlet problem related to a class of Kolmogorov-type equations. Our criterion is inspired by two classical criteria for the heat equation: the Evans-Gariepy's Wiener test, and a criterion by Landis expressed in terms of a series of caloric potentials.
References:
[1] |
C. Cinti and E. Lanconelli,
Riesz and Poisson-Jensen representation formulas for a class of ultraparabolic operators on Lie groups, Potential Anal., 30 (2009), 179-200.
doi: 10.1007/s11118-008-9112-6. |
[2] |
C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, New York-Heidelberg, 1972, With a preface by H. Bauer, Die Grundlehren der mathematischen Wissenschaften, Band 158. |
[3] |
L. C. Evans and R. F. Gariepy,
Wiener's criterion for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314.
doi: 10.1007/BF00249583. |
[4] |
E. B. Fabes, N. Garofalo and E. Lanconelli,
Wiener's criterion for divergence form parabolic operators with C1-Dini continuous coefficients, Duke Math. J., 59 (1989), 191-232.
doi: 10.1215/S0012-7094-89-05906-1. |
[5] |
N. Garofalo and E. Lanconelli,
Wiener's criterion for parabolic equations with variable coefficients and its consequences, Trans. Amer. Math. Soc., 308 (1988), 811-836.
|
[6] |
N. Garofalo and E. Lanconelli,
Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), 775-792.
doi: 10.1090/S0002-9947-1990-0998126-5. |
[7] |
N. Garofalo and F. Segàla,
Estimates of the fundamental solution and Wiener's criterion for the heat equation on the Heisenberg group, Indiana Univ. Math. J., 39 (1990), 1155-1196.
doi: 10.1512/iumj.1990.39.39053. |
[8] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[9] |
R. A. Horn and C. R. Johnson,
Matrix Analysis, Cambridge University Press, Cambridge, 1990. |
[10] |
A.E. Kogoj,
On the Dirichlet problem for hypoelliptic evolution equations: Perron-Wiener solution and a cone-type criterion, J. Differential Equations, 262 (2017), 1524-1539.
doi: 10.1016/j.jde.2016.10.018. |
[11] |
L.P. Kuptsov,
Fundamental solutions for a class of second-order elliptic-parabolic equations, Differentcial'nye Uravnenija, 8 (1972), 1649-1660,1716.
|
[12] |
L.P. Kuptsov,
Fundamental solutions of certain second-order degenerate parabolic equations, Math. Notes, 31 (1982), 283-289.
|
[13] |
E. Lanconelli,
Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl. (4), 97 (1973), 83-114.
doi: 10.1007/BF02414910. |
[14] |
E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63, Partial differential equations, Ⅱ (Turin, 1993). |
[15] |
E. Lanconelli, G. Tralli and F. Uguzzoni,
Wiener-type tests from a two-sided Gaussian bound, Ann. Mat. Pura Appl. (4), 196 (2017), 217-244.
doi: 10.1007/s10231-016-0570-y. |
[16] |
E. Lanconelli and F. Uguzzoni,
Potential analysis for a class of diffusion equations: A Gaussian bounds approach, J. Differential Equations, 248 (2010), 2329-2367.
doi: 10.1016/j.jde.2010.01.007. |
[17] |
E.M. Landis,
Necessary and sufficient conditions for the regularity of a boundary point for the Dirichlet problem for the heat equation, Dokl. Akad. Nauk SSSR, 185 (1969), 517-520.
|
[18] |
M. Manfredini,
The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations, 2 (1997), 831-866.
|
[19] |
V. Scornazzani,
The Dirichlet problem for the Kolmogorov operator, Boll. Un. Mat. Ital. C (5), 18 (1981), 43-62.
|
show all references
References:
[1] |
C. Cinti and E. Lanconelli,
Riesz and Poisson-Jensen representation formulas for a class of ultraparabolic operators on Lie groups, Potential Anal., 30 (2009), 179-200.
doi: 10.1007/s11118-008-9112-6. |
[2] |
C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Springer-Verlag, New York-Heidelberg, 1972, With a preface by H. Bauer, Die Grundlehren der mathematischen Wissenschaften, Band 158. |
[3] |
L. C. Evans and R. F. Gariepy,
Wiener's criterion for the heat equation, Arch. Rational Mech. Anal., 78 (1982), 293-314.
doi: 10.1007/BF00249583. |
[4] |
E. B. Fabes, N. Garofalo and E. Lanconelli,
Wiener's criterion for divergence form parabolic operators with C1-Dini continuous coefficients, Duke Math. J., 59 (1989), 191-232.
doi: 10.1215/S0012-7094-89-05906-1. |
[5] |
N. Garofalo and E. Lanconelli,
Wiener's criterion for parabolic equations with variable coefficients and its consequences, Trans. Amer. Math. Soc., 308 (1988), 811-836.
|
[6] |
N. Garofalo and E. Lanconelli,
Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), 775-792.
doi: 10.1090/S0002-9947-1990-0998126-5. |
[7] |
N. Garofalo and F. Segàla,
Estimates of the fundamental solution and Wiener's criterion for the heat equation on the Heisenberg group, Indiana Univ. Math. J., 39 (1990), 1155-1196.
doi: 10.1512/iumj.1990.39.39053. |
[8] |
L. Hörmander,
Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171.
