Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Hölder regularity for a Kolmogorov equation
HTML articles powered by AMS MathViewer

by Andrea Pascucci PDF
Trans. Amer. Math. Soc. 355 (2003), 901-924 Request permission

Abstract:

We study the interior regularity properties of the solutions to the degenerate parabolic equation, \[ \Delta _{x}u+b\partial _{y}u-\partial _{t}u=f, \qquad (x,y,t)\in \mathbb {R}^{N}\times \mathbb {R}\times \mathbb {R},\] which arises in mathematical finance and in the theory of diffusion processes.
References
  • Antonelli, F., Barucci, E., and Mancino, M. E. A Comparison result for FBSDE with Applications to Decisions Theory. Math. Methods Oper. Res. 2001, 54 (3), 407–423.
  • Antonelli, F. and Pascucci, A. On the viscosity solutions of a stochastic differential utility problem. To appear in J. Differential Equations.
  • Richard Beals, $L^{p}$ and Hölder estimates for pseudodifferential operators: sufficient conditions, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 3, vii, 239–260 (English, with French summary). MR 552967
  • Bramanti, M. and Brandolini, L. $L^{p}$ estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups. Trans. Amer. Math. Soc. 2000, 352 (2), 781–822.
  • G. Citti, $C^\infty$ regularity of solutions of a quasilinear equation related to the Levi operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 3, 483–529. MR 1440031
  • Giovanna Citti and Annamaria Montanari, $C^\infty$ regularity of solutions of an equation of Levi’s type in $\textbf {R}^{2n+1}$, Ann. Mat. Pura Appl. (4) 180 (2001), no. 1, 27–58. MR 1848050, DOI 10.1007/s10231-001-8196-z
  • Giovanna Citti, Andrea Pascucci, and Sergio Polidoro, On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance, Differential Integral Equations 14 (2001), no. 6, 701–738. MR 1826957
  • Citti, G., Pascucci, A., and Polidoro, S. Regularity properties of viscosity solutions of a non-Hörmander degenerate equation. J. Math. Pures Appl. 2001, 80 (9), 901–918.
  • M. Escobedo, J. L. Vázquez, and Enrike Zuazua, Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc. 343 (1994), no. 2, 829–842. MR 1225573, DOI 10.1090/S0002-9947-1994-1225573-2
  • G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
  • Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI 10.1007/BF02392081
  • N. V. Krylov, Hölder continuity and $L_p$ estimates for elliptic equations under general Hörmander’s condition, Topol. Methods Nonlinear Anal. 9 (1997), no. 2, 249–258. MR 1491846, DOI 10.12775/TMNA.1997.012
  • Lanconelli, E., Pascucci, A, and Polidoro, S. Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance. To appear on “Nonlinear Problems in Mathematical Physics and Related Topics Vol. II In Honor of Professor O.A. Ladyzhenskaya". International Mathematical Series, Kluwer Ed.
  • E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 1, 29–63. Partial differential equations, II (Turin, 1993). MR 1289901
  • Alexander Nagel and E. M. Stein, A new class of pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 2, 582–585. MR 487603, DOI 10.1073/pnas.75.2.582
  • Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
  • Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
  • A. N. Kolmogorov, Selected works. Vol. II, Mathematics and its Applications (Soviet Series), vol. 26, Kluwer Academic Publishers Group, Dordrecht, 1992. Probability theory and mathematical statistics; With a preface by P. S. Aleksandrov; Translated from the Russian by G. Lindquist; Translation edited by A. N. Shiryayev [A. N. Shiryaev]. MR 1153022
  • Chao Jiang Xu, Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math. 45 (1992), no. 1, 77–96. MR 1135924, DOI 10.1002/cpa.3160450104
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K57, 35K65, 35K70
  • Retrieve articles in all journals with MSC (2000): 35K57, 35K65, 35K70
Additional Information
  • Andrea Pascucci
  • Affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
  • Email: pascucci@dm.unibo.it
  • Received by editor(s): June 27, 2002
  • Published electronically: October 1, 2002
  • Additional Notes: Investigation supported by the University of Bologna. Funds for selected research topics
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 901-924
  • MSC (2000): Primary 35K57, 35K65, 35K70
  • DOI: https://doi.org/10.1090/S0002-9947-02-03151-3
  • MathSciNet review: 1938738