Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces
HTML articles powered by AMS MathViewer
- by Siva R. Athreya, Richard F. Bass and Edwin A. Perkins PDF
- Trans. Amer. Math. Soc. 357 (2005), 5001-5029 Request permission
Abstract:
We introduce a new method for proving the estimate \[ \left \Vert \frac {\partial ^2 u}{\partial x_i \partial x_j} \right \Vert _{C^\alpha }\leq c\|f\|_{C^\alpha },\] where $u$ solves the equation $\Delta u-\lambda u=f$. The method can be applied to the Laplacian on $\mathbb {R}^\infty$. It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.References
- S. Athreya, R.F. Bass, M. Gordina and E.A. Perkins, Infinite dimensional stochastic differential equations of Ornstein-Uhlenbeck type, in preparation.
- Richard F. Bass, Probabilistic techniques in analysis, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1329542
- Richard F. Bass and Edwin A. Perkins, Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains, Trans. Amer. Math. Soc. 355 (2003), no. 1, 373–405. MR 1928092, DOI 10.1090/S0002-9947-02-03120-3
- Piermarco Cannarsa and Giuseppe Da Prato, Infinite-dimensional elliptic equations with Hölder-continuous coefficients, Adv. Differential Equations 1 (1996), no. 3, 425–452. MR 1401401
- Giuseppe Da Prato, Some results on elliptic and parabolic equations in Hilbert spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 7 (1996), no. 3, 181–199 (English, with English and Italian summaries). MR 1454413
- Giuseppe Da Prato and Jerzy Zabczyk, Second order partial differential equations in Hilbert spaces, London Mathematical Society Lecture Note Series, vol. 293, Cambridge University Press, Cambridge, 2002. MR 1985790, DOI 10.1017/CBO9780511543210
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Gopinath Kallianpur and Jie Xiong, Stochastic differential equations in infinite-dimensional spaces, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 26, Institute of Mathematical Statistics, Hayward, CA, 1995. Expanded version of the lectures delivered as part of the 1993 Barrett Lectures at the University of Tennessee, Knoxville, TN, March 25–27, 1993; With a foreword by Balram S. Rajput and Jan Rosinski. MR 1465436
- Alessandra Lunardi, An interpolation method to characterize domains of generators of semigroups, Semigroup Forum 53 (1996), no. 3, 321–329. MR 1406778, DOI 10.1007/BF02574147
- Paul-A. Meyer, Probability and potentials, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1966. MR 0205288
- Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- P. L. Hennequin (ed.), École d’été de probabilités de Saint-Flour. XIV—1984, Lecture Notes in Mathematics, vol. 1180, Springer-Verlag, Berlin, 1986. Papers from the summer school held in Saint-Flour, August 19–September 5, 1984. MR 876082
- Lorenzo Zambotti, An analytic approach to existence and uniqueness for martingale problems in infinite dimensions, Probab. Theory Related Fields 118 (2000), no. 2, 147–168. MR 1790079, DOI 10.1007/s440-000-8012-6
Additional Information
- Siva R. Athreya
- Affiliation: Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560059, India
- Richard F. Bass
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
- Edwin A. Perkins
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
- Received by editor(s): October 24, 2003
- Received by editor(s) in revised form: February 13, 2004
- Published electronically: March 10, 2005
- Additional Notes: The first author’s research was supported in part by an NBHM travel grant.
The second author’s research was supported in part by NSF grant DMS0244737.
The third author’s research was supported in part by an NSERC Research Grant - © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 5001-5029
- MSC (2000): Primary 35J15; Secondary 35R15, 47D07, 60J35
- DOI: https://doi.org/10.1090/S0002-9947-05-03638-X
- MathSciNet review: 2165395