doi: 10.1007/BF02392081. |
[9] |
R. A. Horn and C. R. Johnson,
Matrix Analysis, Cambridge University Press, Cambridge, 1990. |
[10] |
A.E. Kogoj,
On the Dirichlet problem for hypoelliptic evolution equations: Perron-Wiener solution and a cone-type criterion, J. Differential Equations, 262 (2017), 1524-1539.
doi: 10.1016/j.jde.2016.10.018. |
[11] |
L.P. Kuptsov,
Fundamental solutions for a class of second-order elliptic-parabolic equations, Differentcial'nye Uravnenija, 8 (1972), 1649-1660,1716.
|
[12] |
L.P. Kuptsov,
Fundamental solutions of certain second-order degenerate parabolic equations, Math. Notes, 31 (1982), 283-289.
|
[13] |
E. Lanconelli,
Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl. (4), 97 (1973), 83-114.
doi: 10.1007/BF02414910. |
[14] |
E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63, Partial differential equations, Ⅱ (Turin, 1993). |
[15] |
E. Lanconelli, G. Tralli and F. Uguzzoni,
Wiener-type tests from a two-sided Gaussian bound, Ann. Mat. Pura Appl. (4), 196 (2017), 217-244.
doi: 10.1007/s10231-016-0570-y. |
[16] |
E. Lanconelli and F. Uguzzoni,
Potential analysis for a class of diffusion equations: A Gaussian bounds approach, J. Differential Equations, 248 (2010), 2329-2367.
doi: 10.1016/j.jde.2010.01.007. |
[17] |
E.M. Landis,
Necessary and sufficient conditions for the regularity of a boundary point for the Dirichlet problem for the heat equation, Dokl. Akad. Nauk SSSR, 185 (1969), 517-520.
|
[18] |
M. Manfredini,
The Dirichlet problem for a class of ultraparabolic equations, Adv. Differential Equations, 2 (1997), 831-866.
|
[19] |
V. Scornazzani,
The Dirichlet problem for the Kolmogorov operator, Boll. Un. Mat. Ital. C (5), 18 (1981), 43-62.
|
[1] |
Luigi Ambrosio, Michele Miranda jr., Diego Pallara. Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 591-606. doi: 10.3934/dcds.2010.28.591 |
[2] |
Matteo Novaga, Diego Pallara, Yannick Sire. A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 815-831. doi: 10.3934/dcdss.2016030 |
[3] |
Tanja Eisner, Pavel Zorin-Kranich. Uniformity in the Wiener-Wintner theorem for nilsequences. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3497-3516. doi: 10.3934/dcds.2013.33.3497 |
[4] |
Massimiliano Tamborrino. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. Mathematical Biosciences & Engineering, 2016, 13 (3) : 613-629. doi: 10.3934/mbe.2016011 |
[5] |
Arnaud Debussche, Sylvain De Moor, Julien Vovelle. Diffusion limit for the radiative transfer equation perturbed by a Wiener process. Kinetic and Related Models, 2015, 8 (3) : 467-492. doi: 10.3934/krm.2015.8.467 |
[6] |
Wilfried Grecksch, Hannelore Lisei. Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3095-3114. doi: 10.3934/dcdsb.2016089 |
[7] |
Davide Addona, Giorgio Menegatti, Michele Miranda jr.. $ BV $ functions on open domains: the Wiener case and a Fomin differentiable case. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2679-2711. doi: 10.3934/cpaa.2020117 |
[8] |
Jae Gil Choi, David Skoug. Algebraic structure of the $ L_2 $ analytic Fourier–Feynman transform associated with Gaussian paths on Wiener space. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3829-3842. doi: 10.3934/cpaa.2020169 |
[9] |
Ugur G. Abdulla. Wiener's criterion at $\infty$ for the heat equation and its measure-theoretical counterpart. Electronic Research Announcements, 2008, 15: 44-51. doi: 10.3934/era.2008.15.44 |
[10] |
Seung Jun Chang, Jae Gil Choi. A Cameron-Storvick theorem for the analytic Feynman integral associated with Gaussian paths on a Wiener space and applications. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2225-2238. doi: 10.3934/cpaa.2018106 |
[11] |
Gregorio Díaz, Jesús Ildefonso Díaz. Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021165 |
[12] |
Gisèle Ruiz Goldstein, Jerome A. Goldstein, Abdelaziz Rhandi. Kolmogorov equations perturbed by an inverse-square potential. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 623-630. doi: 10.3934/dcdss.2011.4.623 |
[13] |
Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457 |
[14] |
Luca Lorenzi. Optimal Hölder regularity for nonautonomous Kolmogorov equations. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 169-191. doi: 10.3934/dcdss.2011.4.169 |
[15] |
Yu-Chi Chen. Security analysis of public key encryption with filtered equality test. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021053 |
[16] |
Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial and Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097 |
[17] |
Jiu Ding, Noah H. Rhee. A unified maximum entropy method via spline functions for Frobenius-Perron operators. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 235-245. doi: 10.3934/naco.2013.3.235 |
[18] |
Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 |
[19] |
Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 |
[20] |
Xiaolong Han, Guozhen Lu. Regularity of solutions to an integral equation associated with Bessel potential. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1111-1119. doi: 10.3934/cpaa.2011.10.1111 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]